Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters tu the

editor should be sent tu either of the

editors-in-chief, Chandler Davis or

Marjorie Senechal.

Mathematics and Narrative Marjorie Senechal's article [1 ] is a de­lightful account of a meeting organised by the group Thales and Friends, at­tempting to explore the relationship he­tween mathematics and narrative. What could he more beguiling than dis­cussing this on the Aegean? But it seemed strange to read a paper on mathematics and narrative that doesn't mention the most commercially suc­cessful hooks ever written by a mathe­matician: Alice 's Aduentures in Won­derland and Through the Looking Glass. Neither is there any reference to the most successful paramathematical hook of recent years (and here I'm guessing even more wildly than in the previous sentence), Mark Haddon's The Curious Incident of the Dof!, in the Night-Time. Neither Carroll/Dodgson nor Haddon, who clearly know how to write some­thing that people actually want to read, was trying to put mathematics across. They both produced beautifully written stories with a mathematical theme. Car­roll just couldn't help being playful with mathematical and logical ideas: the mathematics in the background shines through. Haddon, who isn't a mathe­matician, has written a lovely story about a strange boy with some mathe­matical talent that (as it seemed to me) has captured a little of the feeling of what doing mathematics is like.

Thales and Friends' website [2] is also a pleasure to browse through. I was par­ticularly interested in the papers by Mazur [3] and Chaitin [4] . Mazur has made a heroic attempt to classify the different ways in which stories can he used in "mathematical exposition". Rut I think there is a problem here in his use of the word exposition. Stories and exposition don't seem to go together naturally: stories are surely more about exploration than exposition. Carroll is exploring logic in his Alice hooks: set up a crazy situation, apply the rules of logic, and see where we get to. And Haddon is explorinJ< the relationship he­tween mathematics and autism, which, by the way. is exposed by James in [5] . Chaitin, in his paper, contrasts two views of mathematics: Hilbert's attempt to describe it as a closed, formal sys­tem of axioms, rules of deduction, and so on; and the Lakatos-Chaitin approach

to mathematics as quasi-experimental. The Hilbert viewpoint demands expo­sition: here is mathematics all wrapped up in this formal system, now we must expose it. The Lakatos-Chaitin view­point suggests exploration: let's look around us, move off in an interesting direction, and see where it takes us. My suggestion is that, following Carroll and Haddon, you are much more likely to write a readable narrative if you can adopt the Lakatos-Chaitin-exploration point of view.

The contrast between exposition and exploration, between a formal system and a quasi-experimental approach, seems very similar in spirit to the de­sign/evolution dichotomy discussed in [6] . If you take a design point of view, then narrative, if it has any role at all , is merely a pedagogical device to sugar the pill or to set the mathematics in a wider context: a taxonomy for this is very well set out by Mazur. But with an evolutionary/exploration viewpoint, narrative is at the centre of the action. How could one provide any under­standing of an evolutionary world bet­ter than by telling stories' No theory of economic development is going to give one a better idea of what happens in a market-place than the story of the evo­lution of the internet. And Haddon's story offers something that no formal theory of autism will ever give.

Some businesspeople have adopted the idea of writing stories about the fu­ture to help them and their colleagues to understand possible developments in the business environment that may be of vital importance. These stories are called scenarios. (I would prefer to use the simple word story under all cir­cumstances, but I have often encoun­tered resistance unless I adopted more bureaucratic words like narrative, fic­tion, and scenario. ) This approach was pioneered in the energy company Shell International (see [7]) in the early 1970s as a way of opening the minds of se­nior management to the possibility that the price of crude oil might one day rise above $2 per barrel , and has been used in Shell ever since. It has a num­ber of advantages including: • Having more than one scenario re­

minds people that the future is un­predictable. Before reading Chaitin's

© 2007 Springer SCience+ Bus1ness Medle, Inc., Volume 29, Number I, 2007 5

paper, I had thought this might not be relevant to mathematics, but now I'm not so sure. His idea of adding new axioms such as the Riemann Hypothesis (RH) in a quasi-experi­mental way seems rather analogous to the writing of different scenarios to explore the future. In fact I would like to challenge mathematicians to write two narratives about mathe­matics in one of which RH is true and in the other of which RH is false. If this turns out to be not possible, then I suppose the obvious next question is: does RH really matter? (I would also very much like to read a story set in a world in which the Continuum Hypothesis is false.)

• The scenarios help to liberate people from "common sense" and from their prejudices. The different stories allow unusual ideas to be put forward and discussed as pieces of fiction rather than matters of life and death (which they can literally be, for example when this technique was used in a meeting between warring parties in South Africa towards the end of the apartheid era). This idea is reminis­cent of the development of non­Euclidean geometries, a story that often inspires me when I write sce­narios.

• Occasionally, the same outcome crops up in the course of two quite different scenarios. This strengthens belief that this outcome might actu­ally happen. Compare and contrast this with the famous result of Skewes and Littlewood that can be proved in two quite different ways depend­ing on whether RH is true: far from having their belief in the result strengthened, some mathematicians, of course, have refused to accept that this constitutes a proof at all. The most significant benefit of sce­

narios, in my view, is in the under­standing they bring to thinking about developments in the business environ­ment, and hence an enormous im­provement in the quality of discussions about unfolding events. (This is why [8] is called The Art of Strategic Conversa­tion.) As events and trends happen, one can look at them and say, "Ah, yes, this is what one would expect in scenario A, but on the other hand that looks like scenario B". If you can say something like that, you have understood what is

6 THE MATHEMATICAL INTELLIGENCER

going on. Chaitin says that understand­ing means compression, "the fact that you're putting just a few ideas in, and getting a lot more out". I think he's right in this, and that in practice the com­pression is often (always?) in the form of a narrative. When I say that I under­stand why somebody became angry, I mean that there is a narrative starring certain characters and featuring events and motivations, and that the anger fits into this narrative. When we say that we understand why the planets move round the sun in ellipses, we mean that this fits into the narrative of Newton, gravity, cal­culus, and so on. (This narrative will be more or less sophisticated for different people.) Here is an example from math­ematics, which I think is archetypal.

In [9] , Singer says, "We should not be too surprised that mathematics has co­herent systems applicable to physics. It remains to be seen whether there is an already developed system in mathemat­ics that will describe the structure of string theory. [At present we do not even know what the symmetry group of string field theory is.]" Singer is trying to fit string field theory into a narrative. The narrative is called "Symmetry Groups", and Singer might think of this narrative as starring Galois and Einstein and a cast of thousands, or he might think of it, as with Mazur's story about rational points of elliptic curves, as a narrative of ideas. But it's a narrative. And if he finds out what the symmetry group of string field theory is, he will be justified in saying, "Now I understand!" in just the same way as a businessperson faced with the prospect of new environmental legisla­tion can say, "Yes, I understand what's happening, it fits into one of my sce­narios". If the symmetry group of string field theory is never found, then either string field theory will be abandoned or

an entirely new narrative will need to be written.

I can't stop without taking issue with a statement in [1 ] . (I think it is a remark made by a participant at the Thales meet­ing rather than necessarily the opinion of the author. ) The statement is, "Popu­lar math books must not mislead. They must tell the whole truth and nothing but the truth" . If this were taken literally (and I imagine that advocates of the whole truth and nothing but the truth would like to be taken literally) it would simply mean the death of popular math-

ematics. For the whole truth includes all the gory details, technical background, and arcane exceptions. This isn't popu­lar mathematics, it's mathematics. Popu­lar mathematics should certainly not mis­lead, but it can't afford to be cluttered up with the "whole truth" . So what should popular mathematics do? It should be .faitf?ful to the narrative.

The astute reader will have noticed that I have managed to write 1 ,500 words on mathematics and narrative without ever saying what I think a nar­rative is. But I'm in good company be­cause as far as I can tell none of the Thales people has defined a narrative either. (Mazur does partly.) So on the principle of rushing in where angels fear to tread, here is a stab at a definition. "A narrative is a sequence over time of related specific events, emotions, or ideas designed to hold the attention of the reader, listener, or viewer. It is not the laying out of a general situation or

theory in its entirety: but a good narra­tive will help people to gain a better un­derstanding of the general situation. "

Eric Grunwald

Mathematical Capital

1 87 Sheen Lane

London SW1 4 8LE

UK

e-mail: ericgrunwald@aol.com

REFERENCES 1 . Marjorie Senechal, "Mathematics and Nar­

rative at Mykonos", Mathematical lntelli­

gencer, Vol 28 (2), 2006

2. http://www.thalesandfriends.org

3. Barry Mazur, "Eureka' and Other Stories",

June 29, 2005, on [2]

4. Gregory Chaitin, "Irreducible Complexity in

Pure Mathematics", on [2]

5. loan James, "Autism in Mathematicians",

Mathematical lntelligencer, vol 25 (4), 2003

6. Eric Grunwald, "Evolution and Design Inside

and Outside Mathematics", Mathematical

lntel/igencer, vol 27 (2), 2005

7. Pierre Wack, "Scenarios, Uncharted Waters

Ahead", Harvard Business Review, Sep-Oct

1 985, and "Scenarios, Shooting the Rapids",

Harvard Business Review, Nov-Dec 1 985

8. Kees van der Heijden, "Scenarios, The Art

of Strategic Conversation", John Wiley & Sons, 1 996

9. Martin Raussen and Christian Skau, "Inter­

view with Michael Atiyah and Isadore

Singer", EMS, September 2004

The Hidden Mathematics of the Mars Exploration Rover Miss ion

UFFE THOMAS JANKVIST AND 8J0RN TOLDBOD

0 n January 4, 2004, Mars Exploration Rover (MER) A, named Spirit, entered the Martian atmosphere. The spacecraft, weighing 827 kg, was travelling with a

speed of 19,300 km/h. During the next four minutes the ve­locity of the craft was reduced to 1 ,600 km/h at the meet­ing between the Martian atmosphere and the aeroshell of the craft. At this point a parachute was deployed and the velocity decreased to about 300 km/h. At a point 100 m above the Martian surface, the retrorockets were fired to slow the descent, and finally the giant airbags were inflated. The airhag-covered spacecraft hit the surface of Mars with a velocity of ahout 50 krn/h. The airbag ball bounced and rolled for about 1 km on the Martian surface just as Mars Pathfinder had done seven years earlier. When the landing module finally came to a stop its airbags were deflated and retracted and its petals were open. After six months in space the encapsulated rover, Spirit, could at last unfold its solar arrays. Three hours later Spirit transmitted its first image of the Gusev Crater to Earth. On January 15 Spirit left its land­ing module and drove out onto the surface of Mars. Ten days later, on January 25th, the entire scenario was repeated at Terra Meridiani with Mars Exploration Rover B, named Opportunity.1

Introduction In March 2005 we spent a week at NASA's Jet Propulsion Laboratory QPL) as part of our joint master thesis2 at the

Figure I. Opportunitys heat shield and the shield's place of landing as seen from the rover. http: I /marsrovers. jpl.

nasa.gov/gallery/press/opportunity/20041227a/

1NN325EFF40CYLA3P0685L000Ml-crop-B330Rl_br.jpg

mathematics department of Roskilde University, Denmark. The purpose of our stay was to conduct a small investiga­tion of the mathematics in the Mars Exploration Rover (MER) mission being performed at JPL.

Professor Emeritus Philip ] . Davis had more or less sug-gested such an investigation in an article published in 2004:

Consider the recent flight to Mars that put a "laboratory vehicle" on that planet. [ . . . ] Now, from start to fin­ish, the Mars shot would have been impossible without a tremendous underlay of mathematics built into chips

'This information in great part originates from http: I lnssdc. gsfc. nasa. gov ldatabaseiMasterCatalog?sc=2003 -027 A and http: I lnssdcgs fc. nasa.

govldatabaseiMasterCatalog?sc=2003-032A

2The thesis consists of the texts [5], [6], and [7] and can be found in its original Danish version as IMFUFA-text number 449 at http: 1 lrrunf. rue. dkl

imfufateksterlindex.htm

8 THE MATHEMAnCAL INTELLIGENCER © 2007 Spnnger SC1ence+Bus1ness Mecia, Inc

and software. It would defy the most knowledgeable his­torian of mathematics to discover and describe all the mathematics that was involved. The public is hardly aware of this; it is not written up in the newspapers. [2]

Although we are not "the most knowledgeable historians of mathematics," we nevertheless decided to engage in Philip Davis's project. Unfortunately, newspapers are not the only place in which this wasn't written up. In fact, finding ex­tensive literature on the mathematics of the mission was so difficult that we decided to base our investigation on inter­views. Hence the long travel to Pasadena, California. While in the US we decided also to visit Davis at Brown Univer­sity in Providence, Rhode Island, to discuss our pending in­vestigation with him. Davis advised us to tly to gain an in­sight into the employees' personal motivations for working in the aerospace industry, as well as an understanding of the nature of the mathematical work performed at JPL [ 1 ] . We also decided to look at what might be referred to as the ex­ternal influences on the daily work, such as deadlines and economic limitations-the basic work context.

One of the more interesting aspects of our investigation quickly turned out to be the invisibility of the mathematics in­volved in the mission. The fact that the mathematics involved is hidden from the public may seem natural, hut parts of the mathematics of MER are also hidden from the scientists par­ticipating in the mission. In fact the hiding, or invisibility, of the mathematics in MER occurs on several levels, some in­tended and some not. The aim of this a1ticle is to present some of the mathematical aspects of the MER mission and to discuss the way they are hidden in the mission, as well as the effect the work context had on the process5 Much of the ac­count is built by letting the JPL scientists speak for themselves, i .e. , by frequently quoting from our interviews. 1

JPL Scientists at Work We found that, in general, JPL's scientists arc people with the highest educational level who join the institution shortly after completing their university studies. They are driven by a desire to he part of the aerospace industry and a passion for planetary exploration. To some extent, they were also drawn to JPL by a fascination with the mathematical, phys­ical, and engineering problems involved in space explo­ration; hut as a motivating factor this seemed secondary.

Among the first to discuss the mathematical aspects of the work at JPL with us was Jacob Matijevic, a mathemati­cian who had been with JPL for a long time. Particularly we discussed the modelling aspects of the work, which takes place before the actual mission is set in motion.

A mission like MER is to a large extent about being able to predict how the technology onboard the craft is going to behave in space or in the Martian environment. Once the craft is flying it is impossible to make adjustments re­quiring more than a radio signal. Everything must therefore function as expected. Take for instance the Mars environ­ment's influence on the instruments onboard Spirit and Op­portunity. You have to have very precise knowledge about the distribution of heat inside the rover and how this af­fects the instruments. To acquire such knowledge, virtual models of the rovers are built in software so that the ther­mic conditions can be simulated. Such thermic models are typically based on a number of differential equations which are solved within the programs. The work for the JPL em­ployee consists of building the virtual model of the rover. The exact method of solution which the program imple­ments is secondary, as long as it works and is not too slow.

According to Matijevic [ 1 1 ] you also need models of how the environment depends on the seasons on Mars to he able to predict the concrete influence on the instruments.

These models are partly based on data from the different Mars orbiters and partly on concrete measurements per­formed on the Martian surface. The correctness of the sur­face measurements depends on how good the description of the instrument's behavior in the Martian environment is, and it cannot be guaranteed. By comparing the data from the orbiters with the surface measurements, a more accu­rate picture may arise; this may then be used to modify the models, so that they slowly become better and better. All of this is done in software. Regarding the models of how the seasons affect the Mars environment, it is probably fair to compare the work at JPL with that performed by an in­stitute of meterology. Matijevic reported,

When I first arrived here over twenty years ago there were still efforts to hand-implement certain mathemati­cal models for certain applications. And there were spe­cialist applications here for specialists in the applied mathematical sciences who worked here to make those

3An expanded Danish version of this article with a slightly different angle has also appeared in the Nordic mathematical journal Normal (13].

4Transcripts of these interviews in full, along with our conversation with Ph1hp Davis, can be found in (7].

UFFE THOMAS JANKVIST was bom in

Copenhagen. He holds a master's degree from

Roskilde University. He is now a doctoral can­

didate there in the use of history of mathe­

matics in mathematics education.

Roskilde Universitetscenter Universitetsvej I 4000 Roskilde Denmark e-mail: utj@ruc.dk

BJ0RN TOLDBOD is a native of Roskilde, Den­

mark He holds a master's degree from Roskilde

Universrty. He is now, as a conscientious objector

to military service, doing his a�emative service

worl<ing at the computer department of the Royal

Danish Library.

Vesterbrogade I 19 A. 3.tv. I 620 Copenhagen V Denmark e-mail: bjto@ruc.dk

© 2007 Springer Sc1ence+ Bus1ness Med1a. Inc , Volume 29, Number 1, 2007 9

applications possible. But over time much of that has been incorporated in fairly standard and available sim­ulation and modelling packages-computer packages. Expansions have been introduced slowly over time to these packages and that's basically how the engineers here do their job. Instead of going back to first princi­ples they apply these tools . . . the foundation theories are from the eighteenth century to a large degree [ 1 1 ] .

Hence a lot of work involving modelling and simulation is done at JPL, but all of it is done in software packages. This might lead one to suspect that JPL has its own staff of math­ematicians developing such packages, but Matijevic in­formed us that the packages mostly come from commer­cial companies.

A few days after our interview with Matijevic we had the opportunity to interview Dr. Miguel San Martin, an engineer with whom we discussed various aspects of the mission. He told with great enthusiasm about the challenges the scien­tists must overcome to make the rovers able to figure out their orientation on the Martian surface. San Martin explained that the navigation on the surface is based on a well-known technique which sailors have used for thousands of years on Earth-you look at the Sun. Together with a vector of grav­itation, which can be measured by the rover, the position of the Sun in the sky provides the information necessary to determine the rover's position on the surface. San Martin didn't think of this problem as being very mathematical. In fact, he claimed that the majority of the mathematics involved in his work was very simple, and he concluded:

The most important is that you have millions of these little, simple things. And that's the trick; to make them all work, and talk to each other and make sure that no parameter is tightened too much or too little. The com­plexity of the space problem is keeping it simple. [ 1 2] Present at this interview was Dr. William Folkner, a

physicist and our main contact at JPL, and he elaborated on this view:

Well, you hit a lot of mathematics in your descriptions, right, because you need to know the positions of the axes of Mars around the Sun as a function of time and you need

to know what the orbits around the Sun and the Earth were. There is a lot of mathematics hidden in what you just said. [ . . . l We've worked all that out for us in the ta­bles. So to know where the Sun is now, you just look it up. Somebody had to figure it out the first time. [ 1 2] Folkner's answer illustrates why the question of what

mathematics is used in MER is difficult to answer. Knowl­edge that mathematics previously made accessible can over time become such an integral part of our conception of the world that we no longer connect it with mathematics. The trajectory of Mars is a good example of this : Is it mathe­matics to look up a table to see the Sun's position relative to Mars at a given time? Perhaps not, but the making of such a table is a mathematical problem. Thus the mathe­matics at JPL is often disguised as "common knowledge."

MER Work Context Incredibly high reliability is demanded of the work per­formed at JPL. A single mistake in a piece of technology or an algorithm may have serious consequences and in the worst case may result in several years of wasted work for hundreds of people. All of the work being done at JPL is therefore subject to careful development and testing.

We asked Jacob Matijevic about the development of the parachutes for the rovers, partly because we thought there would be little interaction between the parachutes and other devices. That is, we thought this would be a "simple" task. Matijevic, however, revealed more:

We did drop tests. We did wind tunnel tests with the parachutes. But even before this time it was through models of the profiles of these devices that we came up with things like what the entry angles would he, what sorts of release points should we be looking at, as well as designing the algorithm that checks for height above the surface and finding out at which time to deploy the parachute and at which time to fire the rockets for slow­ing the descent. All of this was based on what we ex­pected to be the environmental profile that the vehicle would see as it came down to the surface. So this was all done in simulation. [ 1 1]

Figure 2. Guided tour at JPL. Left: Uffe and Dr. Albert Haldemann, who showed us some of the facilities. Right: Visiting JPL's museum for earlier space missions with Dr. William Falkner.

10 THE MATHEMATICAL INTELLIGENCER

Figure 3. The guided tour takes us by JPL's "sandbox" where rovers are test driven. Left: Bjorn in front of the sandbox. Right:

A replica of a Mars Exploration Rover used for test drives at JPL.

Reliability is paramount for any mission. If the choice stands between two different approaches to a problem, a space scientist will be inclined to choose a well-known, well-tested solution over a new-perhaps more efficient­

solution which has not been thoroughly tested in the con­text of the mission. Our interview with Dr. Jon Hamkins, one of JPL's leading coding theoreticians, also confirmed this for the error-correcting codes used in MER [4] . Mathe­matics and technology which had been onboard an earlier mission is considered to be safer and therefore makes a more attractive choice. This approach is taken in all aspects of the missions. Of course some development takes place from mission to mission hut only at a pace that makes ex­tensive testing possible. Jacob Matijevic called this ··steady progress'' [ 1 1 ] . New ideas introduced into the missions will he at least '5-10 years old at launch time, because they must he laid down when the missions are first planned. In the

case of error-correcting codes, a lot of the mathematics in­volved has to he implemented in hardware for speed. Gen­erally hardware is much more expensive to replace than software, so the gain of introducing a new error-correcting code has to he considerable in order to balance the ex­pense of the substitution .

The grand scale of a project like MER also means that the work performed by different departments must be com­pleted at specific deadlines. Not surprisingly, deadlines may serve as a stop block for the development of new ideas. You cannot promise to use a new method if you are not sure that there will be enough time to test it.

There is another factor working against the introduction of new methods: In recent years .JPL has gone from a small number of large and expensive missions to a large num­ber of small hut less expensive missions. For instance, the Pathfinder mission of 1996 had a total budget of 265 mil-

Figure 4. Left: Wind tunnel test of the MER landing module parachute. http: //www.nasa.gov/centers/ames/images/

content/79641main_picture_2. jpg Right: MER landing rnoduit: airhags. http: //photojournal. jpl.nasa. gov/jpegMod/

PIA04999_modest. jpg

© 2007 Spnnger Sc19nce+Bus1ness Media, Inc. Volume 29, Number 1. 2007 11

Figure 5. Guided tour at JPL. Left: A JPL photo of a MER rover testing prior to launch. Right: One of JPL's laboratories.

lion dollars5 whereas the Viking missions of the seventies had a budget of around 8 billion dollars. The MER mission was more expensive than Pathfinder, hut still nowhere near the Viking budget. This means that if anything from a pre­vious mission can be used again, there are huge amounts of money and time to be saved. Many of the cheaper mis­sions must necessarily rely on reuse from earlier missions. The nature of the space missions in general was summed up in the following way by Falkner:

Everything is a cost-benefit analysis. The whole space system is a cost-benefit analysis. [3] To this point we have mostly focused on how external

factors influence the mathematics of a space mission. We now turn to two examples of concrete mathematical prob­lems in the MER mission. Still the purpose of our selection will be to illustrate common features of the mathematics in a space mission.

Two Selected Mathematical Problems of MER The mathematical emphasis of our investigation was the two coherent mathematical theories called channel coding and source coding, which deal with reliable communica­tion and compression of data (including images), respec­tively. Formal definitions of the specific codes involved would take us beyond the scope of this article. We there­fore merely indicate the areas of mathematics involved.

The signals transmitted to and from Mars are subject to interference during their travel through deep space. Such interference of a binary signal may result in bits becoming altered. The communication between Earth and the rovers needs to be reliable. The problem of interference is solved by way of channel codes, which make it possible to cor­rect altered bits in a message; hence the codes are also called error-correcting codes. The system used in MER de­pends on two different codes used in combination. Jon Hamkins explained:

The majority of the missions flying now are concate­nated. So the data comes in and is Reed-Solomon en-

coded, then it goes through a block interleaver and then it's convolutionally encoded [4] .

Thus MER's coding system consisted of two combined, or concatenated, error-correcting codes; a Reed-Solomon code and a convolutional code.

Reed-Solomon codes are so-called algebraic codes, whose code symbols come from a Galois field. Reed­Solomon codes are linear and cyclic. If the data words to be encoded are of length k and code words are of length n, an (n,k)-cyclic code, with code symbols from the Galois field F q• is a cyclic subspace of the vector space Fq. Reed­Solomon codes can either be defined as special kinds of cyclic codes with specific generator polynomials or by means of Fourier transforms. On the one hand, convolu­tional codes are not as mathematically well understood as the Reed-Solomon codes and not as easily defined either. They can, however, be defined by the use of formal Lau­rent series. On the other hand, convolutional codes are very efficient and therefore very often used in technology. They are excellent for correcting single-bit errors, the kinds of errors which most often occur from interference in deep space. Unfortunately the decoding of convolutional codes6

often results in a run of consecutive errors, so-called burst errors. Fortunately Reed-Solomon codes excel exactly in correcting burst errors, hence the concatenated system. The reason for first encoding the data with the Reed-Solomon code and then with the convolutional code is that the de­coding procedure must be the reverse of the encoding pro­cedure. Block interleaving is a technique used to ensure that the hurst errors from the convolutional decoding are no more severe than what the Reed-Solomon decoder can handle.

One of the main purposes of the MER mission was to take photographs of Mars. Before these photos were trans­mitted to Earth they had to he compressed. The image­compression technique primarily used in MER is called The /Ct-1? Progressive Wavelet Image Compression, in short just ICER, and was developed at JPL by Drs. Aaron Kiely and

5http://nssdc.gsfc.nasa.gov/database/MasterCatalog?sc�1996-068A

6JPL uses the so-called Viterbi algorithm in its convolutional decoder.

12 THE MATHEMATICAL INTELLIGENCER

Matthew Klimesh. The word "progressive" refers to pro­gressive fidelity compression. In such compression a low­quality approximation of the photo is first transmitted . Af­terwards hits are transmitted in such a way that the quality of the photo is gradually improved. When all hits are trans­mitted the reconstructed image equals the original image. By stopping the transmission before it is complete, lossy compression ( i .e . , compression with loss of data) can be obtained. In this way ICF.R supports lossless as well as lossy compression, even though it was entirely used for lossy compression. For lossless compression MER relied on the commercial compression algorithm LOCO. The ICER algo­rithm, like many other image-compression techniques. over­all consists of three stages: preprocessing, modelling, and entropy encoding of data . Kiely explained:

We got data coming in, an image or whatever it is, and then some sort of preprocessing stage, for example a wavelet transform plus quantization or a discrete co­sine transform or something. The goal is that it does­n't perform any compression, and in fact it is often a lossy process, it might throw out some of the data. but the idea is to process the data in a way that makes it more receptive to compression through the entropy en­coder. The entropy encoder is sort of the engine. Given some sort of probabilistic model of the source, it com­presses data or represents it in a more efficient way through something like a variable-length code. That is sort of the big picture of what is going on . So for ex­ample for ICER what is going on is mostly a proba­bilistic transform. For LOCO it is in essence trying to project a probability distribution on the next pixel that it is about to encode based on what it has seen in the nearby neighbors. (9]

ICER uses a wavelet transform that closely resembles a Haar tran.iform, a context model also known as a Markov model, and the majority of the entropy codes used by the entropy encoder are the Golomh codes [R]. The LOCO algorithm is a hit different from ICER in that it does not have a pre­processing stage which is typical for losslcss compression

Figure 6. Spirit's landing module as seen from the rover.

http://www.seds. org/-spider/spider/Mars/Hi-res/

mer a lander. jpg

techniques. It does use context modelling, and it uses both Golomb codes and Huffman codes [ 14] .

The Haar transform i s an invertible transform which makes hoth lossy and lossless compression possible. An image with n X n pixels is naturally represented by a n X n matrix M containing integer values. By the invertible transform, say T: R" � R", which is first performed on rows and then on columns of M, a shift of basis from the euclidian basis to an­other orthonormal basis is made. The elements of this basis are called wauelets. The transform splits the image into a low­frequency suhimage and a high-frequency subimage, con­centrating the majority of the energy in the low-frequency area, thus making efficient lossy compression possible.

In a context or Markov model the likelihood of a sym­bol's being encoded depends on the previously encoded symbols; i . e . , the model is said to have a memory. This is a very common situation in digital images, where the value of a certain pixel may indeed depend on the values of the surrounding pixels. ICER's context model maintains a sta­tistical model with the purpose of estimating the likelihood of the next hit's being a zero bit.

Huffman codes are variable-length prefix codes, which means that they can assign shorter codewords to more fre­quent data symbols and that no codeword is a prefix of another codeword. Golomb codes make up another family of codes which are parametrized by an integer m > 0 and therefore are often written as <§m· A Golomb code encodes integers under the assumption that the larger the integer the less likely its occurrence is. It can be shown that Golomb codes make optimal encodings for geometric probability distributions of non-negative integers, which make them at­tractive in many different contexts.

The above presentation of mathematics in MER is , of course, merely scratching the surface. Other mathematical theories that we came across during our interviews at JPL include the "Lost in Space" problem, pinpointing the craft's position in space Jt a given time; Kalman filtering, esti­mating incomplete and interfered-with data from the craft's different sensors; the Hohmann trajectory, the trajectOiy he­tween Earth and Mars which calls for the least amount of energy at launch; and of course control theory. Discover­ing every little piece of mathematics put to use in the MER mission probably is an impossible task, as Folkner im!i­cated:

There is mathematics in everything. There is control the­ory, aerodynamics, orbital dynamics, Newtonian gravity , bodies going around the Sun. We use general relativity, that's mathematics of physics. [ . . . ] Linear algebra is a field of mathematics we use all the time. Matrices. That's in the control theory all the time. There is Riemannian geometry in the general relativity. Calculus. [ 1 2]

Instead, we shall now turn to the theme of this article, the hidden mathematics of MER.

Hidden Mathematics of MER When you view a space mission from the outside, it is clear that the mathematics involved is to a certain degree hid­den (or invisible). We had imagined that once we entered the JPL facilities the mathematics of MER would suddenly become more visible. Our investigations showed otherwise.

© 2007 Spnnger Science+Bus1ness Media, Inc , Volume 29, Number 1. 2007 1 3

The mathematics of a mission like MER is hidden on several different levels. To most people reading about space missions in a newspaper, the mathematics is hidden in a lot of technology whose mode of operation is seldom dis­cussed. Partly, the media fear boring their readers; but even when they try, it is difficult to communicate abstract math­ematics to the uninitiated. This type of invisibility seems obvious.

More interesting, the mathematics to some extent also is hidden inside the walls of JPL. As we pointed out, a large part of the mathematics for solving the mission's problems is embedded in software packages-and therefore hidden. The packages are often commercial packages, developed out­side JPL. To users of such packages it is important to know how to use the packages correctly, and this most certainly requires mathematical knowledge, but it is not necessary for a user to understand the specific mathematics in detail. One might say that the mathematics is outsourced, and that the implementation in software packages contributes to hiding the mathematics from the people at the ]PL. As the conver­sation between Falkner and San Martin reported above il­lustrates, a consequence of the mathematics being hidden is that to some degree it is being ignored.

The virtual models of the rovers and the models of the Martian environment of which Matijevic speak above rely also on the mathematical technique of modelling. If the sci­entists of JPL who prepare these simulations do not regard the modelling as mathematics-and our investigation sug­gests that to some degree they may not-then this mathe­matics will also be hidden from them.

Anyway, the JPL projects are so vast that a distinction between being inside or outside is somewhat meaningless. A scientist who works with a specific part of the mission of course has an overall perception of the mathematics in­volved in other parts of the mission, but as the following quote from one of our interviews with Falkner shows, the details are sometimes quite hidden. Falkner's first remark was a joke intended to illustrate that the use of mathemat­ics in a mission like MER is so extensive that it would be easier to expose the mathematics not involved.

We don't do any map theory for instance. Four-color map problem. You know that? We don't do any of that. We don't re­ally use any abstract algebra, group theory, and that sort. Except in the channel coding. They use abstract algebra and group theory in that? The Reed-Solomon codes are based on Galois Fields. That's news to me. I didn't know that; all right, inter­esting. [ 12)

Worth noticing is that Falkner says, " They use abstract al­gebra and group theory in that?" This suggests that the dif­ferent departments of JPL may not have a large degree of interaction. In fact William Falkner and Miguel San Martin had never met each other before we interviewed them to­gether. The point here is that the vastness of the project contributes to hiding the mathematics from the employees.

One last kind of invisibility which must be mentioned is that certain areas of the missions are classified: some things are deliberately hidden. Our experience at JPL was that the institution generally was very open to giving in-

14 THE MATHEMATICAL INTELLIGENCER

formation and answering questions, but also that secu­rity, control, and classification were part of a normal work day. Thus when we talked to Mark Maimone about the amount of control theory used for steering the rovers, he replied at one point, "I don't know what has been pub­lished about the details of those algorithms, so I can't tell you anything about that" [ 10). We have not examined the motives for such classification any further, but it seems fair to say that classification makes it harder to discover the mathematics involved and therefore contributes to the hiding of it.

When discussing the hidden mathematics of a project like MER, one question springs to mind: Would the applied scientists of MER have been able to benefit from knowing more about the mathematical tools they use in their work? Perhaps. Our investigation does not imply a unique answer to this question. If we restrict our attention to the model­ling and computer simulation aspects of MER, there might be something to say on this topic. Whenever working with commercial software and standard packages users may have only a general idea of the (mathematical) elements involved. In a large-scale project like MER, different people, and dif­ferent departments work on different aspects of a computer simulation. Indeed, scientists do not always know in detail what is going on 'on the other side of the fence. ' In recent years the engineering industry has been experiencing a 'par­adigm shift' in the way engineers work: the testing of scale models has now been replaced by computer simulations, and the engineer most likely does not know anything about the model(s) on which the simulation is based. Now, this is a scary scenario, for wrong applications or wrong inter­pretations of the simulations may lead to disastrous mis­takes. In such situations the applied scientists would most definitely benefit from knowing more of the ramifications of the (hidden) mathematical tools they use. That they don't is quite paradoxical: the space missions are devoted to min­imizing mistakes, yet the nature of the space missions may itself inflict errors-just because the mathematics in use is hidden.

