The Mathematical Characters of Galileo's Book of Nature

THE BOOK OF NATURE INEARLY MODERN AND

MODERN HISTORY

EDITED BY

Klaas VAN BERKEL and Arjo VANDERJAGT

PEETERSLEUVEN - PARIS - DUDLEY, MA

2006

THE MATHEMATICAL CHARACTERS OF GALILEO'SBOOK OF NATURE*

Carla Rita Palmerino

Introduction

In one of his Academic Lessons, Evangelista Torricelli recalled 'havingheard a great mmd say that God's omnipotence once composed twovolumes. In the one, dixit, et facta sunt, and this was the Universe, in theother, dixit, etscripta sunt, and this was Holy Scripture'.

Torricelli's words summarise the meaning of the old metaphor of the'Book of Nature' very well. The image of a God who created the worldthrough his words is found both in the Old and New Testament: 'And Godsaid, let there be light: and there was light' (Genesis, 1:3); 'By the word ofthe Lord were the heavens made; and the host of them, by the breath of hismouth' (Psalms, 33:4); 'In the beginning was the word, and the word waswith God and the word was God' {Gospel according to St. John, 1:1). Theword, logos, being the instrumental cause of the creation, the world may beconsidered the reification of God's word.

In his Lesson, Torricelli explains that mathematics is necessary todecipher both of God's books. He refers first to Augustine's De doctrinachristiana, where it is said that without arithmetic one cannot understand

many things which 'in the Sacred Pages have been given in a metaphoricaland mystical sense'. Then he adds that

mathematics is also necessary for reading the great tome of the universe (thatis, that book in the pages of which one should study the tme philosophy writtenby God), as will become evident to him who with a lofty mind aspires to a

* Research for this article was made possible through the financial support of the'Borsa Ruberti' and by a VENI grant CNetherlands Organisation for ScientificResearch).' Mi sowiene d'aver sentito dire da un grande ingegno, che I'onnipoteraa di Diocompose una volta due volumi. In uno dixit, et facta sunt, e questo fu I 'Universe.Nell'altro dixit, et scripta sunt, e questofu la Scrittura (Torricelli, Opere, p. 620; mytranslation).

28 CARLA RITA PALMERINO

scientific understanding of the components and the largest parts of that greatbody called 'world'.2

It is not difficult to guess who the 'great mind' is to whom Ton-icelli creditsthese thoughts. The appeal to the exegetical principles of the ChurchFathers, the emphasis on the metaphori6aj character of Biblical language,the image of the Universe as a philosophy book written by God, are allfound m Galileo's writings.

The Book of Nature and the books of men

In the so-called Copernican letters (1613-1615), which represent the mani-festo of his ideas concerning the relationship between revealed and physicaltmths, Galileo repeatedly insists that Holy Scripture does not aim atexplaming the constitoition of the world, but rather at instmctmg the faithfulin matters of faith and morals. Following Augustine's De genesi adlitteram, Galileo claims, moreover, that although the Bible never gives falsedescriptions of the world, it often speaks about physical matters in ametaphorical way, saying things that differ from the 'absolute' fa-uth. Giventhat God's two books cannot contradict each other, whenever a scripturalpassage seems to clash with a demonstrated scientific truth, we must con-elude that we have misinterpreted the Bible. In such a case, it is the task ofthe scriptural exegetes to fmd an alternative reading that takes care of thecontradiction.3

In a passage m the Enarrationes in Psalmos (XL V, 7), a work which isnot quoted by Galileo, Augustine also states, however, that while the pagesofAe Bible can be enjoyed only by those who know how to read, even theilliterate can read the book of the universe. Galileo believes, on the con-

trary, that the Book of Nature is more difficult to decipher than Scripture,because the latter is adjusted to the mtellectual capacities of commonpeople, whereas the former is not. As he explains in the Letter to the GrandDuchess Christina ofLorrain (1615):

2 Cheper legger la Bibbia sieno giovevoli Ie Mattematiche, gia sentiste I'opinione diSant 'Agostino, e d'altri Padri. Che per leggere il gran volume dell 'Universe - doequel libro, ne ifogli del quale dovrebbe studiarsi la verafilosofia scritta da Dio -sieno necessarie Ie Mattematiche, quegli se n 'accorgera, il quale con pensieri mag-nanimi, aspirera alia scienza delle parti integranti, e de i membri massimi di questo

For a comparison of Galileo's exegetical principles with those of Augustine, cf.inter alia McMullin, 'Galileo'; Howell, God's Two Books, pp. 186-196.

THE MATHEMATICAL CHARACTERS OF GALILEO' S BOOK OF NATURE 29

Holy Scripture and nature derive equally from the Godhead, the former as thedictation of the Holy Spirit, and the latter as the most obedient executrix ofGod's orders; moreover, to accommodate the understanding of the commonpeople it is appropriate for Scripture to say many things that are different (inappearance and in regard to the literal meaning of the words) from the absolutetruth; on the other hand, nature is inexorable and immutable, never violates theterms of the laws imposed upon her, and does not care whether or not herrecondite reasons and ways of operating are disclosed to human under-standing.

The claim that natural matters do not 'accommodate themselves to our com-

prehension' is also found in the Dialogue Concerning the Two Chief WorldSystems (1632). Here, however, Galileo adds that 'nature first made thingsin her own way, and then made human reason skillfal enough to be able tounderstand, but only by hard work, some part of her secrets'.5 The capacityto decipher the Book of Nature is in fact what distinguishes a philosopherfrom the average person.

