Review of ‘Loving and Hating Mathematics’to appear in the Mathematical Intelligencer

Jonathan M. Borwein and Judy-anne OsbornCentre for Computer Assisted Research Mathematics and Its Applications

The University of Newcastle, Australia

September 20, 2011

1 Myths & Mathematics

Loving and Hating Mathematics (hereafterreferred to as Loving and Hating) is the childof two passionate scholars: a mathematicianand a social scientist. Reuben Hersh willbe known to some readers for his many ar-ticles in the Intelligencer, as well as earlierbooks such as The Mathematical Experiencecoauthored with Davis and Marchisotto, andWhat is Mathematics Really? The latter hada substantial effect upon the older of the tworeviewers, being, at the time of publication,a welcome blast of mathematical humanism.

The present book, Loving and Hating, iswritten in the same clear gentle style, andhas as its expressed aim the vanquishing offour myths:

1. Mathematicians are different from otherpeople, lacking emotional complexity.

2. Mathematics is a solitary pursuit.

3. Mathematics is a young man’s game.

4. Mathematics is an effective filter forhigher education.

Outline of Loving and Hating

Loving and Hating has chapters addressingmathematical: beginnings, culture, solace,addictive potential, communities, gender andage related issues, philosophies of teaching ofmathematics in Universities, and, last of thenumbered chapters, teaching of mathematicsin schools. At 416 pages, it is as compact asit could be, given the ambitious breadth ofits scope.

Only the last and first chapters of the bookdeal directly with school mathematics. Thismakes the cover design due to Lorraine BetzDoneker, which is clearly school-situated,particularly worthy of mention. The imageis emblematic of a key message of the book,which the authors express as follows as theyconclude their last chapter: Because we lovemathematics, we want to minimize the num-

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ber of those who hate it. The picture is of twoadorable little boys in short-trousered schooluniforms (bringing back the first author’smemories of his own school uniform). Thetwo sit in front of a blackboard, with a silvertrophy between them. One is forward facingand looks quite content, the other notably un-happy and gazes to the side, at the first. Thechildren pictured personify the Loving andHating of the title, as was underlined to theauthors of this review when a young personwho glanced at the book instantly associatedhimself with the sad-looking protagonist.

The book is not structured as a recipe toaddress problems with school experiences ofmathematics, however. Rather it is a tour ofmathematical life in the large, carrying withit a recommendation that design issues relat-ing to the school-level experience of mathe-matics should be addressed in terms of math-ematics in its entirety, and in particular thejoy that its practioners take in the endeavour.

The chapter titles and starting page num-bers are as follows:

Chapter 1: Mathematical Beginnings 9

This title could mean many things - as it hap-pens the authors address how a child becomesengaged in mathematics. To an extent thispicks up on the trophy on the front cover,since it includes a section on mathematicscompetitions. We learn of the childhoodmathematical experiences of famous mathe-maticians such as Terence Tao, Carl FriedrichGauss, Sonia Kovalevskaya and many others;as well as observations of personality and psy-chological issues recurrent in childhood enjoy-ment of mathematics.

Chapter 2: Mathematical Culture 46

This chapter makes clear that mathematics isa subject with a culture reaching back over along long time. The authors’ description en-compasses thoughtful forays into four mainideas: abstraction, aesthetics, belongingnessand the tension between collaboration andcompetition.

Chapter 3: Mathematics as Solace 89

The authors ask: “Is mathematics a safe hid-ing place from the miseries of the world?”in this chapter, and answer that it can be.They illustrate that the meaning of the claimranges from an absorption which temporar-ily keeps the worries of the world at arm’slength, to a means of coping with situationsas extreme as imprisonment.

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Chapter 4: Mathematics as an Addic-tion: Following Logic to the End 106

Mathematical researchers tend to have asense of what ‘mathematics as addiction’means, for better or for worse. Here, we get asample of some of the extremes: after a men-tion of John Nash, whose life and schizophre-nia was the subject of the book and movie ABeautiful Mind, a detailed picture is paintedof the extraordinarily creative and intense lifeof Alexander Grothendieck. Following this,the authors present ‘five cases of actual crim-inal or suicidal insanity in other mathemati-cians’, writing about famous cases includingthe tragic later life and death of the renownedlogician Kurt Godel.

Chapter 5: Friendships and Partner-ships 138

This chapter describes some famous friend-ships between mathematicians: Karl Weier-strass and Sonia Kovalevskaya, the trio ofHardy, Littlewood and Ramanujan, and thefriendship between logician Kurt Godel andphysicist Albert Einstein; amongst others.Mathematical marriages such as between Ju-lia Bowman Robinson and Raphael Robinsonare also described - this relationship is alsodescribed in the book Julia and film JuliaRobinson and Hilbert’s Tenth Problem. Theimportance of friendships and partnershipsin sustaining the individuals involved is de-scribed both in particular and in general.