Conclusions A lot of the mathematics of MER is hidden and not only from the public but even from the applied scientists work­ing on the mission. As briefly sketched above, for the sci­entists, this could be disastrous in a worst-case scenario. The hiding of mathematics, both in our everyday life and within science itself, is a matter not often discussed in pub­lic-which in itself is a disaster, taking into account the consequences the hiding of mathematics might have for the public. We like to think that this article may help let in some light.

Another question raised by our work is that of beliefs in mathematics. Only occasionally are the beliefs of mathe­maticians discussed. We found repeatedly that mathemati­cal elements of MER are not actually considered to be math­ematics among the applied scientists themselves, not on first hand anyway. Is this due to the fundamentally differ­ent views of what mathematics is between applied scien­tists (including engineers) and pure scientists of the 20th century? We do not know.

Finally, we comment on the nature of the mathematics involved in MER. Because of the extreme nature of a Mars mission, one might expect "extreme'' mathematics, mathe­matics developed for the sole purpose of this mission. This does not seem to he the case. We did not come across any basic innovations in mathematics as a result of the MER mission. The MER mission is based on well-established mathematical theories and disciplines. Some of these date back to the 18th century, but a lot of them arc also from the 20th century: the Hohmann trajectory is from 192'i, the convolutional error-correcting codes used arc from 19'i4, and the Reed-Solomon codes are from 1 960, the Lost in Space problem and Kalman filtering arc also from the 1 960s. Image compression relies on the Haar transform from 1 910, Huffman codes from 19'i2, Golomb codes from 196(J, and wavelet theory from the 1990s.

Now, this is not to say that basic research in mathemat­ics is not used. Rather it suggests that technological and math­ematical developments seem to he on different timelincs so that a stop in research in pure mathematics would damage technological development. And since so many of the math­ematical theories in MER are from the 19(JOs, one might think, that the timelines in the case of the space industry arc no more than fifty years apart. Feedback also goes the other way: basic research in mathematics can he inspired by prob­lems in applied sciences and technology [ l J-therefore also from missions like MER and institutions like JPL. But the ma­jor role of JPL seems to he that of consumer of already de­veloped applied mathematics. From start to finish MER is an example of the application of mathematics. But not just any example. It is an extraordinary example of what can he ac­complished with the mathematics at our disposa l .

ACKNOWLEDGMENTS First of all we would like to thank all the employees at JPL who set aside their duties to talk to us, especially Bi l l Folkner. who arranged most of our meetings. We also thank Phil Davis for taking such an extraordinary interest in our investigation and for commenting on this article. Furthermore we thank Tinne Hoff Kjeldsen, Man-Keung Siu, and Chandler Davis for helpful suggestions about how our investigation should he presented here. Thanks also to Anders Madsen and Bernheim

BooiS-Bavnhek for suggestions and comments on the original thesis.

BIBLIOGRAPHY [ 1 ] P. J. Davis. Interview with Professor Emeritus Philip J . Davis, March

6th, 2005. Recorded at Brown University, Providence, R l .

[2] P. J . Davis. A Letter to Christina of Denmark. EMS, pages 21-24,

March 2004.

[3] W. Falkner. Interview with Doctor William Falkner, March 1 7th,

2005. Recorded at JPL, Pasadena. CA.

[4] J. Hamkins. Interview with Doctor Jon Hamkins, March 1 4th, 2005.

Recorded at JPL, Pasadena, CA.

[5] U. T. Jankvist and B. Toldbod. Mathematikken bag Mars-missio­

nen - En em pi risk unders0gelse af matematikken i MER med fokus

pa kildekodning og kanalkodning. Master's thesis, Roskilde Uni­

versity, October 2005. Tekster fra IMFUFA, nr. 449a.

[6] U. T. Jankvist and B. Toldbod. Matematikken bag Mars-missio­

nen - lndfmelse i den grundlceggende teori for kildekodning og

kanalkodning i MER. Master's thesis, Roskilde University, October

2005. Tekster fra IMFUFA, nr. 449b.

[7] U. T. Jankvist and B. Toldbod. Matematikken bag Mars-missio­

nen -Transskriberede interviews fra DTU, Brown University, MIT

og JPL. Master's thesis, Roskilde University, October 2005. Tek­

ster fra IMFUFA, nr. 449c.

[8] A. Kiely and M. Klimesh. The ICER Progressive Wavelet Image

Compressor. lPN Progress Report, 42(1 55) : 1 -46, November 2003.

[9] A. Kiely. Interview with Doctor Aaron Kiely, March 1 4th, 2005.

Recorded at JPL, Pasadena, CA.

[1 0] M. W. Maimone. Interview with Doctor Mark W. Maimone, March

1 7th, 2005. Recorded at JPL, Pasadena, CA.

[1 1 ] J. Matijevic. Interview with Doctor Jacob Matijevic, March 1 4th,

2005. Recorded at JPL, Pasadena, CA.

[1 2] M. San Martin and W. Falkner. Interview with Doctor Miguel San

Martin & Doctor William Falkner, March 1 4th, 2005. Recorded at

JPL, Pasadena, CA.

[1 3] B. Toldbod and U. T. Jankvist. Reportage fra en Mars-mission.

Normat, 54(3): 1 0 1 -1 1 5 and 1 44, 2006.

[1 4] M. J. Weinberger, G. Seroussi, and G. Sapiro. LOCO-I : A Low

Complexity, Context-Based , Lossless Image Compression Algo­

rithm. Proceedings of the IEEE Data Compression Conference,

pages 1 -1 0, March-April 1 996.

"Celebrated Turtle" Plea�ed and proud as \'d.' were to present the mono-monostatic nml • ( ,-ol. 2H, no. "!. 3-1-3R) of Gabor Domokos and P. L. Ya rkon ·i, a nd gratefu l as we are to the artbt for ih beaut ifu l pr •s ·ntation on the cover of th ' issue. we '' cr •n't quite prt'pared for the media frenzy it excited in H ungary and far herond. Th ·re ar • dozens of ar­tid ·s ,1hout it-sec. for example,

http; WW\\ .lahoodlc.hu nc news ne,,-s_archi,-e singlc_page article 1 1 hungarian_ sc ?cHash� i2Sf3fct7b The story has h L'n pick •d u p h} Reuters and broadcast around the world: see. for e. ample,

http:/ WW\\ . msnlx.msn.com id 1 - 1 20210/. And go to

Imp:/ ww\v .sci am .com and d ick on Biolog} .

Gabor comments. "I am tell i ng vou this to sho\\' hem much impa t your \York ha., in u nli ke! corners of the world." \\'e r •ply . with all due t hanks, that the \\'ork '' .1s the authors . not our. , and that Hu ngary, if indeed it is a corner of the world, is a rel::nin:l} l ikely corner.

-The Editors

© 2007 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 29, Number 1 , 2007 15

ijfi(W·£.(.1 David E . Rowe , Ed itor I

Hardy as Mentor MARJORIE SENECHAL

Send submissions to David E. Rowe, Fachbereich 17-Mathematik,

Johannes Gutenberg University,

055099 Mainz, Germany.

Godfrey Harold Hardy ( 1877-1947) is a legendary figure. Arguably England's greatest mathematician

since Newton, he was a number theo­rist and analyst of the first rank, a su­perstar of Cambridge and Oxford be­fore and between the wars, and a gifted writer and editor. Unlike Newton, Hardy was not a loner: his 35-year 100-paper collaboration with]. E. Littlewood is the longest -running in mathematics history; his 5-year collaboration with S. Ramanujan the most dramatic. He was also a dedicated don who "took im­mense trouble with his students whether they were good, bad, or indif­ferent."1 One of those students, later his colleague, was a young woman, Dorothy Wrinch.

Dorothy Wrinch (1894-1976) is a perplexing figure. A Wrangler at Cam­bridge, the first woman to receive an Oxford D.Sc. , and a popular lecturer and tutor, she published 50 papers in pure and applied mathematics and phi­losophy by age 40 and then turned her formidable talents and energies to the nascent science of protein architecture. She's remembered in that field for her elegant but controversial geometrical model (eventually rejected) and the re­search efforts she and it catalyzed on both sides of the Atlantic. Today's pro­tein-folding problem has roots in that catalysis.

Wrinch emigrated to the United States in 1939 and began teaching in the Physics Department at Smith Col­lege a few years later. When I joined the mathematics department in 1966, she was writing on crystal geometry. That was a subject I wanted to learn, so I studied with her informally for sev­eral years. Sharp of eye, mind, and tongue, Wrinch was an inspiring, ex­acting teacher and delightful company. From time to time she dropped hints about her unusual career, but never that I was one handshake away from Hardy, Russell, and Watson.

I learned that after she died, when I read the papers she'd left to Smith's

16 THE MATHEMATICAL INTELLIGENCER © 2007 Spnnger Sc1ence+Business Med1a, Inc

archives: preprints and reprints, note­books and jottings, grant applications, referees' reports, letters she received, and drafts of replies, both sent and un­sent. 2 I pieced together accounts of her work and life and returned to geome­try .3 Over the years, others have added bits to the literature. But something keeps nagging me: none of us got her quite right. So I've returned to her pa­pers (at Smith and elsewhere) to con­nect the dots and place her life in per­spective.4 The picture is complicated, and this is not the place to draw it.S

But Wrinch's correspondence with Hardy is a story on its own.

Wrinch met Hardy circa 1913. As an undergraduate in Cambridge, she at­tended his lectures, and he supervised her first graduate research. Later, in Ox­ford, he tried to help her get back on her feet in a personal crisis. The Hardy in her papers is a Hardy I didn't know. Of course, I never met him in person, but I felt that I had: I'd carried A Course of Pure Mathematics, 1be General The­ory qf Dirichlet Series, Inequalities, An Introduction to the Theory of Numbers, Ramanujan, A Mathematician 's Apol­ogy from office to office since graduate school; I'd read every biographical sketch I could find. 6 I was not surprised to find that Hardy's letters are as clear and graceful as his published writings. In both, he said what he meant and meant what he said. But I was unex­pectedly touched by his patience and humor. Nor did I expect to find a staunch feminist (as he understood the term). I knew he'd directed Mary Cartwright's D .Phil . and encouraged Olga Taussky 7 But Hardy went to bat not only for budding stars but also for Wrinch and for others whose names are forgotten (or illegible in his calligraphic longhand). He thought highly ofWrinch and her mathematical abilities, but he never exaggerated them. "Hardy was a master who knew to an inch which of his work had value and which hadn't," said C. P. Snow.8 He judged the work of others on the same strict scale.

Cambridge and London Dorothy Maud Wrinch was horn in Rosario, Argentina to British expatriate parents, and grew up in Surbiton, near London. In 1913 she won a mathemat­ical scholarship to Girton, one of Cam­bridge University's two ( then unofficial ) colleges for women. ''Contrary to the current opinion we Girton girls did not wear severe shirt blouses with formal ties, nor did we drag hack our hair , " Dora Black, her best friend at Girton, recalled a lifetime later, hut that's how they posed for the official class photo­graph 9 Wrinch is second from the left in the first (full) row.

Lectures and tutorials were held on the premises of this "vast edifice in three main courts-C. S. Lewis once termed it 'The Castle of Otranto' . " 10 The mathematics lecturers and tutors in 1913 were, the Girton Register recorded, "Mr. Berry, Miss Cave-Browne-Cave, Miss Meyer, Mr. Munro, Mr. Nicholson, Mr. Watson, Mr. Webb." Mr. Watson di­rected Wrinch's undergraduate studies; later, she helped proofread the 1920 edition of Whittaker and Watson's A Course qf Modern Ana�ysis. Students also biked into Cambridge for lectures given by "Dr. Baker, Dr. Barnes, Mr.

Bennett, Mr. Beny, Mr. Cameron, Mr. Hardy, Mr. Herman."

Red-haired, freckle-faced Wrinch "was a mathematician of high ability and consequently consumed with ad­miration for Bertrand Russell ," wrote Black, but Russell's name was not on Girton's lists. The undergraduate math­ematics curriculum at Cambridge was geared to the famed and formidable Mathematical Tripos, which did not en­compass logic. Wrinch dutifully put her shoulder to the wheel and earned the coveted laurel of Wrangler (on Part II) in 1916 .

Wranglerhood freed her to study whatever and with whomever she liked-subject to the approval of Gir­ton's Mistress, Katherine Jex-Blake . 1 1 "This morning I had a visit from a Gir­tonian named Miss Wrinch, whom I had never seen before, " Russell wrote to his lover, Bloomsbury hostess Ottoline Morrell, in June, 1 916, 1 2

She wants to learn logic, but Girton doesn't want her to, and threatens to deprive her of her scholarship if she does. She has just finished her mathematical tripos, and is just the sort of person who ought to do it­she is very keen, hut has not a penny

beyond her scholarship, and is at the mercy of Miss ,lex-Blake, a Churchy old fool. It makes me mad. If she could get £80 a year, she would ig­nore the college. But she doesn't see how to. If I had it, I would take her on as my secretary. But I don't see how I can manage it.

Wrinch got her fellowship extended through 1916-1917, enabling her to study for the Logic Tripos Part II . In the meantime, Russell's outspoken defense of pacifists led to his dismissal from Trinity. 1 3 "In fulfillment of his dream of being 'like Abelard' , [he] continued to teach mathematical logic to a small group of students who came to Gor­don Square every week for informal lectures on Principia Mathematica"H ; Wrinch persuaded ,lex-Blake to let her be one of them 1S

Hardy entered the picture the next year, when Wrinch began research . 1(' "Dear Miss ,lex-Blake," she wrote, 17

On the advice of Mr. Russell, I have approached Mr. Hardy, who has consented to direct my studies dur­ing my research work next year, if I am able to come up. The subject I propose for research is the appli­cation of mathematical logic to the

Dorothy Wrinch (second from left, first ful l row) with htT ent<:ring class at Girton, 1913. Photograph courtesy of The Mistress and Fellows, Girton College, Cambridge.

© 2007 Spnnger SCience+ Business Med1a, Inc. , Volume 29, Number 1, 2007 17

theory of sets of points. I have dis­cussed this subject both with Mr. Russell and Mr. Hardy and they agree that it seems to be a suitable one as there is a certain amount of work to be done in it, in putting the mathematical theory on a satisfac­tory logical basis.

But, Wrinch continued, she would need extra funds. The studentship (stipend) she'd received for research paid only 40 pounds; her expenses at Girton for "tuition fees, university and library fees, board and lodging, cabs into Cam­bridge and meals there when neces­sary; Long vacation term fees; personal expenses including books, travelling, etc; dress, etc; doctors and dentist's bills" had averaged £175 a year.

Hardy, Wrinch's nominal adviser, skillfully (and successfully) helped ca­jole Jex-BlakeY'

I have seen Miss Wrinch again, and also Mr. Lloyd, the Secretary of the Committee. I think now that my es­timate of 80 lbs (as the most worth asking for) was under the mark. I have advised Miss W. to apply for 1 20 lbs-it would, I think, be use­less to try for more, but I think she might get this.

I have to approve of the amount: that means in practice that you have to. For naturally I cannot say and would endorse any amount which there was a chance of getting and which you thought reasonable: and I take it she will be by no means well off on 1 20 + 40 lbs a year. The amount given is not supposed to de­pend on the degree of merit of the candidate but solely on financial needs. Miss Wrinch is able and de­termined, and I think she should jus­tify the grant: but of course I can­not say that she is so good that she simply must be allowed to re­search-and therefore it would be very inadvisable to apply for an ab­nonnal grant. It seems that most of their grants are of 100 or 1 10 lbs (not 60 or 80, as I thought). In view of the somewhat higher expenses of Oxford or Cambridge (even for a woman) it would be natural and rea­sonable to ask for the larger sum.

In March, 1918, Russell summarized Wrinch's accomplishments: 19

Miss Wrinch's notes consist in the main of an interesting development

18 THE MATHEMATICAL INTELLIGENCER

of certain ideas suggested by Haus­dorff: they deal with the investiga­tion of series constructed by the "principle of first differences. " There are a number of new results, and the method employed is obviously a fruitful one, giving possibilities of very important theorems. The sub­ject is one upon which, hitherto, not much work has been done; but it is very desirable that it should be in­vestigated carefully, on account of its connection with some of the out­standing problems of the theory of aggregates, notably the comparabil­ity of the cardinal number of the continuum with the Alephs. Miss Wrinch's classification of the ele­ments and gaps in various very gen­eral types of series is original and valuable. Moreover it points the way to a whole field of new research. Two months later Russell was sent

to Brixton Prison for sedition. Wrinch, now his assistant (running errands, in­dexing books), sent reading material to her incarcerated mentor and wrote him long newsy letters. Hardy was urging her to write up some papers but blanched at the prospect of reading them.20

August 20 . . . . [Whitehead] has had an ms of mine for 10 weeks now and has not done anything about it yet. It is a little paper on the condi­tions under which a series pQ is Ded. you remember I disagreed with your result in Vol. 3 (of Principia Mathe­matical . Hardy urged me to write it out at length, saying he would pub­lish it in the Camb. Phil. Soc. Proc. and so please the Research Dept. by showing them something of mine cut and dried. But his heart failed him I suppose, and he sent it to W. Where it will remain I expect. W. does not seem very keenly interested in the topics treated in the first half of Vol. 3 and will be frightfully bored by this paper of mine.

August 23 . . . . I really must go and tear my ms. from Whitehead as I want it. Have to write a thing for the Res. Dept. by October 18th. It is rather annoying as the more inter­esting things I have come across are still unfinished and not yet ready to face the light. Does the notion of mediate cardinals of various degrees please you?

September 3 . . . . You will be glad to hear that ms. of mine is found! I am so relieved about it. Sup­pose I were to be able to get cer­tain short papers of mine [2 or 3 on serial types (1 of these you saw in a ragged state in March) and one on mediate cardinals and perhaps one other] thro' to you , would you be so kind as to glance at them? I don't like sending to Hardy because he doesn't like the (??) of reading them-as he has to get up all the symbols each time-or he sends them to Whitehead which is worse as they never come back-before he can pass them for the Research Dept. I suppose they will have to be read by somebody. If you would not mind reading them, could you send me a report on them addressed to Hardy?

From 1918 to 1920, Wrinch held a re­search fellowship at Girton and taught at University College London concur­rently21 ; she received an M.Sc. from the University of London in 1920 for her thesis on hypergeometric functions and a D.Sc. a year later. She also began a productive collaboration on probability theory and scientific method with a young colleague, Harold Jeffreys. "I should like to put on record my ap­preciation of the substantial contribu­tion she made to this work, which is the basis of all of my later work on sci­entific inference, " Sir Harold wrote af­ter her death 22

Oxford After the war, the Cambridge constel­lation faded. Hardy moved to Oxford as Savilian Professor of Geometry in 1919; Russell married Dora Black in 192 1 . Wrinch married John Nicholson, a Fellow of Balliol College, in 1922 and began teaching at Oxford's women's colleges a year later. Soon she was Di­rector of Studies for the mathematics students at all five of them, a post she would hold for fourteen years. "I have seldom known a teacher who aroused so much interest in her pupils ," wrote the Principal of Lady Margaret Hall, Lynda Grier. "They have again and again spoken to me with enthusiasm of their work with her, and their enthusi­asm has not been confined to what they say of their work, but has extended to her wide range of interests, and her

knowledge of subjects artistic and l it­erary extends far beyond the scope of her specialized work. ''25

Wrinch was very active in mathe­matical circles at Oxford . She no doubt met Hardy often at lectures, meetings, and professional and social events, though she left no accounts of these encounters. From referees' reports on her papers, which she did save, we glimpse Hardy as an editor 24 Whether the news was good or had, he played the role of a neutral observer, hut not an indifferent one. "For myself, I have no particular opinion, as it is a ques­tion about which I have no competence to speak," he concluded one negative report. "But I can only advise you to withdraw it, as I feel that the Council has really made up its mind against it . "

Hardy and Wrinch were Oxford's delegates to the 1928 ICM meeting in Bologna; she spoke "On a method for constructing harmonics for surfaces of revolution. " (Nicholson telegraphed daily to report on their one-y<:'ar-old daughter. )2s The next year, Wrinch sub­mitted fifteen closely related papers-a third of her then-published oezwr�to the Board of the Faculty of Physical Sci­ences for consideration for an Oxford D.Sc. The University appointed G . B. Jef­fery, a professor of mathematics at UCL, and A. E. H. Love, Sedlerian professor of Natural Philosophy at Oxford, as judges. 26 In an addendum to their posi­tive report Love singled out two papers, ''On the asymptotic evaluation of func­tions defined by contour integrals"27 and "Scl!ne boundary problems of mathe­matical physics" ,2H for special praise.

Wrinch had boundary problems of her own: for better or worse ( as it turned out) , in her published work she knew none. "On some aspects of the theory of probability,'' "On the expo­nentiation of well-ordered series , " "On the structure of scientific inquiry . " "The relation of geometry to Einstein's the­ory of gravitation, '' "A generalized hy­pergeometric function with n parame­ters , " "On the lateral vibrations of bars of a conical type" are the titles of a fev.: of her earliest papers. Scientific method led her to sociology; the paper on con­ical bars was sparked by questions on spicules in sponges (her interest in morphology would steadily grow) . Wrinch also wrote a hook o n the lo­gistical nightmares of British homes, the

dearth of child-rearing services, and a utopian design for a better future. Wisely, she published this under a pseudonym. 29

As the 1920s drew to a close, things fell apart. John Nicholson, F.R.S . , "an influential member of many scientific bodies, including the Royal Astronom­ical Society (a fellow from 191 1 ) , the Rhntgen Society (president 1921-1922), and the British Association committee on mathematical tables (chairman 1920-1930) , '' imploded. His family and friends had seen it coming. '' In the later twenties Nicholson published little and his teaching deteriorated. In May 1930 he was treated in Norwood Sanatorium for alcoholism. Shortly afterwards a conjugal separation was agreed.''5°

Desperate for a change of scene, Wrinch applied for fellowships that would take her and Pamela abroad. One of these was a Rhodes Traveling Fellowship, designed to permit "a num­ber of Fellows or Tutors in residence

at Oxford to travel and study or con­duct research overseas, particularly in the countries from which Rhodes Schol­ars are elected." Oxford's leading math­ematicians, Hardy, Love, and Milne, and the Principals of the women's col­leges, supported her application with enthusiasm.

"The Fellowships will be open to graduates whether men or women," the Rhodes announcement continued, hut H. A. L. Fisher, a Rhodes Trustee and Warden of Hardy's Oxford college, made sure no woman would receive one. Hardy put up a fight. "Dear Mrs. Nicholson, '' he wrote in December, 1929,31

I got at the Warden on Friday and had a long argument with him, hut a very unsatisfactory one. . . . My disappointment was that I just failed to get any sort of sympathy out of him. He did seem to be definitely anti-feminist: he said for example that their procedure now was to tour

Hardy's Testimonial for Wrlnch's application for a Rhodes Travelling Fellowship

New College, Oxford. March 26, 1930

�Irs. D . 1 . \\ rin h 1 hols m tells me that he int nds to appl} for one ol t he Hhod � I rm · l l ing I ellm\ I up . and I am ' c r\ glad to h,l\ e the op­pmtunH} of supporting h 'I .tpplt ation

1\l r i hoi t n 1s ' cr\ '' e l l kno\\ n 1n cient1fk urd • for her origin.tl " ork .1 ,1 I ctur ·r . • md t a h<..·r .tnd for her •eneral .th1lrt} . he h.t .m unu ual rang • of mtercsl'i, and her l:t ntnbut 1011 to mathematiCs < ' er a ' CIJ " rd • field, from .tppli d m:.lth ·mattu; to .m.t l} 1s .tn d mathc1n.tttatl phtlo� ph} I km '' from rn O\\ n e penence that he is m excellent le -turer .mel th.tt she ha pi 'nt\ to o;,t\ for her If about h ·r O\\ n subJeCt or .t h ut the \ orl I m rcn r:al.

'JlH:t • 1s no doubt that .,h ' ould he .1 \ Cl) mter t i ng addttion to the '>ociet) of a m l n t \ er I t) m t h l nrt d 1..'\te of m ·n 1 or el C\\ here.

Sh • b ,tl o \ 1) k ·nl} mtcrcsted both 111 edu .ttum gcneralh and m \meri :an duc.ttt n in p rtr ular, a nd 1 .tn 10u to .1 ertam for her.;elf

'' h.tt rngl r h ;.tnd \men an unr\ •rs1t 1cs ha\ · to lc.tm from one an ther I think that she '' ould b in mam '' •'} a partrcularl} good person to end on this ort of duc-ational mi 1011 he ' auld not he preo upied too L' d wm cl } \\ i th the intt:re t ot re ear h as 1 inC\ Itablc \\ lth an� one \\ ho c e p ·rience of \rn ·ric.t 1 g.unc I, l ike rn} 0\\ n . • t an e change profcsso1 On the other hand h �r JUdgment \\ auld not he clouded, as that of o man} Lnglish \1.: itor to \mcrica sc:cm to me t b •. h an pr JUdt c \\ lth rc gar I to the xt ord .tnd C,mthndge } tem.

For th e r ·.tson . then I am onfid Ill that un \\ hose appointment \\ ould he p.trticularl

po,.,es of the I nt tees

(,, H l iard

© 2007 Springer Sc1ence+ Bus1ness Media, Inc., Volume 29, Number 1, 2007 19

the men's colleges in a fixed rota­tion. He obviously had the idea that you wanted the position simply for leisure for research, and that (while tolerant of research) they regarded it as quite secondary. Of course, Fisher is an awkward customer when (??) at him at the wrong an­gle. Anyhow, I was just discouraged and ultimately shut up.

All I can say is this: if you, Miss Jun or Miss (??) can get yourself of­ficially in posttton, either for 1930-31 , or for any later time, I will do what I can. It won't be very ap­preciable, hut I will say what I can as strongly as I can. It does seem to me quite obscene that your claims are, at the lowest, a good deal stronger than those of some people who have had them. Also, our beloved Warden did put my back up very much, and the less he wants you to have one the more I do. But it's no earthly good my resuming the topic directly with him.

Wrinch reapplied for the Rhodes. Among her papers I found "Agenda for GHH March 5 30: AMERICA How far are the following typical and typically different? New York Univ, Los Angeles, Michigan; Where are the intelligent people in USA to be found? Quite un­officially, where are the best schools of mathematics in USA? work in general analysis?" According to her notes, Hardy suggested "autumn in the east, visiting Princeton, Johns Hopkins, Yale; to stay in New York or Harvard and Mass Institute of Tech; to visit the Rice Institute of Texas (Evans and Van Diva), the University of Wisconsin at Madison or Ann Arbor Univ. of Michi­gan, and to spend the winter in Berke­ley and at the In. of Tech. Pasadena (Bateman). " He drafted a letter to the Rhodes Trustees on her behalf (see boxed statement) , and sent it to her for approvalY

Here is the testimonial : of course it is very difficult to produce exactly the sort of pomposity required, and I will alter, in reason, anything you would like altered. Is p 2 tactful or not? you may think it dangerous, hut on the whole I am disposed to think that the suggestion that it isn't de­sirable that all Rhodes Fellows should be too exactly like Ridley is more likely to advance your

20 THE MATHEMATICAL INTELLIGENCER

prospects than to prejudice them. You see, if you try to pretend that you are exactly like Ridley, you are attempting the impossible, and it is better to make a virtue of necessity. Ask Miss Fry if you are doubtful.

Still Fisher didn't budge. Wrinch also applied for a Rocke­

feller Research Fellowship.-'�3 She wrote to Hardy,54

With regard to the Rockefeller busi­ness; to spend a year quietly in Got­tingen in doing mathematics is of course far nearer my heart than stumping America, much as I want to do that. I cannot easily overstate the intensity of my desire to have leisure and opportunity to do some serious mathematics, so I shall be thankful to you for the rest of my life if you are able to help me in get­ting a Rockefeller Fellowship.

Now as regards my original sug­gestion to you that I should ask for it in order to study. I feel on second thoughts that this would be putting myself outside the field covered by the Rockefeller trustees who always talk so much about Research in their reports. Also I am still very keen and much interested in my current re­searches. It seemed [?] therefore to ask for a Fellowship for "study and research" to spend part of my time on continuing the present work, which is at present badly held up for want of time . . . . In these re­marks I am referring to the type of Fellowship which was secured by our friend Miss Y . . . .

I must say again how frightfully grateful I am for your help. Thank you very much indeed.

Hardy judged the Rockefeller a very long shot: Wrinch's varied accomplish­ments did not add up to any one big thing; the breadth of her research would count against her. But he would support her.3o

What I say now is in substance what I said in conversation, but it is eas­ier to be precise on paper. The peo­ple I have recommended before have all been 100% specialists in analysis. In every case I have been able to say-there are the works, perfectly clear cut, just what I say they are, and I can challenge any expert to say the contrary. My only rebuff has been over Walfisz, a Pol-

ish Jew who, for some reason I do not know, seems to have made a disagreeable personal impression. He was, beyond all question, a very fine analyst, who knows his special field better, perhaps, than anyone in the world bar Landau-so you can judge from that that the present stan­dard is a very high one . Take Zyg­mund, for example: he knows the theory of trigonometrical series bet­ter than Littlewood and I do.

Now I don't say that an applica­tion put on different grounds might not be successful. If it were a 'na­tional' Fellowship, as opposed to an international one, no doubt the chance would be a very good one­you would certainly have good rea­son for complaint if you couldn't get what Miss Y got! As it is, I should be prepared to try, but I should not be at all sanguine of success. In any case the only chance would be to put all the cards on the table and trust to finding them in a sympa­thetic mood; one wouldn't be deal­ing with a bunch of pomposities like the Rhodes Trustees, but with peo­ple like Birkhoff, Veblen, and it would be no use at all trying to put an impossible bluff across, even if I were prepared to try. That is to say, I should have to put the case broadly as you put it to me: you put it fairly and reasonably, and I agreed with what you said; but I am afraid the odds are 5 to 1 that they would re­ply, not that the case was not a good one in its way, but that is was not the sort of case they required.

Hardy was indefatigable on Wrinch's behalf. He advised her, again,36

. . . As regards Rockefeller: I have written to Tisdale for the necessary forms and will act with Love. Re­member

(1) the parallel with Miss Y is not a sound one: she had a National Fellowship (for which the standard is a good deal lower)

(2) each Board (nat. and Inter­nat. ) meets at stated intervals, and applications can only be dealt with then. I know the National Bel. meets in April (I was told so a day or two ago by an American on behalf of whom I had to write a letter): whether the other does also I don't know; there may be some sort of

joint meeting of trustees. To get an application in in time for an April meeting may he impossible, will cer­tainly be difficult: no harm in trying, of course

(3) It is essential to have a letter from someone in the place where you are going (e.g . , Gottingen) , say­ing that he will be glad to have you. Under whom does Rellich work, or what is his status? Is Courant the right man to ask? Landau would nat­urally say that he didn't know the subjects you proposed to work in

(4) they do not make any sharp distinction between 'study' and 're­search' . What they mean is that these Fellowships are not meant for 'stu­dents' in the ordinary sense. hut for people of a more mature type who have shown what they can do, and have a definite programme. It is therefore no good saying that you want to learn a subject in the vague hope that you might find something in it to do.

In your case there is no sort of difficulty of principle: it is merely a question whether what you have ac­tually done will prove strong enough to overcome the competition. We will of course make out as strong a case as we can; but the mere fact that Love and I are supporting you will be no guarantee of success.

[PS] An application to Courant should, I think, go through Love or me, unless you happen to know him personally. So far as the Board is concerned; you can do nothing.

And againY ( 1) Yes, I can manage. Subject 'Some integral equations' . Will you let me know of any other commu­nication to be made, so that I know about how long to talk?

(2) I hear from Dr. Tisdale to the following effect

(a) It will be necessary for him to visit England to interview you, Love, and me

(b) As the semester in Germany ends in July, and the winter one doesn't begin till Nov . , and as 'he is sure his Committee would not be willing to have her go to Germany and spend the summer vacation there' , he 'would hesitate to present the request as now made, unless we understand between ourselves that

it was to be made effective at a later date . '

(c) that being so, he doesn't mean to come to England until he has other business to do here at the same time.

This position does not seem to me unreasonable and (my reading of his language being that he means what he says) I can only advise you to agree. It is of course not unnat­ural that they should jib at paying a stipend for study in Gottingen over a period when you obviously could not do so effectively. It woul d he fa­tal to put their hacks up at the he­ginning by trying to dispute this .

lPS] I got a letter from Courant all right. so the application is now in perfect order.

Except that it wasn't . A few days later. his patience sorely

tried, Hardy spluttered,3H I h:Ive had a letter from Dr. Tisdale to which I have found it extremely difficult to make any satisfactory re­ply. You never warned either Love or me that you were proposing to make a second application in an en­tirely different field. We put the case (as I told you l was bound to) on the ground that, while it was im­possible to put the work you had actually done on a level with that of (say) P(>lya or Ostrowski, there were considerable possibilities if only you had a chance of turning your self

into a ' 100 p/c' mathematician in a definite field; and that that was what you were determined to do. It was the only way of putting forward the application with a one in twenty chance of success. Now Dr. Tisdale retorts that apparently you arc equally willing to desert mathemat­ics altogether-and what on earth is the reply? I have replied as best I could . but he has us waving our hats vaguely-it looks as if neither you nor Love nor I had any conception of what the standard of the fellow­ship is; and, however one looks at it , the chances of success are very gravely prejudiced.