Eugenio Garin has once remarked that 'for those who want to penetratethe value of the symbol of the Book of Nature, the cmx of the problem mayturn out to be the question of the characters in which the book is presumedto have been written'.7 It is well known that for Galileo the Book of Natureis written in the language of mathematics and that its characters are geo-metrical figures. This famous claim is made in the Assayer (1623), whereGalileo reminds his Jesuit opponent Aat philosophy is not

a book of fiction created by some man, like the Iliad or Orlando Furioso -books in which the least important thing is whether what is written in them istrue. Well, Sarsi, that is not the way matters stand. Philosophy is written in thisgrand book -1 mean the universe - which stands continually open to our gaze,but it cannot be understood unless one first learns to comprehend the languageand interpret the characters in which it is written. It is written in the language ofmathematics, and its characters are triangles, circles, and other geometricalfigures, without which it is humanly impossible to comprehend a single wordof it.8

Letter to Christina, in: Finocchiaro, The Galileo Affair, p. 93.5 Galilei, Dialogue, pp. 264-265.In the dedicatory epistle of his Dialogue Galileo writes: 'He who looks the higheris the more highly distinguished, and turning over the great Book of Nature (whichis the proper object of philosophy) is the way to elevate one's gaze' (ibidem, pp. 3-^

Garin, La culturafilosoflca, pp. 451-465.8 Galilei, The Assayer, pp. 183-184.


As James Bono has recently observed, in these lines Galileo 'rhetoricallywithdraws himself from the scene of writing, from active agency in theproduction of the text of philosophy'. Instead of comparing his own phi-losophy with that of the scholastics, he contrasts philosophy as a creation ofGod to philosophy as a creation of human beings. In other words, Galileochooses to present himself not as an author, but rather as the reader of theuniverse, the philosophy book already written by God.

Galileo also uses the unage of the Book of Nature to affirm his ownlibertas philosophandi with respect to the Aristotelian tradition. Late in life,m a letter to Fortunio Liceti, he writes:

If philosophy is that which is contained in Aristotle's books, you would be thebest philosopher in the world ... But I really think that the book of philosophyis that which is perpetually open to our eyes. But being written in charactersdifferent from those of our alphabet, it cannot be read by everyone; thecharacters of this book are triangles, squares, circles, spheres, cones, pyramidsand other mathematical figures, the most suited for this sort of reading.

So far we have seen how Galileo contrasts the Book of Nature to works of

fiction, such as the Iliad or the Orlando furioso, which do not have tmth astheir goal, to the Aristotelian texts, which tell questionable truths, and toscriptural books, which do tell the truth, but a truth that is often only 'adiun-brated'.n These three categories of books have one thing in common: theyare all written in human, verbal language.

9 Bono, The Word, p. 195.10 E quando la fllosofia fosse quella che ne i libri di Aristotele e contenuta, V.S. permioparere sarebbe il maggior filosofo del mondo ... Ma io veramente stimo, il librodelta filosofw esser quello che perpetuamente d sta aperto innanzi a gli occhi; maperche e scritto in caratteri diversi da quelli del nostro alfabeto, non puo esser datutti letto: e sono i caratteri di tal libro triangoli, quadrati, cerchi, sfere, coni, pira-midi et altre figure matematiche, attissime per tal lettura (Galileo to Liceti, January1641, in: Galilei, Le Opere, vol. XVIII, p. 295; my translation). In his Letters onSunspots, Galileo had ab-eady castigated his Peripatetic opponents for believing that'philosophizing is and can be nothing but to make a comprehensive survey of thetexts of Aristotle ... as if this great book of the universe has been written by natureto be read by nobody but Aristotle, and his eyes had been destined to see for allposterity' (Drake, Discoveries, pp. 126-127). In his Life of Galileo, Niccolo Gherar-dini chose to emphasise this particular use of the metaphor of the Book of Nature:Hebbe pochissima quantita di libri e lo studio suo dependea dalla continua osser-vcizione, con dedurre da tutte Ie case che vedea, udiva, o toccava, argomento difilo-sofare; e diceva egli ch 'il libro net quale si dcwea studiare era quello della natura,che sta apertoper tutti (Galilei, Le Opere, vol. XIX, p. 646)." For Galileo's theory of adumbratio, cf. Stabile, 'Linguaggio', pp. 54-56.

THE MATHEMATICAL CHARACTERS OF GALILEO'S BOOK OF NATURE 31

Indeed, Galileo repeatedly opposes the ambiguity and the vagueness ofverbal languages to the strictness and precision of mathematical language,the language of God's Book of Nature. Although it is essential to the under-standing of his exegetical principles, Galileo's philosophy of language hasbeen strangely neglected by scholars. In the following, I shall therefore firstdiscuss Galileo's ideas concerning the different properties of verbal andmathematical language. Next, I shall examine Galileo's use of the metaphorof the Book of Nature in the light of his views concerning the ontologicaland gnoseological value of mathematical tmths.