Chapter 6: Mathematical Communities176

Communities of mathematicians that haveformed spontaneously or in organized ways,to meet the needs of the groups that com-prise them, are described. Examples rangefrom over a century old to the present,including the faculty at the University atGottingen in Germany (1890’s-1930’s), thefamous French group Bourbaki which beganin the 1930’s, the short lived Jewish People’sUniversity (1978-1983) in Moscow, and con-temporary examples such as the Associationfor Women in Mathematics (AWM) and theweb-supported Polymath Project. The waysin which the communities support their mem-bers, in which the communities themselvesdie or flourish, and in which these groups to-gether are part of a larger mathematical com-munity, are described.

Chapter 7: Gender and Age in Mathe-matics 228

This chapter addresses mathematical lifethrough the lens’ of gender and of aging. Theexperiences of many famous women mathe-maticians are described, including historicalexamples in which being female was a consid-erable impediment to mathematical life, suchas experienced by Sonia Kovalevskaya andEmmy Noether. Notably absent amongst thehistorical examples is Lady Ada Lovelace whois famous for her work on algorithms and in-formation, in connection with Babbage’s an-alytical machine.

The varied experiences of contemporary

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women mathematicians such as Karen Uhlen-beck, Joan Birman and Fan Chung are de-scribed. Again, this is a selection — pleas-ingly there are now too many accomplishedwomen in the profession to be comprehen-sive — and does not include other equallynotable women such as Cathleen Morawetz,nor any non-Americans. The experience ofbeing a mathematician and getting older isalso described in its variousness in some de-tail, reprising the results from an earlier pub-lished survey conducted by Hersh as well ascomments from other surveys.

Chapter 8: The Teaching of Mathemat-ics: Fierce or Friendly? 273

The focus in this chapter is upon just twoUniversity level examples: the systems exem-plified by Robert Lee Moore (Moore Method)and Clarence Francis Stephens (PotsdamModel). These examples are extremes points,both from the USA, rather than the sort ofbarycentric averages that may be commonpractices now inside and outside of the USA.As the authors of Loving and Hating write,the two models ‘embody two different, op-posed strains in American Education: theegalitarian versus the elitist; the cooperativeversus the competitive; the heritage of theDeclaration of Independence versus the her-itage of the Confederate States of America.’

Chapter 9: Loving and Hating SchoolMathematics 301

The final chapter begins with observationsabout the effects of school education in math-

ematics upon the feelings that adults havetowards mathematics, including the obser-vation that these feelings often include the‘hating’ of the book’s title. The chapter in-cludes relatively little description of math-ematics in the classroom as experienced byschool students. There is reference to mathe-matical learning in a variety of contexts, suchas the shopping contexts investigated by an-thopologist Jean Lave. The chapter includesmany suggestions for reform in the teachingof mathematics, with reference to trial pro-grams in various contexts. A thread underly-ing many of these suggestions is the idea thatpeople have multiple different kinds of intel-ligences, and that teaching generally shouldnot privilege mathematical thinking or evenspecific kinds of mathematical thinking.

End matter

Following the numbered chapters are fivepages of conclusions, nine pages of ‘literaturereview’ listing other popular books on math-ematics, and thirty four pages of paragraphlong biographies of mathematicians. The lastmentioned compendium was of particular in-terest to the (mathematician) husband of oneof us, who picked up the book upon its arrivalin the household, and when he discovered thebiographies at the end, sequestered it untilhe had read them and the rest of the bookthrough. This biographical ‘digestif’ to thebook with its overview of the lives of bothwell known and less known mathematiciansmay be one of the highlights for those readerswho are themselves part of the mathematicalcommunity.

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To summarize, Loving and Hating is asweeping survey of mathematical life, intowhich the four myths and the antidotes theauthors provide are woven. We structure theremainder of our review around the followingtwo sets of questions, which arose for us inthe reading.

About the myths

1a. Are the four claims actually myths?

1b. Who believes them?

1c. Are these myths about mathematicians,or about broader groups?

About the audience

2a. To whom is the book addressed by theauthors?

2b. To whom would the book be useful?

2c. Will the book Loving and Hating find itsaudience?

About us

We are both research mathematicians, withinterest and experience in communicatingmathematics with the general public as wellas students and peers, one of us late-careerand the other early-career. Laureate Profes-sor Borwein has been involved in mathemat-ics outreach in four countries on three con-tinents over a period of nearly forty years.Dr Osborn has led community-building andoutreach activities at the Australian National

University and the University of Newcastle.Both of us are currently employed at the Uni-versity of Newcastle in Australia, in the cen-tre for Computer Assisted Research Math-ematics and its Applications (CARMA), ofwhich Professor Borwein is the director.