No, Wrinch had not told Hardy or Love about the sociology application th:It Grier had submitted on her behalf. (Nor had she told Grier about theirs . ) She apologized profusely to Love, with a copy to Hardy,-19

I am fearfully sorry there is this mess. Had it not been that I have been entirely enveloped in the seri­ous illness of John-which by no means started when Oxford knew about it but had been worrying me distracted all the v:Ication-1 should no doubt have been able to keep calm. As it was, it was all I could do to keep my work going and fever­ishly to take every possible chance of arranging next year and to assume that the R. people would consider each application on its merits.

_:.;,.-TH E O.u E E N 's H O TEL ,

/ k . ..- Pe-N z AN C E

.,� k- �� ... . �A � �

1K(7;;·k A.A ..: o-..� loW1- � h- .. IIA.l � n. �- , ;v.. ' 0 � � � (.nu Jr-�� k- k

Postcard from Hardy to Wrinch, spring 1930. Courtesy of the Sophia Smith Collec­tion, Smith College.

© 2007 Spnnger Sc,ence+Bus1ness Med1a, Inc . Volume 29, Number 1 , 2007 21

The position with regard to the points raised by Tisdale is as follows; what you said to T. was of course the truth, in that I am determined to become a proper mathematician with a definite field. What you said about the work I have already done (what this was I don't know but the argument is the same whatever it was) is totally unaffected by the fact that for years I have worked on Sci­entific Method and for some time on the application of S. M. to the socio­logical problems mentioned in the other application. Tisdale cannot properly deduce from the existence of the other application that I am 'equally willing to desert mathemat­ics altogether. ' I want the chance to go to G6ttingen next year and I am determined to get it somehow-if not now, then later. However if the chance does not come to go to G6t­tingen now, then I propose if I can get the R. Social Sciences Fellowship to do the sociological research now. I know it is odd to have a serious interest in two fields, but I just have it. I have always had it. . . .

If, in spite of the trouble I have given you, you can see me through this crisis, I shall be deeply grateful .

Love replied graciously; he and Hardy had seen Tisdale; Tisdale would bring her application to the Board.

To everyone's disappointment but no one's surprise, the answer was no. "Dear Mrs. Nicholson, " Hardy informed her,40

I am sorry to say that I have had a letter from Dr. Tisdale which means-when the official language is translated-that the application is useless. He was over here about a fmtnight or 3 weeks ago, and Love and I had a series of interviews and did all we could for you; but I was practically certain when he left that we had failed, and so it proves.

There is no doubt that the dou­ble application prejudiced them very seriously; and, taking their principles and standards as they are (and they are not bad ones), I am not prepared to condemn them for it altogether. If I had been told, when asked about supporting Zygmund, that he was also a candidate in history or eco­nomics, the first result would cer­tainly have been to make me feel ex-

22 THE MATHEMATICAL INTELLIGENCER

tremely susp1c1ous. However, quite apart from this, success was from the beginning, as I told you, very doubt­ful-if it had been for one of their American Fellowships, such as Miss Y held, the standard of competition would have been quite different. I was told the names of some of the candidates, and they were-so far as I knew them-very formidable rivals indeed.

They have also-and admit to having-a certain definite anti-femi­nist leaning-though they have given a few to women. Here of course I am definitely against them. In any case I am of course person­ally very sorry that you should be disappointed.

Epilogue Over the next few years, fellowships at Cambridge and Oxford enabled Wrinch to visit Menger in Vienna and Hadamard in Paris. A snapshot in Pulya 's Picture Album shows her with Menger at the 1932 ICM in Zlirich, where she spoke on "Harmonics associated with certain inverted spheroids. " Wrinch may have snapped this famous photo of Hardy then and there 41 But her transition to the life sciences was already underway.

The time was not ripe.42 In the so­ciology application, also not funded, Wrinch had proposed to investigate "Rationalisation in the Professions" : the

Dorothy Wrinch and Karl Menger at the

1932 ICM. Reprinted, with permission,

from the P6lya Picture Album.

Hardy at the 1932 ICM. It is thought that

Wrinch took this photograph. Reprinted,

with permission, from the P6lya Picture Album.

conditions of entry into and expulsion from the professions; training, salaries and employment conditions; opportu­nities for in-service training, and "the attitude of established practitioners to­ward free access to the profession of those with the necessary intelligence and training. " She would grapple with these issues for the rest of her l ife-as a participant, not an observer.

Hardy returned to Cambridge in 1931 . In the shadow of world depres­sion and looming war, he "secured jobs for many young people, and from 1933 he was deeply concerned with the fate of his fellow mathematicians on the continent, and directed attempts to find places for those whom persecution had driven out. "4:l We've glimpsed the en­ergy, dedication and humanity he would bring to the task.

NOTES AND REFERENCES 1 . J. Tattersall and S. McMurran, "An inter­

view with Dame Mary L. Cartwright, D.B.E. , F.R.S . , " The College Mathematics Journal,

32, no. 4 (2001 ) , 242-254.

2. Wrinch left her papers, comprising 30

boxes, to the Sophia Smith Collection at

Smith College, Northampton, Massachu­

setts.

3. M. Senechal, "A Prophet Without Honor,"

Smith College Alumnae Quarterly, 1 978;

M. Senechal (ed.), Structures of Matter and

Patterns in Science, Schenkman Publish­

ing Co. , 1 978.

4. Pun intended -Wrinch was called "Dot" by

family and friends.

5. See, eventually, M. Senechal, The Grammar

of Ornament (working title), in preparation.

6. See, for example, C. P. Snow, "Hardy", Va­

riety of Men; Bela Bollobas's biographical

essay in the Oxford Dictionary of National

Biography; and Part Four of Robert

Kanigel 's biography of Ramanujan, The

Man Who Knew Infinity. (Bollobas is writ­

ing a full-length biography.)

7. Re Cartwright, see note 1 ; re Taussky, see

note 24.

8. See note 6.

9. Dora Russell, The Tamarisk Tree, G. Put­

nam and Sons, New York, 1 975.

1 0. M. C. Bradbrook, ' That Infidel Place', a

Short History of Girton College, 1869-

1969, revised edition, Girton College, 1 984.

1 1 . Cambridge Colleges are headed by a Mas­

ter or Mistress, Oxford's by a Warden or

Principal. Katharine Jex-Biake was Mis­

tress of Girton College from 1 9 1 6 to 1 922.

When a frightened student apologized for

sitting in her chair, Jex-Biake retorted, "all

the chairs are my chairs . " (This anecdote

is told in http://www.girton.cam.ac.ukl

staff/scholarshipsarts. html.)

1 2 . Bertrand Russell to Ottoline Morrell , June

8, 1 91 6 . N . Griffi in, ed. , The Selected Let­

ters of Bertrand Russell: the public years,

19 14-1970, London, Routledge, 2001 .

1 3 . See G. H. Hardy, Bertrand Russell & Trin­

ity, Arno Press, NY, 1 977 (a reprint of a

l imited edition printed by Cambridge Uni­

versity Press in 1 942).

1 4. Ray Monk, Bertrand Russell: the spirit of

solitude (1872-1921) , New York, The Free

Press, 1 996.

1 5. The other students were Jean Nicod and

Victor Lenzen.

1 6. Hardy, a good friend of Russell 's, was also

interested in his work: see I. Grattan-Guin­

ness, "The Interest of G. H . Hardy, F .R .S. ,

in the philosophy and history of mathe­

matics, " Notes. Rec. R. Soc. Land. 5 (3)

41 1-424, 2001 .

1 7 . Dorothy Wrinch to K. Jex-Biake, undated.

Quoted with permission of The Mistress

and Fellows, Girton College, Cambridge.

1 8 . G. H. Hardy to K. Jex-Biake, undated.

Quoted with permission of The Mistress

and Fellows, Girton College, Cambridge.

1 9. Bertrand Russell, March 1 7 , 1 91 8. Quoted

with permission of The Mistress and Fel­

lows, Girton College, Cambridge.

20. The following excerpts are quoted from let­

ters from Wrinch to Russell, Bertrand Rus­

sell Archives, McMaster University.

2 1 . Watson had left UCL for Birmingham; the

male mathematicians being occupied with

the war in one way or another, he recom­

mended that Wrinch succeed him.

22. Obituary, Nature, val . 260, April 8 , 1 976,

p . 2 1 1 .

23. Lynda Grier, November 1 7 , 1 930, The

Wrinch Papers, Sophia Smith Collection,

Smith College.

24. J. A Todd, "Hardy as Editor," The Mathe­

matical lntelligencer, val. 1 6, No. 2 (1 994),

32-27.

25. Lady Bertha (Swirles) Jeffreys, unpublished

memoir, cited with permission of The Mis­

tress and Fellows, Girton College, Cam­

bridge.

26. As a student, Hardy wrote in his Apology,

he considered mathematics a competitive

sport; he credits Love, one of his teach­

ers, with showing him its true nature.

27. D. M. Wrinch, "On the asymptotic evaluation

of functions defined by contour integrals,"

Amer. Jour. of Math . , 50 (1 928), 269-302.

28. D. M. Wrinch, "Some boundary problems

of mathematical physics , " Proc. Land.

Math. Soc. 24 (1 924), 204-224.

29. Jean Ayling, The Retreat From Parent­

hood, London, Kegan Paul, 1 930, 293 pp.

30. John Jones, Dictionary of National Biogra­

phy. "On 1 2 October 1 930 John Nicholson

was removed from Balliol to the Warneford

Hospital in Oxford, where he was confined

as a certified lunatic. " He died there, all but

forgotten , a quarter century later.

31 . Hardy to Wrinch, December 1 929. The

Wrinch Papers, Sophia Smith Collection,

Smith College.

32. Hardy to Wrinch, undated. The Wrinch

Papers, Sophia Smith Collection, Smith

College.

33. See Reinhard Siegmund-Schultze, Rocke­

feller and the Internationalization of Mathe­

matics Between the Two World Wars, Sci­

ence Networks, val. 25, Basel, Birkhauser,

2001 .

34. Wrinch to Hardy, March 24, 1 930. The

Wrinch Papers, Sophia Smith Collection,

Smith College. An American, Miss Y- not

her real name-spent a post-doctoral year

(1 925-26) studying with Hardy at Oxford

on a prestigious National Research Coun­

cil Fellowship. In witholding her name, I fol­

low Todd (note 24).

35. Hardy to Wrinch, undated. The Wrinch Pa­

pers, Sophia Smith Collection, Smith Col­

lege. This excerpt continues the letter cited

in note 32.

36. Hardy to Wrinch, undated. The Wrinch

Papers, Sophia Smith Collection, Smith

College.

37. Hardy to Wrinch, undated. The Wrinch

Papers, Sophia Smith Collection, Smith

College.

38. Hardy to Wrinch, undated. The Wrinch

Papers, Sophia Smith Collection, Smith

College.

39. Wrinch to Love, May 26, 1 930. The Wrinch

Papers, Sophia Smith Collection, Smith

College.

40. Hardy to Wrinch, July 24, 1 930. The

Wrinch Papers, Sophia Smith Collection,

Smith College.

41 . See note 24.

42. E. F. Keller, Making Sense of Life: Explain­

ing Biological Development with Models,

Metaphors, and Machines, Cambridge,

Harvard University Press, 2002.

43. Bollobas; see note 6.

© 2007 Spnnger Sc1ence+Business Media, Inc , Volume 29, Number 1, 2007 23

Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters to the

editor should be sent to either of the

editors-in-chief, Chandler Davis or

Marjorie Senechal.

Opinion Survey: The Best Mathematical Books of the Twentieth Century A recent Mathematical Intelligencer in­cluded a fascinating article on collecting mathematical books [1] . The paper fea­tures eye-watering amounts of money, such as $3'56,000 (Newton's Principia), $ 1 1 2 ,000 (a 1 '544 edition of Archimedes), and even $8,000,000 for an entire col­lection. Naturally, all these books, as well as being unaffordahle, are rather old.

But what about more recent hooks? A present-day mathematician might not only conceivably have the financial clout to build up a decent collection of twen­tieth-century mathematical books, but might also enjoy exercising the judgement required to pick out the truly important books of the last hundred years. After all, it's not too hard to look back and see that anything by Archimedes or Newton is likely to have been influential on the de­velopment of mathematics. But to pick out the more recent books over which collectors will be fighting each other in a hundred years' time-that needs skill.

So I invite you to help draw up a list of the most important mathematical books written since 1 900. You are in­vited to submit up to ten hooks in each of the following two categories: Category A: 'Straight' mathematics hooks.

Such books would expound mathe­matical theories, possibly but not necessarily original to the author or to the book. Textbooks would be in­cluded in this category.

Category B: Books about mathematics or on the fringes of mathematics. This category would include: histories of mathematics; the philosophy of math­ematics; biographies and autobiogra­phies of mathematicians; popular mathematics; mathematical treatments of other fields; hooks making con­nections between mathematics and apparently non-mathematical subjects; novels centered on mathematicians. The idea is to find, in each category,

the hooks written since 1 900 that Intel­ligencer readers think will prove in the long run to have been the most influ­ential. In other words, which hooks will the equivalents of Alexanderson and Klosinski be writing about in the Intel­ligencer in the year 2 1 00?

24 THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+Bus1ness Med1a. Inc

The results of this survey will he published in a forthcoming issue. Please would you send your entries (up to twenty books in total) to me by e-mail (preferably) or by snail mail by Sep­tember 1 , 2007. Please use the format "author's name; book title; publisher and date of first edition (if known)" .

Please bear the following points in mind. 1 . Category B is intended to be very

wide. I hope that the results will mir­ror the vast sweep of twentieth-cen­tury mathematics and its influences. So if you're not sure whether one of your favourite books is really suffi­ciently mathematical to be in cate­gory B, include it.

2 . Don't worry if you don't know whether to put a book in category A or B. No doubt the boundary is blurred. If people submit the same book in both categories, I will sim­ply combine the votes and put the book in the category into which more have placed it.

3. Despite the title of this article, please don't exclude books written since 2000.

4. I would have liked to ask the ques­tion, "Which books will have the highest monetary value in 2100?" This would have been both testable and measurable, though admittedly not for nearly a hundred years. But the price of a second-hand book de­pends on factors other than its im­portance-such as its scarcity-so I settled on the vaguer formulations of "most important", "most influential", and "most fought-over by collectors".

'5. Don't feel restricted to books written in English.

6. You are not asked to place your choices in any order.

REFERENCE [ 1 ] G L Alexanderson and L F Klosinski, Math­

ematicians and Old Books, Mathematical

lntelligencer 27:2 (2005), pp. 70-79.

Eric Grunwald

Mathematical Capital

1 87 Sheen Lane

London SW1 4 8LE United Kingdom

e-mail: ericgrunwald@aol.com

M athemati c a l l y Bent

The proof is in the pudding.

Opening a copy of The Mathematical

Intelligencer you may ask yourself

uneasily, "What is this anyway-a

mathematical journal, or what?" Or

you may ask, "Where am I?" Or even

"Who am I?" This sense of disorienta­

tion is at its most acute when you

open to Colin Adams's column.

Relax. Breathe regularly. It 's

mathematical, it's a humor column,

and it may even be harmless.

Column editor's address: Colin Adams, Department of Mathematics, Bronfman

Science Center, Wil l iams College,

Wil l iamstown , MA 01267 USA

e-mai l : Col in .C .Adams@wi l l iams.edu

M ore from the M athematica l Eth ic ist

COLIN ADAMS

Dear Dr. Brad,

I spent every Saturday night of the last nine years building a proof of the Ichi­hara-Warner Conjecture, the greatest outstanding question in Combinatorial Logic. The resulting paper has been honed and perfected and was ready to go out to the Annals ol Mathematics, where it most assuredly would have been accepted, when my 1 4-year-old nephew happened to pick it up off my coffee table. He pointed out that one of the inequalities in a later lemma was obviously backward. This tiny defect has the effect of destroying my life's work . Now the entire problem boils down to a single inequality sign, \vhich easily could point the other direction. It is highly unlikely that anyone would notice the problem. Chances are the ref­eree will either he lchiha ra or Warner. Ichihara himself often has his inequali­ties backward, and Warner dropped out of math and owns a car wash in lncli­ana somewhere. So, don't you think, just this once. I could ignore th is tiny little error, and go ahead and publish the result� The nephew is no longer a problem.

Aaron Alkanacia University of Northampton

Dear Aaron,

Unfortunately. the truth of the matter is that the bigger the result, the more care­fully i t is combed for errors. If this were

some minor conjecture, there is a good chance you would get away with it. But lchihara-Warner? The greatest conjec­ture in Combinatorial Logic? I don't think so. Graduate students from around the world would be studying your paper in seminars that would spring up overnight. One of them would be presenting it and would get into trouble over this lemma. Everyone in the seminar would make fun of him, giggling over his ineptitude, assuming the error was on his part, not on the author of this famous paper in the An­nals. Then they would slowly come to realize where the error lay. One person after another would stop tittering until finally the entire seminar room would he bathed in cold stony silence. A year later, you would be living in the street, scraping by on pennies earned as a tu­tor for students who confuse 6's and 9's.

The sad truth, my dear friend, is that you cannot publish this work as the so­lution of the Ichihara-Warner Conjec­ture. No, you must find an obscure con­jecture, preferably in an unrelated field, and publish your work as the solution to that conjecture instead. Then you can sleep easily, knowing that no one will ever bother to check the details of a conjecture that no one cares about in the first place. Good luck!

Dr. Brad

Dear Dr. Brad,

I have a student who was forced to miss the midterm because of the death of his grandmother. I always let students out of an exam in cases such as these, hut for this particular student, this is the third time it has happened. What should I do?

Eleanor Oxentable Southsoutheastern University

Dear Eleanor,

The likelihood that this student has three grandmothers, all of whom died this semester, is extremely low . Most

© 2007 Spnnger Sc1ence+Bus1ness Med1a, Inc , Volume 29, Number 1 , 2007 25

likely, he is lying about the deaths of his grandmothers to avoid taking the exams. Of course, as mathematicians, we know that being extremely unlikely does not rule out the possibility that it could happen. If the parents have di­VC)!"Ced and remarried, it could even be the case that he has four grandmothers, the fourth of whom will kick the bucket right before the final . In fact, it is not unheard of for elderly relatives to in­tentionally time their deaths so as to aiel a granddaughter or grandson t1ounder­ing in a math course. This phenome­non has been studied in the psychol­ogy literature. (See "The Final Exam: More Final Than We Thought?" Psy­chology Today, (2004) 321-342 and "The Evolutionary Advantage of Ex­tended Family Morbidity During Times of Crisis" Misbehavioral Digest (2005) 222-223 . ) There is at least one case where an entire extended Turkish fam­ily died, all of different causes, within three days of the son's final. Ironically, he chose to take the exam anyway, and did surprisingly well. It was obvious that his family had underestimated his abil­ities.

Given that there is some probability that your student is telling the truth and three of his grandmothers did die, it is incumbent on you to be sympathetic. Think how you would feel. Give the kid a break.

Dr. Brad

26 THE MATHEMATICAL INTELUGENCER

Dear Dr. Brad,

How do you become a math ethicist? What's it like? Would I like it?

Carson Braintree Age 10

Dear Carson,

Technically, I shouldn't address your questions, as they are not about mathe­matical ethics but instead, about a math­ematical ethicist. However, since you are just a kid, and since I believe my read­ership is at least as interested in these questions as you are, I will indulge you.

I did not choose mathematical ethics as a profession. It chose me. From a very early age, I found myself in a va­riety of dilemmas of a mathematically moral nature. While still a toddler, I claimed I was a year older than I was by keeping track of the year of life I was in rather than the number of years already completed. Other children were upset by the advantage this provided me, and the resulting altercations were often settled on the playground rather than in scholarly debate.

Since then, I have continued to con­front ethical dilemmas of a mathemati­cal nature on an almost daily basis. When I receive incorrect change, should I point out the error? Should it matter if the error is in my favor? Should it matter if the error is in my favor and is in the amount of $3,407.45?

Over time, I came to realize I had considered these issues much more deeply than most people do. It seemed entirely appropriate that I share my knowledge.

And what is it like to be a mathe­matical ethicist? It is amazing. Thou­sands of mathematicians around the world rely on me to be their moral com­pass. A word here or there by me can make or break a career. Lives hang in the balance. It is an awesome respon­sibility, but a responsibility I take on willingly, and even relish. Having that kind of power is a rush.

Would you like it? That's like asking me if you like french fries at McDon­ald's. I don't know you, so how could I possibly know whether you would like it? But speaking probabilistically, as with the french fries, almost certainly you would. It is just that cool.

Dr. Brad

And here is a special private response to AR at UT with regard to the case of TD: Ixnay on the enuretay.

Well, here we are again at the end of our time together. I hope you have found this column as ethically inspiring to read as I have found it to write. And remember, numbers don't lie, people do. Keep those letters coming! See you next time!

Turing : A B it Off the Beaten Path DENNIS A . HEJHAL

Mathematician, logician, World War II codebreaker, father of modern computer science, one of Time magazine's Top 20 scientists and thinkers ot the

twentieth century. Most readers of the Intelligencerwill prob­ably already have at least some degree of familiarity with Alan Turing and his life story. Turing died in 1 954, just a few weeks shy of his 42nd birthday, of what a coroner's in­quest concluded was a self-administered dose of potassium cyanide.

Andrew Hodges explores the frustratingly ambiguous cir­cumstances surrounding Turing's death in his excellent 1983 biography of Turing ( [HJ: see also [Hi, p . 25]) . (A partly eaten apple found on the table beside Turing's bed has generally been assumed to have had something to do with matters.) The Postscript on page 'i29 of Hodges's hook concludes with a rather forlorn-sounding, "There is no memorial . ..

A couple of years ago, I was pleasantly surprised to learn that this last assertion is no longer correct. Since June 2001 , a poignant, life-like, life-size sculpture of Turing has resided in a small public park in central Manchester, England, located just a few minutes' walk from the University of Manchester's Sackville Street campus. If my survey of about twenty col­leagues and collaborators provides any indication, this re­markable grass-roots memorial to Turing could benefit from a hit more publicity within the mathematical community.

Strangely enough, I myself only learned of the sculpture when, in a Minneapolis coffee shop, I happened to pick up a copy of one of our smallest alternative newspapers and was startled to encounter a picture of it prominently fea­tured in one of the articles. ( I seem to have somehow missed any news of the thing in the "more standard" media I nor­mally scan.)

In June 2004, I had the opportunity to make a short math­ematical visit to Manchester as part of my three-week stay at the Isaac Newton Institute in Cambridge (and their " trav­elling lecturer" program).

I had never been to Manchester before and thought it would he interesting to give a lecture there and to take at least a brief look around. (It was in looking around. in fact,

that the idea of possibly preparing a tourist section essay along the present lines first hit me. ) The program at the New­ton Institute that I was participating in had as its focus recent connections between random matrix theory and zeta func­tions. Perspective-wise, I think it's fair to say that people come to Turing along a variety of avenues, some more common than others. With me, the desire to "pay Mr. Turing a visit"

Figure I. The Alan M. Turing statue in Manchester's Sackvi lle Park.

© 2007 Springer Sc1ence+Bus1ness Med1a, Inc., Volume 29, Number 1, 2007 27

was prompted not only by the coffee shop article that I'd seen (which was excellent) hut also, scientifically, by a shared in­terest in computing, more specifically, zeros of zeta-functions.

The Dynamical Systems Seminar in Manchester, at which I was invited to speak, met on Wednesdays. To avoid any conflict with the Newton Institute workshop schedule, my hosts proposed that I give my talk on Wednesday, June 23. I agreed and sent back a short reply by e-mail. I could only smile when, working in the Institute library a few hours later, I noticed that the 23rd was Turing's birthday.

Turing and Zeta Turing had wide interests. Highlights of Turing's work on the Riemann zeta-function C(s) can be found in several of the recent series of popular hooks on the Riemann Hy­pothesis; e .g . , [D,Du,Ro,Sa] . l Though by no means his most famous area of work, the "zeta area" was clearly one that held a certain fascination for Turing.

Turing's 1 953 paper [T] gives a very picturesque account of the first calculation of zeros of CCs) ever made by an elec­tronic computer; viz. , the Mark 1 , at Manchester University in 1950. Turing's "hands on" approach to zeta actually be­gan about a decade earlier with his attempt to build an ana­log computer specifically intended for calculating values of CCs) . See [H, pp. 1 40-14 1 , 1 55-1 57] . (Hodges remarks that a 1 939 visitor to Turing's room found it to contain a veritable "jigsaw puzzle" of brass gear wheels scattered across the floor.) The 1 953 paper and Turing's gear wheels strike a note of warm familiarity in anyone, such as myself, who has ever worked seriously with computational aspects of zeta functions and their zeros.2

Though with the advance of technology, the range of s­values over which the Riemann Hypothesis has been veri­fied has grown astronomically (see, e .g . , [O,G]), the aspect of [T] that has been found to have lasting value is Turing's elegant result (theorem 5) to the effect that, in checking R.H. over 7] � Im(s) � Tz, it suffices to examine tC+ + it), more particularly the modified function Z(t), for real values of t only.3

"Success" is achieved by exhibiting an appropriate num­ber of sign-changes of Z(t) on { 7i � t � T2) and over cer­tain small, two-sided neighborhoods of 7i and 72 (the lat­ter sign-changes needing to also be reasonably well-spaced). See [E,L,Ru,Bo] for improvements and additional details.

Turing asserts in [T, p.99] that his calculations were done "in an optimistic hope that a zero would be found off the critical line." (I .e . , off u = +: italics mine). Whether this con­trarian wish was expressed (at least partially) tongue-in­cheek, we will never know; but it does bring up an inter­esting point.

In Turing's theorem, the part about the two-sided neigh­borhoods is present because the total number of zeta zeros in { 7] � Im(s) � T2) is known "a priori " only up to a term which is effectively" the difference 7T- l arg cc+ + it)JR . Ful­fillment of the neighborhood condition has

-the effect of

causing both "arg" terms in the difference to become neg­ligible, thus enabling one to know precisely the total num­ber of zeta zeros which need to be accounted for. Though arg cc_l + it) has mean value 0 as t---'> 00 (and "starts out" mod­estly �no�h), its normal order-of-magnitude is cYJog log t (c = 1/V2). A deep theorem of A. Selberg ([S,§2] ) says that the values of arg cc+ + it)!cv'log log t catso log Icc+ + it) l ! cYlog log t) are distributed like a standard Gaussian nor­mal i\(0; 1 ) in the limit of large t. Very loosely put, this property is a manifestation of the fact that Jog cc+ + it) can be approximated by a [lengthy] superposition of the com­plex exponentials p- it, which, in turn, mimic independent random variables as t---'> oo. (Here p denotes a prime. ) Nor­mality then follows from a suitable form of the Central Limit Theorem familiar from basic probability theory (see [F1 , pp. 229, 173] for the standard version, and [B,F2] for something closer to what's involved here).�

For this and several other reasons (see [S2]), there is a sense that, in order for putative counterexamples to R.H. to appear "with any kind of inherent regularity," the value of (c/7T)Ylog Jog t needs to be fairly big. The extremely slow rate of growth of �gt makes the detection of any such counterexamples highly problematic: at t = 103, (c/?r)Ylog log t is approximately . 3 1 ; at 1 0100, about .52 . To reach even the lowly value of 1 . 52 , the base 10 logarithm of t would need to be about 2.7R X 1019. (Compare: [D, p.358] . ) The laws of physics being what they are, it is clear that-even with the fastest computers imaginable-there is very little prospect of carrying out any computational ex­periments with cc+ + it) in the vicinity of this last t, at least not in this universe.

The only real hope for R.H. "negativists" thus lies more in the direction of sporadic-zype counterexamples 6

1 See [E,Co] for additional background on /;'(s). The Riemann Hypothesis, which dates back to 1 859, is the conjecture that every nonreal zero of !;'(s) (i.e., nonreal root of /;'(s) = 0) lies along the vertical line {<T = �); here s = 0' + it = Re(s) + ilm(s) . The R.H. is one of the seven Clay Institute Millennium Prize Problems [CI] and is impor­tant because of the pivotal role that nonreal zeta zeros play in determining the finer distribution properties of ordinary prime numbers p in the limit of large p. (Foot­notes 1 -6 are appended in the hope of making this and the next section more accessible to general readers. Minimal loss in comprehension occurs if these two sec­tions are read just cursorily.) 2Turing's more theoretical 1 943 paper [T2] on !;'(s) (submitted 3/39) may well have been conceived partly with an eye toward some sort of eventual machine imple­mentation, specifically in regimes of "intermediate-sized" lm(s). See [T2, p. 1 80 (lines 9-1 2)] and the curiously open-minded statement on p. 1 97 (line 1 6).

3The function Z, which has the advantage of being real-valued for real t, is simply /;'(� + it) multiplied by a certain elementary phase factor exp�IJ(t)]. See [E]. (The ex­ponential factor is of course never zero.) 4Recall here that log w = loglwl + i arg(w) and that, in complex function theory, the number of zeros of a function f�) on a region R is intimately linked with the change manifested by the logarithm of f�) as z traverses R's boundary. 5A bit more work shows that, for any fixed h * 0, the arg(l)-values taken at t and t + h actually become statistically independent as t --> oo; likewise for the values of arg(l) and log!� taken at the same t. (The log log t factor arises from the formal variance lp-1 , wherein p ranges over those primes ;;:; t.)

6There is some reason to believe ([Ts,Fa]) that the pertinent "instability indicator" here may take the less egregious form bViOQI(Iog log t)w, where uJ' = 1 /4 and b is some modest positive constant. The waiting time for this expression to become big is, of course, substantially less than that for (cl7r)Yiog log t.

28 THE MATHEMATICAL INTELLIGENCER

A More General Setting Though stated for ((s), it is known that Selberg's result ac­tually holds for a wide class of number-theoretical zeta func­tions (or Euler products ), L(s). See [S,BH]. For each such L, there is an analog of R.H.

Prompted partly by Turing's contrarian wish, I thought it would be fun for my Manchester lecture to highlight a frame­work in which there definitely were zeros off the critical line, and in which one could say something about their dis­tribution. For sums of zeta functions ( more precisely: linear combinations �>i' h1 L1 with N?; 2 and nonzero h1 ) , it turns out that the Selberg theorem for log l l, essentially allows this. See [He1-3l. This being the case, I decided to call my talk "Multivariate Gaussians for L-functions with Applications to Zero Counting'' and to spend the last 1/3 or so of my lecture time discussing off-line zeros of 'ihJ1 .

Nearly Faded Footprints This was actually my second visit to the Newton Institute in Cambridge. Because I run several times each week, the In­stitute housing office placed me (as in 1997 > in a sublet apartment situated very close to the Haling Way (towpath/ bike route) paralleling the River Cam from Jesus Green in Cambridge out to Waterbeach, a distance of approximately six miles. (The university crew teams row along the same stretch of river, hut typically turn around just past the half­way point, near Baits Bite Lock; see [RM] . )

Since my Manchester and Newton Institute talks had con­siderable overlap, I decided to prepare both lectures simul­taneously, working at the desk in my apartment over a se­ries of evenings. One evening. after running a bit further than normal (to a point about 1 kilometer past Waterbeach), I came home too spent to work and decided to simply read a hit in Hodges [Hl, a copy of which I had fortuitously found earlier among the apartment owner's many hooks. Though I knew [T] well, it had frankly been more than 1 S years since I had last even browsed through IHJ .

.._ __ -""'---'

DENNIS A. HEJHAL eamed his doctorate

at Stanford University in 1 972. He is a fellow in the supercomputing inst1tute at Minnesota. and is now a professor at both Minnesota and Uppsala. He worl<s in analytic number theory

and automorphic fonms. While he was prepar-ing this paper for publication, he was awarded the 2005 Girding Prize for some of the same

work on off-line zeros that he had descnbed at Manchester. He is a longtime running enthusiast.

School of Mathematics

University of Minnesota

Minneapolis, MN 55455

USA

Mathematics Department

Box 480 Uppsala University

S-75 I 06 Uppsala

Sweden e-mail: hejhal@math.umn.edu e-ma1l: hejhal@math.uu.se

I soon got two very pleasant surprises: 1 . The Central Limit Theorem, the zeta-theoretic coun­

terpart of which lay at the heart of both my talks, was ac­tually the subject of Turing's first research paper; viz . , his 1934 King's College Dissertation at Cambridge (unpub­lished, copy available at [T3]) . Turing effectively m:liscov­ered the now standard Lindeberg-Feller version [Li,F2] of the theorem; I never knew this. (See [H, pp. 87-88 and 941 . Also [Z] . )

2 . Even more relevantly given my tired feet, I was sur­prised to learn that, while he was Fellow at King's College (both before and after World War II) , Turing often took long afternoon runs along the riuer, sometimes going even as far as Ely and hack (a destination about twice as far as Water­beach). What this meant was clear: in all likelihood, out to Waterheach at least, Turing must have followed the same, very scenic, Haling Way path that I and numerous other runners enjoyed using half-a-century later in getting "our own breaths of fresh air. " ( See [H, pp. 96, 372], and, for run­ning enthusiasts, also [Bu,CW]. As an undergraduate Turing rowed for King's College; he was thus well familiar with the Haling Way. )

I felt energized learning these small bits of history and looked forward to visiting Manchester.

Sackville Park Following my arrival by train at Manchester's Piccadilly Sta­tion (on June 22nd) , I picked up a map at the information center to locate Sackville Park-where I knew the statue was situated-and a good walking route to my accommo­dations at the University's business school. My map showed the city center divided into 10 subdistricts; Sackville Park lay just off the intersection of Whitworth and Sackville Streets, approximately 1 I 4 mile from the train station, along the outer edge of the city's "Gay Village" and (as it turned out) directly across Whitworth Street from the Sackville Street Campus's main building. I decided to stop by the park for a quick look prior to continuing on to the busi­ness school.

The photos in Figures 1, 2 and 4, 5 were taken by me in essentially that order. (To improve readability, my ver­sion of Figure 3 has been replaced by a sharper one from the Web.) The last two photos, Figures 6 and 7, were taken later the same afternoon while I was out doing some addi­tional sightseeing.