The strict language of the Book of Nature and the ambiguous language ofthe Bible

In his Copernican Letters, Galileo repeatedly notices that if one sticks to the'pure', 'naked', 'limited', or 'apparent' meaning of the words, manypassages of the Bible will seemingly say false thmgs. Only the expertexegete can explam the hidden, 'true meaning' of such passages and 'in-dicate the specific reasons why it is expressed in such words'.12 In his lettersto Benedetto Castelli (1613) and Christina ofLorrain (1615), Galileo wearsthe cloak of such an exegete when he analyses the famous miracle of Joshua(10:12-13), which in his days was usually interpreted by theologians in anti-Copemican terms. What meanmg should one attribute to the biblicalaccount of Joshua extending the duration of a day by ordering the Sun tostop? First, Galileo suggests that Joshua, 'knowing that his words werebeing heard by people who perhaps had no other knowledge of heavenlymotion except for the greatest and most common one from east to west',described the miracle from the view point of those who witnessed it andwho saw the Sun stop;13 but then, he also tries to discover if this passagecan be understood literally.14 The conclusion he reaches is that even ifinter-preted according to fheir strict sense, Joshua's words are more compatiblewith the heliocentric than with the geocentric hypothesis. Galileo explainsthat in the Ptolemaic system the altemation of day and night is not causedby the Sun, but rather by the primum mobile, which rotates from east towest. This implies that 'in order to make the Sun stay for some time in thesame place above the horizon without moving to the west, it would benecessary to accelerate its motion so as to equal the motion of the PrimeMobile, which would be to accelerate it to about three hundred and sixty

Galileo, Letter to Christina, in: Finocchiaro, The Galileo Affair, p. 92.nlbidem,pp.l\5, 107-108.14 Ibidem, pp. 114-115. Cf. also Galileo, Letter to Benedetto Castelli (ibidem, pp.52-54).


times its usual motion'.15 In other words, if Joshua had wanted to prolongthe length of the day, he would have had to order the Sun to speed up ratherthan to stop.

Joshua's words taken literally, however, also allow for another interpre-tation which is also fully compatible with the heliocentric system. If oneassumes that God interrupted not the daily but the monthly rotation of theSun, which Galileo identifies as the cause of the orbital motion of the Earthand of the oflier planets, then he would have also

[

stopped all other turnings, so that the earth as well as the moon and the sun(and all the other planets) remained in the same arrangement; and during thatwhole time night did not approach and day miraculously lenghtened. In thismanner, by stopping the sun, and without changing or upsetting at all the waythe other stars appear or their mutual arrangement, a day on earth could havebeen lengthened in perfect accord with the literal meaning of the sacred text.16

Many scholars believe that by attempting to reconcile the Joshua passagewith the heliocentric hypothesis, Galileo violated his own exegeticalprinciples.17 Why - they ask - did he first claim that the description of theuniverse was not among the objectives of the Bible and that it was legi-timate to use a demonstrated scientific truth to re-mterpret scriptural pas-sages in a non-literal manner, and then use a literal reading of the Joshuapassage to prove the superiority of the Copemican theory over the Pto-lemaic one? It seems to me that, far from violating his exegetical principles,Galileo is trying to defend them against those theologians who, to useLudovico delle Colombe's phrase, maintain that 'when Scripture can beunderstood literally, it must never be interpreted in any other way'.18 Thisposition appears overly simplistic to Galileo, who believes that the task ofthe biblical exegete cannot exhaust itself in the choice between a literal anda non-literal interpretation of the Scripture. By offering two alternativeliteral readings of the Joshua passage, Galileo wants to show that the verynotion of literal interpretation is problematic, for verbal language is ambi-guous by its very nature.

As to the other scriptural passages which seem to contradict this position [=theCopemican one], I have no doubt that, if it were known to be true and

15Ibidem,pp. 115,53.w idem,p.ll7.d. inter alia Rossi, La sciema, pp. 82-84; Pesce, 'Una nuova versione'; McMul-lin, 'Galileo'.18 Galilei, Le Opere, vol. Ill, p. 290.

THE MATHEMATICAL CHARACTERS OF GALILEO'S BOOK. OF NATURE 33

demonstrated, those same theologians who consider such passages incapable ofbeing interpreted consistently with it, (as long as they regard it as false) wouldfind highly congenial interpretations for them; this would be especially tme ifthey were to add some knowledge of the astronomical sciences to their exper-tise about Holy Writ. Just as now, when they consider it false, they think thatwhenever they read Scripture they only find statements repugnant to it, so ifthey thought otherwise they would perchance find an equal number of passagesagreeing with it.

Galileo seems to be saying that the interpretation of the Joshua miracle andof other similar passages given by theologians is, as it were, theory-laden.Exegetes believe in the tmth of the Ptolemaic system and therefore read theBible in the light of it. But if they could be convinced of the validity of theCopemican system, they would find ways of hannonising it with theScriptures.

The real aim of Galileo's exegetical experiment is, I believe, not tooffer a theological argument in favour of the Copemican hypothesis, butrather to show that where physical matters are concerned, 'the intention ofthe Bible' can sometimes be understood only if one reads it in the light of ademonstrated natural tmth.20 Nevertheless, Galileo is also wilTing torecognise that

even in regard to those propositions which are not articles of faith, the authorityof the same Holy Writ should have priority over the authority of any humanwritings containing pure narration or even probable reasons, but no demon-strative proofs; this principle should be considered appropriate and necessaryinasmuch as divine wisdom surpasses all human judgment and speculaticn.21

As far as natural philosophy remains a bookish activity, which consists incommenting and discussing the works of the Aristotelian corpus, itshyPotheses and conclusions can never have priority over the testimony ofHoly Scripture. For the word of God, even when it is expressed in'thelanguage of men, is always more reliable than a human conjecture. But ifphilosophising means decoding the Book of Nature which God wrote in hisown language, then a natural tmth 'which is placed before our eyes by sen-

M Lf^r,tochristiw' m: Finocchiaro, The Galileo Affair, p. 118.Stabile ('Linguaggio', p. 63) has convincingly Mgued that the aim of Galileo's

interpretation of the Joshua passage 'non e tanto quello di poggiare concor-disticamente 1'effetto di natura (il sistema coperaicano) sull'autoritadeliaScrittura,quanta viceversa di ribadire la validity del suo principio esegetico, ritorcendo sui^eologi ci6 che volevano dimostrare con una testimonianza letterale'.