As reviewers of Loving and Hating, we findourselves largely in agreement; this reviewbeing a snapshot of our discussions. For muchof the review we write in explicit dialogue(JB for Jon Borwein and JO for Judy-anneOsborn), to clarify our different perspectivesand occasional disagreements. Where wewrite in one voice, we agree with each other.

2 The Myths one by one

Myth 1

Mathematicians are different fromother people, lacking emotionalcomplexity

Is this a widely held belief? Does it havea basis in fact? Hersh and John-Steiner have

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run two claims together here, and we wonderif the result is obfuscation?

JB: If Myth 1 is meant to say that mathe-maticians are ‘a bit odd’, then its widelybelieved, and often true. Otherwise itmight not be true. In my father’s houseI grew up around many mathematicians,who ranged from the urbane and articu-late to the seemingly mute.

JO: Yes - indeed quite a few anecdotes inLoving and Hating reinforce rather thandiminish a perspective of eccentricity.For instance in Chapter 2: Mathemat-ical Culture, R H Bing is described asdriving colleagues to a conference, andwhen the windscreen fogged up, used it

to draw mathematical diagrams on,rather than wiping it clean.

JB: Films like A Beautiful Mind pick up onand emphasize the idea of the eccen-tric or insane mathematician. It is amyth that being crazy helps a person todo good mathematics (or much anythingelse): it doesn’t. As Michael Crichtonhad said, however, “All professions lookbad in the movies - why should scientistsexpect to be treated differently?”

JO: Evidence countering the second part ofthe claim - that mathematicians lackemotional complexity - is overwhelming,as reflected by Hersh and John-Steinerin story after story. Emotions suchas attachment, affection, joy, courage,fear, empathy, anxiety, sorrow, indigna-tion, depression and wonder, are relatedin many accounts of discoveries, friend-ships, prison terms, politics, competi-tion, collaboration and every-day life. Iwas struck with a sense of recognitionwhen I read in Chapter 2 of the joy thatProfessor Jenny Harrison finds in nature,exploring paths through the woods; andlooking at mathematical landscapes withsomething of the same feeling.

JB: Yes, the sense of wonder is palpable inGrothendieck’s description of his feelingswhen he switched mathematical fieldsfrom analysis to geometry:

It was as if I had fled the harsharid steppes to find myself sud-denly transported to a kind of‘promised land’ of superabun-dant richness, multiplying outto infinity wherever I placedmy hand on it, either to searchor to gather ...

I was also taken by the descriptionof Chandler Davis’ response to his sixmonths in prison, which resulted fromhis refusal to cooperate in some Mc-Carthy era questionings by the Commit-tee on Anti-American Activities. Davis’

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sense of humor, and courage, is ex-pressed in a footnote to one of his sub-sequent papers:

Research supported in part bythe Federal Prison System.Opinions expressed in this pa-per are not necessarily those ofthe Bureau of Prisons.

We are told that the delicate wording ofthis gem was suggested to Davis by afriend. The story is an instance of howmany things can happen to a principledperson in a long life.

JO: In terms of who the audience for thisbook is, this myth under discussionmakes me think that it must be aimedat the general public, for surely the be-lief that ‘mathematicians lack emotionalcomplexity’ could only be held by mem-bers of the public who don’t happen toknow any mathematicians?

JB: I am not so sure. The part onGrothendieck’s work, for instance, is fartoo technical for the lay-reader.

JO: I agree that the detail on Grothendieck’swork is very technical. For myself as amathematician I found it required moreconcentration than I was willing to giveat the time. Yet surely it gives a fla-vor of a kind of mathematics, in a waythat would be impossible otherwise, andlets people who have never trodden thehalls of a University Mathematics de-partment have a sense of what it is like,

even without understanding details. Ithink that Hersh and John-Steiner haveaimed to write a book which has some-thing for both lay-readers and mathe-maticians alike.

JB: Much of what is being put forward inLoving and Hating as being unique tomathematicians, applies to any group ofpeople who pursue a life of the mind.As for characteristics such as madnessor emotional range, I see no differencewith Physicists or with English scholars,for instance. (It may be that there isan autistic tendency in mathematicians- quantitative studies could measure this,and maybe even distinguish mathemati-cians from physicists.) However in themain, just as mathematicians are por-trayed as mad in the movies, the ideaof the ‘mad poet’ is a romantic concept.There is no reason to think that fewerPhysicists or Writers go off the deep end.

JO: Famous physicist Ludwig Boltzmannand famous author Virginia Wolfe?

JB: Yes, those instances and more.

JO: Surely there’s no harm in just focus-ing on mathematicians, in a book aboutmathematical life?

JB: Loving and Hating would be better formore situating. It is misleading, for in-stance, to write about the Unabomberas a mathematician, in the chapter onMathematics as Addiction, without talk-ing about other scientists who also did

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crazy violent things. I searched the Uni-bomber’s massive manifesto in Altavistaat the time it was published. I found theword mathematics only occurred in thephrase “science and mathematics” andthat only three times.