A few comments on the various pictures . . . The statue was unveiled on June 23, 2001 , Turing's 89th

birthday. The sculptor, Glyn Hughes, writes on his website [Hu]:

The form of the statue, more like a piece of real l ife frozen into bronze than the commonplace attempt to he grand or clever, was intended to invite the visitor to touch, and to perhaps sit next to Mr Turing; while there is sufficient, slightly puzzling, imagery, to prompt them to investigate further. To my delight, this has proved to be the case-there are sometimes even queues!

A bit earlier, Hughes noted: "I chose rather to present Tur­ing as a very small and ordinary man . . . "

The "visitor-friendly'' aspect is visible in Figure 4, which appears courtesy of a young woman on her way to a class

© 2007 Spnnger SC1ence+Bus1ness Media, Inc., Volume 29, Number I, 2007 29

Figure 2. Another view (with a hit of wear-and-tear showing

on the face).

in a nearby building. (Also visible in Figure 4 is the under­stated aspect of the work-my height being 5 '9" .) The bench's hack-boards, which the statue and I are obscuring, contain the inscription:

Alan Mathison Turing, 1912-1954 IEKYF ROMSI ADXUO KVKZC GUBJ

The (Enigma-) encrypted portion is reputed to say "Founder of Computer Science"; see [Si] .

Figure 3. The plaque on the ground in front of the sculpture. ( Image courtesy of en.wikipedia.org.)

30 THE MATHEMATICAL INTELLIGENCER

Figure 5 shows the wording on the informational sign lo­cated off to one side of the statue. Due to weather and what not, parts of the sign have become faded; to improve read­ability, the contrast has been enhanced somewhat through­out the pictures. Intended for passers-by in the park, the in­formation given about Turing is clearly rather concise-and, stylistically, perhaps just a bit stark. There's no mention, for instance, of either running or zeta function gear wheels . . . . Yet, in a way, its net effect is perhaps "to prompt visitors to investigate further"; e .g . , via [H].

In this spirit, two further items-of-background concern­ing the memorial itself may be of interest.

First, as mentioned on the sign, the British Society for the History of Mathematics contributed to this effort. A re­port on the unveiling ceremony can be found in their (not­so-readily-available) newsletter [Ch]. One thus learns [Ch, pp. 9-10] that:

The [unveiling] ceremony marked the fulfilment of four years of tireless effort by the Alan Turing Memorial Fund Committee. The Committee was set up in 1997 by Richard Humphry, a barrister in Manchester, and Glyn Hughes, an industrial designer, sculptor, and computer buff from Lan­cashire, who had both been drawn to Turing. They found themselves questioning the lack of a memorial to a man who had made such immense and lasting contributions to mathematics, computing, and national security during his brief life. As non-mathematicians coming fresh to the story, perhaps they were especially well placed to appreciate the loss of such a prodigious talent.

The idea that Sackville Park would (by virtue of its loca­tion) be an ideal spot for placement of a memorial to Tur­ing was suggested to the Manchester City Council inde­pendently by both Hughes and Humphry. The Council saw potential in the idea, and the two were put in touch with one another.

Figure 4. AMT and the author.

@ N

§ (f) 3 � g' (D

§ + � 5 � ,;: � !" 1l C'i � N so z � cr "'

§

A Alan Hathi�on Turing is now recognised as one of the greatest thinkers of the modern age. A mathematician and _logician. he was born in London a.nd educated at Sherborne and King's College, Cambridge. In 1936 whilst at King's, he published "On Computable Numbers" i11 which he conceived of "a universal computing machine" which would be able to carry out thought processes using numbers and is thus regarded as the father of computer �cience.

During World War II he was recruited by the Government to the Bletchley Park Code and Cypher School and there devised the Bombe machines that eventually broke the German �aval Enigma code, an event which proved a turning point that directly led to the Allied vi�tory and saved hund�eds of thousands of lives. After the war he became involved with the Manchester University team that was responsible for the most import�nt breakthroughs in the development

�

of the electronic computer. His most significant contribution before he was prevented from continuing with his work was a surprisingly readable manual on computer programming.

There�fter �po�itiv� vetting status was removed a�d he was . excluded frol!'·lhe work to which he had devoted his life.

In 1 954 he viis found dead at home. having taken a bite out of a_ri apple poisoned with 9'3nide . .His work on artificial •

intetligence .a nd !he mathem�tical basis 1.. brological forms - ,.

llltillllllelllr ............. limll�

al to commem(}rate the man and his · contribution to his country �nd fint-fnrm�u, made at a meeting of interested persons at Cafe Metz, Canal Street in

s finaUy set up in November 1998 and after just over-two years •

een raised, including contributions .from the public, a gtant from the City Council, a sigmficant don_ation froin the Briti� Society For The His!ory Of Mathematics and the proceeds of fundraising"-evenu held in the'Village, to have the statue cas't by the Arts.�nd Crafts foundry in Tranjin, .China. Oesprte many approaches, the major computer companies failed to give any practical support.·

The memorial was unveiled at ) ceremony on _23rd june 2001 -Turing'1 89th birthday - by his bjographer, Or Andrew Hodges, and Or J .V. Fjeld, past president of the BSHM. •

Always open about his sexuality, in 1952 he mentioned his relationship with a Manchester man to a ·detective .. investigating a burgl!l"f at his home in Dean Row, Wilinslo� and was subsequently presecuted and required _... tO aCCep��OfmO� injectiOnS tO CUre hiS "perversion". r..t r 'liM

_ ..... ... ........,. fiii:UII rr.t iii:IIIIIIIWIIIIil liy olll:k•

-

c.l Figure 5. (Slightly cropped and juxtaposed) picture of the text on the informational sign . ....

Figure 6. A slightly different perspective . . .

After the initial wave of enthusiasm [however] , Humphry and Hughes soon realised that if anything was actually going to happen, they were going to have to do it them­selves . . . . The most striking feature of the early phase of their campaign was an almost total lack of interest from the computer industry, the academic establishment, and the press. Fortunately, Manchester City Council were very supportive of the scheme, and interest was even­tually generated by a website and an article in the Daily Telegraph.

Momentum picked up as various luminaries signed on, in­cluding Sir Ian McKellen and Sir Derek Jacobi. (Jacobi starred in Hugh Whitemore's 1986 play, Breaking the Code, and be­came patron of the fund.) Though the projected minimum cost of the statue was £46000, only about £20000 could be raised: £1000 from BSHM members, £5000 from fund-rais­ers in the Gay Village, and £14000 from other (mainly in­dividual) contributors. Because of this financial shortfall, it was decided to have the sculpture cast at a foundry in China. (Though the bulk of the financial contributions for the statue came from individuals, Andrew Hodges notes in [H2, page 31 that £2500 was donated to the project by Manchester City Council; this was later supplemented by an additional £1000 for the unveiling ceremony.)

The second item is a point I decided to check with the sculptor about after reading something about it on the Web. In his e-mail reply, Hughes confirmed that his early-1980s­era Amstrad personal computer is indeed buried beneath the sculpture's plinth in tribute to Turing. (On a more tech­nical level, I also learned from Hughes that the sculpture is made out of gunmetal bronze, and that, in parts of the cast­ing, there is some contamination with iron and free carbon which may eventually become visible.)

The memorial seeks, of course, to convey something of Alan Turing, the man. When, following my sightseeing, I re­turned to the park later that afternoon to take the pho-

Figure 7.

tographs in Figures 6 and 7, I again sat for a spell on the park bench. Influenced perhaps by the "mood" (and mat­ter-of-factness) of the sign, I found myself vaguely remem­bering a short passage I had seen earlier in Sara Turing's book [Tu] about her son. Originally penned in connection with his running, her words ([Tu, p . 1 1 1]) struck me as sin­gularly appropriate here at this understated, grassroots, park bench memorial:

The club [the Walton Athletic Club] comprised men from all walks of life-road-sweepers, clergymen's sons, den­tists, clerks, and so forth-he was always at ease among them and made them feel at ease . After an interval of some years they still talk about him. In a subsequent telephone conversation with Hughes

concerning some of the Committee's fundraising efforts, I learned that the second sentence in this quotation is not just "some nice-sounding words. " The overwhelming ma­jority of contributions for this memorial were relatively small ones (£5-10) , from individuals. A significant number of these individuals were senior citizens, who often took the time to enclose notes expressing gratitude to Turing for the contributions he made during World War II in helping to defeat Nazism?

000

7Somewhat ironically, Turing died on June 7, 1 954, just one day after the 1 0th anniversary of D-Day (a wartime turning point whose strategic success hinged in no

small way on cryptological breakthroughs achieved at Bletchley Park [Wi, Hin]).

32 THE MATHEMATICAL INTELLIGENCER

As one might expect of any statue sitting in a center city park, this one has been subjected to its share of wear and tear, e.g. , in the form of artistry with felt-tipped pens and what not. Sad-to-say, the "artistry" has most recently come to include splattered red paint accompanied by a gutter­level slur scrawled on the sculpture in heavy black ink (these two latest "contributions" remaining there largely unex­punged, I am told, but by weather effects, for at least sev­eral months' time during late 2005 and early 2006) s

Figure 7 shows an augmentation of an entirely different kind. Whether the streaks are the byproduct of metallic con­tamination, rainy weather, or simply some residue from cleaning off grime and artwork, they do add a certain sub­tle something to the "freezeframe" quality of the bronze, particularly when viewed up close. One of my Manchester hosts reports (6/2006) that, apart from a general lightening in their color and a bit of fading, the streak marks have not really changed that much since the time the photo was taken in June 2004.

Some Further Comments and Personal Impressions I enjoyed my Manchester visit very much, even though­being largely on foot as I was-I didn't get much beyond the general area of the park and university.

The Turing Memorial struck me as excellent: not only ap­propriate (after so many years), but also something different (in terms of both form and location). I'm glad I took the op­portunity to visit "in real life" , as opposed to just virtually.

Though, for the most part, matters now tend to speak for themselves, real-life visits do have a way of bringing out ''aspects and angles" of more subtle types not so readily captured, for instance, by a camera's eye. Drawing the reader's attention to several such in an article of this kind seems only reasonable:

1. A Nice Touch. In my case, it was only in sitting qui­etly next to Mr. Turing for a minute or two that the artistic effectiveness of the statue's understated size hit me on a hu­man level. One got the sense of looking at a real person "through the lens of time. " No longer here perhaps, but still relevant today (as a thinker and human being) . The apple only accentuated this thought-provoking effect.

I would wager that I'm far from the only one to have

experienced this sense while seated on the bench next to Turing.

2. Food for Thought. For most people, the background sign in Figure 5 will be an integral part of the memorial. Its description of Turing-though succinct-is not unreason­able 9 Subsequent to that description, at the end of the ac­knowledgments, a sentence occurs that seems likely to give most on-site visitors pause; viz . ,

8The slur is now gone after a telephone call t o the city parks department.

Despite many approaches, the major computer compa­nies failed to give any practical support.

I found this line unsettling. (It is poignant to turn around and see a life-size sculpture sitting there in silent repose but several steps away. I remember thinking to myself at the time, "Doesn't this dark cloud stuff ever stop?")

A short time after my trip ended, however, two things dawned on me that seem well worth mentioning here, if only for the additional context-and food-for-thought-that they offer.

The first of these is that some computer companies seem not to have had any "issue" with Turing. The ACM Turing Award, occasionally referred to as the "Nobel Prize in com­puter science, " has been awarded annually to international researchers since 1966. Checking back a bit, I found that in the 1990s, the award was funded first by AT&T ('91-'95); then its spin-off, Lucent ('96-'99), in the amount of $25,000/year. 10 More recently, 1 1 Intel has taken over, and the amount of the award has risen to $100,000.

The second fact concerns geography and is simply the ob­servation that Turing actually worked at Manchester Univer­sity's Oxford Road campus, approximately one mile south of Sackville Park ([H, p .394]). It was only in 2004 that Manchester University (formally, the Victoria University of Manchester) and UMIST (the University of Manchester I nstitute of Science and Technology) merged to form The University of Man­chester. Prior to that time, the buildings on the Sackville Street campus, i .e . , basically across the street from Sackville Park, belonged to UMIST -not Manchester U. This fact hints at a possible reason for some of the academic establishment's Jack of interest cited earlier in connection with [Ch].

In the case of the computer companies, this factor has less significance and the reason(s) for any "funding regrets" must presumably be sought elsewhere. Exactly where is un­clear. 12 The matter seems likely to remain something about which one can at best only speculate, and perhaps that's just as well. As Albert Einstein said, "In the midst of diffi­culty lies opportunity."

3 . Adieu. Following my lecture, I thought it would be nice to stop for awhile in Sackville Park prior to continuing on to catch my early-evening train back to Cambridge.

My stop was a brief one. In looking again at the statue and sign, I immediately noticed the positive energy that I felt fol­lowing my lecture less than 40 minutes earlier gradually "mor­phing" into a kind of quiet sadness, a sadness that Turing never got the chance to see how computers-and even the central limit theorem-would ultimately come to have important ap­plications in the study of zeta functions ("the zeta enigma")­e.g. , in opening up new links with random matrix theory.

As I stood there and looked again at the apple, I noticed something curious about the grip of the hand and angle of

9(although some may wish for a better word or two in places; cf, e.g. , [Hi, pp. 23,251 and [H, p. 400 (top)]). 101 991 was the first year the award had corporate sponsorship. I am told that, though the Manchester Committee knew of the Turing Award, it was not familiar with

any of its funding details. [The committee was composed solely of non-academics.) 1 1 apart from two years, 2000-01 , when ACM and lntertrust Technologies funded it at its previous level 12My Minnesota colleague, Andrew Odlyzko, recently commented that he was inclined to think that the lack of corporate support was chiefly due to expectations of

low visibility of the statue and lack of any connection with current cutting-edge research.

© 2007 Sprrnger Scrence+Busrness Media, lnc., Volume 29, Number 1 , 2007 33

the lower arm. An ambiguity as it were. The grip was a bit "cupped and left-rotated," not so much like one used in eat­ing (or deep philosophic introspection), but closer to one that might be seen just prior to a person rising up . . . and simultaneously flinging a small object off to one side.

It may have been my mood, but I found the suggestion­intentional or not-a compelling one. I regretted very much that the "lens of time" prevented its fulfillment.

For my part, I would like to have seen Turing go for dis­tance. Maximal distance.

Afterword In October 2004, a second statue of Alan Turing was un­veiled (by H.R.H. Prince Edward, The Earl of Wessex), on the campus of the University of Surrey. The work, which is modelled after a 1934 snapshot of Turing (cf. [H, photo­graph 6]) , was sculpted by John W. Mills [M] and stands 8' 4" tall. See [H2] or UJ for a picture of it. It seems fair to say that the two statues have a completely different "feel. "

Two months earlier, in August 2004, an Alan Turing In­stitute began operations at the University of Manchester as one of their Centres of Excellence. The institute's research theme ([TI]) can be loosely described as mathematical as­pects of enabling technologies. According to a university public relations official, there is no connection, however, between the Turing Institute (or any other university "unit") and the memorial in Sackville Park.

See [H2] for a list of some further tributes to Turing, and [Wh] for some additional views of Sackville Park. The gar­goyle of Turing mentioned in [H2] on the campus of the University of Oregon is discussed at greater length (together with a similar one of ]. von Neumann) in [Hi, p. 24].

Readers interested in learning more about Turing's math­ematical work will find the combination of]. Britton's intro­duction to [T4] and Max Newman's 1955 Royal Society Mem­oir (reprinted in [T5, pp. 268-279]) an excellent starting point. Turing's MacTutor entry is also quite helpful; see [Bu] for its URL. (Added in proof: Reference [N], which appeared just recently, furnishes additional information-and places things in a broader context.)

ACKNOWLEDGMENTS While preparing this article, I was fortunate to receive com­ments and encouragement from a number of people, includ­ing G. Alexanderson, L. Arkeryd, P. Hilton, D. Huylebrouck, A. Marden, A. Odlyzko, M. Senechal, and R. Sharp. Warm words of thanks also go out to M. Pollicott (for his invitation to speak in Manchester) , to the Isaac Newton Institute (for its financial support and hospitality), and to DeAnna Miller (whose serendipitous article sparked my interest in this matter) .

REFERENCES [B] P. Billingsley, Probability and Measure, 2nd edition, John Wiley,

1 986, especially pp. 407 (ex 30. 1 ) , 408-4 1 0.

[BH] E. Bombieri and D. Hejhal, On the distribution of zeros of linear

combinations of Euler products, Duke Math. J. 80(1 995), 821 -862.

(See Theorem B.)

[Bo] A Booker, Artin's conjecture, Turing's method, and the Riemann

Hypothesis, Experimental Math. 1 5(2006), 385-407 . See also pp.

1 208-1 21 1 in [N].

34 THE MATHEMATICAL INTELLIGENCER

[Bu] P. Butcher, Turing as a runner, see l ink at A M. Turing's Mac Tu­

tor History of Mathematics entry (www-history.mcs.st-and.ac.uk/

Biographies/Turing.html).

[CW] Cambridge-Waterbeach running map, available at (www.srcf.

ucam.org/cuhh), website of the Cambridge University Hare & Hounds

running club. Turing was a member of H & H during 1 947-48. (See

www.go4awalk.com/userpics/sharahiggins5.php for a glimpse of the

route.)

[Ch] F. Chalmers, Unveiling the Alan Turing Memorial (Manchester, 23

June 2001 ), BSHM Newsletter 44(2001 ), 8-1 1 .

[CI] Clay Institute website, see (www.claymath.org/millennium/).

[Co] J. B. Conrey, The Riemann Hypothesis, Notices of the Amer. Math.

Soc. 50 (2003), 341 -353.

[D] J. Derbyshire, Prime Obsession, Plume (Penguin Group), 2004.

[Du] M. DuSautoy, The Music of the Primes, Perennial (HarperCollins),

2004.

[E] H. Edwards, Riemann's Zeta Function, Academic Press, 1 974, es­

pecially pages 1 9, 1 1 9, 1 59, 1 64, 1 72-1 75. (Turing's method is dis­

cussed on 1 72-1 75.)

[Fa] D. Farmer, S. Gonek, and C. Hughes, The maximum size of

L-functions, preprint, August 2005, available at www.arXiv.org

(math. NT /050621 8).

[F1 ] W. Feller, An Introduction to Probability Theory and Its Applica­

tions, val. 1 , 2nd edition, John Wiley, 1 957.

[F2] W. Feller, An Introduction to Probability Theory and Its Applica­

tions, val. 2, 2nd edition, John Wiley, 1 971 , especially pp. 260(th. 2),

51 8-520, 544-545, and 262-264(e�. 269(b).

[G] X. Gourdon, The 1 01 3 first zeros of the Riemann zeta function, and

zeros computation at very large height, available at (numbers.

computation. free. fr/Constants/Miscellaneous) .

[He1 ] D. Hejhal, On a result of Selberg concerning zeros of l inear com­

binations of L-functions, International Math. Research Notices (2000),

no. 1 1 , 551-577.

[He2] D. Hejhal, On Euler products and multi-variate Gaussians,

Comptes Rendus Acad. Sci. Paris (I) 337(2003), 223-226.

[He3] D. Hejhal, On the horizontal distribution of zeros of linear combi­

nations of Euler products, Comptes Rendus Acad. Sci. Paris (I)

338(2004), 755-758.

[Hi] P. Hilton, Working with Alan Turing, The Mathematical lntelligencer

1 3(1 991 ), no. 4, 22-25. This paper is an important supplement to

Hodges's book.

[Hin] F. Hinsley, The influence of Ultra in the Second World War, in

Codebreakers: The Inside Story of Bletchley Park, Oxford University

Press, 1 993, pp. 1-1 3, especially 1 0 (top) and 1 2 (bottom hal�.

[H] A Hodges, Alan Turing: The Enigma, Simon and Schuster, 1 983.

[H2] A Hodges, www.turing.org.uk/turing/scrapbook/memorial .html

[Hu] G. Hughes, www.btinternet.com/�glynhughes/sculpture/turing.

htm

[J] D. Jefferies, images at (www.ee.surrey.ac.uk!Personai/D.Jefferies/

turing/).

[L] R. S. Lehman, On the distribution of zeros of the Riemann zeta­

function, Proc. Land. Math. Soc. (3) 20(1 970), 303-320.

[Li] J . Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der

Wahrscheinlichkeitsrechnung, Math. Zeit. 1 5(1 922), 21 1 -225.

[M] J . Mills , www.johnwmills.com.

[N] A tribute to Alan Turing, in Notices of the Amer. Math. Soc. 53 (Nov.

2006), pp. 1 1 79, 1 1 86-1 206, 1 208-1 2 1 9 .

[0] A Odlyzko, The 1 022-nd zero of the Riemann zeta function, i n Oy-

namical, Spectral, and Arithmetic Zeta Functions (ed. by M. van Franken­

huysen and M. Lapidus), AMS Contemporary Math. Vol. 290, 2001 , pp.

1 39-1 44. See also: (www.dtc.umn.edu/�odlyzko/unpublished).

[Ro] D. Rockmore, Stalking the Riemann Hypothesis , Pantheon, 2005.

[RM] River Cam Rowing Map, available at (www.firstandthird.org), web­

site of First & Third Trinity Boat Club. See also (www.cucbc.org), and

(maps.google.co.uk) under "Fen Ditton. "

[Ru] R . Rumely, Numerical computations concerning the ERH, Math. of

Camp. 6 1 (1 993), 41 5-440, especially §3.

[Sa] K. Sabbagh, The Riemann Hypothesis , Farrar, Straus, and Giroux,

New York, 2003.

[S] A. Selberg, Old and new conjectures and results about a class of Dirich­

let series, in Collected Papers, vol. 2, Springer-Verlag, 1 991 , pp. 47-63.

[S2] A. Selberg, The zeta-function and the Riemann Hypothesis, in Col­

lected Papers, vol. 1 , Springer-Verlag, 1 989, pp. 341 -355, especially

352-355(note 1 ) .

[Si] S . Singh, www.sirnonsingh.net/Turing_Mernorial.htrnl

[Ts] K. Tsang, The large values of the Riemann zeta-function, Mathe­

matika 40(1 993), 203-21 4, especially 205(rniddle).

[T] A. M. Turing, Some calculations of the Riemann zeta-function, Proc.

London Math. Soc. (3) 3(1 953), 99-1 1 7 .

[T2] A. M . Turing, A method for the calculation of the zeta-function,

Proc. London Math. Soc. (2) 48(1 943), 1 80-1 97.

[T3] A. M . Turing, On the Gaussian error function, King's College Fel­

lowship Dissertation, 1 934, i i + 60 pp. (Available at the Turing Digi­

tal Archive, www.turingarchive.org, item AMT/C/28.)

[T4] A. M . Turing, Collected Works: Pure Mathematics (ed. by J. L. Brit­

ton), North-Holland, 1 992.

[T5] A. M . Turing, Collected Works: Mathematical Logic (ed. by R. Gandy

and C. Yates), North-Holland, 2001 .

[TI] Turing Institute Website, at (www.knowledgehorizons.manchester.

ac.uk/centresofexcellence).

[Tu] S. Turing, Alan M. Turing, Hefter, Cambridge, 1 959.

[Wh] Online encyclopedia article at (www.answers.com/Whitworth_

Gardens).

[Wi] F. Winterbotham, The Ultra Secret, Harper & Row, 1 974, espe­

cially pp. 2 (middle) and 1 90 (lines 21 -24). The significance of the

work done at Bletchley Park is described quite compellingly by Gen.

Eisenhower in his letter on p . 2. (Much of this work was underpinned

by methods or ideas that originated with Turing.)

[Z] S. Zabell, Alan Turing and the central limit theorem, American Math.

Monthly 1 02(1 995), 483-494.

Scient ificWorkP iace· The Gold Standard for Mathematical Publishing

Scientific WorkPlace 5.5 makes writing, baring, and doing mathematic easier. A click:

of a button allows you to typeset your documents in � "fEX. Scientific WorkPlace enables both profe ional and upport taff to produce tunning books and article .

Mathematical Word Processing • L\TEX Typesetting • Computer Algebra

- T- ­Te.-e .n ..... M tuN ,._

T� • ..-saon ln - ora.o ....-... l V\th .. .....-.on poritn .. .... SIOI\

Opon ... - l'ropon!H - ond, ... ... .... -

Animate, Rotate, Zoom, and Ay New in Version 5.5

• Compute and plot using the MuPM)8 3 computer algebra engine

• Animate 20 and 30 plo using MuPAD s VCAM

• Rotate, move, zoom in and out, and fly tbrougb 30 plo with new OpenGL 30 graphics

• Label 20 and 30 plots so that the label move when you rotate or zoom a plot

• Create 30 implicit plots

• Import IJ.lfX files produced by other programs

• U e many new IJ."fEX packages

• lect and right click: a word for links to internet search options

• Get h lp from the n w, exten ive troubl hooting secti n of 'fYpesetring Documents with Scientific WorkPlace and Scitmtific Word, Third Edition

www.�.com/111i • Emal: Wllo@rnad<icNn.cam Tol-free: sn-724-9673 • Fu: 3�39+6039

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 1 , 2007 35

1Mftjii§!r@hl¥11#fii§#f"11rri§•IN M ichael Kleber and Ravi Vaki l , Editors

S udol<u's French Ancestors CHRISTIAN BOYER

This column is a place for those bits

of contagious mathematics that travel

from person to person in the

community, because they are so

elegant, suprising, or appealing that

one has an urge to pass them on.

Contributions are most welcome.

Please send al l submissions to the

Mathematical Entertainments Editor,

Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA

e-mai l : vakil@math.stanford .edu

Entertainment Editor's Note. For any of our readers unfamiliar with

the phenomenally successful logic puzzle discussed below: please type

"sudoku" into your favorite web search engine, immerse yourself,

and come back tomorrow. Christian Boyer's article "Les and!tres

fran�·ais du sudoku" appeared in the june 2006 issue of "Pour La

Science. " We are pleased to reprint this revised and enhanced version;

the solutions to the nine problems will appear in the next issue.

Translation by Chandler Davis.

S ince 2005, the game of Sudoku has been a success worldwide. It is indeed a fascinating game, and

once one has started to solve a grid, it is hard to break off until it is completed.

Who invented Sudoku? The history is now known [7JlHJ [10]. The first Su­dokus were published in 1979 by the American, Howard Garns. then, in the 19ROs and 1990s, they appeared in Japanese journals, where they got their name. They took off internationally, thanks to the New Zealander, Wayne Gould. who published them in the Times of London beginning in Novem­ber 2004. There followed the Daily Mail, then the Daily Telegraph, then many other newspapers all over the world. A brilliant success!

But were there precursors before 19791

The Swiss mathematician Leonhard Euler did not invent Sudoku, hut he did invent Latin squares, in an article in French published in 17H2, "Recherche sur une nouvelle espece de quarres magiques" 191 .

An n X 11 Latin square is a square having all the numbers from 1 through n appearing in each row and in each column. Eve1y Sudoku is a 9 X 9 Latin square, but the converse is not always true: a 9 X 9 Latin square may fail to be a Sudoku. for it may not he composed of 3 X 3 suhsquares, each containing all the integers from 1 to 9. In particular, the 9 X 9 Latin squares published by Euler were not of this sort, as for ex­ample Figure l . Euler also did not make a game of it by leaving the reader to fill in missing entries.

-Michael Kleber

So did nothing happen between 17H2 and 1979? We will see that much did.

1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 3 4 5 6 7 8 9 1 2 4 5 6 7 8 9 1 2 3 5 6 7 8 9 1 2 3 4 6 7 8 9 1 2 3 4 5 7 8 9 1 2 3 4 5 6 8 9 1 2 3 4 5 6 7 9 1 2 3 4 5 6 7 8

Figure I. This very simple 9 X 9 Latin

square published by Euler in 1782 is not

composed of the Sudoku 3 X 3 sub­

squares.

9 x 9 Grids with 3 x 3 Subsquares At the end of the 19th century, French dailies and magazines were overflowing with games that involved completing magic squares. I cite Le Sif?cle, La France, Gil Bias, L 'Echo de Paris, Les Tablettes du Chercheur, Revue des jeux. Several numbers of the square would he given, and the reader was to fill in the miss­ing entries. All sizes of square were used, in particular the size 9 X 9, which we are focusing on. Problem 1 (below), which appeared in 1888 in the daily Le Siecle, is one example among many.

The distinguishing feature of a Su­doku is its subdivision into 3 X 3 squares. Among the variants appearing

© 2007 Springer SC1ence+Bus1ness Media, Inc., Volume 29, Number 1 , 2007 37

R E V U E D E S J EU X

• de papi r,

eures l'alphal>tl le dea •5 premitn

millions cl'a'­' par a: , produi·

'oos de ltllres im-

Ia feuiUe de pepitr

tn11Ji001 --- ,

sur£ ee de la Ture,

Die, par Pierrot.

� de /11 mu el de

: 450 .. 1 om Ires, Ia .mm 1 nord, l Au­

DI d' l.lnl de Dun-

, par Luc. ita He.

deus ann4 d:o nales do carr<! de 9 de·tronl lemeo1 donnu ul.

f..a superposaat les rm 1-2•3, ··�6. ;·8-9 00 · ···7· t-5-8, �. oa obti ndra des cubes mo "'ues donoenl

•

u

=� ••

- - - - -.. ..

.. =II-- - - - - -

6

-H-

- - -

-�-=I - --1- - -9

8 une COD lOlii de 1 13 dans lea aeuf Yertical tl danl I di•·hu 1 hori:ontales, mail non pas dans lea qualre

gnndes dta onalea des cubes. Les 1 nombres places dod dtt rmintnt compli-

lemcnt le carri. F.n o�ran1 por compl meo11 l u3 tl k >, e1 per s perpotiuon on peul, sans le moindre t11onnement, recon Uluer le carri ci-deaus.

N 182.-Polygraphiedu Oanlier en une ohalne OUTerte, par M. L. C.

Figure 2. Revue des Jeux, 1891 . The first 9 X 9 grid with 3 X 3 subsquares. (See Problem 3 . )

in these old journals, some used pre­cisely this feature. Perhaps the first 9 X 9 grid to be completed which had this

breakdown into 3 X 3 subsquares was one published by H. Mary in the Revue des jeux in August 189 1 , reproduced here as Problem 3 (see also Figure 2). We may wonder if Colette (the famous author of the Claudine series, Cheri, and Gigi) attempted this problem, for she had a regular gossip column in the same weekly.

Mary had earlier published a similar problem in the same journal, hut it was misstated, and no initial array was given. Afterward, many other 9 X 9 grids with 3 X 3 subsquares were pub­lished by various authors in several pe­riodicals. Such arrays were often called "squares with compartments . "

Grids Decomposable into Two Sudokus A bimagic square is a magic square which remains magic when all its en­tries are squared. After publishing the first-ever bimagic square, which was H X 8 [1 ] [2][5] [ 1 1 ] , the Frenchman G.

38 THE MATHEMATICAL INTELLIGENCER

Pfeffermann carried on and published the first 9 X 9 bimagic square in 1891; see Problem 2 below. Like the other

squares, these first himagic squares were published as games in which missing en­tries had to be filled in. Once Pfeffer­mann had opened the way, numerous other 8 X 8 and 9 X 9 bimagic squares appeared in various periodicals, for ex­ample, Problem 7 by Bmtus Portier. A most interesting hidden feature of this example: unlike the predecessor by Pf­effermann, is that it can be decomposed into two Sudokus, and this even helps in solving it! But a square need not be bimagic to be decomposable into two Sudokus, as witness Problem 4 from Le Siecle. This surprising decomposibility comes from the squares being eulerian squares, also called Greco-Latin Squares.

Gaston Tarry, who met Brutus Portier in Algeria, became interested in magic squares and Latin squares. In 1900, Tany gave the first proof [13] of the impossibility of the famous problem of 36 officers posed by Euler [9] in his 1782 paper: arrange a delegation from six regiments, each of which supplies

six officers of different ranks, in a 6 X 6 square in such a way that each row and each column contains one of­

ficer from each regiment, and one of each rank. It is now known that for each n2 other than 4 and 36, there are solu­tions to the problem of n2 officers, known as n X n eulerian squares; see for example Figure 3.

Not every 9 X 9 eulerian square is obtained from two Sudokus-i.e . , has the subsquare stmcture. But the solu­tion of Problem 4 or Problem 7, when put in terms of two Sudokus, becomes a 9 X 9 eulerian square, a solution of the problem of 81 officers. Though this relationship was not mentioned in the

2.1 3.3 1 .2 1 .3 2.2 3.1 3.2 1 . 1 2.3

Figure 3. A 3 X 3 Eulerian square, or so­lution of the problem of 9 officers. An Eulerian square is a combination of two Latin squares, one for the ranks and one for the regiments.

Gaston Tarry (Villefranche de Rouergue 1834-Le Havre 1913) .

statement of these two problems in the 1890s, it was well known and discussed, and in fact was the subject of Problem 6 by A. Huber. Its solution, published in 1894 in Les Tahlettes du Chercheur, brings this out clearly: the 9 X 9 square is obtained from what we would call today two Sudokus, printed so as to dis­play their 3 X 3 subsquares, each con­taining numbers 1 through 9.

It is probably by analysing the euler­ian structure of these bimagic squares published by his friend Brutus Portier and others that Tarry thought of his marvellous general method of con­structing multimagic squares [5]; Tany's first note on his method [14] was pre-

sented by Henri Poincare in 1906. The book of General Cazalas [6] from 1934 gives details of the method and many constructions of 8 X 8 and 9 X 9 bimagic squares using it, with slight im­provements by Cazalas.

An important remark which neither Tarry nor Cazalas made: all 9 X 9 bimagic squares constmcted with the Tarry-Cazalas method are combinations of two Sudokus, fully organized into sub­squares [5] . Of course, all the pairs of Su­dokus don't produce a bimagic square, and all the bimagic squares (those not constructed by the Tarry-Cazalas method, like Pfeffermann's Problem 2 here) can't be generated by a pair of Sudokus.