Letter to Christina, in: Finocchiaro, The Galileo Affair, p. 94.


sory experience or proved by necessary demonsti-ations should not be calledin question, let alone condemned, on account of scriptural passages whosewords appear to have a different meaning'.22 For, he adds, the language ofnature is bound by strict laws, whereas the language of Scripture is not.

It should be clear by now that the only way Galileo has to affmn thepriority of natural truths over revealed truths is to msist on die diversity oftheb- respective languages. Revealed truths have been dictated by God m theambiguous language of humans, whereas natural truths have been writtenby God in the precise language of mathematics: 'God arranged all things bynumber, weight and measure', Galileo writes in one of his fragments,quoting the Bible (Wisdom, 11:20-21).23

The mathematical language of God's first book is economical, unam-biguous, self-explanatory. Each sign carries one and only one meaning, andthe propositions we can constmct out of these signs can but be true or false.The case is different for verbal language, which can be tuned and mani-pulated in many ways. Words allow one to be succinct, or prolix todescribe real, possible, or imaginary facts, to tell the truth, or to lie, but alsoto say things that can be tme or false according to the way in which weinterpret them. As Galileo explams in The Assayer, verbal language is thelanguage of persuasion, while mathematical language is the language ofcertainty: 'For a person who wants to convince others of something which,if not false, is at least very questionable, it is a great advantage to be able touse probable arguments, conjectures, examples, analogies and other so-phisms'. It is far more difficult, Galileo adds, 'to reduce oneself to the rigorof geometrical demonsti-ations', where there is no middle way between acertain conclusion and an unforgivable paralogism, just as 'in thingsthemselves (exparte rei) there is no middle way between true and false'. Ageometrical demonstration does not allow one to 'invoke limitations,distinctions, verbal distortions, or other fireworks', because 'one must withbut a few words and at first assault become Caesar or nobody'

The arbitrary link between names and words

Galileo abhorred the lack of criteria for settmg a scholastic dispute. In hispolemic with the Aristotelian philosopher Giorgio Coresio, Galileo de-fiantly declares that he was not able to dispute problems ad utramquepartem, because his own method consisted in 'studymg the Book of Nature,

22 Ibidem, p. 93.23 Posuit Deus omnia in numero, ponders et mensura (Galilei, Le Opere, vol. FV, p.52).24 Galilei, The Assayer, p. 252.


where things are written in one way only'.25 In his works, he also frequentlymsists on the superiority of strict and succinct mathematical demonstrationsover the verbose causal explanations of the Aristotelians, which pretend toreveal the hidden essence of things, but are mostly mere tautologies. When,in the Dialogue, Simplicio confidently attributes the cause of free fall togravity, Salviati reminds him that to know the name of something is nottantamount to knowing its essence:

You are wrong, Simplicio; what you ought to say is that everyone knows that itis called 'gravity'. What I am asking you for is not the name of the thing but itsessence, of which essence you know not a bit more than you know about theessence of what moves the stars around. I except the name which has beenattached to it and which has been made a familiar household word by the con-

tinual experience that we have of it daily. But we do not really understand whatprinciple or what force it is that moves stones downward any more than weunderstand what moves them upward after they leave the thrower's hand, orwhat moves the moon around.26

Contrary to mathematical signs which were written by God into the thingsthemselves, names have merely been imposed on them by humans. Theytherefore say more about our experience of physical objects than about theiressence. 'Names' and 'things', 'attributes' and 'essences' are four key-words in Galileo's speculations about language. In his First Letter on Sun-spots, he observes that 'names and attributes must be accomodated to theessence of things, and not the essences to the names, since things come firstand names afterwards'.2 The fact that men have imposed names on whatthey perceived through the senses explains for Galileo why words moreoften than not reflect the appearances rather than the essences of things.This is demonstrated by the fact that things that are the same in essence butlook different are sometunes called by various names, and that things thatare different in essence but look alike sometimes bear the same name.

In the famous passage of The Assccyer in which he explains that tastes,odors and colors do not reside in the perceived objects but only in the per-ceiving subject, Galileo says that these sense impressions are 'mere names'.Only because these names are different from those of the real attributes ofbodies, i.e. shape, position and motion, do we believe - wrongly - that they

Galileo, il quale, essendo usato a studiare sul libro delta natura, dove Ie casesono scritte in un modo solo, non saprebbe disputar problema alcuno ad utranqwpartem (Galilei, Le Opere, vol. IV, p. 248; my translation).26 Galilei, Dialogue, p. 234.Drake, Discoveries, p. 92.


are 'tmly and really different from the latter'.28 But in fact there is nothingin the real world that corresponds to our sensing qualities.

Whereas in The Assayer Galileo castigates the unnecessary multipli-cation of names, in the Dialogue we find a case where he criticises theexcessive economy of verbal language. He maintains that it is only becauseof 'a single and arbitrary name' that people are reluctant to accept thenotion that our planet is a large loadstone. For if instead of using the name'earth' to refer concomitantly to the soil which we plow and to the globe weinhabit, people had decided to call our planet by the name of 'stone',speaking of the world as a loadstone would have seemed a much more na-tural step. In Galileo's view, therefore, the unprecision of verbal languagenot only reflects the unprecision of our knowledge of the physical world,but it is sometimes also an impediment to the development and acceptanceof new ideas.