JO: I asked my sister, as a person outside ofacademia, whether she thinks that Math-ematicians are different from other peo-ple, lacking emotional complexity? Mysister replied that she does not thinkthat most people think mathematicianslack emotional complexity. She said thatshe thinks that the general public thinkof mathematicians as people who spendtime writing strange complicated thingson bits of paper.

JB: That’s an interesting reply.

JO: Yes, so was Myth 1 ever a myth at all?

JB: It has the weakest claim of the four tobe a myth. What is true is that manymathematicians have interesting stories.However, any reader of Loving and Hat-ing who begins the book with the viewthat mathematicians are uniquely andexclusively passionate about their par-ticular field, and idiot savant in every-thing else, will come away knowing thatis not the case.

Myth 2

Mathematics is a solitary pursuit.

JO: It is not very surprising that peopleoutside of mathematical research wouldthink of it as a solitary pursuit, since it isoften depicted in the movies as being thework of a lone genius in their own privateivory tower replete with chalkboard.

JB: The movies aren’t the only source forsuch a view. It is a default perceptionof creative work, if there isn’t reason tothink otherwise - we also think of novel-ists as working in solitude.

JO: Yet in mathematics, the idea of solitarycreation is only a half-truth. Excellentmathematical work can be done in soli-tude (for instance as the half dozen orso examples of mathematicians in prisonin the book testify), but this conditionis not the norm. Interaction between

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mathematicians, both planned and ac-cidental, with sharing of ideas and mu-tual inspiration, is typically important ornecessary in mathematical creativity.

JB: I think that we’re agreed that Myth 2is a myth: a belief both widely held,at least outside of mathematical circles,and in the large not true. That said,it is still true that we have no technol-ogy for telepathy, so that the creativethinking that is essential to mathemat-ical progress is still necessarily an indi-vidual part of the activity.

JO: Yes, indeed, and Hersh and John-Steinerdo a good job of rebutting the mythin the large. In Chapter 5 on Friendsand Partnerships, I was fascinated toread of the friendship between Hilbert,Minkowski and Hurwitz, and how thelatter two helped Hilbert plan his fa-mous ‘23 important open problems’ talkwhich he gave in Paris in 1900, whichhas shaped much of mathematics for thefollowing hundred years.

JB: A story I find resonant is that ofMassera, since my colleague at CarnegieMelon, Massera’s student Shaeffer, washeavily involved in the campaign to freeMassera when he was interned in prisonin Uruguay for 9 1/2 years. Massera wasa collaborative and generous person bothin and out of prison. As a prisoner hewas involved in circulating forbidden pa-pers on dialectic, logic and mathematicsthat helped the prisoners keep their spir-its up. It is a fine example of humans

supporting ideals and each other in hardtimes.

JO: Another interesting instance of a cooper-ative spirit, in an entirely different con-text, is Timothy Gower’s 2009 PolyMathproject, described in Chapter 6, in whicha major open mathematics problem wasposed on the web, a collaborative effortinspired, and the problem solved in theastonishingly short time frame of a fewweeks by “the shared effort of over twodozen contributors from several coun-tries.”

JB: The internet has made an astonishingdifference to the practicality of multi-location collaboration. In Chapter 1Hersh and John-Steiner comment thatstudents receive guidance and inspira-tion not only face to face but also at adistance. I found it curious that theychose to illustrate this with a century-old example of Birkhoff in the USA be-ing inspired and learning from Poincarein Paris. Yes, collaboration and mentor-ship has been important in mathematicsthroughout the ages, but there are manymore contemporary examples — such asthe role of the arXiv — that they couldhave drawn upon.

For instance, when I was still at SimonFraser University in Vancouver in 2003,the one of the biggest power blackoutsfor a generation in Northern America,occurred. In our research centre we knewinstantly that there was a problem, whennone of our contacts east of the Missouri

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would come up. In fact, we realized thatwe were more quickly aware of a prob-lem affecting our distant colleagues inour discipline, than we would have beenif there had been major trouble in partof our own physical campus.

JO: The support that mathematicians cangive each other is beautifully describedin a quote about Chern in his capacityas a thesis supervisor, who

... conveyed the philosophythat making mistakes was nor-mal and that passing frommistake to mistake to truthwas the doing of mathemat-ics. And somehow he also con-veyed the understanding thatonce one began doing mathe-matics it would naturally flowon and on.

JB: About tenn years after my thesisI started collaborating seriously withother scientists, and discovered that isnot just from other mathematicians thatinspiration can be found. I have cometo the realization that how good you areat formalizing 20th century rigorous ex-pressions is not a complete measure ofyour mathematical worth. I would of-ten rather collaborate with a Physiciston a poorly posed problem, than witha mathematician close to my own field,because there may be more undiscoverednuggets in the bringing together of twomore different perspectives.