The Viricel-Boyer method, which we use for tetra- and pentamagic squares (squares remaining magic when entries are raised to powers up to 4, resp. , 5) , as described in [1 ] , is much inspired by the Tarry-Cazalas method.

All the Ingredients of Sudoku Were There All the problems discussed thus far are magic squares, and although they use the underlying idea of Sudokus, they don't directly ask the reader to deal with a simple Latin square involving only the numbers 1 through 9-which is what makes the charm and attraction of Sudoku. Nevertheless this purely Latin square feature was in use at that time. Thus Problem 5 by the same Bru-

Figure 4. L'Echo de Paris, 1894. Is this a Sudoku? (for

details, see Problem 8).

tus Portier is a 9 X 9 Latin square to be completed, but it fails to be a Sudoku because it lacks the notion of 3 X 3 subsquare. Problem 9, from the daily La France, maybe comes closest: it is a 9 X 9 Latin square to be completed, and each of the 3 X 3 subsquares does have all the integers 1 through 9. Isn't it too bad that the author, B . Meyniel, didn't mention it! The array appeared in print without any indication of these subsquares-which, however, is done in other problems we have seen.

This partitioning into 3 X 3 sub­squares is used, by the way, in another problem not really related to Sudoku: Problem 8, a little goody from L 'Echo de Paris.

I have given some examples, among the most pertinent I have been able to find, of the incredible proliferation around 1890 of diverse problems close to Sudoku: • 9 X 9 arrays with 3 X 3 subsquares • entries to be filled in with numbers • and sometimes only a single use of

the numbers 1 through 9 in each row, each column, even each subsquare.

Absolutely all the ingredients are there. In the wonderland of squares conceived by many different authors over several years, it is astonishing that none of them thought of combining all the ingredi­ents and proposing the game of Sudoku exactly as we know it today. They came so close!

© 2007 Spnnger Sc,ence+Business Media, Inc . . Volume 29, Number 1, 2007 39

Not that my research has been ex­haustive. There are so many grids in so many dailies from the end of the 19th and beginning of the 20th cen­turies in France. Maybe some lOOo/o Su­dokus are hiding there somewhere? Readers who enjoy rummaging in li­braries, take note.

But consider how very close these problems are to Sudokus. Thus if you decompose Problem 4 or 7, or mark the 3 X 3 subsquares on Meyniel's Problem 6, you solve them just as you would a present-day Sudoku, yet they offer a taste of something created more than a century ago . . . and then forgotten!

1 2 3

3 4 5

3 4

5 1

2 3 Figure 5. l'andiagona l Latin squ:trl'.

E:Kh broken diagona l . l ikl.' the one in·

dkat ·d, indudcs .til the numbers.

Is it Possible to Construct a Pandlagonal Sudoku? I t b p< •. siblc to con strw .. t .1 diagonal Sudoku : that is. one '' hose main d i­agona ls abo indude al l the number:-.:

such is, for c ·ample, Prohkm 9. It is also p< ssil>lc to construct .1 panuiag­<m:t l l.at in square, that ts, one in '' lm.h

all t he hmken Jiagonals indude a l i the numb ·r-.: such is, for c. ·ample, Figure

5. But can th ·r · he a panc.liagonal Su· c.loku?

� hen he im L'nll'U Latin squ;tres, Lulcr h:�d alread} lum ec.l l9l thc im­po sibilit} of a pandi.tgonal Latin squ.tre \\ hose -.ide b e\ en or .1 mu lt i­ple of 3. In the e.trl) 20th t" ·ntuf} , the llunganan c.eorge Pol} ,t -.h< m cd [ 1 2]

the unpossibilit} of the equh alent

problem of 11 queen-. < capturing pan­

diagonally) on .111 11 X 11 che..,slx >ard i

40 THE MATHEMATICAL INTELLIGENCER

REFERENCES The various French newspapers and magazines published at the end of the 19th century and the beginning of the 20th century. And:

[1 ] Christian Boyer, Les premiers carres

tetra et pentamagiques, Pour La Science,

No286, August 2001 , 98-1 02

[2] Christian Boyer, Some notes on the magic

squares of squares problem, The Mathe­

matical lntelligencer, 27(Spring 2005),

52-64

[3] Christian Boyer, Letter- Magic squares,

Mathematics Today, 42(April 2006), 70

[4] Christian Boyer, Les ancetres fran<;ais du

Sudoku, Pour La Science, N°344, June

2006, 8-1 1 & 89

[5] Christian Boyer, Multimagic squares web

site, with some Sudoku pages, www.

multimagie.com/indexengl .htm

[6] General Cazalas, Carres Magiques au de­

gre n, Hermann, Paris, 1 934

(7] Jean-Paul Delahaye, Le tsunami du Su­

doku, Pour La Science, N°338, December

2005, 1 44-1 49

[8] Jean-Paul Delahaye, The science behind

Sudoku, Scientific American, 294(June

2006), 70-77

[9] Leonhard Euler, Recherches sur une nou­

velle espece de quarres magiques, Ver­

handelingen uitgegeven door het zeeuwsch

Figure 6. I hh 21 , 21 �udoku I'> .1 p:mJt:lgonal l.:l lin �quan.: L1<.h ... ubsquarc,

W\\ , (·olumn, ll lagon:tl , and broken d ia�on.tl i ncludes a l i the numb ·rs l through

2'>. There is no pandiagon. I Latin -.qu.ue of a non·pritnl' ordl'r smal ler than 21.

n = 21.! or 9�. Bcl.ttts · 9 1s a multiple

of j, ther · can h • no pandiagunal 9 X

9 Latin quarl..'. and a jinti()ri no pan·

diagonal 9 X 9 Sudoku Like\\ he a pandi:tgonal 16 X 16 udoku is ruled out. But noth ing cems to lorh1d .1 pan-

diagonal 21 X 21 Sudoku , that is, a p.tnd iagona l 2'> X 25 Lat i n square h.l\ ­i ng 111 adclltioll the , udoku feature th.lt

') X S suh'>qu.t res lOnt.t in a l l 2') num­b ·rs. Is it r�:all} po sibk-? Yes. Figure 6 sh<m -. th · ftr'>t kn1m n example 131.

Genootschap der Wetenschappen te

V/issingen, 9(1 782), 85-239 (reprint in

Euler Opera Omnia 1-7, 291 -392)

[1 0] Brian Hayes, Unwed numbers, American

Scientist, 94(January-February 2006), 1 2-1 5

[1 1 ] G. Pfeffermann, Probleme 1 72 - Carre

magique a deux degres, Les Tablettes du

Chercheur, Paris, Jan 1 5th 1 891 , 6 , and

Feb 1 st 1 891 , 8.

[1 2] George P61ya, Uber die doppelt-periodis­

chen Lbsungen des n-Damen-Problems,

in W. Ahrens, Mathematisch Unterhaltun­

gen und Spiele , Vol .2 , 2nd edition , Berl in,

1 91 8, 364-374

[1 3] Gaston Tarry, Le probleme des 36 of­

ficiers, Comptes-Rendus de /'Association

Franr:;aise pour I'Avancement des Sci­

ences, Congres de Paris 1 900, 1 70-204

<> <> <>

[1 4] Gaston Tarry, Sur un carre magique,

Comptes-Rendus Hebdomadaires des

Seances de I'Academie des Sciences,

Paris , 1 42(1 906), 757-760

53, rue de Mora

95880 Enghien les Bains

France

e-mail : cboyer@club-internet.fr

Illustrative Recreational Problems from France, 1888 through 1895

or the sake of authenticity, the problems are given in their orig­inal wording (directly translated)

and format. Thus if you see the 3 X 3 subsquares outlined with a darker line here as in contemporary Sudokus, this means they were that way originally.

After the statements appear com­mentaries in italics giving more detail. Other comments are in the body of the article where the problems are referred to. Solutions will appear in the next is­sue of The Mathematical Intelligencer.

The nine problems are given in chronological order.

Problem 1. Pandiagonal 9 x 9 magic

square

In Le Siecle, 23 June 1888, by Luet (pseu­donym of Major Coccoz).

Complete the square below by in­serting the first sixteen numbers, in such a way as to get a diabolic square:

It is known that in a ''diabolic square" all the conditions have to be fulfilled, that is, adding the numbers making up a horizontal row, or making up a ver-

1 7 24 31 38 54 61 68 75

33 37 53 63 67 74 23

62 69 73 1 8 22 32 39 52

27 41 48 34 71 78 55

40 47 36 70 77 57 26

72 76 56 25 42 46 35

21 51 28 44 81 58 65

50 30 43 80 60 64 20

79 59 66 1 9 49 29 45

tical column, or making up a main di­agonal, always gives the same total; moreover, the square remains magic if one moves a row or a column to the end.

Comment. Diabolic squares are now called pandiagonal magic squares, meaning that the broken diagonals, as well as the main diagonals, are magic.

Problem 2. The first 9 x 9 bimagic

square

In Le Siecle, 27 June 1H9 1 , by G. Pfef­fermann.

Complete the square below with the first 81 numbers, in such a way that one gets the same sum, 369, adding the nine vertical columns, the nine horizontal rows, and the two main diagonals.

One must also find all sums equal to 20049 when one adds the squares of the numbers horizontally, vertically, and diagonally.

Our warm compliments to the au­thor of this new and interesting prob­lem. We recall that Mr Pfeffermann was also the first to find a magic square of

62 43 69 50 73 30

66 40 36 79 47 59

33 76 56 21 72 53

38 57 54 70 77 31

80 48 41 60 34 64

67 51 74 28 63 44

75 29 61 45 68 49

58 35 46 65 42 81

52 71 32 78 55 39

3844 1 849 4761 2500 5329 900

4356 1600 1 296 6241 2209 3481

1089 5776 3136 441 5184 2809

1 444 3249 2916 4900 5929 961

6400 2304 1681 3600 1 1 56 4096

4489 2601 5476 784 3969 1936

5625 841 3721 2025 4624 2401

3364 1225 2116 4225 1 764 6561

2704 5041 1 024 6084 3025 1521

size 8 to two degrees. When will there be one of size 10?

Comment. Magic squares to two de­grees are now called himagic squares. Pfeffermann had indeed already found the first known himagic 8 X 8 square [I ll. A. Feisthamel, the editor of this col­umn, ':A problem a day" in Le Siecle, asks when we would have one of size 10. It is now known that the wait would he more than a century, for the first 10 X 10 himagic square was constructed in 2004 hy Fredrik Jansson of Finland.

But why are the magic sums 369 and 20049? Unlike a Sudoku or a 9 X 9 Latin square, which uses on�y the inte­gers 1 through 9 (so that each row adds to 45), a normal 9 X 9 magic square has to have all the integers 1 through 81 . What is the magic sum? The sum of the integers from 1 through n being n (n + 1 )/2, the sum of all the numbers used is 81 · 82/2 = 3321 . Every line has the same sum, the magic sum, so it must he 3321/9 = 369. By the same reason­ing, using the fact that the sum of the n

© 2007 Springer Sc1ence+Business Med1a, Inc , Volume 29, Number 1 , 2007 41

squares 12 through n2 is n · (n + 1) ·

(2n + 1)/6, we see that the magic sum of a bimagic square when the entries have been squared is 20049.

Problem 3. 9 x 9 magic square with

three 3 x 3 x 3 cubes

In Revue des jeux, 21 August 189 1 , by H . Mary (see Figure 2).

Using the first 8 1 numbers, complete the square below in such a way that every small square gives 1 23 horizon­tally and vertically.

2

14 40 54 59 39 80 3

4 6

7 9

8

The diagonals of the small squares directed the same as the two main di­agonals of the square of size 9, must also give 1 23 .

Superposing the squares 1 -2-3, 4-5-6, 7-8-9, or 1-4-7, 2-5-8, 3-6-9, one gets magic cubes giving the same constant, 123, in the nine verticals and the eigh­teen horizontals, but not along the four main diagonals of the cubes.

The six numbers filled in in the figure completely determine the square. Using complements to 123 and to 82, and by superposition, but without any trial-and­error, one recovers the square to be given.

Comment. The first known appear­ance of a 9 X 9 grid composed of3 X 3 subsquares!

Problem 4. 9 x 9 magic square

based on two Sudokus

In Le Siecle, 19 November 1892, by Ma­jor Coccoz.

Complete the square at the top of the next column using the first 81 num­bers, in such a way that the nine squares have their nine numbers adding up to the same total, 369. In addition, in the whole square, each main diagonal, each

42 THE MATHEMATICAL INTELLIGENCER

1 7 20 5 55 79 51 36 39 70 73 42 8 11 45 33 2 26 64 19 16 41 29 63 78 47 32 81 10 22 75 72 13 1 35 50 59 27 1 5 37 34 6 18 28 77 62 31 43 46 80 56 68 12 24 9

horizontal, and each vertical must give 369.

Comment. The special interest of this square is that, unlike the previous ones, it can be decomposed into two Sudokus using base 9. This reduces the given problem to that of solving these two Su­dokus:

Here is how to decompose, say, the entry 67 into 8 and 4 in the two grids: • subtract 1: 67 - 1 = 66;

2 3 1 7 9 6 4 5 8 9 5 1 2 5 4 1 3 8 3 2 5 4 7 9 6 4 9 2 3 9 8 2 1 4 6 7 3 2 5 4 1 2 4 9 7 4 5 6 9 7 8 2 3 1 8 2 5 1 7 6 9 3 7 1 6 8 2 9 6 2 8 1 1 7 5 2 9 6 2 5 9 1 4 3 9 4 1 8 5 5 9 6 1 7 6 9 1 5 8 4 7 1 8 2 5 3 6 9

• write the number in base 9, so here 66 = 7 · 9 + 3 becomes 73 in base 9;

• add 1 to each of the two digits, so 73 yields 8 and 4.

When these two Sudokus have been solved, it is easy to reverse the steps and recover the desired square.

The problem can also be solved di­rectly from the given array without this decomposition.

Problem 5. 9 x 9 Latin square

In Le Siecle, 31 December 1892, by B. Portier.

Complete the square below using the first nine numbers, each one nine times, in such a way that the horizontals, the verticals, and the two main diagonals give, when added, the same total.

�8 6 4 1 3 5 3 7 6 4 5 4 6 3 7 2 8 9 1

Comment. The problem statement might have specified that every row and every column is to include all the inte­gers from 1 through 9, like a Sudoku. But this is not a Sudoku: the subsquares of this latin square do not each include all the numbers just once, as one sees in the two 2s given in the upper left sub­square.

Problem 6. The problem of 81 officers, and a 9 x 9 bimagic

square, based on 2 Sudokus

In Les Tablettes du Chercheur, 1 April 1894, by A. Huber.

Arrange in a square 81 officers of nine ranks and coming from nine dif­ferent regiments, in such a way that in any row, in any column, in any diago­nal, and even in any compartment or 9-entry square, there are no two offi­cers of the same rank or from the same regiment. In addition the diagonals must remain good if three columns are

moved to the right or to the left, or three rows to the top or to the bottom.

1 4 5 9 7 72 48 24 1 0 36

30 4 48 9 1 2 4 27 56 35

1 6 63 36 20 1 8 35 24 1 2 1

32 36 5 6 1 8 4 40 21 63

28 1 8 72 42 25 1 2 2 8 1 8 3 21 1 0 54 8 56 45 1 6 1 2

8 1 32 1 4 1 5 28 30 6 3 1 6

1 5 6 7 64 42 9 8 54 20

6 40 24 8 2 27 49 45 24

The magic square above, made with the Pythagorean table of 9 elements, has 24 constants equal to 225, plus the 9 compartments; it is consequently 1/3 di­abolic, it can be rolled over if three columns or three rows are taken at a time.

This square gives the solution of the problem of 81 officers and of several other difficult problems: magic squares of two degrees, magic squares coming from the products of two magic squares, etc.

For the special case of the problem of 81 officers, it suffices to decompose each number as a product of two fac­tors in such a way as to get two magic squares, each made with the first nine numbers taken nine times.

Thus:

14 = 7 . 2. 5 = 5 . 1 . 9 = .� . 3 , 7 = 1 . 7, 72 = 8 . 9, 48 = 6 . 8, 24 = 4 . 6, 10 = 2 . 'i , 36 = 9 · 4, etc.

Comment. Tbe given square, using the numbers occurring in the multiplica­tion table of 1 through 9, is therefore not an ordinary magic square, as some numbers occur in it repeatedly. That is how it happens that the magic sum (the "constant" in the statement) is 225 and not 369. Every entry represents an offi­cer. The difficulty in this problem is

jznding the factorization of each num­ber and the order of the two factors which work. One then has the rank (a number from 1 through 9) and the reg­iment (a number from 1 through 9) of each o,( the 81 officers.

Tbe solution will thus he in the form of two Sudokus, one giving the ranks, the other giving the regiments. One can

then get a himagic square by putting everything back into base 10, just asfor Problems 4 and 7, which are different solutions o.f the problem o.f 81 o,fj!cers.

Problem 7. 9 x 9 magic square

based on two Sudokus

In La France, 16 June 1894, by B. Pmtier. Magic square of size 9 to two de­

grees. With the rest of the first 8 1 num­bers each used just once, fill out the square below in such a way as to ob­tain the constant 369 in the horizontals, verticals, and diagonals.

5 21 49 38

42 9 1 1 59

65 32 43 1

51 37 1 8 68 57

52 41 30 25 1 4 64 45 31

81 39 50 1 7

23 71 73 40

44 33 61 77

The numbers appearing in 1° every diagonal, 2° every big row in the cen­tral cross, and 3° the central square give equality to three degrees.

One can move a rectangle of three columns or rows to the end of the two others and the square will remain magic.

Comment. As this square is stated to be hi magic ("to two degrees

., J, the magic sum when the numbers are squared has to be 20049. Tbe two magic sums cif its sub­squares also have to be 369 and 20049.

Tbis can also be solved h,v decom­posing it into two Sudokus, as in Prob­lem 4. Tbe d!flerence is that this square is also himagic, and even partiafZy trimagic ( "equali�y to three degrees •:..__ i.e., we find the same total if we sum the cubes of the entries of either diagonal, the row or column through the central cell, or the central 3 X 3 suhsquare).

Problem 8. The macaroons

In L 'Echo de Paris, 10 July 1894, by Pic de Brasero (see Figure 4) .

The figure at the top of the next col­umn represents, if you wish, one of those sheets of macaroons which you may win at certain games.

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

"Listen carefully!" says the merchant to the kids who are crowding around his stand: "listen to this: whichever of you can take away 45 of these maca­roons in the way I tell you gets the whole lot for nothing! You see there are 9 compartments.

"Take away one macaroon wher­ever you like, two somewhere else, then 3, 4, 5, . . . 9, always changing squares. If you've done it right, there will be 4 macaroons left in each verti­cal, horizontal, and diagonal line . Also, what is left in the nine dozens will make up a magic square of size 3. Okay, kids! who is first? The losers pay ten sous for a sheet, the sharp ones get to eat free!"

The poor children all were stuck paying ten sous. And yet the problem is solvable, but how?

Comment. A little error in the state­ment: o,( course they aren 't dozens in the compartments but nines. Tbe solution to the problem is not unique.

A mere three weeks later, another publication, Les Tablettes du Chercheur, posed the same problem, as well as an­other problem in which 36 macaroons rather than 45 are to he taken away: zero in one compartment, then one, then 2, 3, 4 , . . . 8 and 5 macaroons are left in each vertical, horizontal, or diagonal line. Can you solve this vari­ant too? (les Tablettes spoke of spice cakes instead o,( macaroons, hut the mathematics is not a,.fj'ected.)

Problem 9. 9 x 9 diagonal Latin

square, diagonal Sudoku

In La France, 6 July 1895, by B. Meyniel (see Figure 4) .

Complete the square at the top of the next page using the first nine num-

© 2007 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 29, Number 1 , 2007 43

7 8 9 1 2 3 4 5 6 3 4 8 5 9 1

8 3 4

1 2 3 4 5 6 7 8 9 6 7 2

9 1 5 2 6 7 4 5 6 7 8 9 1 2 3

bers nine times each, in such a way that adding the horizontals, the verticals, and the two main diagonals all give the same total.

The square must be diabolic; that is,

44 THE MATHEMATICAL INTELUGENCER

it must remain a magic square if one moves a row or a column to the end of the others.

Comment. if on this grid you draw in borders of the 3 X 3 subsquares, you will have a Sudoku proposed for solu­tion in 1895.

Not only the rows, columns, and sub­squares, but also the two diagonals must include all the numbers.

And the solution must also be a pan­diagonal ("diabolic ") magic square, in the sense that the sum of every broken diagonal is 45. But it is not a pandi­agonal Latin square, for not every bro­ken diagonal includes all the numbers 1 through 9, and moreover it could not be: a 9 X 9 pandiagonal Latin square is impossible (see the section titled "Is It Possible to Construct a Pandiagonal Sudoku?").

� Springer the longuoge of •clence

Springer Onl ine Journa l Archives

Bringing Yesterday's Masters to Today's Minds Springer unloc:ks the gateway to historiCal

scientific research of the past century wrth the

IntroductiOn of the Online Journal Archives.

Available through Spnngerllnk researche� will

be able to retrieVe Information makong search

resuhs more efficient and productiVe and save

valuable research time The archiVes will add

more than 2 million scientific artiCles available

electronle.llly-rlght to your desktop.

VIsit springer.com for more Information

or go to sprlngerlink.com to browse

current and archive articles.

'""

On the True Nature of Turbu lence Y. CHARLES ll

In this article, I express some of my views on the nature of turbulence. These views are mainly drawn from my re­cent results on chaos in partial differential equations [ 12] . Fluid dynamicists believe that Navier-Stokes equations

accurately describe turbulence. A mathematical proof on the global regularity of the solutions to the Navier-Stokes equations is a very challenging problem. Such a proof or disproof does not solve the problem of turbulence. It may help in understanding turbulence. Turbulence is more of a dynamical system problem.

I will explain my reasons for believing that turhulence cannot he averaged. 1he hope is that turhulence can he con­trolled.

The Governing Equations of Turbulence Fluids are ubiquitous. The common forms of fluids are gas and liquid-for example, air and water. Fluid flows are clas­sified into two categories: laminar flow and turbulent flow. Laminar flow is smooth and mild, while turbulence is rough and violent.

Because of the rough nature of turbulence, some peo­ple even doubted whether the commonly used governing equations of fluids, the Navier-Stokes equations, are accu­rate in describing turbulence. It has now been over­whelmingly accepted by fluid dynamicists that the Navier­Stokes equations are accurate governing equations of turbulence. Delicate experimental measurements on turbu­lence have led to such a conclusion.

Let me summarize this development. A simple form of the Navier-Stokes equations, describing viscous incom­pressible t1uids, is

( 1 . 1 ) U1,t + u1u;.j = -p, , + Re - 1 u,,11 + ./i, uu = O :

where u/s are the velocity components, p is the pressure, f's are the external force components, and Re is the Reynolds number. (A subscript preceded by a comma de-

notes a partial differentiation; when a term has an index repeated, it is to be summed over that index . ) This form of the Navier-Stokes equations is relevant to fluids like wa­ter, which is almost incompressible. In contrast to water, air is more compressible, and the corresponding Navier­Stokes equations take a more complicated form.

There are two ways of deriving the Navier-Stokes equa­tions: ( 1 ) The fluid dynamicist's way of using the concept of a fluid particle which can be viewed as a small cube of fluid. moving like a particle and (2 ) The theoretical physi­cists's way of starting from the Boltzmann equation. Ac­cording to either approach, one can replace the viscosity term Re- 1 ui.if by, for example,

( 1 . 2 ) Re- 1 u1..zt + aui.Jfkk + · · · .

According to the fluid dynamicist's way, the viscosity term Re- 1 u1,11 was derived from a principle proposed by New­ton that the stress is proportional to the velocity's deriva­tives (strain, not velocity). Such fluids are called Newton­ian fluids. (Of course, there exist non-Newtonian fluids like volcanic lava, for which the viscosity term is more compli­cated and can be nonlinear.) According to the theoretical physicist's way, the viscosity term was obtained from an expansion which has no reason to stop at its leading-order term Re- I ui.if' There are two aspects of doubt about the Navier-Stokes equations: (a) because turbulence is rough, the higher-order derivatives in ( 1 .2) should be important and (b) turbulent solutions may not have second deriva­tives and may be solutions only in a weak sense, so that ( 1 . 1 ) might better he replaced by an integra-differential equation [8] . On the other hand, fluid experiments show that turbulent solutions do have finite second derivatives, and in fact are not even rough enough that the higher­order derivatives in ( 1 .2 ) are important. Nowadays, fluid dynamicists are focusing more on directly solving the Navier-Stokes equations on computers.

© 2007 Spnnger Sc1ence+ Bus1ness Med1a, Inc., Volume 29, Number 1 , 2007 45

Global Weii-Posedness of the Navier-Stokes Equations It is well known that the global well-posedness of the Navier-Stokes equations 0 . 1 ) has been selected by the Clay Mathematics Institute as one of its seven million-dollar prob­lems. Specifically, the difficulty lies at the global regularity [8] : the key is that

I I u u · dx dt l�J l�J

being bounded only implies

I U· ·U · · dx l,J l,J

being bounded for almost all t. Leray was able to show that the possible exceptional set of t is a compact set of mea­sure zero. There have been a lot of more recent works de­scribing this exceptional set [3]. A proof that this possible exceptional compact set is actually empty would imply global regularity and solve the problem. The hope for such a proof seems slim.

Even for ordinary differential equations, often one can­not prove global well-posedness, yet their solutions on computers look perfectly globally regular and sometimes chaotic. Chaos and global regularity are compatible. The hallmark of chaotic solutions is their sensitive dependence on initial conditions. That is, a small change in the initial conditions may lead to a huge change after a sufficiently long time, while all the solutions remain in a bounded re­gion in the phase space. This leads to the unpredictability, not irregularity, of chaotic solutions. The fact that fluid ex­perimentalists quickly discovered shocks in compressible fluids-and never found any finite-time blow-up in incom­pressible fluids-indicates that there might be no finite-time blow-up in the Navier-Stokes equations (or in the Euler equations, obtained by dropping the viscosity term Re- 1 u,,11 in ( 1 . 1)) . On the other hand, the solutions of the Navier-Stokes equations can definitely be turbulent.

Replacing the viscosity term Re-1 U;,ti by higher-order de­rivatives (1 .2), one can prove global regularity [6]. This leaves the global regularity of ( 1 . 1) a more challenging and inter­esting mathematical problem. Assume the unthinkable, that someone proves the existence of a meaningful finite-time blow-up in ( 1 . 1) ; then fluid experimentalists need to identify such a finite-time blow-up in the experiments. If they fail, the choice will be whether or not to replace the viscosity term

Y. CHARLES Ll received his B.Sc. from Peking

Universrty and his Ph.D. from Princeton. He is

known especially for the study of chaos in par­

tial differential equations. His worl< extends be­

yond the theory of p.d.e. and turbulence to nan­

otechnology and mathematical biology. He is

co-editor-in-chief with S.-T. Yau of the joumal

Dynamics of Partial Differential Equations.

Department of Mathematics University of Missouri Columbia, Missouri, 652 1 I USA

e-mail: cl@math.missouri.edu

46 THE MATHEMATICAL INTELLIGENCER

Re-1 U;,ti in the Navier-Stokes equations ( 1 . 1 ) by higher-order derivatives like (1 .2) to better model the fluid motion.

Even after the global regularity of (1 . 1) is proved or dis­proved, the problem of turbulence is not solved, although the global regularity information will help in understand­ing turbulence. Turbulence is more of a dynamical system problem. Often a dynamical system study does not depend on global well-posedness; local well-posedness is often enough. In fact, this is the case in my proof on the exis­tence of chaos in partial-differential equations [12] .

Chaos in Partial-Differential Equations Ever since the discovery of chaos in low-dimensional sys­tems, people have been trying to use the concept of chaos to understand turbulence [19] . As mentioned before, there are two types of fluid motions: laminar flows and turbulent flows. Laminar flows look smooth, and turbulent flows are non-laminar and look rough. Chaos can be made more pre­cise, as in the example below. On the other hand, even in low-dimensional systems, there are solutions which look chaotic for a while, and then look non-chaotic again. Such a dynamic is often called a "transient chaos. "

As mentioned before, the signature of chaos i s sensitive dependence on initial data. Often the word "sensitive" is used too loosely. For any fixed large time, the chaotic so­lution still depends continuously on its initial condition. It is the infinite time that destroys this.

Low-dimensional chaos is the starting point of a long journey toward understanding turbulence. To have a bet­ter connection between chaos and turbulence, one has to study chaos in partial-differential equations [ 12] . Take the simple perturbed sine-Gordon equation, for example [ 1 1] [ 17] :

(3. 1 ) uu = CZ.u= + sin u + E [- au1 + cos t sin3 u],

subject to the periodic boundary condition

u(t,x + 21T) = u(t,x) ,

and even or odd constraint

u(t, -x) = u(t,x) or u(t, - x) = - u(t,x) ,

where u i s a real-valued function of two real variables ( t,x), c is a real constant, E 2: 0 is a small perturbation parame­ter, and a > 0 is an external parameter. One can view (3. 1 ) as a flow defined in the phase space

(u,u 1) E H1 X L2,

where H1 and L 2 are the Sobolev spaces on [0,21Tl . A point in the phase space corresponds to two profiles

(u(x), ulx)) .

In the phase space, homoclinic orbits and heteroclinic cycles are often responsible for chaotic dynamics. A ho­moclinic orbit is an orbit that approaches the same point in both forward and backward infinite time limits. A hete­roclinic orbit will approach two different points in forward and backward infinite time limits. A heteroclinic cycle is formed when two heteroclinic orbits form a closed loop which looks like a eat's eye. For the system (3 . 1 ) , one can prove that there exists a homoclinic orbit ( u, u 1 ) = !:X_ t,x)

asymptotic to ( u, utJ = <0 ,0 ) [ 1 1 ] [ 1 7] . To establish the exis­tence of chaos, let us define two orbit segments,

T'Jo : ( U, Ut ) = (0,()) , T'/1 : ( u, u1 ) = h(t,x),

t E. [ - T, TJ . t E [ - T, T l .

When T is large enough, T'Jl is almost the entire homoclinic orbit (chopped off in a small neighborhood of ( 11, u1) = ( 0,0)). To any binary sequence

(3 .2) a = I · · · a�2a- , Cl(), a , a2 · · · I . ak E. 10, 1 } .

one can associate a pseudo-orbit

The pseudo-orbit T'Ja is not an orbit but almost an orbit. That is, the finite-time orbit segment starting from any point on T'Ja stays close to T'Ja· The concept of a pseudo-orbit was introduced by Anosov [2]. The goal is to prove the exis­tence of a true orbit around the pseudo-orbit. This is known as the shadowing lem ma.

Recently, such a tool has proved to be fundamental in proving the existence of chaos [12] . In the problem above, one can prove that for any such pseudo-orbit T'Ja, there is a unique true orbit in its neighborhood [ 1 1] [ 17] . Therefore, each binary sequence labels a true orbit. All these true orbits to­gether form a chaos. In order to talk about sensitive depen­dence on initial data, one can introduce the product topol­ogy by defining the neighborhood basis of a binary sequence

as all the sets

nN = I a : an = a;, , l nl ::; N) .

The Bernoulli shift acting on the binary sequence (3 .2 ) moves the comma one step to the right. Two binary se­quences in the neighborhood nt, will he of order n , away after N iterations of the Bernoulli shift-sensitive depen­dence on initial conditions. Since the binary sequences la­bel the orbits, the orbits will exhibit the same feature. In fact, the Bernoulli shift is topologically conjugate to the per­turbed sine-Gordon flow.

Replacing a homoclinic orbit by its "fattened'' version­a homoclinic tube, or hy a heteroclinic cycle, or by a het­eroclinically tubular cycle-one can still obtain the same Bernoulli shift dynamics [9] [ 10 ] [ 1 1 ] [ 17] . Adding diffusive perturbation Ehu1xx to (3 . 1 ) , one can still prove the exis­tence of homoclinics or heteroclinics, hut the Bernoulli shift result has not been established [ 1 1 ] [ 17 1 .

How close does this bring us to tluid problems' So far, we have been able to study a water wave problem-the so­called Faraday wave. By taking a bucket of water and shak­ing it vertically, one can generate persisting waves, a phe­nomenon first observed hy Faraday [')]. Recent experiments [20] have generated a variety of symmetric Faraday wave pat­terns: strips, squares, hexagons, de. These Faraday waves can he effectively described by an amplitude equation, the so­called complex Ginzburg-Landau equation [ 13] [14] [1 "il,

(3 .3) iqt = q"'. + 2 l1 q! 2 - w2] + iE [q, , - aq + {3q] ,

where q is a complex-valued function of two real variables U,x) , ( w,a,/3) are positive constants, and E 2: 0 is a small

perturbation parameter. We posit the periodic boundary condition and even constraint

q( t,x + 2 7T) = q( t,x), q(t, -x) = q(t,x) .

In this case, one can prove the existence of homoclinic or­bits [ 1 3] [ 1 ')] . But the Bernoulli shift dynamics was estab­lished under generic assumptions [ 14] [ 1 S ] . This is the first time that one can prove the existence of chaos in water waves (under generic assumptions) . Proving the existence of chaos for the full Navier-Stokes equations is still open.