Conscious of the arbitrariness of the link between names and things,Galileo often criticises the essentialist definitions used by his Peripatetic op-ponents. In The Assayer he reminds Grassi (alias Sarsi), who calls thecomet a quasi-planet, that 'opinions' and 'voices' do not 'have the power ofcalling into existence the things they name'. And in the Third Letter onSunspots he warns Scheiner, alias Apelles, who believes that the allegedsunspots are, in fact, stars, that 'they essentially have properties that differfrom the tme stars'. At the same time, however, he recognises that it is notworth 'quibbling over names', given that 'anyone may impose them to suithunself. So long as a man does not think that by names he can conferinherent and essential properties on things, it would make little differencewhether he calls these stars'.

On the basis of what has been said, then, it is evident that Galileo'sproposal of various readings of the Joshua passage in his CopemicanLetters fonns part of a coherent strategy. In so far as it is the word of God,Holy Scripture can but speak the tmth; but in so far as it is expressed inhuman language, this word can sometimes sound ambiguous, vague, andindetenninate.

Biblical exegesis and natural philosophy are hence two utterly differentpursuits. While it is the task of the exegetes to interpret the polysemous lan-guage of the book of Scripture, it is the task of the philosopher to decodethe strict language of the Book of Nature.

28 Galilei, TheAssayer, p. 309.29 Galilei, Dialogue, p. 403.30 Drake, Discoveries, p. 253.31 Ibidem, p. 139. An analogous argument is found in Galilei, Le Opere, vol. IV p.631.


How to read the Book of Nature

If it is true, as Galileo says, that the language of the Book of Nature is notadjusted to the limited capacities of the human intellect, how can the naturalphilosopher aim at discovering the constitution of the universe? A firstanswer to this question is provided by the famous ending of the first day ofthe Dialogue, where Galileo explains that taken extensively (extensive), thatis to say, with regard to the infmite multitude of intelligible objects, thehuman intellect is nothing in comparison to the divine intellect. Taken in-tensively (intensive), however, human intellect does understand some pro-positions as perfectly as God does, and these are the mathematical propo-sitions. Admittedly, God knows the entire infmity of mathematical propo-sitions, whereas the human mind, which works not in an intuitive, but in adiscursive manner, can only grasp a few of them. But 'its knowledge equalsthe divine in objective certainty, for here it succeeds m understanding ne-cessity, beyond which there can be no greater sureness'.32

If m spite of our lunited intellectual skills we can decipher variouschapters of the Book of Nature, this is so only because this book waswritten in the simplest possible manner. In a letter to Gallanzone Gallanzoniof July 1611, Galileo warns against applying human criteria of order andperfection to the evaluation of God's acts. One might wonder, for example,why the circle was made in such a way that the proportion between thecircumference and the diameter cannot be determined. Why could not asimple ratio of three to one have been established between these two mag-nitudes? Galileo explains that if the latter had been the case, the circlewould have lacked many other simple and admirable properties.33

Two elements stand out in this argument. First, Galileo seems to assignto mathematical objects the ontological status of created bodies. Second, heassumes that God's acts, even when they look irrational to us, always fol-low the rules of mathematical order and simplicity.34 But Galileo also ad-mits that numerous accidents complicate the course of events, and that inorder to find the sunple laws that rule upon the natural world one mustabstract from these accidents. In his Two New Sciences (1638), he writes:

32 Galilei, Dialogue, p. 103.33 Cf. Galilei, LeOpere, vol. XI, pp. 149-150.

Cf. Galluzzi, 'II tema', pp. 254-255: 'pur ammettendo che avrebbe potuto dar vitaa infiniti altri tipi di ordinamento dell'universo, Galileo lascia tuttavia chiaramenteintendere che Dio non puo fare a meno di operare secondo il criterio della massimarazionaliA'

3 8 CARLA RITA PALMERINO

No firm science can be given of such accidents ofheaviness, speed, and shape,which are variable in infinitely many ways. Hence to deal with such mattersscientifically, it is necessary to abstract from them. We must find and demon-strate conclusions abstracted from the impediments, in order to make use ofthem in practice under those limitations that experience will teach us/

These famous lines are complementary to a less often quoted passage ofThe Assayer, in which Galileo writes:

Regular lines are called those which, havmg a single, firm, and determinatedescription, can be defined and whose accidents and properties can bedemonstrated ... But the irregular lines are those which, not having any deter-mination whatsoever, are infinite and casual, and thus indefinable, and of whichtherefore no property can be demonstrated and nothing, in sum, can be known.To say that 'this accident happens according to an irregular line' is for this rea-son the same as to say 'I do not know why it happens .

Galileo's linguistic consistency, which is always admirable, is particularlyimpressive in this instance: m both The Assayer and the Two New Scienceshe'describes the natural philosopher as capable of reaching a Turn' knowl-edge of the finite, but hopelessly lost in the face of the mfmite, a categorythat includes irregularly variable accidents.

In an influential article, Noretta Koertge has maintained that Galileo'sconcern with the 'problem of accidents' - a problem for which he developsincreasingly sophisticated solutions in his writings - shows that TheAsscyer's image of the universe as a book written m the language ofgeometry should not be 'taken as a significant or careful statement ofGalileo's philosophical views'.37 Koertge's conclusion is based on theassumption that Galileo regards mathematical laws as idealisations whichdo not have a counterpart in the actual world, and that he acknowledges theexistence of a discrepancy between the abstract and the concrete.