JO: That’s something that doesn’t shinethrough so much in Loving and Hating- the rich mathematical opportunitiesfor mathematicians in collaboration withother scientists.

JB: Yes, I’d like to see a book which doesfor the sciences what Loving and Hatingattempts to do for mathematics. Whatwould such a book look like? The closestthat I can think of at the moment is APassion for Science, by Alison Richardsand Lewis Wolpert. I am coming tothink that there may be more differ-ences within mathematics, than thereare across the sciences in their entirety.

Wolpert interviewed Christopher Zee-man who told a story about one ofhis (non-mathematician)) administra-tors who helped run an annual summerconference series in Warwick, which ro-tated every three years between the threemathematical areas of Algebra, Topol-ogy/Geometry, and Analysis. She saidthat after a while she could tell whichwas the year’s theme without looking atthe program, but just by observing thebehaviour of the participants. The Al-gebraists were punctual, organized andthrifty. They wanted single cheap roomsand arrived by train when they said thatthey would. The Topologists wanted bighouses, they brought their families andwanted to stay the whole week. The An-alysts were predictably unpredictable,promising to turn up Tuesday with theirpartner and arriving on a different dayfrom somebody else.

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Maybe the main cleavages are within ourdiscipline? There are differences in levelsof rigour. Perhaps there are differencesin levels of embodiment? The book, inits refutation of the myth of solitude andmore generally, would be better for morecontextualizing.

JO: I am reminded of the characteristicsthat Hersh and John-Steiner describe inChapter 1 as reflective of the whole tribeof mathematicians: curiousity, determi-nation, willingness to spend time doingmathematics, not minding being aloneso much as others might, cherishing in-dependence, and having a love for sym-metry or logic or pictures or, sometimes,how things work. It strikes me that thecharacteristic of willingness to be aloneis the grain of truth in the myth, thatgives it some of its traction.

If Hersh and John-Steiner had includedall the contextual comparison with therest of science that you want them to,it would likely have been a much biggerbook, whereas currently it is just a com-fortable size for carrying and reading onthe train.

JB: The characteristic of being willing tospend time alone thinking isn’t special tomathematicians or even scientists moregenerally. It is a property of leading alife of the mind, and that is a point worthmaking in a book which endeavors, atleast in part, to introduce the wide pub-lic to what a mathematical life is like.

Myth 3

Mathematics is a young man’sgame.

JO: Hardy wrote that ‘Mathematics is ayoung man’s game’ when he was in hissixties and in a self-confessed melan-choly mood, writing about mathematicsinstead of doing mathematics, which hewould have preferred.

Hersh and John-Steiner address thismyth in their Chapter 7: Gender andAge in Mathematics, where they makeit rightly clear that Hardy would havemeant ‘person’ by ‘man’, and that he wasmaking a claim about age, not women.Their chapter deals with issues of bothgender and age; here I focus on ‘age’.

Is it widely believed that mathematics is‘a young person’s game?’ And is it true?

JB: This is widely believed, both within andoutside of the mathematical community.

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JO: I agree with you that this view is preva-lent inside the mathematical community.I disagree about outside of it. For manyof the general public, the only personthat they envisage as a mathematicianis Einstein, and their picture is of a bril-liant old man with a shock of white hair.I think that the public myth of “genius”is of “Age and Wisdom”.

JB: If that’s the case, then the public haveforgotten what a media star Einsteinwas when he burst onto the scene in hisyouth.

JO: Is it true that mathematics is ‘a youngperson’s game’?

JB: There’s an aspect of truth to it in manyfields, not just mathematics. There areyoung geniuses and old masters, as isrelated in the piece on artistic geniusin my book with David Bailey aboutExperimental Mathematics. The differ-ence is like that between an early Picassocubist work and a Renaissance master-piece. Breakthroughs tend to be madeby the young.

JO: It is related in Loving and Hatingthat Hardy’s own long-time collabora-tor Littlewood was to become a counter-example to Hardy’s claim, publishing a“monster” paper with Mary Cartwrightthat “was recognized as an early break-through in the discovery of chaos”, whenLittlewood was in his seventies.

JB: Yes, if you maintain a passion for yourfield, as I do, and reasonable health then

you can continue to make fine contribu-tions as you age. I can still do goodmathematics in my sixties. My fatherenjoys doing mathematics at much at 87as he did at 18.

The myth with the strongest credoswithin the mathematics community isreally:

Doing first rate research math-ematics is something that youhad better hurry up and do be-fore you’re 35 or 40.

It is very unusual to find someone whohas been a toiler in the ‘mathematicalvineyards’ who suddenly has a huge re-sult at age 50.

JO: The mathematician Mary Ellen Rudenis quoted in the book as stating in inter-view

I don’t think most people’s bestwork will be done by the timethey’re thirty, and certainlymy best work wasn’t done untilI was fifty-five years old.