The complex Ginzburg-Landau equation is a parabolic equation, which is near a hyperbolic equation. The same is true for the perturbed sine-Gordon equation with the diffu­sion term Ehu1xx added. Both contain effects of diffusion, dis­persion, and nonlinearity. The Navier-Stokes equations are diffusion-advection equations. The advection term is missing from the perturbed sine-Gordon equation and the complex Ginzburg-Landau equation. But the modified KdV equation does contain an advection term. In principle, the perturbed modified KdV equation should have the same features as the perturbed sine-Gordon equation. Turbulence happens when the diffusion is weak, i .e . , in the near-hyperbolic regime. One should hope that turbulence should share some of the fea­tures of chaos in the perturbed sine-Gordon equation.

There is a popular myth that turbulence is fundamentally different from chaos because turbulence contains many un­stable modes. In both the perturbed sine-Gordon equation and the complex Ginzburg-Landau equation, one can incor­porate as many unstable modes as one likes; the resulting Bernoulli shift dynamic is still the same. On a computer, the solution with more unstable modes may look rougher, but it is still chaos. So I think the issue of the number of unsta­ble modes between turbulence and chaos is an illusion.

Turbulence is any flow that is non-laminar. Sometimes, turbulence can happen in a localized spot of a fluid do­main, or during a finite period of time. These are not chaos. I have a favorite simile to represent the situation: One can think of turbulence as marbles; and those flows for which the existence of chaos can he rigorously proved, as dia­monds. Marbles are everywhere, whereas diamonds are rare. Understanding diamonds can help in understanding marbles. Diamonds are precious, while marbles are realis­tically useful in engineering.

A simple setup for studying the chaotic nature of tur­bulence is posing the Navier-Stokes equation 0 . 1) on a spatially periodic domain, with a temporally and spatially periodic external force. In this case, one can take advan­tage of Fourier series. One can show that there are well­defined invariant manifolds [16] . A thorough numerical in­vestigation of this dynamical system should be significant for a better understanding of turbulence.

Control of Turbulence When dealing with random solutions to a stochastic equa­tion, researchers are not content with the random solutions as they are. Various averagings will be conducted to gain more certain quantifications of the random solutions, since uncertainty is never preferred to certainty. Fundamentally encouraging to such thoughts is that these averagings are very successful in describing the random solutions.

© 2007 Spnnger Sc1ence+Bus1ness Med1a, Inc., Volume 29, Number 1, 2007 47

When dealing with Navier-Stokes equations, which are nonlinear deterministic equations, fluid engineers are very happy with laminar solutions as they are, but not the tur­bulent solutions. They have been trying hard to quantify turbulent solutions with averaging techniques. Reynolds en­visioned a relatively long time-averaging to the turbulent solutions, discussed below.

From what we learn about chaos in partial-differential equa­tions, turbulent solutions not only have sensitive dependences on initial conditions but also are densely packed throughout a domain in the phase space. Far from fluctuating around a mean, they wander around in a fat domain rather than a thin domain. Here averaging makes no sense at all; one has to be content with turbulent solutions as they are.

In real life, turbulence often represents unpleasant or disastrous events. When an airplane meets turbulence, the passengers do not feel comfortable and the airplane can be damaged. The question is whether turbulence can be con­trolled. Here the word "control" represents a wide spec­trum of actions: Taming turbulent states into laminar states [ 1 ] , reducing turbulent drag [ 18] [7] , enhancing turbulent mixing [ 18] [7] , gearing a turbulent orbit to a specific target [4] , etc. The final motto that I am aiming at is this:

Turbulence cannot be averaged, but can be controlled. Specific control tools have been developed. These are sen­sors and actuators placed in flow fields. These sensors and actuators can perhaps be replaced by MEMS (Micro­Electro-Mechanical-System) technology in the future to ob­tain more effective control.

One can re-interpret the Reynolds averaging as a con­trol taming turbulence into a laminar flow. According to Reynolds, one splits the variables in (1 . 1) into two parts:

U; = u, + U; , p = P + p ,

where the capital letters represent relatively long time­averages which are still a function of time and space, and the tilde-variables represent mean-zero fluctuations,

U; = (u;), (u ;) = o , P = (p) , (p) = o .

A better interpretation is by using ensemble average of repeated experiments. One can derive the Reynolds equa­tions for the averages,

( 4 . 1 ) U;,t + f1U,J = -P, + Re-1 U;.ff - (u;u},1 + /; , U;,; = 0 .

The term (u ,u1) is completely unknown. Fluid engineers call it Reynolds stress. The Reynolds model is given by

(4.2)

where R is a constant. There are many more models of the term (u;uj). But none leads to a satisfactory result.

One can re-interpret the Reynolds equations ( 4 . 1 ) as con­trol equations of the original Navier-Stokes equations ( 1 . 1) , with the term (u;, u1),1 being the control of taming a turbu­lent solution to a laminar solution (hopefully nearby). The Reynolds model (4.2) amounts to changing the fluid viscos­ity, which can bring a turbulent flow to a laminar flow. This laminar flow may not be anywhere near the turbulent flow, though. Thus, the Reynolds model may not produce satis­factory comparison with experiments. Fluid engineers grad­ually gave up all these Reynolds-type models and started di­rectly computing the original Navier-Stokes equations ( 1 . 1)

48 THE MATHEMATICAL INTELLIGENCER

An advantage of the control theory is that it can be con­ducted in a trial-correction manner without detailed knowl­edge of turbulence. Of course, better knowledge of turbu­lence will help the control. In a sense, locating chaos and controlling chaos are intertwined. For example, using the so-called Melnikov integral , one can predict the existence of chaos at some values of the external parameters [ 12] ; at the same time, one predicts the non-existence of chaos when the values of the parameters are changed.

REFERENCES [ 1 ] N. Alexeeva, et a/., Taming spatiotemporal chaos by impurities in

the parametrically driven nonlinear Schrodinger equation, J. Non­

linear Math. Phys. 8, suppl. (2001 ) , 5-1 2 .

[2] D. V. Anosov, Geodesic flows on compact Riemannian manifolds

of negative curvature, Proc. Steklov lnst. Math. 90 (1 967).

[3] L. Caffarell i , R . Kohn, L. Nirenberg, Partial regularity of suitably

weak solutions of the Navier-Stokes equations, Comm. Pure Appl.

Math. 35 ( 1 982), 771-831 .

[4] G. Chen, X. Yu, Chaos Control, Lecture Notes in Control and In­

formation Sciences, vol. 292, Springer-Verlag, Berlin, 2003.

[5] M. Faraday, On a peculiar class of acoustical figures, and on cer­

tain forms assumed by groups of particles upon vibrating elastic

surfaces, Phil. Trans. R. Soc. Lond. 1 21 (1 831 ) , 299-340.

[6] N. Katz, N. Pavlovic, A cheap Caffarelli-Kohn-Nirenberg inequality

for the Navier-Stokes equation with hyper-dissipation, Geom.

Funct. Anal. 1 2, no.2 (2002), 355-379.

[7] J. Kim, Control of turbulent boundary layers, Physics of Fluids 15, no.5 (2003), 1 093-1 1 05.

[8] J . Leray, Sur le mouvement d'un liquide visquex emplissant

l 'espace, Acta Math. 63 (1 934), 1 93-248.

[9] Y. Li, Chaos and shadowing lemma for autonomous systems of

infinite dimensions, J. Dynamics and Differential Equations 15, no.4 (2003), 699-730.

[1 0] Y. Li, Chaos and shadowing around a homoclinic tube, Abstract

and Applied Analysis 2003, no.16 (2003), 923-931 .

(1 1 ] Y. Li, Homoclinic tubes and chaos in a perturbed sine-Gordon

equation, Chaos, Solitons and Fractals 20, no.4 (2004), 791 -798.

[1 2] Y. Li, Chaos in Partial Differential Equations, International Press,

Somerville, MA, 2004.

[1 3] Y. Li, Persistent homoclinic orbits for nonlinear Schrodinger equa­

tion under singular perturbation, Dynamics of POE 1, no.1 (2004),

87-1 23.

(1 4] Y. Li , Existence of chaos for nonlinear Schrodinger equation un­

der singular perturbation, Dynamics of POE 1, no.2 (2004),

225-237.

[1 5] Y. Li, Chaos in Miles' equations, Chaos, Solitons and Fractals 22, no.4 (2004), 965-974.

(1 6] Y. Li, Invariant manifolds and their zero-viscosity limits for Navier­

Stokes equations, Dynamics of POE 2, no.2 (2005), 1 59-186.

[1 7] Y. Li, Chaos and shadowing around a heteroclinically tubular cy­

cle with an application to sine-Gordon equation , Studies in App/.

Math., 1 1 6 (2006), 1 45-1 7 1 .

[1 8] J . Lumley, P . Blossey, Control of turbulence, Annu. Rev. Fluid

Mech. 30 (1 998), 3 1 1 -327.

[ 1 9] D. Ruelle, F. Takens, On the nature of turbulence, Comm. Math.

Phys. 20 (1 67-192), 23 (243-244), ( 1971 ) .

[20] M. -T. Westra, et a/., Patterns of Faraday waves, J. Fluid Mech.

496 (2003), 1 -32.

la§iji§'.Jtj Osmo Pekonen , Ed itor I

Feel like writing a review for The

Mathematical Intelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or,

if you would welcome being assigned

a book to review, please write us,

telling us your expertise and your

predilections.

Column Editor: Osmo Pekonen, Agora Centre, 40014

University of Jyvaskyla, Finland

e-mai l : pekonen@mit.jyu.fi

Arnold's Problem edited by Vladimir I. Anzold

BERLIN-HEIDELBERG-NEW YORK, SPRINGER­

VERLAG, & MOSCOW, PHASIS, 2005. xv + 639 p., €53.45 SOFTCOVER. ISBN: 3-540-20748-1

REVIEWED BY SERGEI TABACHNIKOV

his is a very unusual book, and I think it is appropriate in this re­view to put things into perspec-

tive and to provide some background information.

V. I. Arnold is a famous mathemati­cian who belongs to the generation of Russian mathematicians now approach­ing their 70th birthdays. I will not list here Arnold's numerous awards and prizes: such lists are easily available on the Internet. I wil l , however, mention a most unusual distinction: a small planet, Vladarnolda. discovered in 1981 and registered under #10031 , is named for him.

This generation is extremely rich with talent and includes such towering figures as D. V. Anosov, A. A. Kirillov, Yu. I. Manin, S. P. Novikov, and Ya . G. Sinai. One cannot help noticing that these mathematicians were born about 19."17, the year of Great Terror in the Soviet Union. I once asked V. Arnold why, in his opinion, this generation was so exceptional. He replied that the pre­ceding generation of Soviet mathemati­cians largely perished in the Second World War. As a result, his generation had more freedom to develop inde­pendently, growing like trees on a cleared land, not in the shade of taller trees in a forest. One should also take into account that this generation came of age in the relatively liberal time of the post-Stalin "thaw".

Moscow was, and still remains, one of the leading mathematical centers of the world (I am talking only about Moscow, and only about the period from the late 1 9'SOs, because that is where and when most of the activity of V. Arnold and his seminar took place). Mathematical life revolved around a

number of seminars, some of them quite legendary. Mathematics of the highest quality and the charismatic per­sonalities of the organizers made these seminars strong attractors for budding mathematicians. It is, of course, beyond the scope of this review to describe Moscow mathematical life of the period in any meaningful detail; the interested reader can consult Golden years qf Moscow mathematics [9].

Arnold's Seminar began in the 1960s, a 2-hour weekly event. The format is different from the majority of mathe­matical seminars in the West that last an hour. This makes a substantial dif­ference in the style of presentation: a 2-hour talk is likely to include proofs. Traditionally, the first meeting of every semester was devoted to Arnold's talk on open problems. These problems, along with comments, comprise the book under review. The problems were addressed to all the participants of the seminar: undergraduate students, grad­uate students, and established re­searchers. (It should be noted that un­dergraduates specialized in their major from clay one, taking a large number of mandatory courses in mathematics. Starting with the 3rd year, undergradu­ate students of mathematics at Moscow State University had to choose an advi­sor, and by the end of the last year, the 5th, to produce a research thesis. It was not unusual for undergraduate students to have published results in refereed journals.)

When, about 1990, the Iron Curtain fell, many Soviet mathematicians left, temporarily or permanently, for the West. V. Arnold accepted a position in Paris for the spring semester every year; the Paris branch of Arnold's Seminar started to work in 1993. This signifi­cantly expanded the geography of the Seminar: many of its former partici­pants, now working in various coun­tries of the world, continue to follow Arnold's problem lists, now available on the Internet (see www.institut.math. jussieu .fr I semina ires/ singularites/).

Before reviewing the content of the book, a few words about V. Arnold's

© 2007 Springer Sc1ence+Bus1ness Med1a, Inc., Volume 29, Number 1, 2007 49

views on how to practice and teach mathematics are in order; this philoso­phy is implicit in the book and is ex­plicitly expressed in interviews Arnold gave to various mathematical maga­zines. I will quote from those sources.

What is mathematics? "Mathe­matics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap" [ 1 ] . "It seems to me that modern science (that is, theoretical physics, along with mathematics) is a new kind of religion: the cult of truth founded by I. Newton 300 years ago" [5] (my translation into English) .

On unity of mathematics: "Jacobi noted the most fascinating property of mathematics, that in it one and the same function controls both the presentation of an integer as a sum of four squares and the real move­ment of a pendulum" [ 1 ] .

About fashion in mathematics: "The evolution of mathematics re­sembles fast revolution of a wheel, so that drops of water fly in all di­rections. Fashion is the stream that leaves the main trajectory in the tan­gential direction. The streams of im­itation works are most noticeable; they constitute the main part of the total volume, but they die out soon after parting with the wheel. To keep staying on the wheel, one must apply effort in the direction per­pendicular to the main flow" [5] (my translation into English).

It is worth mentioning that Arnold himself has created fashions in mathematics more than once!

On errors in mathematics: "Mis­takes are an important and instruc­tive part of mathematics, perhaps as important a part as the proofs. Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of words" [2].

On applications: "A remarkable property of mathematics, which one cannot help but admire, is the un­reasonable effectiveness of its most abstract, and at first glance com­pletely useless, branches, provided

50 THE MATHEMATICAL INTELLIGENCER

that they are beautiful" [8]. "Accord­ing to Louis Pasteur, there exist no applied sciences-what do exist are applications of sciences" [3] .

On advising students: "A student is not a sack to be filled but a torch to be lit . " [5] (my translation into English) .

And finally, from the author's preface to the Russian edition of Arnold's Problems (2000; reprinted in the book under review): "I . G. Petrovskii, who was one of my teachers in Mathematics, taught me that the most important thing that a student should learn from his su­pervisor is that some question is still open. Further choice of the problem from the set of unsolved ones is made by the student himself. To se­lect a problem for him is the same as to choose a bride for one's son . " Arnold 's Problems consists of two

parts: the first third consists of formu­lations of the problems, and the rest comprises comments on them. The problems are grouped by years, from 1956 to 2003 . The total number of prob­lems is 861 ; some of them repeat, usu­ally, with variations (in the case of such "twin" problems, only one comment is given, with reference to other similar problems). In the late 1950s and the 1960s, the problems were not collected systematically, and many were lost. From 1970 on, the lists are more-or-less complete. The number of problems per year changes significantly, with the an­nual average in 1970-2003 of about 24. Some of the problems were posed by other mathematicians, in which cases the name of the proposer is indicated.

Problems are usually not formulated in the most general form, but instead consider the first non-trivial case of a general phenomenon; to discover and describe this general phenomenon is a substantial part of the problem (V. Arnold refers to this as the "Russian style" of posing problems) .

Working on this review, I tried to classify the problems according to the areas of mathematics they represent. This attempt of "subject classification" proved fruitless. It is fair to say that the majority of problems reflect long-term research interests of V. Arnold, and these interests span a sizable part of contemporary mathematics. Many prob-

!ems concern dynamical systems, in particular, the KAM (Kolmogorov­Arnold-Moser) theory, local and global singularity theory, real algebraic geom­etry, symplectic and contact geometry, classical mechanics, topological hydro­dynamics. Numerous problems from the late 1980s and 1990s concern geometri­cal and topological, in particular, sym­plectic and contact, generalizations of the classical four-vertex theorem (a plane oval has at leastfour curvature ex­trema; this result, due to S. Mukhopad­hyaya, 1909, has generated a huge lit­erature) , and a substantial part of the problems in recent years belongs to number theory, in particular, to the theory of multi-dimensional continued fractions.

A typical formulation of a problem is one paragraph long-in some cases, just a sentence-but there are notable exceptions to this rule, and all the prob­lems in 2003 have much more detailed formulations, sometimes several pages long (but they are not commented on). Instead of continuing with this general description, necessarily boring, let me give a few examples.

The first three are examples of prophetic vision: these questions and their generalizations and ramifications significantly influenced the develop­ment of contemporary mathematics.

PROBLEM 1958-1. Let us consider a partition of the closed interval [0; 1] into three intervals 111 , 112, 113 and rearrange them in the order 115, 112, 111 . Explore the resulting dynamical system [0; 1] --4 [0; 1 ] : is it true that the mixing rate and similar ergodic characteristics are the same for almost all lengths (11 1 , 112, 113) of the partition interval? An analogous question may he asked for n intervals and for arbitrary permutations as well (changing the orientation of some in­tervals also being allowed) .

The theory of interval exchange transformations and the related subjects of flat surfaces, quadratic differentials and billiards in rational polygons have become fast-growing and important areas of research, rich with deep and beautiful results.

PROBLEM 1963-1. Is there true insta­bility in multidimensional problems of perturbation theory where the invariant tori do not divide the phase space?

This problem concerns the phenom­enon known as "Arnold diffusion" in the theory of small perturbations of inte­grable Hamiltonian systems (KAM the­ory). A breakthrough was recently made by J. Mather, who proved the existence of Arnold diffusion in some cases.

PROBLEM 1966-4. Let a diffeomor­phism A : q � q + fCq) of the torus T2 = { (q1 ,q2) mod 27T) preserve the mea­sure dq1 1\ dq2 and the center of mass:

Prove that A has at least 4 fixed points counting multiplicities and at least 3 geometrically distinct fixed points.

This is one of the celebrated Arnold conjectures on fixed points of exact symplectomorphisms; it was proved by Conley and Zehnder in 1983 . Today symplectic topology is a large and rapidly developing field, one of the main achievements in mathematics of the last 20 years.

Let me give a few more samples of problems, from very broad to fairly con­crete.

PROBLEM 1979-4. Construct a ''com­plexification" of the homology theory (replacing a boundary with a two-sheet branched covering) . What is the com­plexification of orientation? ( Appar­ently, it assigns an element of Z =

7T1( U( n) ) to a loop. )

PROBLEM 1987-14. Do there exist smooth hypersurfaces in R11 (other than the quadrics in odd-dimensional spaces), for which the volume of the segment cut by any hyperplane from the body bounded by them is an algebraic func­tion of the hyperplane? For these quadrics the volume is an algebraic function (Archimedes), and the area ol segments of plane cumes is never al[!,e­hraic (Newton).

PROBLEM 1989-18. The sequence of meandric numbers 1 , 1 ,2,3,8, 14,42,81 , . . . i s defined a s follows. Suppose an infinite river running from south-west to north-east intersects an infinite straight road going from the west to the east under n bridges numbered 1 , . . . , n in the order from west to east . The or­der of the bridges along the river de­termines a meandric permutation of the

numbers 1 , . . . , n. The meandric num­ber Mn is the number of meandric per­mutations on n elements. Meandric numbers possess many remarkable properties; j(;r example, M11 is odd iff n is a power of 2 (S. K. Lando). Find the asymptotics of M11 as n ---'> oo. It is known that c4" < Mn < C16n jbr some con­stants c, C

PROBLEM 1994-17. Find all projective curves equivalent to their duals. The an­swer seems to he unknown even in RP2

I would like to add, in case the reader has doubts, that there exist even smooth convex curves, other than conics, pro­jectively equivalent to their duals!

PROBLEM 2000-7. There are observa­tions that the number of species (of an­imals, insects, birds, . . . ) on an island of area S is proportional to S 114

, whereas the number of cell types in an organism with a genome of N genes grows with N like N112 How can one explain these exponents( Compare with the Kolmogorov law, according to which the radius of the minimal but still typical brain or computer of N elements grows like NL 2 ( rather than like N1d, as the volume argument suggests) .

0 0 0

I cannot resist mentioning the very first problem in the book, #1956-1 , "The rumpled dollar (originally, rouble) problem": is it possible to increase the perimeter of a rectangle by a sequence of foldings and unfoldings?

Like many problems in the book, this one comes with an interesting story. Also known as the "Margulis napkin prob­lem", it has become part of mathemati­cal folklore. The answer to the question is yes: see [7, 6] . It is interesting that the problem was solved by origami practi­tioners long before it was posed (as early as 1797, in the Japanese origami book "Senbazuru Orikata' ' ) , see [10]. One can­not help but agree with M. Berry's Jaw: Nothing is G>uer discovered for the first time (posted on his web site)!

The comments presented in the rest of the book vary from detailed surveys several pages long, equipped with ex­tensive bibliography, to very brief, one­paragraph, references to the literature. Sometimes a comment simply states: ''Nothing is known" . Each comment is marked 'iJe or m, indicating a historic or

research comment. Some problems are commented on more than once, by a number of authors.

The list of authors of the comments consists of 59 names, twice the number in the 2000 Russian edition. An author index for comments is provided; all but one (J. Lagarias) are current or former participants in Arnold's Seminar, his stu­dents in the general sense of the word (this includes me); one of the most pro­lific commentators is V. Arnold himself. Many of the commentators contributed to (partial) solutions of these problems (according to Arnold, the average half­life of a problem is 7 years).

This genre of book is rather rare; "The Scottish Book" comes to mind (a col­lection of about 200 problems, with com­ments, composed by a group of Lvov mathematicians in 1935-41 ; this group included Banach, Mazur, Steinhaus, Ulam and others.) . Surely, many mathe­matical problem lists are known, some of them very influential (for example, Hilbert's problems), but none come close in their sheer volume, width, and breadth to this one. Arnold's problems remain today as inspiring and stimulat­ing as ever. The book belongs in evety mathematical library and on the book­shelf of every research mathematician.

The authors, editors, and publishers of the hook have done an excellent job. I hope that this is an ongoing project, and that there will be further editions, with new problems and new comments. I would like to suggest that the new edition(s) have a broader commentator base; the creators of the book may con­sider having a designated website for downloading comments on the prob­lems. It would be very helpful to have some kind of a problem index and their rough classification by topics, hut as I already mentioned, this is a very non­trivial task.

In conclusion I would like to refer to another review of the same book [4] , written by an editor of the Russian edi­tion of "Arnold's Problems" and a ma­jor contributor to the comments, M. Sevryuk.

REFERENCES [ 1 ] V. Arnold. On teaching of mathematics.

Russian Math. Surveys 53 (1 998), 229-

236.

[2] V. Arnold. Polyrnathematics: is mathemat­

ics a single science or a set of arts? Math-

© 2007 Spnnger Sc1ence+Bus1ness Media, Inc., Volume 29, Number 1 , 2007 51

ematics: frontiers and perspectives, 403-

41 6, Amer. Math. Soc., Providence, Rl ,

2000.

[3] S. H. Lui. An interview with Vladimir Arnold.

Notices Amer. Math. Soc. 44 (1 997), 432-

438.

[4] M. Sevryuk. Arnold's problems, book re­

view. Bull. Amer. Math. Soc. 43 (2006),

1 01 -1 09 .

[5] S. Tabachnikov. Interview with V . I . Arnold.

(in Russian) Kvant 1 990, No 7, 2-7, 1 5 .

[6] A Tarasov. Solution of Arnold 's "folded

rouble" problem. (in Russian) Cheby­

shevskii Sb. 5 (2004), 1 74-1 87.

[7] I . Yashenko. Make your dollar bigger

now!!! Math. lntelligencer 20 (1 998), no. 2,

38-40.

[8] S. Zdravkovska. Conversation with Vladimir

lgorevich Arnold. Math. lntelligencer 9 (1 987), no. 4, 28-32.

[9] Golden Years of Moscow Mathematics.

American Mathematical Society, Provi­

dence, R l , 1 993.

[1 0] www.origarni .gr.jp/Modei!Senbazuru/in­

dex-e.htrnl

Department of Mathematics

Pennsylvania State University

University Park, PA, 1 6802

USA

e-mail: tabachni@math.psu.edu

Imagining N umbers

(particularly the

square root of

minus fifteen) by Barry Mazur

NEW YORK, FARRAR, STRAUS, GIROUX. 2003.

270 pp. , US $22.00, ISBN 0 374 17469 5.

REVIEWED BY PAMELA GORKIN

A mathematical colleague of mine once said that when a poet or artist asked him if he could ex­

plain what "he did," his answer was "no. " The question is not really what "he" does, but rather what "we" do. What do mathematicians do? This is a question every mathematician should be able to answer. Yet it is a difficult

52 THE MATHEMATICAL INTELLIGENCER

task if we cannot use the vocabulary we find so natural, if we have no groundwork on which to build our an­swer and, in the words of a colleague of mine, if we are "trying to sell math­ematics to people who don't buy it."

In the book Imagining Numbers (particularly the square root qf - 15), Barry Mazur sets himself the difficult task of explaining to a humanist what mathematics really is. And, for human­ists who are determined to understand our work and our motivations, Mazur's book will help them do so. Furthermore, a mathematician who invests the time and effort it takes to digest the literary references will be handsomely rewarded with some interesting new ideas.

So what do mathematicians do? The answer, not to ruin the suspense, is that a mathematician does what a poet does . . . sort of. The goals are the same: to be creative in a way that no one has been before, to push the boundaries of knowledge, and to convince the reader with carefully chosen words. Mazur ar­gues that poets and mathematicians rely on "economy of expression, " but for dif­ferent reasons. He makes his case, doc­umenting his argument with references from poetry, literature, art, and the White Flower Farm catalogue.

This book encourages us to imagine "the yellow of the tulip" before turning to imagining numbers. But how do we imagine numbers and why focus par­ticularly on the square root of minus fif­teen? The imagination, Mazur argues, may do its work immediately (as yours might have done imagining the yellow of the tulip) . Sometimes, however, it takes time and effort for the imagina­tion to work. To illustrate this point and to preface the discussion of square roots of negative numbers, he turns to a dis­cussion of why v2 is irrational. Once the case for studying v2 is made, he moves on to the case for studying v"=l. Those who chose to ignore such con­cepts, Mazur tells us, did so "at the price of limiting their power as algebraists" (p. 36). This brings him to the story of Girolamo Cardano, to whom he attrib­utes t�uote "You will have to imag­ine Y - 1 5" (p. 39).

Cardano's story leads naturally to an even more interesting tale: that of a "pub­lic problem-solving contest" between Antonio Maria Fiore and Niccolo Fontana (also known as Tartaglia) (p. 108). Three

problems are presented, the third of which is to find all solutions of X3 + 1 = 3X In this way, Mazur transports the reader from familiar ground, solving qua­dratic equations with the quadratic for­mula, to ground that is, perhaps, unfa­miliar. Considering a relative of this equation, X3 = 3X - 2, the reader is led throu?h a convincing argument that X = \1=1 + \1=1 (p. 1 21) . But clearly, X = 1 and X = -2 are solutions. How can this be? If the reader is a mathe­matician, he or she will see where Mazur's book is headed. On the other hand, the humanist will surely be taken aback and, we would hope, delighted.

Imagining Numbers chooses its ref­erences from many different fields, and the book's blend of mathematics, history, philosophy, language, and literature can be overwhelming. At times it seems that there might be too much focus on math­ematics for the humanist and too much focus on the humanities for the mathe­matician. References to Wittgenstein, "the Stoic Chrysippus", "the early twelfth­century Sufi Ibn al-'Arabl, " Yeats, John Livingston Lowes, Coleridge, the poet Stephen Dobyns, Rainer Maria Rilke, and the poet Franz Kappus appear in the space of two pages in the book, only to be replaced by Thomas Harriot, A. L. Cauchy, and Augustus De Morgan in the pages that follow.

It is important to keep in mind that although this book presents a point of view that is helpful for any mathemati­cian trying to make the case for his or her subject, this is a book aimed at the poet, the artist, and the historian. It is easier than you might think to throw someone who is unaccustomed to thinking mathematically off the track: One of my colleagues in the humani­ties who was adventurous enough to read this book (and who thanked me for introducing him to it) pointed out that he was confused by the quadratic formula. He recalled the formula hav­ing an "a" in it, one that has been re­placed here by the constant 1 . While a mathematician might not even notice that the "a" is "missing, " humanists might be confused by this change.

The literary references, the focus on a geometric point of view, and the clear and extensive explanations of things mathematicians take for granted are surely helpful . Mazur has a keen eye for deciding what to explain. I might not

have realized, for example, that the placement of i in the polar coordinate system is something that should be so carefully motivated and justified. The balance, however, is sometimes difficult to maintain: while I would guess that every mathematician would grow impa­tient with the extended argument of why a negative times a negative is a positive, my colleague in the humanities felt the discussion could have gone on longer. And if humanists need the lengthy ex­planations of where i should he placed in the plane and why a negative times a negative is positive, what would a sen­tence like "Cauchy's first memoir on contour integrals in the complex plane was published, ushering in the immense 'literature of use' of the geometric view­point" mean to them? (p. 213 )

In the end, however, !lnaginin.g Numbers succeeds not only in con­vincing its readers that we can imagine imaginary numbers, hut in convincing us that we should imagine them. In Smilla 's Sense (�! Snuu• the smart and strong heroine of the novel, Smilla Qaavigaag Jaspersen, describes the foundations of mathematics. She tells us about the creation of natural numbers, negative numbers, rational numbers, real numbers and, finally, complex numbers. Smilla says, "we expand the real numbers with imaginary square roots of negative numbers. These are numbers we can't picture, numbers that normal human consciousness cannot comprehend. '' Smilla, Ima.ginin.g Num­bers is the book for you .

Department of Mathematics

380 Olin Science Building

Bucknell University

Lewisburg, PA 1 7837

USA

e-mail: pgorkin@bucknell .edu

Le Cas de Soph ie 1<. written and directed hv

jean-Fram;uis Peyret

AVIGNON FESTIVAL, VILLENEUVE LEZ AVIGNON,

LA CHARTREUSE, JULY 9TH 2005 TO JULY 24TH

2005

PARIS, THEATRE NATIONAL DE CHAILLOT,

APRIL 26TH 2006 TO JUNE 27TH 2006

REVIEWED BY ALAIN JUHEL

e are all aware of the formi­dable complexity of the Three­Body Problem in Celestial

Mechanics. By splitting Russian mathe­matician Sofia Kovalevskaya ( 1850-1891 ) into three characters, the French playwright and director Jean-Franc;:ois Peyret evokes both the surprising, sometimes even baffling complexity of the uncommon female scholar and one of her masterpieces in the field of dy­namical systems. This was the work for which she was rewarded in 1886 with the Prize of the French Academie des Sciences and in 1889 with the Prize of the Swedish Academy in Stockholm.

The title of the play sets up these multiple views: 1be Case of Sofia K. refers of course to the mathematical problem, that builds on earlier work of Euler and Lagrange, in which a system of differential equations is solved in closed form by means of elliptic func­tions. But it may also be understood as a human story, an exploration into the life and mind of a fascinating woman, and this is mostly what Peyret's play is about. Mathematics necessarily brought Sofia face to face with feminism (she had to assert herself in a world of men that was not quite free of preconcep­tions, despite the early and generous support of Weierstrass, Hermite, and Mittag-Left1er), hut there are far more surprising facts for those unfamiliar with her biography. Between the time she attended Weierstrass's private lessons and completed her Ph.D. , Sofia traveled to Paris in 1871 with her elder sister Anna and enlisted as a nurse in the ranks of the rebels fighting for the fa­mous Fre11ch Commune. The two sis­ters looked toward unconsummated marriages, as a means of freeing them­selves: Anna set her heart on a pale­ontologist named Kovalevsky, who happened to prefer Sofia . Marriages of convenience were widespread among the Nihilists, a growing movement among the young Russian Intelligerzt­siya; Kovalevskaya was a sympathizer, albeit a non-violent one. Last hut not least , she was a lover of art and litera­ture; she wrote A Nihilist, a mostly au­tobiographical novel, published post­humously. Vera, the heroine, and Sofia's alter-ego, spends a lot of time looking for ways to serve her Great Cause be­fore she finds her own: she will sacri­fice herself and marry a rebel sentenced

to life imprisonment in Saint-Peter and Saint-Paul's fortress in Saint Petersburg, the most awful jail at that time. Thus she will save him and have his sentence commuted (so to say softened!) to de­portation to Siberia. It was the custom for wives to follow their husbands and stay with them until their sentences were served, so Vera's choice had to be considered as a martyr's, and this was precisely the way she looked at it.

There are three Sofias in the story, and three fine actresses on stage: Olga Kokorina, Elina Lowensohn, Nathalie Richard. Do not believe, however, that each is restricted to a single face of Ko­valevskaya: this would have been an oversimplification, like limiting the study to steady-state solutions, as seen by a mathematician. The way the three women share the feelings and writings of Sofia could rather he compared with the basins of attraction of a discrete dy­namical system with three attractors. (Thinking of a pendulum oscillating over three magnets provides a good pic­ture . ) Such dynamics do not show their wonderful and sophisticated portraits at once, they are not drawn in a linear way. This play delivers an image of Sofia in a similar manner: we get the pieces of a puzzle mixing childhood recollections, letters, mathematical writ­ings (Kovalevskaya, Poincare). It inter­laces the real Sofia and the Nihilist Vera she dreams of. This duality was under­stood by Sofia's friend and biographer, Anne-Charlotte Leffler ( Gosta Mittag­Leffler's sister): an internal and restless struggle between two directions of her mind. Should she dedicate herself to Mathematics or . . . to the Revolution? In contrast to a wavering Hamlet, Sofia never remains undecided for long: she has energy and resolution, but as she knows that her entire being yearns for an exclusive commitment, the choice she makes at one particular moment never seems to be the right one . . . .