35 Galilei, Two New Sciences, p. 225. I have modified Drake's translation, writing'accidents' instead of 'events' to render Galilei's term accidenti.36 Chiamansi linee regolari quelle che, avendo la hr descrizzione uno, ferma e de-terminata, si possonodefmire, e di loro dimostrare gli accidenti e proprieta .. Ma Ielinee irregolari son quelle che, non avendo determinazion veruna, sono infinite ecasuali, epercio indefinibili, ne di esse si puo, in conseguenza, dimostrar proprietaalcuna, ne in somma saperne nulla. Si che il voter dire: 'II tale accidente accademerce di una linea irregolare ' e it medesimo che dire 'lo non so perch6 ei s accag-gia' (Galilei, Le Opere^vo\. VI, p. 244). I offer my own translation as Drake's is un-reliable here.37 Koertge, 'Galileo', p. 402.

THE MATHEMATICAL CHARACTERS OF GALILEO' S BOOK OF NATURE 3 9

This interpretation seems mcorrect to me because it attributes an onto-logical meaning to considerations that are in fact epistemological. WhenGalileo says that 'no fimi science' can be given of physical accidents thatare 'variable in infinitely many ways', he does not mean, I think, that theseaccidents are non-mathematical but that their random variations are too

complex to be captured by the natural philosopher, who can only under-stand what is mathematically regular and simple. In Galileo's eyes theproblem is not - as many have wrongly believed - that the world is atvariance with mathematical truths but that its mathematical set-up is most ofthe time beyond the reach of our intellectual skills.

This point is evinced in a passage of the Dialogue, where the Aristo-telian Simplicio criticises the demonstrative methods used by the GalileanSalviati, arguing that mathematical tmths lose their validity when applied tophysical matters. As an example of the opposition between mathematical'abstractness' and physical 'concreteness', Simplicio invokes theproposition: 'The sphere touches the plane at a point'. Simplicio is heresiding with Renaissance Aristotelians like Alessandro Piccolomini and Be-nito Pereira, who in the debate de certitudine mathematicarum had invokedthis very proposition as the typical example of a mathematical truth thatfails to work in the physical world.

In his response, Salviati first demonstrates by means of mathematicalreasoning that two spheres must necessarily touch each other at a singlepoint. He shows his Aristotelian interlocutor that if two spheres (and hencealso a sphere and a plane) were to touch each other, not at a point, but alonga line, one of Euclid's postulates would be violated.39 At this point,Simplicio intermpts Salviati, reiterating the objection that this proof is validonly in mathematics, but not in physics, because 'material spheres aresubject to many accidents to which immaterial spheres are not'. Theporosity of matter, to mention just one reason, makes the existence ofaper-feet plane and of a perfect sphere very unlikely.

But Salviati is not willing to accept such an opposition between mathe-matical and physical truths. Instead, he introduces an epistemological dis-tinction between 'simple' and 'complex' and a logical distinction between'true' and 'possible'. Smce this double distinction has so far been ignored inthe literature, it will be explamed here.41

38 Cf. e.g., Butts, 'Some Tactics', p. 81; Giorello, 'Galileo', p. xxix.39 Galilei, Dialogue, p. 206WIbidem.41 My analysis of Salviati's response to Simplicio disagrees with Butts' ('SomeTactics', pp. 70-79), Pitt's ('Galileo', pp. 190-191) and Feldhay's ('The Use', pp.127-129).


The first step in Salviati's answer consists in claiming that if a materialsphere and a material plane do not touch each other at one pomt only, this isnot because they are physical and 'concrete', but because they are 'unper-feet'. But, he adds,

even in the abstract, an immaterial sphere which is not a perfect sphere cantouch an immaterial plane which is not perfectly flat in not one point, but over apart of its surface, so that what happens in the concrete up to this point happensthe same way in the abstract.

Although in these lines Salviati adopts Simplicio's terminology, it is clearthat he gives a different meaning to the words 'perfect' and 'imperfect'.43For him the two attributes should be used not to demarcate the reahn of

mathematics from that of physics but only to distinguish what is simple, orregular, from what is complex, or irregular. If it is possible to conceive ofan irregular, abstract figiire, why should one not grant the right of existenceto a perfectly regular concrete object? Why should one exclude apriori thatin the concrete world there exists a sphere that can touch a plane at onepoint only? Indeed, Salviati admits that the actual behavior of physicalbodies does not always exactly respond to sunple mathematical law, but heinsists that ifthefilosofo geometra wants

to recognise in the concrete the effects which he has proved in the abstract, [he]must deduct the material hindrances, and if he is able to do so, I assure you thatthings are in no less agreement than arithmetical computations. The errors,then, lie not in the abstractness or concreteness, nor in geometry or physics, butin a calculator who does not know how to make a tme accounting.44

The 'calculator who does not know how to make a tme accounting' isclearly the human intellect which, as we have ah-eady explained, onlygrasps what is mathematically simple and regular. When Galileo says thatthe philosopher must abstract from the 'material hindrances' in order to'recognise in the concrete the effects which he has proved in the abstract',he does not mean, I believe, that the concrete behavior of physical bodies isnon-mathematical, but only that most of the time it is complicated by

42 Galilei, Dialogue, p. 207.Galileo often criticises the Aristotelian notion of perfection, cf., e.g., Galilei, LeOpere, vol. IV, p. 446; vol. VI, pp. 319-320; vol. VII, p. 35 and vol. XI, pp. 149-150.44 Galilei, Dialogue, pp. 207-208.


accidents of which, to use The Assayer's expression, we cannot have 'fumscience', for they happen 'accordmg to an irregular Ime'.