Might the myth you state be a self-perpetuating myth? For instance onlymathematicians not over 40 years of ageare eligible for the Fields Medal, which isthe mathematical equivalent of a NobelPrize.

JB: In fact the Fields medal is awarded forthe contribution that the person hasmade and is expected to continue tomake!

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JO: Oh that’s interesting, and it fits with theresponses to Hersh’s survey of his fellowmathematicians. Many say that theycontinue to make good progress, in theirlater years, though their mix of skills andstrategies tends to change.

JB: Yes, some mathematicians make moreand more impact as they get older,whilst others burn out. Again, lookat other creative fields - the syntheticrather than the accretive ones. Look at‘geriatric rock’. The Rolling Stones arestill energetic, creative musicians.

JO: We’ve discussed how the book and ourlife experience, especially yours, topplesthe myth that excellent mathematics isonly done by young people. But thebook also tackles the claim that if you’reany good at all as a mathematician thatits going to show up before you turn 40,and intriguingly, much of the data refersto women.

For instance, I quote from Chapter 7:

Joan Birman, a topologist atColumbia University-BarnardCollege, did not get her PhDuntil she was 40 years old.Birman focused better onmath after the issues of mar-riage were sorted, her childrenolder, etc. “I think doingmathematics when you’reenthusiastic is important –not your age.”

If it is true that women often make their

first good achievements later on, it isconsistent with what my (male) thesisadvisor used to tell me. He didn’t be-lieve that mathematicians become no-tably worse as they age, but rather thatthere are more other responsibilities anddistractions that tend to get in the wayof creative achievement. As I remem-ber it, he told me that if you can keepyourself clear of the ‘crap’, you can stillcreate.

JB: Yes and yes. I learned to keep the‘crap’under control from my father who wasa research active mathematics Depart-ment head for a very long time. Mymother who got an anatomy PhD theyear before I got mine was a wonderfulexample of deferred female achievement.

Myth 4

Mathematics is an effective filter forhigher education.

JO: It is said that the door to Plato’sAcademy was engraved with the phrase:

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Let none ignorant of geometryenter here.

Whether or not this fable is true, thespirit of the idea is consistent withPlato’s conception of mathematics as ameans to train the mind, an idea whichhas been embedded in western thoughtever since.

JB: Yes, historically in Cambridge the earli-est, and for along time only, examineddegree awarded was in Mathematics. Itwas called the Tripos. Keynes did theTripos, as did Airy, Herschel, and manyother famous people in and out of theprofession of mathematics.

JO: Such as the economist Thomas Malthus;astrophysicist Arthur Eddington; discov-erer of argon Lord Rayleigh; founder ofthe theory of electromagnetism JamesClerk Maxwell; philosopher, logician,mathematician, historian, and socialcritic Bertrand Russell?

JB: Indeed - there is an illustrious history.At much the same time the degree ofequivalent standing at Oxford Univer-sity was called Greats (or Classics), anarchetypal humanities degree, emphasiz-ing literature, language, philosophy, his-tory and art, and was the course takenby aspiring ministers of the church, so-cial thinkers, writers, politicians andcivil servants.

JO: So the special status of the subjectsMathematics and English in schools

through-out the English speaking world,is inherited from the two great EnglishUniversities of the Middle Ages?

JB: In many ways, yes. Even now, althoughindividuals claim to ‘hate math’, or notbe able to do it, nobody doubts its im-portance.

JO: I am not so sure.

JB: No serious business at any serious levelis functioning these days without anenormous amount of mathematics. Thetrend is reported on in publications suchas Business Week, with cover storieswith titles like, “Top Mathematiciansare becoming a new global elite” in Jan-uary 2006.

Successful tech companies, such as Nor-tel (originally Northern Bell) was adecade ago, have been successful partlybecause of employing mathematicallytrained people. At the height of itssuccess, the only factor that they couldidentify to distinguish successful fromless successful research groups was thatthose with mathematicians in them tendto be more productive.

JO: That understanding may not be as broadas you think, particularly in schools.Students are asking, What is the point?and, When am I ever going to use this?

JB: Yes, in Chapter 9 of the book, the math-ematician Underwood Dudley is quotedas arguing that algebra and calculus areseldom used or needed by most people.

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JO: Hardy made the same point with moreeloquence in his famous ‘Apology’:

... some mathematics iscertainly useful in this way;the engineers could not dotheir job without a fair work-ing knowledge of mathemat-ics, and mathematics is begin-ning to find applications evenin physiology. ... It is use-ful to have an adequate supplyof physiologists and engineers;but physiology and engineeringare not useful studies for ordi-nary men.

JB: Since Hardy’s day (he was born in 1877and died shortly after the end of the sec-ond world war), the professions that relyon mathematical thinking have multi-plied. I think he would have been greatlyamused at the number of number theo-rists who work for the security agenciesaround the world such as CSIS or NSA.That said, I do not believe that we teachthe right mathematics to the right stu-dents. The problem is that we are badat identifying the right students.