Jean-Fran<;ois Peyret has devised a setting that enhances this feeling of complexity: the three Sofias appear in turn on a large screen in the back­ground. Sometimes these are pictures filmed live by a cameraman (who re­mains on stage throughout the play), sometimes these fragments of movies were recorded before . Sometimes we see the actress who is actually speak­ing, sometimes another one who is lis-

© 2007 Spnnger Science+Business Media, Inc., Volume 29, Number 1 , 2007 53

tening to her, but always in the fore­ground. I must confess that I felt the pure-video introduction a bit lengthy (was I so impatient to see the actresses come in? Did Peyret want to whet my appetite?), but then I became comfort­able with that mix of on-stage play and movie. I wished the cameraman had not been so close to the three women: he succeeded in being inconspicuous, but I could never put him out of my mind. There is a pianist on stage, too, but his physical presence did not bother me at all: he was a few meters away to the left, and the music, a mix of repertoire and improvisation, was quite in har­mony with the acting.

My wife and my daughter went with me to the theater that evening. They have nothing to do with mathematics, they had never heard of Sofia before. As the play began, I felt anxious about their reactions: would they be discon­certed by such a fragmented portrait? One confessed she had mixed feelings; but the play appealed to the second one's fancy. I regard her applause a bet­ter proof than mine of Peyret's success! Choosing such a topic-an unexpected encounter, he told a reporter-was ob­viously not risk-free; it was all the more daring because he refrained from high­lighting a romantic vision of the math­ematician. On the other hand, he showed great professionalism at work in reading, with all his team, books that try to explain how a mathematician's mind works (Henri Poincare's Science et Methode, Jean-Pierre Changeux and Alain Coones's Matiere a Pensee) . He also managed to meet Kovalevskaya's French biographer Jacqueline Detraz, and invited a well-known specialist of Dynamical Systems, Michele Audin, to a rehearsal so that the actresses could learn Everything They Always Wanted to Know About Maths But Were Afraid to Ask. We may consider the result as a success in its display of mathematics to the non-professional, and perhaps also as a victory if it does away with a narrow-minded view of this science. Let us hope that people will walk out of the theater with a recollection of young Sofia wondering at the amazing wall­paper in her bedroom:

The room remained unfinished for quite a long time, there was just some paper laying on the walls. It hap-

54 THE MATHEMATICAL INTELLIGENCER

pened to be folios of Ostrogradski 's Differential and Integral Calculus Course; it had been bought by my fa­ther when he was a young student.

or these words by Poincare (Science et Methode), spoken by one of the Sofias:

The scientist does not study Nature because it is useful; he studies it be­cause he gets pleasure from it, and he does because Nature is beaut[ful. If it was not, it would not be worth studying, and l[fe would not be worth living.

Charles Hermite was known to be a stern-looking person with conservative views. But he could hardly resist Sofia's charm-intellectually speaking. As a testimony, here is an excerpt from a let­ter he wrote to Mittag-Leffler (June 19, 1882) telling about the very moment when he heard Sofia say that Lindemann had proved that 1T is a transcendental number: "I met Mrs de Kowalewsky at her house several times: she adds a charming grace to her extraordinary talent as a geometer. "

Jean-Fran<;:ois Peyret's play will surely impress every spectator and com­pel him or her to find out more about Sofia's life (and the political context of those troubled times) and about her works (either A Nihilist or Dynamical Systems). Time passed rapidly, and the play lasted only an hour and a half. Many questions remained unsolved, as Peyret must have felt himself . . . He might want to carry on with this work in progress begun at the celebrated A vi­gnon Theater Festival and complete a larger portrait. But there is a more ur­gent task to deal with: translating this play into other languages than French.

REFERENCES (1 ) Michele Audin, Le cas de Sophie K. i n

Gazette des Mathematiciens, no1 06, Octo­

ber 2005.

(2) Michele Audin, Man choix de Sophie, Feb­

ruary 2006.

(3) Jacqueline Detraz, Kovalevskaia, !'A venture

d'une Mathematicienne, Belin, Paris, 1 993.

(4) Ann Hibner Koblitz, A Convergence of Lives,

Sofia Kovalevskaia, Birkhauser, Boston 1 983.

(5) Pelageya Kochina, Love and mathematics.

Satya Kovalevskaya, Mir Publishers, Moscow,

1 985.

(6) Sofia Kovalevskaya, A Russian Childhood,

Springer, Berl in, 1 978.

(7) Sophie Kovalevska"ia, Une Nihiliste, Phebus,

Paris, 2004.

N.B.: Michele Audin's papers are available at her

web page: http://www-irma.u-strasbg.fr/-maudin/

publications.html

Lycee Faidherbe

Lille, France

e-mail: ajuhel@nordnet.fr

Real ity Conditions:

Short Mathematical

Fiction by Alex Kasman

THE MATHEMATICAL ASSOCIATION OF AMERICA,

2005, 247 PP., US $29.95, ISBN 0-88385·552·6

REVIEWED BY MARY W. GRAY

hatever its deficiencies, C. P . Snow's Two Cultures [1 ] con­cept of mutually incompre­

hensible dialogue between scientists and others lives on. For as intriguing as the collection of short fiction in Reali�y Conditions might be to mathematicians, it is difficult to believe that many of the stories would appeal to, or indeed be understood by, those on the "other side" of the cultural divide. In one story, the eponymous Topology Man relates what may be music to a mathematician's ears but is unfortunately probably just jargon to the rest of the world:

In a Hausdorff space, like the one we live in, the bad guy . . . can al-ways avoid the superhero . . . be-cause he can get into a disjoint neighborhood out of his reach. But if this was the right sort of non-Haus­dorff space, he wouldn't be able to avoid me. But the villain Homotopy responded

to this maneuver by growing the bas­ketball at our hero's feet, twisting it around him, and inverting it to im­prison the Topology Man. The re­sourceful topologist reacted:

Endowing the basketball with the topology of a Klein Bottle, I was able to escape . . . . The floor of the bas­ketball court began to deform again. It returned to being flat, but now

each point was continuously shifting and moving around, making it al­most impossible to stand.

Invoking Brouwer's Fixed-Point Theo­rem, the Topology Man continues:

I leapt into the air, did a back flip and landed smugly on the unique fixed point.

While there is a lot of argot in the hook, many of the best lines are like the per­haps non-politically correct joke: "A Unitarian is one who believes that there is at most one god. " All the words are known to everyone, but generally only the mathematicians laugh. Not that Re­alizy Conditions will provide much hearty laughter; more likely, some self­satisfied smiles.

The publisher's blurb asserts that the entire hook could help form the basis of a creative course on mathematics in fic­tion. Maybe so, hut the audience would have to be pretty sophisticated mathe­matically. The main effect might other­wise be to reinforce the nerdy image many have of mathematicians; some of Kasman's mathematicians are pretty weird. An adolescent desire to emulate Superman or Xena is unlikely to be re­placed by a yen to become the Topol­ogy Man or his colleague The Category Girl. In spite of the patronizing "girl" here paired with "man," there is a satisfying number of women mathematicians in the collection, but frankly they are not likely to be seized upon as role models by to­day's students. Especially not the closet mathematician whose work is uncovered in "Murder, She Conjectured"' In this, the only mystery among the 16 stories (re­grettably in my view). a psychologist and a mathematician pair up to try to dis­cover the story behind the strangling of a woman in London in 1870.

What Kasman is very good at is cre­ating the world in which mathematicians work. From the anxious graduate stu­dent, to the novice Ph.D . , to the disil­lusioned professor, to the distinguished expert, and from the isolation of a shabby office, to the interchanges at a coffee bar, to the atmosphere at MSRL mathematicians will recognize them­selves, their colleagues. and their sur­roundings. The title story is perhaps the best in its ability to portray the "reality conditions" in which many of us labor. Here we see how hope can spring eter­nal for those who believe their work is

just a little bit away from fame. That the hero of "Reality Conditions" is called Gil­gamesh is a nice touch.

The author, a faculty member at the College of Charleston who has worked in algebraic geometry with applications to physics and biology, knows his math­ematics and knows his colleagues in mathematics and in academe in gen­eral. It is easy to identify with the meet­ing of a mathematician with a board of education in "Another New Math,'' and even with the most bizarre of his char­acters and situations. Admittedly, how­ever, a ransom note in "The Math Code" containing

I proved a new theorem. It was that for every pair L and B in Minkowski space, there is an A so that B times L tensor A is a subspace of Hilbert space.

does seem a hit contrived. In notes at the end of the hook Kasman points out that mathematical symbols are not in­tended to obscure ideas from outsiders hut to communicate ideas among math­ematicians. Well , yes, but what about communicating with others?

"The Legend of Howard Thrush'' ex­alts the purported inventor of subscript notation in response to the Great Vari­ahle Shortage. This saga of the migra­tion of mathematicians from Princeton to Berkeley has the hero escaping from a tight situation, tied up in a room with thick cement walls and no windows:

What could he do' Well, the first thing he did was a proof. He showed that if f(z) = z + a2z2 + a'z" + . . . is univalent in the unit elise then la2 1 :S 2. The point is, as we all know now, this is a shmp inequality' So, he used it to cut the rope.

Now, the room was practically empty. All that was in it was a big, old desk. And all that was in the desk, in the hack corner of one drawer, was an old, broken pencil! But, of course, in the hands of Howard Thrush, a pencil is a pow­erful tool .

What he did, quietly so that the rustlers wouldn't catch on, is to tri­angulate the walls, ceiling and floor of the room ( standing on the desk when necessary). He just covered the room with a thousand triangles. But, you see, he did it in a clever way so

that there were exactly 1 502 vertices, 2504 edges and 1000 triangle faces. Then, since 1502 - 2504 + 1000 = -2, this made for two holeS: One of them turned out to be too small, hut the other was just big enough that he could squeeze through it and get back to safety.

If only we all could escape from un­pleasant situations so easily.

Another of my favorites was "Un­reasonable Effectiveness," which offers an answer to the question of why so much elegant abstract mathematics is later found to have real-world applica­tions. The stories are of mixed quality; some are real gems, others are labori­ous to read through. Also, I am not sure that the fact that there is mathematics in each story really makes it "mathe­matical fiction . " Perhaps "fiction with some mathematics" might be more de­scriptive. To me, any genre fiction has its key concept at the heart of the story. From Agatha Christie to Dorothy Sayers to P . D. James, there is little doubt that their fiction is "mystery fiction . " What then is mathematical fiction? I'd cite The Parrot's Theorem [2] or Fractal Murders [3] . The former is really ahout mathe­matics (maybe too much so), and in the latter mathematics is essential to the plot. In most of the stories in Reality Conditions the mathematics, while not itself fictional, is really nothing more than a device.

Kasman has a creative imagination and the ability to evoke characters and their surroundings effectively. Although it probably won't have the wide appeal claimed for it, Reali�y Conditions makes good reading for those who glory in ab­sorption in mathematical ideas.

REFERENCES [ 1 ] C. P. Snow, The Two Cultures, Reissue of

1 959 edition, London: Cambridge Univer­

sity Press, 1 993.

[2] D. Guedj, The Parrot's Theorem, New York:

Thomas Dunne, 2001 .

[3] M. Cohen, The Fractal Murders, Boulder

CO: Muddy Gap Press, 2002.

Department of Mathematics and Statistics

American University

Washington DC 2001 6-8050

USA

e-mail: mgray@american.edu

© 2007 Springer Science+Business Media, Inc . . Volume 29, Number 1, 2007 55

John Pe l l and H is

Correspondence

with Sir Charles

Cavendish hy Noel Malcolm and

Jacqueline Stedall

OXFORD, OXFORD UNIVERSITY PRESS,

2004. HARDCOVER 664 pp. £90.00

ISBN-10: 0-19-856484-8

REVIEWED BY VICTOR J. KATZ

he obvious question on taking up this 650-page hook is why any­one would be interested in the

life, mathematics, and letters of John Pell. Pell has always been a somewhat mysterious character, mentioned in passing in various accounts of English mathematics of the seventeenth century. Most mathematicians and historians of mathematics would he hard pressed to name any mathematical result or, in­deed, any mathematical idea that Pell contributed, except that Leonhard Euler, having come across a discussion of the equation :i' + N.Y. = I , for integers N, x, and y, in John Wallis's Treatise ()/Al­gebra, erroneously named it in Pell's honor. (The only relationship of Pell to this equation seems to he that he took William Brouncker's solution and rewrote it in his own three-column method. ) Yet after reading the detailed biography of Pell by Noel Malcolm and the analysis of his mathematics hy Jacqueline Stedall, which precede t he edition of the letters between Pell and Cavendish, an important point becomes dear: the progress of mathematics does not depend only on the "geniuses" who create the mathematical ideas, hut also on those who study their works and communicate them widely.

As Malcolm informs us, many of the biographical details of Pell's life in stan­dard reference works are erroneous, so he went to considerable trouble to ver­ify and correct the details, citing nu­merous references for virtually every important item. Thus, we learn that Pel! was born on 1 March 161 1 . that he ma­triculated at Trinity College, Cambridge in the Easter term of 1624 (at the pre-

cocious age of 1:) ) , that he received his BA in 1629, and that shortly thereafter he took a position as a school teacher. Pell married Ithamara Reginalds in 1632; they ultimately had eight children. Sometime in 1629, Pel! made the ac­quaintance of Samuel Hartlib, in whose circle Pell developed his long-term in­terest in questions of pedagogy and, in fact , not only in reforming entirely the teaching in the schools hut also in the reformation and perfection of all human knowledge. It was during the 1630s that he first made grandiose plans to pub­lish . hut he never finished the work. For example. as he wrote in 1638, "no one. so far as I know, has dared to attempt thh task. namely . to register in a cata­logue a l l the simple sounds of human language, and all the simple concepts of the mind. and to express the com­plications of the latter by means of the combinations of the former. May God

. permit me to begin, complete, and publish that work, for the glory of his name and the unspeakable benefit of the human race. " But this project, like so many others he conceived, never came to fruition.

It was also in the 1630s that Pell stud­ied mathematics seriously, including the works of William Oughtred, Thomas Harriot, Bartholomew Pitiscus, Albert G irard, and Franc,;ois Viele. As he stud­ied these works, he made notes for him­self and planned to organize these into a new and improved text on algebra. especially on "this whole new doctrine of Aequations. " Over the years. h is friends and acquaintances begged him to complete and publish such a work, hut. as usual, he did not accommodate t hem. In addition, Pell busied himself with producing tables, primarily long ta­bles of logarithms. In 1634, he made plans to produce a book on logarithms and trigonometry, including "the Loga­rithms of every number under 100 thou­-�and and of y· chords, Sines, Tangents and Secants for every Degree and cen­esim of degrees to a Radius of 10.000,000,000 particles . " But though his manuscripts contain probably thou­sands of pages of tables. no such book ever appeared.

Pel! \Vas also interested in various questions on optics, and it was through that interest that he was introduced to Sir Charles Cavendish. who would he­come, during the 1640s, Pell's patron

and avid supporter. Cavendish had an interest in the design of telescopes, and, as his early correspondence with Pell demonstrates, asked for assistance in designing and manufacturing the best lenses for this purpose. But evidently, he also employed Pell to teach him mathematics, and constantly awaited a book from Pell which would encom­pass the new algebra. Thus, as early as 1641 , Cavendish would write, "I confess I expect not an exact booke of anali­ticks [ i .e . , algebra] till you perfect yours . '' This sentiment would be a con­tinuing theme of Cavendish's letters over the next decade-since Pell would never, in fact, "perfect" his algebra.

Despite his lack of publications, however, Pell was well-regarded in mathematical circles. Thus it was in 1644 that, with the help of friends, he secured an appointment to the Chair of Mathematics at the Athenaeum in Am­sterdam, what one might today call a "preparatory school'' for the higher uni­versity faculties of law, medicine, and theology. Two years later, he moved on to another similar institution in the Netherlands, the Illustre School te Breda, where he remained until 1652.

Most of his remaining years were spent in England, with the exception of a two-year diplomatic stint in Zurich. And with the restoration of the monar­chy in 1660, Pell was able to secure a Church living, as a rector in Essex. He also became one of the earliest members of the Royal Society. And it was in 1668 that virtually the only mathematical work written by Pell appeared in print, his comments on and additions to the Eng­lish edition of Johann Rahn's Teutsche Algebra. Unfortunately, although Pell found patrons during the last years of his life, much of that time he was in very straightened circumstances, in part be­cause he had a son-in-law and grandson who were spendthrifts.

During his long life Pell , who died in 16H'), was always regarded as a math­ematician. For many years he was em­ployed as a teacher of mathematics, but what contributions did Pell make to mathematical knowledge? As Stedall re­marks. "for all the prodigious time and effort Pell expended [on mathematics], one is left with an overriding sense of incompleteness . " As she notes, Pel! left thousands of pages of assorted notes on mathematics, most of which have been

© 2007 Spnnger Science+ Bus1ness Media, Inc , Volume 29, Number 1, 2007 57

randomly bound into over thirty vol­umes that are now held in the British Library in London. However, his publi­cations are very few and even his as­sorted notes do not show evidence of great mathematical creativity. Neverthe­less, he did contribute to mathematics.

One major part of Pell's mathemati­cal opus deals with algebra. He had read and absorbed the work of Viete, Harriot, Oughtred, and Descartes, among others, and attempted to de­velop their work further and system­atize it. In fact, he planned to design an "algebra of knowledge, " in which com­plex ideas could be built from simple notions: "If we had all our simple no­tions set downe, we had as perfectly all the thoughts of men as if we had all our simple sounds, we have perfectly all y words & speeches of men po­tentially. " But such an algebra was clearly not possible in practice.

What Pel! did accomplish was a method of attacking algebraic problems in a standardized way, his method of three columns. In this method, the mid­dle column simply contains the line numbers. The left-hand column begins with the list of unknown quantities and then continues with instructions on ma­nipulating the various lines to get new lines. Finally, the right-hand column be­gins with the known relationships, writ­ten in algebraic notation, and then fol­lowing the instructions on the left, gives the various changes in the relationships until, at the end, the solutions to the unknowns appear. Among Pell's own notational innovations were the division sign ( ..;. ) and the use of lower case let­ters for unknown quantities, replaced by capital letters as soon as the nu­merical solution was found. For exam­ple, if in a particular problem, lines 7 , 8, and 9 were 4a - b - c = 400, - a + 3b - c = 300, and - a - b + 2c = 200, respectively, then with the instruction for line 10 reading "7 + 8 + 9", line 10 itself reads 2a + b = 900. And, again in this problem, the final line, 19 , reads C = 5300/13 . In general, Pel! followed Harriot in his algebraic notation, using juxtaposition for multiplication and just using repetitions of a letter for powers, rather than exponents. But he did rep­resent "equal" by our modern sign ( = ) rather than by Harriot's own symbol.

While Pel! was in Zurich in the mid 1650s, he taught his methods to Johann

58 THE MATHEMATICAL INTELLIGENCER

Heinrich Rahn, who in 1659 published his Teutsche Algebra, clearly based on Pell's tutoring and displaying the three­column method. Several years later, the book was translated into English by Thomas Brancker, who then asked Pell to make additions and improvements. Pell acceded to this request and, al­though it took several years to com­plete, the new version appeared in 1668. This book contains some of the basics of the theory of equations, in­cluding the results that an equation has as many roots as the highest power and that the second highest power of an equation can always be removed by ap­propriate substitution. But the major part of the book consists of problems with solutions, both "arithmeticall and geometrical! , " most of which Pell added to Rahn's work. The first problem is, given any two of the quantities a + b, a - b, ab, alb, aa + bh, aa - hh, to find the remaining four.

One other work that Pel! published (1647), after spending what seems an inordinate amount of time on the proj­ect, was a brief book in which he re­futed the work of C. S. Longomontanus, who claimed that the ratio of the cir­cumference of a circle to its diameter was \118,252 : 43. Pel! refuted this by using repeated bisection to find the length of a circumscribed polygon of 256 sides, applying a new double­angle formula for the tangent to aid in the calculation, a formula equivalent to the modern

2 tan (} tan 2 (} = ------,:--

1 - tan2 ({ In fact, Pel! used the formula back­wards, starting with the tan 45° = 1 and then finding the tangent of the half an­gle until, after six bisections, he arrived at the tangent of 0°42 3/16 ' . He was certainly correct in his refutation, but he delayed the publication until he could get several mathematicians to endorse his methods. In fact, some of these mathematicians gave a proof of the double-angle formula, whereas Pel! himself did not produce one.

In his correspondence with other mathematicians, Pel! solved numerous other problems, mostly algebraic or trigonometric, and evidently a good bit of what he knew made its way into John Wallis's Treatise qf Algebra. But since Pel! often promised to write up pieces

of work, "when he had leisure" and did not do so, it is difficult to know exactly what he had in mind. In addition, al­though Pel! was certainly current in the algebraic developments of the first half of the seventeenth century, the work in analysis of Fermat, Descartes, Roberval , and ultimately Newton had little effect on him. Nevertheless, he maintained his reputation as a mathematician, not be­cause of any major contribution, but be­cause he had a general understanding of at least a part of the mathematics of his time that was better than that of most of his contemporaries.

0 0 0 The second half of the book under re­view, the complete correspondence be­tween Pel! and Cavendish, provides a fascinating view into the intellectual lives of two-seventeenth century per­sonalities. There is only a little mathe­matics in these letters, mostly Pel! ex­plaining some ideas to Cavendish or Cavendish writing up something that he does not quite understand. But we also learn a good deal about how intellec­tual work was accomplished before easy communication was available. Each correspondent comments on the work of others; each reports on who he had visited with recently, and when he would expect a new work of another to be published. In their letters they let each other know whose book has ap­peared when, suggesting frequently that the other try to find a copy locally.

Given that Pell was certainly the bet­ter mathematician of the two, it is com­forting to see that he is very gentle in his mathematical criticisms of Cavendish. He tells Cavendish when his proofs are circular and suggests ways of improv­ing them. He asks Cavendish to attempt to solve problems and then either com­pliments him on a correct solution, or gives him hints as to how to improve the solution. And he patiently explains to Cavendish how the method of analy­sis works and why one needs to invert the analysis argument to get a correct synthetic proof. But the theme that most pervades these letters, particularly the more mathematical ones, is Cavendish's constant plea to Pel! to "publish his an­alyticks" so that the world will be able to learn from him. Pel! always replies that he does not have the leisure to do this.

It is difficult to know if the history of mathematics would be diflerent had Pell "published his analyticks. " Cer­tainly, if he had done so, the work of Thomas Harriot would have reached the public in a form much closer to Har­riot's idea than the version that ap­peared in the Praxis. And then perhaps Harriot and Pell rather than Descartes, would receive credit for our modern al­gebraic notation. Yet even without this publication, there is good evidence of Pell's influence through his correspon­dence and even through his limited teaching.

Although Malcolm and Stedall have combined to give us a well-rounded picture of Pell's life and his mathemat­ics and Oxford University Press has pro­duced a virtually error-free book, it is unfortunate that few people will read it, given its price of nearly $300. I fail to understand how a press can expect even many libraries to spend that kind of money, particularly for a book about a subsidiary figure in the historical de­velopment of mathematics. I would hope that OUP will consider putting out a paperbound edition at an affordable price.

University of the District of Columbia

4200 Connecticut Avenue NW

Washington, DC

e-mail: vkatz@udc.edu

Constantin

Caratheodory:

Mathematics and

Pol itics in

Turbu lent Times hy Maria Georgiadou

BERLIN-HEIDELBERG-NEW YORK, SPRINGER,

2004, HARDCOVER €89.95 XXVI I i + 651 pp. , 87 fig. ISBN 3-540-44258-8

REVIEWED BY REINHARD SIEGMUND-5CHULTZE

onstantin Caratheodory (1H73-1950) , born in Berlin of Greek parents, was one of the most im-

portant and influential mathematicians living in Germany in the first half of the 20th century. This makes Caratheodory, described on the book's cover as "the greatest Greek mathematician of recent times, " a most desirable object for mathematical historiography.

In many respects, Caratheodory's life and career were unusual: he was an en­gineer trained in Brussels, who came to mathematics only at the age of 27; a polyglot mathematician fluent in French, English, German, Greek, and even in Turkish (because his father was a diplo­mat in Turkish service) in a period of widespread nationalism; a pioneer of German-American mathematical ex­change (being guest professor in the US several times); and a "pure mathemati­cian'' contributing to thermodynamics, quantum mechanics, relativity, and other parts of mathematical physics. with par­ticular emphasis on the axiomatic point of view.

At the same time, Caratheodory co­operated with many of the leading fig­ures of mathematics and physics of the times ( Hilbert, Schmidt, Zermelo, Ein­stein ) , worked in the main centers of German science (particularly Gottingen, Berlin, and finally Mi.inchen ) , and con­tributed considerably to mainstream mathematical research in the calculus of variations and in complex and real func­tion theory. Caratheodory's specific "di­rect method" and "field the01y'' in the calculus of variations, his simplification of the so-called main theorem of con­formal representation, and his theories of measure and integral were very orig­inal and modern results which owed considerably to his historical sense and to his perfect command of the interna­tional literature.

This voluminous book, which is beau­tifully printed and presents interesting documents pertaining to Caratheodory, invites consideration of the function and possible readership of scientific biogra­phies and of the prerequisites to be de­manded of an author. Historians of mathematics profit the most from bi­ographies based on new, unpublished sources (as Georgiadou's book is, to a considerable extent). The average reader might need more basic literature first. At the same time, professional his­torians are likely to be most critical, be­cause they are able to compare the sources with others and the author's

conclusions with the existing secondary literature. In addition, professional his­torians of mathematics, as any other scholars, have certain expectations of a scientific biography, such as previous research articles by the author on the history of the scientific theories dis­cussed, and a bibliography of the pub­lications of the scientist described (both absent in this book). If these admittedly high standards-which may call for decades of preparation and a mastery of the problems of the historical epoch-are not met, the biography can­not be considered "complete" in the sense of research in the historiography of mathematics. In my own view, how­ever, such a book may still be a con­siderable and important step toward a complete biography and may serve many purposes for different kinds of readership, including mathematicians/ scientists or general historians. And it might still be a treasure-trove for the historiography of mathematics.

The author, Maria Georgiadou, is Greek and a physicist by training. Her most original contribution is, indeed, her discussion of Caratheodory's Greek roots and connections, for example Caratheodory's role in the expansionis­tic foundation and subsequent failure of the Greek university in Smyrna (today Turkish Izmir) in 1920-1922 (Chapter 3 : "The Asia Minor Project" , pp . 137-1Hl) . Caratheodory's elitist and missionary zeal, revealed in this project, helps ex­plain his later stiff insistence on and at­tachment to the German ideal of sci­ence, even after it had been discredited and misused by the Nazis.

Georgiadou has worked conscien­tiously and presents the archival mate­rial with care and not without critical distance. Given the lack of extant per­sonal documents (estate, personal let­ters, political commentary), there is no danger that the author or the reader will empathize or identify with the mathe­matician. The theories are for the most part clearly discussed, without being too technical. The presentation is fre­quently supported by long quotes from letters, reviews, or introductions to text­books, without comment by the author (particularly striking for its almost total lack of commentary is Caratheodory's correspondence with T. Rad6 on area theory, pp. 1 1 3-1 16) . These quotes, which contain many historical ret1ec-

© 2007 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 29, Number 1 , 2007 59

tions by their various authors, are mostly interesting, although not always indu­bitable or convincing (at least in the case of minor authors such as Caratheodory's student N. Kritikos, p. 1 10). However, I doubt whether in the future the history of mathematics can be written in this manner, because many modern re­search mathematicians do not seem as prone to historical reflections.

Archival material is usually scrupu­lously quoted and annotated in this book. However, in some instances (for exam­ple, in the discussion of Caratheodory's mediating role in the nationalistic con­troversy about the International Mathe­matical Congress in Bologna of 1928, pp. 235ff.) the existing literature is not fully used, and statements which are well-known from secondary sources are established once again by some new or even known primary sources. Gener­ally, the use of secondary literature seems a weaker point of Georgiadou's book, including crediting other authors. This is reflected both in the rather slim bibliography (where, for instance, the work by R. Bulirsch of 2000 on Caratheodory is missing and which does not include many publications previ­ously mentioned in the notes even if they are quoted there as many as ten times), and in some unsatisfactory sum­maries. On Birkhoff's 1938 address to the AMS, the author writes that he ex­pressed there the "opinion . . . the main focus of mathematical research shifted from France and Germany to the USA at the beginning of the century" . (224) For all his nationalistic sentiment in the speech, Birkhoff was not that stupid.

The obvious historical core questions of the biography concern Caratheodory's remaining in Nazi-Germany between 1933 and 1945, in spite of existing al­ternatives for emigration in his case; his arrangement with the rulers, in spite of his undeniable contempt for the anti­intellectual mood of the regime; and the options at that time for international communication on his part. As the sub­title of the biography suggests, the au­thor does not evade these questions, and Georgiadou is not uncritical vis-a-vis the behavior of her compatriot. She rightly deems Caratheodory's and B. L. van der Waerden's formal and indifferent reac­tions to a desperate cry for help by the Polish mathematician P. Schauder­who was soon to be murdered-as

60 THE MATHEMATICAL INTELLIGENCER

SUBSCRIPTION INFORMATION: The Mathematical Intelligencer is pub­

lished by Springer Science+ Business Media, Inc. winter, spring, summer and

fall. Volume 28 (4 issues) will be published in 2006. ISSN: 0343-6993 COPYRIGHT: Submission of a manuscript implies: that the work described

has not been published before (except in the form of an abstract or as part

of a published lecture, review or thesis); that it is not under consideration

for publication elsewhere; that its publication has been approved by all coau­

thors, if any, as well astacitly or explicitly6by the responsible authorities at

the institution where the work was carried out. Transfer of copyright to

Springer becomes effective if and when the article is accepted for publica­

tion. The copyright covers the exclusive right (for U.S. government employ­

ees: to the extent transferable) to reproduce and distribute the article, in­

cluding reprints, translations, photographic reproductions, microform,

electronic form (offline, online) or other reproductions of similar nature.

All articles published in this journal are protected by copyright, which cov­

ers the exclusive rights to reproduce and distribute the article (e.g., as

reprints), as well as all translation rights. No material published in this jour­

nal may be reproduced photographically or stored on microfilm, in electronic

data bases, video disks, etc., without first obtaining written permission from

Springer. The use of general descriptive names, trade names, trademarks,

etc., in this publication, even if not specifically identified, does not imply that

these names are not protected by the relevant laws and regulations.

An author may self-archive an author-created version of his/her article on

his/her own website and his/her institution's repository, including his/her fi­

nal version; however he/she may not use the publisher's PDF version which

is posted on www.springerlink.com. Furthermore, the author may only post

his/her version provided acknowledgement is given to the original source of

publication and a link is inserted to the published article on Springer's web­

site. The link must be accompanied by the following text: The original pub­

lication is available at www.springerlink.com. Please use the appropriate

URL and/or DOl for the article (go to Linking Options in the article, then to

OpenURL and use the link with the DOl-). Articles disseminated via

www.springerlink.com are indexed, abstracted and referenced by many ab­

stracting and information services, bibliographic networks, subscription

agencies, library networks, and consortia.

While the advice and information in this journal is believed to be true and

accurate at the date of its publication, neither the authors, the editors, nor

the publisher can accept any legal responsibility for any errors or omissions

that may be made. The publisher makes no warranty, express or implied,

with respect to the material contained herein. Springer publishes advertise­

ments in this journal in reliance upon the responsibility of the advertiser to

comply with all legal requirements relating to the marketing and sale of prod­

ucts or services advertised. Springer and the editors are not responsible for

claims made in the advertisements published in the journal. The appearance

of advertisements in Springer publications does not constitute endorsement,

implied or intended, of the product advertised or the claims made for it by

the advertiser.

Special regulations for photocopies in the USA. Photocopies may be made

for personal or in-house use beyond the limitations stipulated under Section

107 or 108 of U.S. Copyright Law, provided a fee is paid. All fees should be

paid to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,

MA 01923, USA (WEB: http://www.copyright.com; FAX: 978-646-8600; TEL:

978-750-8400), stating the ISSN 0343-6993, the volume, and the first and last

page numbers of each article copied. The copyright owner's consent does

not include copying for general distribution, promotion, new works or re­

sale. In these cases, specific written permission must first be obtained from

the publisher.

The Canada Institute for Scient�fic and Technical Information (CISTI)

provides a comprehensive, world-wide document delivery service for all

Springer journals. For more information, or to place an order for a copyright­

cleared Springer document, please contact Client Assistant, Document De­

livery, CISTI, Ottawa KIA OS2, Canada. TEL: 613-993-9251; FAX: 613-952-8243; E-MAIL: cisti.docdel@nrc.ca. You may also contact INFOTRIEVE (WEB:

http://www.infotrieve.com/; E-MAIL: orders @infotrieve.com; fax: 310-208-5971; TEL: 800-422-4633), UnCover: http://www.ingenta.com; UMI: http://www.

umi.com/; lSI: http://www.isinet.com

RESPONSIBLE FOR ADVERTISEMENTS:

Printed in the United States on acid-free paper.

© 2006 Springer Science+ Business Media, Inc.

Periodicals postage paid at New York, New York and additional mailing

offices. POSTMASTER: Send address changes to: The Mathematical Intelli­

gencer, Springer, 233 Spring Street, New York, NY 10013, USA.

irresponsible and, as the reader feels, cowardly (pp. 380-383).

As in any book, there are mistakes in some details. The word "Philister" (p . 226) is not meant disparagingly, as the author assumes, but denotes a certain member of student fraternities. The em­igration of Richard von Mises from Berlin in 1933 and his policies for the appointment of a successor are mis­represented (p. 286) . Jean Cavailles is misspelled as Jan Cavailler (p. 369) both in the text and in the index, Erwin Schrbdinger's attempt to go along with the new Nazi rulers in Graz is not men­tioned (p. 565). These inaccuracies seem partly due to the author's having to acquaint herself with a new research area.