Salviati's reply to Simplicio does not end here. It contmues with the ar-gument that even if in the material world there existed neither a perfectlyregular sphere nor a perfectly smooth plane, it would nevertheless bereasonable to assume that an irregular material sphere and an irregularmaterial plane existed such that they would touch each other at a singlepoint. For the physical existence of two siu-faces that fit each other perfectlyfor a certain extension is far less probable than fhat of two irregular surfacesthat touch each other only at a single point.

The wise Sagredo now joins the discussion. In reaction to Simplicio'sclaim that no perfectly spherical material body can be found in nature, hedemands to know why it should be more difficult to obtain from a block ofmarble a perfect sphere or pyramid than a perfect horse or grasshopper.Isn't it tme that irregular fonns are more difficult to produce than regularfonns? And 'if of the shapes which are irregular, and hence hard to obtain,there is an infinity which are nevertheless perfectly obtained, how can it beright to say that the simple and therefore the easiest of all is impossible toobtain?' At this point Salviati decides to put an end to the discussion. Sa-gredo's questions, which remain unanswered, serve Galileo to suggest thatm principle all geometrical figures can be found in the concrete world, andalthough a material body is more likely to have an irregular rather than aregular shape (because irregular lines are more numerous), the probabilityof the occurrence of regular shapes is increased by the fact that they aremore easily realised than irregular ones.

I think that Galileo wishes to apply the same conclusion to all physicalphenomena. Even if material hindrances often complicate the course ofevents, one must never exclude on a priori grounds that an exact mathe-matical mle is realised in the concrete world. For nature is intrinsicallyrational and is generally governed by the most simple laws.

The fact that Galileo's discussion of the sphere-cum-plane argumentbegins with a geometrical proof and ends with a probabilistic argument is, Ibelieve, emblematic of his overall views of the relationship betweenmathematical and physical truths. For Galileo, a mathematical falsehood ne-cessarily implies a physical falsehood, whereas a mathematical truth is notnecessarily equivalent to a physical truth but only to a physical possibility.Put differently, what is tme in physics miist be true also in mathematics,whereas what is tone in mathematics is possibly true in physics. But of

45 Ibidem, p. 210.4 Galileo's idea that nature always follows the easiest path is expressed in ibidem,pp. 60, 117,123.


course the natural philosopher who researches the 'real constitution of theuniverse' will not content himself with probable conclusions. He aims atcertainty, and certainty can only be assured by a well-pondered muture ofsensory experiences and necessary demonstrations. Only sensory expe-nences can in fact help us decide which of the perhaps infinite mathemati-cal possibilities God actually chose when he wrote tiie great Book of Na-ture.

Conclusion

The difficult and much debated issue of how sensory experiences andmathematical demonstrations interact in Galileo's scientific practice willnot be discussed here. The question I wish to address by way of conclusionfor its relevance to the present context concerns the relationship between thesubjective language of experience and the objective language of thephysical world. It is obvious that the natural philosopher has no directaccess to the mathematical characters of the Book of Nature, but obtainsthem in the translation of the language of perception. The question thereforearises concerning Ae reliability of sense perception and its relation to themathematical Urtext.

As we have seen, Galileo establishes a direct link between the arbi-trariness of names and the illusory character of sensations. He finds thathuman perceptions like verbal language are riddled by synonymies andhomonymies. The famous fable of sounds, told in The Assayer,is nothingelse tha" a way of proving that 'the bounty of nature in producing hereffects' is such that 'our senses and experience' sometimes judge asiden-tical phenomena that are in fact produced by utterly different causes.47

Galileo's cosmological discourse is based on the recognition that oursenses are not tmstworthy, given that they would not be able to detect themotion of the Earth. In the Dialogue Salviati repeatedly praises Copemicusfor having given preference to reason over perception and for having stub-bomly believed 'what sensible experience seemed to contradict'.48 Galileoadmits that he himself would have been reluctant to accept the Copemicansystem, had it not been for the existence of 'a superior and better sense thannatural and common sense', which cooperated better with reason.49 By this

cGalilei' The Ass.aye''^:.236:lcalmot a8ree with Wqjciehowski (Old Masters, p.154), who thinks that Galileo did not realise that 'the Book of Nature, as translatedthrough the language of empirical data, is complicated by figuration',just asscriptural language.48 Galilei, Dialogue, p. 339; cf. also ibidem, p. 328.

'Ibidem,p.32S.


superior sense, Galileo most likely means the telescope, which allowed himto gather new evidence in favour of the heliocentric system.50 Copemicus'hypothesis that Venus rotates around the Sun, for example, seemed to clashwith appearances until the telescope showed the planet looking 'homedwhen beneath the sun' and changing 'its shape in exactly the same way asthe Moon'.51

Yet, the fact that telescopic observations were compatible with theCopemican system was for Galileo still not enough to conclude that the sys-tem was tme. Galileo commented on a letter by Cardinal Bellarmine toPaolo Antonio Foscarini, a philo-Copemican Carmelite theologian, asfollows:

It is tme that to show that the appearances are saved by the mobility of the earthand the stability of the Sun is not the same thing as to demonstrate that thishypothesis is really true in nature. But it is equally or even more true that theother commonly accepted system - the Ptolemaic system - is not able to givereasons for these appearances. The latter is undoubtedly false, just as it is clearthat the former, which corresponds to the appearances perfectly, could be tme.No greater truth can be, or ought to be, sought for in a position than that itcorresponds to all the particular appearances.