As an eleven year old in Britain I expe-rienced something called the ‘11 plus’.It was an examination taken by allBritish children, which determined atthat young age the rest of my academicfuture by specifying what kind of sec-ondary school I was admitted to. Thepressure on my eleven-year old self washorrendous, even though I was capable

in doing well in that exam and did so.I do not advocate the ‘11 plus’ or any-thing like it, yet in its absence, I chal-lenge anyone to find a diagnostic forthose who will need mathematics beyondarithmetic better than letting everyonetry it.

JO: You’re saying that we need engineers andso forth, and that they need mathemat-ics, and - I’m inferring here, since math-ematics is cumulative and takes a longtime to learn - we had better ensure ev-eryone does mathematics up to a certainlevel.

JB: Yes. There is an analogy with learn-ing a foreign language. One of the fewthings that linguists agree about, is thatlearning a language before age eight isdifferent to learning it afterwards. Inthe same way, it is very hard to laydown high-end mathematical skills af-ter school; to say, ‘I’m going to catchup at University’. I do know somecounter-examples: one of my teachersMichael Dummett, the renowned Britishphilosopher, taught himself mathematicsso that he could understand Frege - butsuch counter-examples are rare.

JO: What about the claim that mathematicsshould be used as a filter to entry to pro-fessions that don’t need it, as describedin Loving and Hating?

JB: That is very different. However, a be-lief in mathematics as a general filter has

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had considerable staying power in Uni-versities. For instance, by and large, astudent may not fail much else in a Busi-ness Degree, but he or she will need toget through Calculus 100 and somethingActuarial.

JO: I am puzzled by what is meant by thephrase, an effective filter? Does it meanthat a moderate proportion of studentstend to fail first year Calculus, so thatthis can be used as a sieve independentlyof whether it is sieving on some sensiblecriteria?

Or does effective filter mean that an abil-ity to do mathematics courses is beingused as a proxy for general intelligence?People sometimes say to me, “Oh youdo maths; you must be so smart”. Itstrikes me as an odd comment, comingas it often does from people who do workwhich seems to me much more subtleand difficult, like the clinical practice ofmedicine. Perhaps the belief that math-ematical capacity implies intelligence iswidely held? I don’t know about theother way around.

The authors of Loving and Hatingcounter the myth by quoting Gardner’stheory of multiple intelligences, describ-ing mathematical abilities as being dis-tinct from and not necessarily relatedto other equally important intelligencessuch as linguistic, musical and interper-sonal.

JB: We filter inappropriately. Many of thepeople who come to University would be

better off in a more technical training,but as long as there is a (completely un-justified) status gulf between universityand technical college, it won’t happen.This is managed better in Germany Ithink.

JO: What is University for?

JB: Respected social scientists like Christo-pher Jenks would say that, in its degree-awarding capacity, a University is notso much about what students learn asit is about what employers learn aboutthe graduand: that you’re reasonablywell socialized, that you can follow otherpeople’s rules. This is in contrast withthe liberal arts idea of education. De-grees like Classics (Greats) at Oxfordand the Tripos in Mathematics at Cam-bridge used to be a measure of generaleducational attainment, and people werehired on that basis for careers that hadnothing to do with the subject matter.Even in the seventies you could see indi-viduals advertising for a job in the Lon-don Times with a qualification of ‘BAOxon (failed).’

I had an old friend who tried to losehis history thesis draft on the London toBristol train the day after he was hiredby the British Home Office. He wanteda defensible excuse not to finish it! Like-wise, most people coming to Universitiesto study these days are not coming witha passion for some given subject. Thereis however, an important minority whodo. This is mixing chalk and cheese.

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JO: I am deeply imbued with the liberal artstradition, whose values are expressedbeautifully in Alfred North Whitehead’s1930 essay on the ‘Aims of Education’, inwhich he wrote The university impartsinformation, but it imparts it imagina-tively. ... A university which fails inthis respect has no reason for existence.Based on my experience of learning andof teaching, I expect that students maybe transformed and delighted by whatthey learn at University, as well as get-ting a qualification that may lead to ajob, even if their original aim was themore tangible one only.

JB: We conflate the idea of the ability to domathematics well with the ability to ap-preciate it. We don’t make the same mis-take with Shakespeare, or sports.

What I would like to be able to do, whenteaching mathematics, is to ask studentsin the class who is there for apprecia-tion, and who needs mastery for subse-quent professional use. Those in the firstgroup could take the course as Pass/Fail,and those in the second for a numeri-cal grade. I don’t mind which group in-dividuals are in. Both are worthwhile.With modern technology there is moreand more capacity to meet both kinds ofneeds in the same classroom.