But because Georgiadou has done this with great conscientiousness, and because the new material displayed and the mathematician who is the focus of the book have great independent im­portance, Constantin Caratheodory: Mathematics and Politics in Turbulent Times is a huge step toward a "com­plete" scientific biography (in the sense described above) and a significant con­tribution to the historiography of math­ematics in the 20th century.

Agder University College

Faculty of Science

4604 Kristiansand

Norway

e-mail: reinhard.siegmund-schultze@hia.no

Mathematicians

under the Nazis by Sanford L. Segal

PRINCETON, NEW JERSEY, PRINCETON UNIVERSITY

PRESS, 2004 XXII + 530 PP. US$85.00

ISBN 0-691-00451-X

REVIEWED BY GERALD L. ALEXANDERSON

The plight of the Jews in Nazi Ger­many is well-documented. Math­ematicians who fled Germany and

those countries under German domina­tion came largely to the United States, with some going to England, Turkey, Russia, Palestine, and the countries of South America, among others. Stories of many of these were told in a series of four articles by Maximilian Pin! in issues

of the ]ahresbericht der Deutschen Mathematiker Vereinigung ( ]DMV) be­tween 1969 and 1976. The influence these refugees had on American math­ematics in particular is often cited and almost certainly accounts for the dom­inance of American mathematics in the period following World War II.

What is less well-known is the fate of those mathematicians who either could not escape or, for one reason or another, chose not to leave. Those who could not emigrate often ended up in death camps or came to other tragic ends. Felix Hausdorff was Jewish, but he mistakenly assumed that because of his many honors and his standing in the scientific world, he would somehow be exempt from deportation to the con­centration camps. He was wrong, of course, and ended up, with his wife and his wife's sister (who were Lutherans), committing suicide in their apartment in Berlin in 1942. Richard Courant, also a Jew, naively assumed that his service on the front in World War I and his standing as a mathematician would save him, but when he became aware that it would not, he left Germany. Others stayed on and some prospered under the new regime. Less has been written about these and it is their story that is told by Sanford Segal in this much­needed account. Though several stud­ies of this period have appeared in re­cent years , most are in German. (Some evidence that the preponderance of ma­terial on this subject is coming out of Europe is that only 40% of Segal's ref­erences are in English.)

Most mathematicians of absolutely first rank escaped one way or another, but still some highly respected and pro­ductive mathematicians stayed on, for whatever reasons. Some were indis­putably quite reprehensible in their be­havior. George P6lya, who knew both Wilhelm Blaschke and Ludwig Bierber­bach, contrasted their behavior during the War by claiming that Blaschke didn't really believe in the Nazi cause but used its political machinery to get back at his enemies. By contrast Bieber­bach was a true believer in Nazi prin­ciples. Both were creative mathemati­cians. How do we decide which is more culpable? Oswald Teichmiiller's behav­ior, for example, seems entirely inde­fensible. On the other hand, Helmut Hasse's name still provokes a good deal

of debate about his level of complicity with the regime when he succeeded to the position of Director of the Mathe­matical Institute in Gbttingen. His cul­pability is even now not entirely clear. Some of these debates will not be set­tied until the Holocaust archives are opened, if then.

How could intelligent, thoughtful mathematicians become Nazis? Did this happen to academics in other disci­plines, like those in the empirical sci­ences? (Some other fields, like physics­two German physicists who were No­bel laureates were Nazis-and chem­istry, have been more extensively stud­ied.) Is the purity and cerebral nature of mathematics a factor? Segal ponders these questions.

The parallels between mathematics and music are interesting in this con­text. Some musicians who did not leave Central Europe for the Allied countries during the war had their reputations re­habilitated, up to a point, in the post­war period-Kirsten Flagstad, Walter Gieseking, Wilhelm Furtwangler, Elisa­beth Schwarzkopf, for example. They continued with their careers after the war, visiting the Allied countries to per­form, though sometimes facing hostile audiences. Richard Strauss's position was among the most ambiguous. He re­mained in Germany, becoming Presi­dent of the Reichsmusikkammer and conducting his compositions for top­ranking Nazi officials. At the same time, his correspondence with his former li­brettists, Stefan Zweig and Josef Gregor, both Jews, shows no hint of anti-Semi­tism. In fact the letters are full of refer­ences to hoped-for future collaborations after the war. The problem for Strauss was that his daughter-in-law, and there­fore two grandsons, were Jewish. It is easy to argue that he had to cooperate with the regime in order to save his family. It is a persuasive argument. Of course, it does not explain why he did not choose to take his family and leave Germany during the early 1930s. But it is probably a natural human reaction to such monstrous events to hope that the situation will not grow worse and that somehow one can survive without a major uprooting of a whole family oth­erwise loyal to their country. So it is probably too facile to condemn people quickly and without all the evidence . Incidentally, Strauss claimed that he was

© 2007 Springer Sc•ence+Bus1ness Med1a, Inc . Volume 29, Number 1, 2007 61

not consulted about his appointment to the Reichsmusikkammer position.

Drawing a parallel between mathe­maticians and musicians may not be far­fetched. Segal quotes Bentley Glass, the American biologist, who said, "as for music, it is audible mathematics . " Segal adds: "Perhaps the traditional musical aesthetic is the one most closely re­sembling the mathematical; here, too, given the underlying assumptions, there is a purity of form that is part of the notion of beautiful ."

Segal is to be congratulated for tak­ing on such an overwhelming task, try­ing to make sense of this very complex social and ethical scene. It must have taken years and an extraordinary effort to dig up whatever primary source ma­terial is available. The breadth of schol­arship needed is sobering. For the non­specialist, this book is probably not something one picks up and reads from cover to cover, with all the footnotes and long quoted passages, unless one has a strong interest in Germany or the plight of mathematicians of the period. Segal is a working mathematician, not a professional historian or philosopher of mathematics, though he clearly knows a lot about German culture and politics. The book is strongest when conveying stories about people and events. Whether these stories would be as interesting to members of the gen­eral public as they are to experienced mathematicians, it is hard to say. My guess is that most readers will be math­ematicians or scientists with a general knowledge of the mathematics dis­cussed. There are enough good anec­dotes and astounding facts to keep readers turning the pages.

Segal starts off his book with some reflections on the nature of mathemat­ics and observes that it could be sur­prising to some that a subject as pure and abstract as mathematics should be connected in some way to politics. The Nazis "argued that exactly the apparent culture-free nature of mathematical ab­straction and mathematical causality makes mathematics the ideal testing­ground for theories about racially de­termined differences in intellectual atti­tudes. " He points out that however abhorrent the actions of the Nazis were, the party had developed intellectual un­derpinnings for its policies, however

62 THE MATHEMATICAL INTELLIGENCER

bizarre they may be by most people's standards. Of course, mathematics was relevant to Nazi actions because of its applicability to science, though the leaders of the movement may not have appreciated this. Segal quotes Hans Heilbronn, "The application of mathe­matics to military problems was ne­glected in Hitler's Germany, certainly by comparison with England and the U.S."

Bieberbach and Theodor Vahlen, a much inferior mathematician but one well-connected politically, together formed the infamous mathematical jour­nal, Deutsche Mathematik, based on the theory that there is a difference between Nordic mathematics and "French or Jewish mathematics" (an interesting jux­taposition). This theory was supported by the view that "because Jews thought differently, and were 'suited to do math­ematics in a different fashion', they could not be proper instructors for non­Jews"-whence the firings of Jews from university positions throughout the 1930s. Of course, such ideas existed outside Germany as well. The Ameri­can psychologist A. A. Roback and oth­ers claimed to show that Jewish stu­dents demonstrated a different (and superior) writing style from that of other students, for example.

In his second chapter, Segal looks at what he calls a crisis in mathematics go­ing on at the time-the earlier inven­tion of set theory setting the stage for disputes over intuitionism, the conse­quences of the Axiom of Choice, the development of "modern algebra" and what some considered an excess of abstraction, the new integration of Lebesgue, and so on. As soon as one admits there is a certain subjectivity about how mathematics might be done, some will try to put together an elabo­rate construct of how it should be done. For example, for Bieberbach, there were two definitions of 71': the ratio of the circumference of a circle to its di­ameter, or twice the smallest positive zero of the cosine function (itself de­fined as a power series). The latter was actually used by Landau. For Bieber­bach the second definition was "typi­cally Jewish . . . . This ideological Nazi point of view is that Jews, for example, attempt to 'create' mathematics, whereas the true German 'discovers' mathemat­ics . . . . Not only is this old distinction

in philosophical points of view, ac­cording to them, racially determined, but also the 'creation' point of view is perverse-at best, undeutsch, and so unsuited to the instruction of Germans."

Segal describes in Chapter 3 the aca­demic climate during the last years of the Weimar Republic and what the uni­versity faculties were like in the 1930s. Professors were a privileged class and, it is significant to note, worked for the state, since the universities received their support from the government. So the professors were high-class civil ser­vants. Though they enjoyed freedom to pursue their own studies, they did not enjoy freedom of speech as it is widely understood today. They were not free to speak out on political matters. Over­all they tended to be conservative: they resented the Treaty of Versailles and some saw Hitler as a hope for Germany to undo some of the indignities im­posed on the German State. Professors like Hasse claim to have been "na­tional ," not "Nazi. " Anti-Semitism was ever-present in the universities where it was often harder for Jews to be ap­pointed or get promoted. This problem was not confined to German universi­ties, of course.

In Chapter 4, titled "Three Case Stud­ies, " the author (1) chronicles a project for reprinting mathematical texts that were viewed as militarily important, and the conflicts between the two principals involved, Wilhelm Suss, head of the ]DMV, and Gustav Doetsch, described by a colleague as " l l Oo/o Nazi"; (2) de­scribes the replacement of Max Winkel­mann at Jena in 1938; and (3) tells the story of the appointment of Helmut Hasse as director of the Gottingen In­stitute. All are fascinating accounts of the Byzantine politics in the mathemat­ical community of the Third Reich, hut it is Hasse's case that is perhaps the most interesting, in part because of the time he spent in the United States after the war, where he lectured at various universities. His bete-noir at the Insti­tute was Werner Weber, who was act­ing director after the resignation of Her­mann Weyl in 1933 and the short term of Franz Rellich. Weber recommended Hasse to be director. Whatever Hasse's reservations may have been about the Nazi movement, Weber apparently had none. He had joined the SA early on

and firmly believed that Hitler would restore "German national prestige and honor, and . . . [he] would complain about other party members in whom the pure flame did not burn sufficiently brightly."

Segal points out that Hasse jointed the DNVP (Deutschnationale Volk.\partei), "a right-wing somewhat aristocratic party," and although it was "in 1929 . . . closed to Jews . . . it continued to have Jew­ish adherents until 1933. Despite his DNVP membership, Hasse was no anti­Semite . " Weber recommended Hasse hut without enthusiasm. He thought Hasse had standing in the mathemati­cal community and that all other can­didates available and with a compara­ble reputation in mathematics may not have been sufficiently committed to the Nazi cause. Weber soon discovered that Hasse was not as malleable or commit­ted to the cause as he had assumed, so he shifted over to opposing the ap­pointment. And if Hasse could not be stopped then there should be someone else appointed at Gcittingen who would he a sufficiently committed Nazi to pro­vide a counterbalance to the less-reli­able Hasse . This turned out to be Er­hard Tornier. Hasse succeeded to Hermann Weyl's chair, Tornier to Ed­mund Landau's (at least temporarily). But Hasse soon made it clear that he was in charge and Tornier left for Berlin, where he survived various scandals. Al­ways interested in probability, after the war he moved into parapsychology! Se­gal says: "Thus Weber, Teichmi.iller, and Tornier, the radical Nazis at Giittingen, all of whom initially were prominently anti-Hasse, came correctly to view him as thoroughly assimilated to the Na­tional Socialist state. Indeed, Hasse seems to have been the sort of man whose first academic loyalty was cer­tainly to mathematics, hut who had no objections to operating within the Nazi state. " Having reassured his friend Harold Davenport that he would stand up for mathematics in the political ma­neuverings in Gottingen at the time, Hasse nevertheless accommodated the political forces when necessary.

It was later disclosed that Hasse was " 1/16 Jew as a consequence of a bap­tized great -great -grandmother. " (Be­cause of widespread anti-Semitism in Europe, it was common for Jews to

"convert" in order to be eligible for appointments and promotion.) Thus Hasse was a member of the Nazi party and technically, according to the intri­cate method for deciding such matters, Jewish! This gives some idea of how very complex and fascinating these case studies are.

Chapter 5 covers the gradual but sys­tematic means by which the government drove Jews from their jobs in German universities. It is not pleasant reading. Some of the worst aspects of campus politics in any university were magni­fied to a grotesque level in the Third Reich. Of course, the German system may have been particularly vulnerable to these excesses, with the Teutonic concern for bureaucratic procedures, rigid hierarchy, and a tendency to con­fuse incessant activity with accomplish­ment. But it went far beyond that.

There were compulsory ten-week "field sport camps" that helped senior faculty and administrators choose assis­tants and instructors "according to their qualities of character and will. Bodily deficiencies can he very well compen­sated for by joy in service and specially exemplary comradely behavior." (The Nazis developed their own specialized and euphemistic language. "Joy in ser­vice" recalls to mind the infamous "Ar­beit macht frei (Work makes [one] free)," the sign over the entrance to Auschwitz.) No new faculty member was allowed to begin teaching without going through one of these field sport camps. A pamphlet of that time, titled "Mathematics-For What?" asked "Is it really German, to do something-this something-for its own sake?" So much for pure mathematics.

The pressure on faculty came not only from the government and other faculty hut also from students. Some of this even survived the war when there was a "call from a German student or­ganization against the 'un-German spirit': 'A Jew cannot write German. Were he to write German, he is lying. We demand that Jews write only in He­brew.' " Does this sort of nonsense ever end?

The description of the decline in German mathematics during the Third Reich goes into the next chapter with a discussion of mathematical institu­tions-professional societies, journals,

and even mathematics in the concen­tration camps. There is too much even to summarize here, hut I shall just men­tion a couple of items that deserve at­tention. The Mathematische Annalen was (and is) a journal of the highest rank. But in 1928 the Annalen was in­volved in the struggle between Hilbert, the formalist, and Brouwer, the intu­itionist. (It was reported that the only thing Hilbert and Brouwer had in com­mon was they both disliked Paul Koebe!) Hilbert tried to remove Brouwer from the editorial board of the Annalen (of which Hilbert was a senior editor), and this prompted Brouwer to suggest that Hilbert was of "unsound mind." Einstein said that "Brouwer was an in­voluntary proponent of Lombroso's the­ory of the dose relation between ge­nius and insanity. " This little episode is more entertaining than ominous. But the law of August 17 , 1938, is appalling. It decreed that "Jews were only allowed to have distinctively Jewish first names, and those who did not had to append the name Israel if a male, or Sara if a female, to their names." This appears in the hook in a discussion of the Annalen.

Whatever the problems with the A n­nalen they were nothing compared to those of the]ahresbericht, where Bieber­bach was one of the three editors (the others were Hasse and Konrad Knopp). Segal provides a long and fascinating ac­count of Bieberhach's contentious rela­tions with eminent mathematicians of the day over his contributions to the JDMV: Harald Bohr, Gabor Szego, G. H. Hardy, Oswald Veblen, Carl Ludwig Siegel, and more.

Other sections in this chapter are in­teresting, but the topics are perhaps bet­ter known than some because of recent works written about them: the found­ing of the Institute at Oherwolfach; the establishment of Deutsche Mathematik; and the conflict surrounding the Inter­national Congress in Bologna in 1928. Chapter 7 is entirely devoted to matters relating to Bieberbach.

The eighth and final chapter consists of individual sections on some of the German mathematicians who played a role in this stark and frightening story. Some of these names are known to every graduate student of mathematics; others are relatively obscure: Wilhelm Blaschke, Heinrich Behnke, Erich

© 2007 Spnnger Sc1ence+Business Media, Inc., Volume 29, Number 1, 2007 63

Heeke, Oswald Teichmi.iller, Ernst Witt, Richard Courant, Edmund Landau, Fe­lix Hausdorff, Ernst Pesch!, Paul Riebe­sell, Helmut Ulm, Alfred Stohr, Ernst Zermelo, Gerhard Gentzen, Hans Pe­tersson, Erich Kahler, and Wilhelm Suss. The names on this list range from the committed Nazis, through non-Nazi right wing nationalists, to naive, other­worldly men who seemingly didn't know much of what was going on, to those who stumbled into situations that were beyond their control. The stories are gripping. In assessing guilt the reader is hard pressed to decide with any certainty which people fall into which categories. Teichmi.iller comes off very badly, for example. Segal sug­gests that he in fact came up with the theory that Aryan mathematics was dif­ferent and superior to "Jewish and French" mathematics, a notion usually credited to Bieberbach. Teichmi.iller claimed the student revolt against courses taught by Landau was not anti­Semitic but pro-German. He was a bril­liant mathematician but a Nazi fanatic. He volunteered to go fight on the Russ­ian front and was killed there at the age of 30. One should probably keep in mind that because of his early death, he, unlike many others, never had a chance to try to redeem his reputation during the denazification period after the war.

In this last chapter, the reader can­not help regretting that certain other mathematicians were not included for this more expansive treatment: Hans Zassenhaus, Gustav Doetsch, Georg Hamel, Helmuth Kneser, or Erhard Schmidt, for example. Perhaps Segal felt that their involvement was adequately described elsewhere in the text.

Of those treated, some behaved well, some badly. All were competent math­ematicians; some were giants. Their mathematics, however, did not save some of them from being monsters.

Department of Mathematics and

Computer Science

Santa Clara University

Santa Clara, CA 95053-0290

USA

e-mail: galexand@math.scu.edu

The Mathematical

Theory of

Information hy]an Kahre

BOSTON, KLUWER. 2002. 520 PP. , US$50.00

ISBN: 1-4020-7064-0

REVIEWED BY CRISTIAN S. CALUDE

ndoubtedly, the title of the book was well chosen: it is provoca­tive, promising, and full of infor­

mation. Syntactically, the title can be viewed as a variation on the titles of both the seminal paper [ 1 1] ("a" is re­placed by "the") and the book [12] ("Communication" is replaced by "In­formation"). It provocatively questions Shannon's theory; according to [1 ] (page 215), "no prophet remains unchallenged for ever". And it promises "a new math­ematical theory of information, built on a single powerful postulate: The Law of Diminishing Information."

The book was praised-"a hold new approach to classical information the­ory"-by von Baeyer [1 ] , who dedicated a special chapter of his book to it; de­tails about Kahre and the fascinating au­tonomous islands Aland on which he lives are presented too.

Do we need a new information the­ory? Unsurprisingly, there is no one sin­gle theory of information, but several theories: semantic theories [2] , algorith­mic information theory [5,4], logic of in­formation [7] , information algebra [9] , philosophy of information [8], informa­tion flow [3] , quantum information the­ory [10] , evolutionary information [13] , to name just a few (a workshop de­voted to various theories of information was recently held in Mi.inchenwiler) . Each theory focuses on some specific aspect of information, and overlaps are minimal. There is little evidence that the existing theories will converge towards a single, unified theory of information, so, indeed, there is ample room for (even a partial) unification.

The book discusses information from various angles, with interesting ideas and many examples. Bits and entropy

are used for quantitative problems, while hits (the number of correct clas­sifications), reliability, and nuts (von Neumann's utility) appear in more qual­itative analyses.

Although the author's ambition is to develop a (if not the) "mathematical the­ory of information, " the embodiment is pre-mathematical. It is neither a naive mathematical theory (as in naive set the­ory) nor is it abused mathematics (in the sense of mathematics applied in mean­ingless ways). However, the mathemat­ical formalism is too rudimentary for a theory; I illustrate this point with two examples, the definitions of probability and algorithmic complexity. The proba­bility P(a) is a real number that satisfies the following three axioms (pages 25-26): probability cannot be a negative number, the probability of something that must occur is 1 , and the probabil­ity that a or h will occur is the sum of their probabilities provided that a and h cannot both occur. 1 Kolmogorov com­plexity is defined (page 234) as the length l(aJ of the shortest algorithm generating a given hJ- In both cases the intuition is correct; even if some facts can be deduced from those definitions, there is still a long way to a satisfactory mathematical presentation.

The book is rather firmly based on Shannon's probabilistic view of in­formation and entropy; the standard books [12 , 6] are frequently used and cited. The information measure used in the book is defined by inf(B@A) = the information B gives about A (author's notation). Here inf(B@A) is a real func­tion satisfying the Law of Diminishing Information (or, simply, the Law, as it is referred in the book): Compared to di­rect reception, an intermediary can onzy decrease the amount qf information. If A � B � C denotes a transmJssJon chain, then the Law reads: inf(C@A) s

inf(B@A). The "theory of information" developed in the book is based on prob­ability (as defined below) plus the Law (page 14):

[The Law] will be used as the fun­damental axiom of the mathematical theory of information. The Law is the pruning knife of information theory: we will argue that the Law is the necessary and sufficient con-

1 1t takes no fewer than 235 pages to realise that probabilities, defined in this way, apply only to finite sets (see section 8.4).

64 THE MATHEMATICAL INTELLIGENCER

clition for a mathematical function to be accepted as an information mea­sure, i .e . qualify as inf(B@A). The Law is easy to understand and

informally seems correct (for example, to anyone who has played the Tele­phone game in which one person chooses a sentence, whispers it into the ear of her left neighbour, who in turn whispers it into the ear of her left neighbour, and so on down the line) . It ties in well with other principles such as the second law of thermodynamics, the data processing inequality [6] , and the invariance of algorithmic complex­ity under computable transformations [4]. Moreover, it is not difficult to see that there are infinitely many functions sat­isfying the Law (trivially, each constant function satisfies the inequality). Fi­nally. a weaker form of the Law has al­ready been discussed in [6] (page 32) as a consequence of the data process­ing inequality.

According to the book the Law is ubiquitous. It makes physics possible: "systems are forgetting their past as they reach equilibrium, or rather, the initial conditions can be eliminated from their description. Otherwise . physics would be complicated beyond comprehen­sion" . It also explains evolution. It ap­plies to information technology, game theory, legislation, logic of research, al­gorithmic information, chaos theory, control engineering, medical tests. It can even be used as a legitimacy test: any acceptable information measure must satisfy the Law.

Is the Law true and should it be adopted? First, there are exceptions. The author himself discusses one: the Chi­nese paper. Assume that the channel A � B � C consists of A = an English­man tries to read an article in Chinese, B = an interpreter translates the article into English, C = the English translation of the article. Clearly, inf(C@A) > inf(B@A), hence the Law fails. Second, similar but less subjective violations of the Law can be easily constructed us­ing algorithmic complexity. This signals a problem: which restrictions should be imposed?

The book, which covers more than 'iOO pages, discusses a wealth of topics grouped in 14 chapters, from specific information measures. statistical infor­mation and algorithmic information to control and communication, informa-

tion physics and quantum information and applications. Some topics are bet­ter presented than others. The chapter on algorithmic information, which is close to my expertise, is far from satis­factory, as one can see by browsing the paragraphs 2, 3 , and 4 on page 238. The main aim, a grand unification theory of information, is certainly not achieved. Despite this, the book, written by an original thinker, contains a number of intl'resting ideas which may inspire mathematically oriented readers to con­tinue the project.

REFERENCES 1 . H. C. von Baeyer. Information: The New

Language of Science, Harvard University

Press, Cambridge, Mass. , 2003 (paper­

back edition 2004).

2. Y. Bar-Hil lel (ed.). Language and Informa­

tion: Selected Essays on Their Theory and

Application, Addison-Wesley, Reading,

Mass, 1 964.

3 . J . Barwise and J . Seligman. Information

Flow: The Logic of Distributed Systems,

Cambridge University Press, Cambridge,

1 997.

4 . C. S. Calude. Information and Random­

ness: An Algorithmic Perspective, 2nd Edi­

tion, Springer-Verlag, Berl in, 2002.

5. G. J. Chaitin. Algorithmic Information The­

ory, Cambridge University Press, Cam­

bridge, 1 987.

6. T. M. Cover and J . A. Thomas. Elements

of Information Theory, Wiley, New York,

1 991 .

7. K. J. Devlin . Logic and Information, Cam­

bridge University Press, Cambridge, 1 991 .

8. L. Floridi . What is the philosophy of infor­

mation? Metaphilosophy 33(1 -2) (2002),

1 23-1 45.

9 . J. Kohlas. Information Algebras: Generic

Structures for Interference, Springer-Ver­

lag, London, 2003.

1 0. M. A. Nielsen and I . L. Chuang. Quantum

Computation and Quantum Information,

Cambridge University Press, Cambridge,

2000.

1 1 . C. E. Shannon. A mathematical theory of

communication, Bell System Technical

Journal 27(1 948), 379-423, 623-656.

1 2 . C. E. Shannon and W. Weaver. The Math­

ematical Theory of Communication, Uni­

versity of I l l inois Press, Urbana, Il l inois,

1 949 (paperback edition 1 963; special fifti­

eth anniversary edition in 1 999).

1 3 . T. Stonier. Information and Meaning: An

Evolutionary Perspective, Springer, Heidel­

berg, 1 997.

Department of Computer Science

The University of Auckland

Private Bag 920 1 9

Auckland

New Zealand

e-mail: cristian@cs. auckland .ac. nz

Negative Math :

How Mathematical

Ru les Can Be

Positively Bent by Alberto A . Martinez

PRINCETON UNIVERSITY PRESS, 2006, 267 PP.,

ISBN·13: 978·0·691·12309-7

REVIEWED BY ERIC GRUNWALD

xtricating the book from its pack­aging, I was greeted by a picture of a large spoon on the dust wrap­

per. What could this mean? Some sort of reference to spoon-bending in the last word of the subtitle? Is the reader going to be spoon fed? Surely the book isn't written in spoonerisms. Immedi­ately after the title page, came the fol­lowing:

You can use a spoon to drive a screw into a wall. With practice, you can become skillful at it. You can also learn many juggling tricks with the spoon, and thus impress and be­wilder people who don't juggle spoons. And you can make all of this more puzzling by calling the spoon a 'fork' . And you can write books about it and form societies with other people who also juggle spoons called forks . And even then, sure. you can use a spoon to drive screws into a wall . But a screwdriver is better. And even if you've never seen a screwdriver, you can just as well invent one. It might resemble the spoon in some ways though not in others. So you can keep your spoon as wel l ; for eating soup, for juggling, or even, occasionally, for driving screws into walls. At least until you have more skill with a better tool.

© 2007 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 29, Number 1, 2007 65

Is that clear to you? Me neither. My heart sank; what had the Reviews Ed­itor let me in for? But I soon discov­ered that my first impression was mis­leading; this is a serious-minded and interesting book. Martinez intends to show that mathematics does not nec­essarily emerge from our daily experi­ence, and that we can create it to suit our purposes. He writes in a deliber­ately informal style: as he says, "plenty of seemingly incomprehensible books sit on library shelves, enough to last anyone a lifetime . . . So let this one read easily . " On the whole he is as good as his word and writes fluently, with only the occasional clumsy con­struction, including, curiously, the last sentence of the book, "It is thus easy to confuse a tool that serves well to predict the outcomes of certain kinds of physical interactions with a means of physical representation."

The first part of the book, which I enjoyed immensely, is a history of the struggles of mathematicians to cope with the idea of negative numbers. It is enormously encouraging to anyone wrestling with a difficult concept that such a fine mathematician as Hamilton could, as late as 1833, write thus about the foundations of algebra; "it requires no peculiar skepticism to doubt, or even to disbelieve, the doctrine of Negatives . . . that a greater magnitude may he subtracted from a less, and that the re­mainder is less than nothing; that two negative numbers . . . may be multi­plied the one by the other, and that the product will be a positive number. . . . " Hamilton contrasted this state of affairs with the principles of Euclidean geom­etry, which "no candid and intelligent person can doubt". Martinez discusses the changing roles of algebra and geom­etry; where once algebra needed to be justified in terms of its geometrical in­terpretation, "as faith in the certainty of traditional geometry eroded, algebra emerged by default as an alternative on which to place one's faith in mathe­matical analysis . "

As Martinez says, mathematicians came to terms with the difficulties in

66 THE MATHEMATICAL INTELLIGENCER

placing physical, or geometrical, inter­pretations on such concepts as negative numbers, imaginary numbers, and quaternions by "redefining the nature of mathematics" into something highly ab­stract and not necessarily physical at all. In the section From Hindsight to Cre­ativity, described by Martinez as "the heart of the book, " he puts forward his thesis that, in contrast to most mathe­maticians before the 1840s, we now re­gard the principles of mathematics as open to modification to suit our pur­poses. It was refreshing to view the his­tory of this transformation through the lens of algebra, and in particular negative numbers, rather than the lens of non-Euclidean geometries, through which it is more commonly presented.

Much of the book is taken up with the construction of various systems of algebra in which standard concepts no longer apply-for example, ones in which the product of two negative num­bers is negative. I suspect this would not appeal much to professional math­ematicians, who must know perfectly well that one can invent new rules if one wants to; the more interesting ques­tion would be, which systems of axioms are the most useful? But non-mathe­maticians who are interested in this kind of thing might well find the invention of new symbols and rules quite an eye­opener.

The rest of the book reads a bit like a manifesto for Martinez's philosophy that mathematics is created, not dis­covered. Indeed, Reuben Hersh says on the dust wrapper, "The author's point, that mathematics is constructed accord­ing to our judgment of what will serve us, is very important and little under­stood." But at various points I found myself mentally shouting, "Yes, yes, I agree with you! Who doesn't?" It was, at times, difficult to hear myself think for the rustling of straw men being knocked down. When I read that " . . . the notion that mathematical principles are discovered is still nowadays em­ployed much more often than any no­tions that some of those principles are created, decided, or contrived," I want

to see some evidence to back this up. After all, even G. H. Hardy, who claimed to be dogmatic in his view that "mathematical reality lies outside us, that our function is to discover or ob­serve it," also wrote in the same book (A Mathematician 's Apology of course) that "A mathematician, like a painter or a poet, is a maker of patterns. " We don't observe them, we make them. So Hardy, despite claiming dogmatism, is ambivalent, as all sensible people should be on a subject like this.

Reading this intriguing and provoca­tive book, I would have appreciated a deeper, nonpartisan discussion of the relationship between mathematics and physics; the beautiful, fruitful tension between mathematics as the servant of science and mathematics as an abstract system of ideas. When Martinez says, "The physical origins of basic mathe­matical rules were cast aside, obscured, and relegated to history . . . Mathemat­ics hence came to resemble less a phys­ical language or science, and more an art or logic referring not to empirical re­lations but to abstract or transcendental principles, " he seems to be adopting a strangely old-fashioned point of view, bearing in mind, for example, the de­velopment of algebraic topology over the last thirty or so years. The Unrea­sonable Effectiveness Question (to para­phrase Wigner, surely the author of the paper with the most-quoted title and least-quoted content in the history of mathematics) is alive and kicking and could do with more discussion, which I suggest Martinez could handle very well.

And the spoons? They don't appear in the book itself, but Martinez's un­usual interest in kitchen utensils resur­faces on page 226 in a paragraph about orange peelers, no doubt a metaphor for something or other. If he can con­fine the kitchenalia to a drawer, I will gladly read his next book.

Mathematical Capital

1 87 Sheen Lane

London SW1 4 8LE

e-mail: ericgrunwald@aol .com

.. j£'1.1.19.h.i§i Robin W i l son I

The Ph i lamath' s Alphabet-N

Napier: In 1614 the Scottish laird john Napier designed his logarithms as an aid to calculation for navigators and as­tronomers, replacing lengthy multipli­cations and divisions by easier addi­tions and subtractions. The 'Law of Napier', as portrayed on the stamp, would not have been understood by Napier.

Newton: Isaac Newton (1642-1727) was Lucasian Professor of Mathematics at Cambridge University, a post now held by Stephen Hawking. While in

Cambridge, he obtained the general form of the binomial theorem, studied power series, explained the relation­ship between differentiation and inte­gration, and analysed cubic curves.

Newton's gravitation: Seeing an ap­ple fall, Newton asserted that the force pulling it to earth is the same as that which keeps the planets orbiting around the sun, and that they are gov­erned by a universal inverse-square law of gravitation. In Principia mathemat­ica (1687), he deduced Kepler's laws of planetary motion and accounted for cometary orbits, the variation of tides, and the flattening of the earth at the poles.

Nightingale: Florence Nightingale ( 1820-1910) was strongly influenced by the work of the Belgian statistician Quetelet. Remembered as the 'lady with the lamp' who saved lives during the Crimean War, she was also a fine statistician who collected and analysed

mortality data from the Crimea and dis­played them on her 'polar diagrams' , a forerunner of the pie chart.

Non-Euclidean geometry: For over two thousand years mathematicians failed to deduce Euclid's fifth postulate from his other postulates; this is be­cause there are 'non-Euclidean geome­tries' , discovered by Gauss, Lobachevsky, and Bolyai, satisfying the first four pos­tulates but not the fifth. Given any line I and any point p not on it, these geome­tries have infinitely many lines parallel to I passing through p.

Nunes: Pedro Nunes ( 1502-1594), Royal cosmographer and the leading figure in Portuguese nautical science, applied mathematical techniques to cartography. He constructed a 'nonius' that measured fractions of a degree, and his 1537 treatise on the sphere showed how to represent each rhumb line, the path of a ship on a fixed bear­ing, as a straight line.

Napier's law Newton Newton's gravitation Nightingale

Please send al l submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, M i lton Keynes,

MK7 6AA, England

e-mai l : r.j .wilson@open.ac.uk Non-Euclidean geometry

68 THE MATHEMATICAL INTELLIGENCER © 2007 Spnnger Science+Bus1ness Media, Inc

Nunes