Eman McMullin has expressed his bewilderment with regard to thispassage:

Granted that these notes are no more than jottings, it is still disturbing to findGalileo so uncertain regarding the principal philosophical issue separatingBellarmine and himself... The most that can be said of a hypothesis (like thatof Copemicus) that 'fits the appearances perfectly' is, apparently, that it cmildbe true. But this is far too weak to carry any weight in the face ofBellarmine'sobjection.53

But, quite to the contrary, Galileo's 'jottings' reflect his epistemologicalviews very accurately. We have seen how Galileo maintained, in the Dia-logue, that while a self-contradictory mathematical proposition must befalse for the reabn of physics, a mathematical truth implies not necessarily a

50 This interpretation is favoured by most but not by Wisan, 'Galileo's ScientificMethod', p. 50, note 20. On Galileo's celebration of the 'superior sense' afforded bythe cannocchiale, see Luthy, 'Atomism', pp. 6-11.51 Galilei, Dialogue, p. 335'." Quoted from Blackwell, Galileo, p. 271.53 McMullin 'Galileo', pp. 285-286.


physical tmfh but at least a physical possibility. His notes on Bellarmine'sletter convey an analogous idea: whereas a mathematical model that cannotsave the appearances is 'undoubtedly false', a model that fits the appear-ances is possibly, but not necessarily true. There may in fact always existalternative hypotheses that save the phenomena equally well.

Astronomical observations collected over the years convinced Galileothat the geocento-ic hypothesis did not fit the appearances, whereas the helio-centric hypothesis did. We know, however, that Galileo was unwilling tolimit himself to the role of a 'pure astronomer', who could provide 'reasonsjust for the appearances of celestial bodies'. Instead, he wanted to be an'astronomer-philosopher', that is to say, someone who attempted to read theBook of Nature so as to fmd the 'real constitution of the universe'.55 Giventhat the telescope could not help him attain this goal, Galileo somehow had,as it were, to defeat the principle of relativity of motion and fmd a terrestrialproof of the validity of the Copemican system. Precisely this is the functionof the problematic and ultimately fallacious explanation of the tides, whichtakes up the Fourth Day of the Dialogue. By maintaming that the flux andreflux of the sea cannot be brought about by any other cause than by theEarth's double motion, Galileo tries to transfonn what was a probablehypothesis into a demonstrated scientific truth, the only tmth capable ofoutrivalmg the authority of the book of Scripture. The language ofmathe-matics, although essential for the construction of physical demonstrations,was by itself not sufGcient to formulate certain proofs.

By way of conclusion we should therefore note that despite his call onphilosophers to read the book of philosophy written by God, Galileoauthored philosophical texts just like his scholastic opponents. His rich andhumorous prose indicates that, afiter all, he took pleasure in translating themathematical characters of the Book of Nature into the ambiguous languageof words. This may be the reason why, in the Dialogue, he calls thealphabet the most admirable invention of mankind.

54 Galilei, Dialogue, p. 341.55 Ibidem. Cf. also Drake, Discoveries, p. 97.56 Ibidem, pp. 105, 109.

CONTENTS

Preface and Acknowledgements

Into-oduction

Klaas van Berkel andArjo Vanderjagt

Contributors

Vll

DC

Xl

The 'Book of Nature' and Early Modem SciencePeter Harrison

The Mathematical Characters of Galileo's Book of NatureCarlo Rita Palmerino 27

Reading the Book of Nature in the Seventeenth-Century Dutch RepublicEric Jorink 45

George Berkeley's 'Universal Language of Nature'Costica Bradatan 69

Nature in Defense of Scripture. Physico-Theology and ExperunentalPhilosophy in the Work of Bernard NieuwentijtRienk Vermij 83

'The Law of Nature is a Lamp unto your Feet'. Frederik Adolfvan derMarck (1719-1800) on the Book ofNature and RevelationHenri Krop 97

Sermons in Stone. Johann Jacob Scheuchzer's Concept of the Book ofNature and the Physics of the BibleMichael Kempe 111

Nature Writing and the Book of Nature. From Taxonomy toNarrative Truth

Johanna Geyer-Kordesch 121

Children's Walks in the Book of Nature. The Reception ofJ.F. Martinet's Katechismus derNatuur around 1800

Arianne Baggerman 141

Deus sive Natura. J.G. Herder's Romanticised Readmg ofSpmoza's Physico-TheologyDetlev Patzold 155

A Scientific Bible. Novalis and the Encyclopedistics of NatureDavid Wood 167

Readmg Nature in the Light of Scripture. The case ofGeorges Cuvier (1769-1832)

Jitse M. van der Meer 181

The Word and the Works. Concordism in American EvangelicalThought

Edward B. Davis 195

Reading the 'fme print' m the Catskills. John Burroughs Remterpretsthe Book of Nature

Charlotte Zoe Walker 209

The Moral Significance of 'Life's Splendid Drama'. From NaturalTheology to Adaptive ScenariosPeter J. Bowler 227

Improving Nature. Victor Westhoffand Dutch Nature Conservationin the Post-War Years

Bert Theunissen 243

Reading the Book of Nature through American LensesRonald L. Numbers 261

Comets, Necessity, and NatureJohn D. North 275

Bibliography 299

Index 329

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The Mathematical Characters of Galileo's Book of Nature