JO: That’s an arrestingly interesting idea. Itis also, in hindsight, compatible with thesuggestion by Hersh and John-Steinerthat

There is a wrenching strain be-tween opposing pressures: acontinuing demand for enoughsophisticated math specialists,with a shrinking need for tra-ditional math skills in the gen-eral population. ... Studying[math(s)] should continue tobe required, but not in such amanner that students remem-ber it with antagonism andloathing.

JB: Many of the points made in the lastchapter are about what could or shouldbe, as opposed to what is. This is a shiftin style from the earlier chapters, whichwere more driven by personal experienceand anecdote.

JO: Yes, there is an enormous idealism in thefinal chapter, which collects together lotsof stories of ideas that individuals havefor trying to begin to make a differenceto the experience the school-childrenhave of mathematics classes. The sug-gestions are less developed in this finalchapter on school-level teaching, thanare the two striking cases studies of theegalitarian methods of Stephen and eli-tist methods of Moore, presented in thepenultimate chapter on University-levelteaching.

Whether in the school or the Universitysetting, personally I find myself drawnto both Moore’s method (minus theracism and anti-feminism, of course!) inwhich he believed immensely in individ-

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ual students’ capacity to discover thingsfor themselves; and Stephen’s PotsdamModel based on his fundamental beliefthat

[A]ny college student whowanted to learn college math-ematics could do so if thelearning environment wasfavorable.

Based on my experience of learning andof teaching, I find resonance in Stephen’soft-repeated phrase, go fast slowly.

Conclusions

Loving and Hating is a book full of gems. Wefound that we could open it on any page andfind something interesting. It is imbued withthe authors’ love of mathematics and respectfor people. The message that mathematicsis a fundamentally human activity, in whichpeople can find meaning and joy, is clearlyconveyed.

The book has flaws. We liked the partseach in turn more than the whole. Whilst allmathematicians of a generally philosophicalnature are likely to enjoy browsing in Lovingand Hating, we are less sure if the order inwhich the material is presented is of service tothe rest of its potential readership, includingteachers, policy-makers, and general public.

We find ourselves much in agreementwith Kevin McConway, whose review inPlus Magazine: http://plus.maths.org/

content/loving-hating-mathematics con-cludes that,

This is a complex book, which doesnot entirely achieve what it set outto do, and which does not entirelyhang together as an organised whole.Sometimes the richness of the anec-dotes and case studies gets in theway of the overall messages. Thestories are not going to convincesomeone to continue their mathe-matical studies if they are worriedabout becoming different from theirnon-mathematical friends. But it isworth reading for the admirable waythat the stories and anecdotes hu-manise mathematics - and becausemany of the anecdotes are very goodstories in their own right.

We conjecture that amongst potentialreaders who are not professional mathemati-cians but who are interested in mathematics,the chapter on school mathematics may beplaced rather late on the scene, given thatfor this group of potential readers one of themain touchstones of mathematical experiencemay have been in school. That said, the con-text in which the chapter on school math-ematics is saved for last is that of makingsuggestions for improvements to the schoolexperience of mathematics, where those sug-gestions are made on the basis of connecting‘school math’ with the discipline at large de-scribed so well in earlier chapters. The im-mersion in mathematical life with its joys ofdiscovery, which the earlier chapters provide,gives a fresh mindset for thinking about im-proving the experiences in the school class-room.

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We do feel that this book may be of spe-cial interest to graduate students in mathe-matics, as part of an introduction to the sto-ries and culture of the community that theyare joining. The first author has a prac-tice of recommending that his new gradu-ate students read Lakatos’ Proofs and Refuta-tions, Medawar’s Advice to a Young Scientist,Davis and Hersh’s The Mathematical Experi-ence and Yandell’s The Honors Class ; andis tempted to add Loving and Hating Mathe-matics to that list.

JB: Will the book find its audience?

JO: It is readable, informative, interesting: Ithink so.

JB: A school girl asked to review a book onpenguins wrote:

This book told me a great dealmore about penguins than I re-ally wanted to know.

If mathematicians are penguins and thereaders are all marine ornithologists, thisis great. Sadly, I am unconvinced that abroader audience will take the time toread this book. As a professional pen-guin, I think that this is a pity.

JO: Your point is so beautifully made thatit is an exquisite pain for me to differ,but I do. I don’t think that you have tobe a bird-specialist to love penguins, or,for instance, to enjoy a well-made DavidAttenborough special on them.

Maybe Hersh and John-Steiner are notquite David Attenborough, but they’re

close. The descriptions in Loving andHating are sympathetic and understand-able.

The lives that Hersh and John-Steinerhave led have allowed them to get up-close and personal with a species (math-ematicians, and more generally peoplewhose work is creative thinking) whoseworld many people don’t ordinarily getto see, and may welcome a window into.

Acknowledgements

The second author thanks her sister and hus-band for helpful discussions and her formerPhD supervisor for allowing his advice to bequoted. All the drawings in this article wereproduced by the second author.

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