platonic hypotheses: galileo’s “analytic method” and the...

Draft: please do not cite or circulate Marco Romani Mistretta

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[email protected] Harvard University, Department of the Classics

Boylston Hall, 1350 Massachusetts Avenue 02138 Cambridge, MA (USA)

Platonic Hypotheses: Galileo’s “Analytic Method” and the Phaedo

Introduction

Galileo often resembles an epistemological Proteus. Because of his unquestionable status

as one of the ‘founding fathers’ of modern science, his methodological tenets have not only been

constantly bent to support whichever theory of natural science is en vogue at any given moment,

but his fundamentally ‘Classical’ allegiances have accordingly experienced major shifts.1 Is Galileo

a ‘Platonist’ or an ‘Aristotelian’? Is he a ‘realist’ or an ‘instrumentalist’? Does he ever prove to be

an ‘Archimedean’ scientist? Or perhaps none of the above? What ancient sources, if any, inform

his conception of the workings and methods of physics?

The dichotomy of ‘Platonism’ and ‘Aristotelianism’, in particular, seems to dominate

early- and mid-twentieth-century Galilean scholarship. A. Koyré's seminal article (1943),2 for

instance, establishes a polarizing contrast between ‘Platonic’ and ‘Aristotelian’ natural science,

based on the role assigned to mathematics in making sense of the physical world. Koyré argues

that, whereas Aristotle conceives of the workings of nature as being intelligible in purely empirical

terms, without the need for mathematical formalization, Plato (in the Timaeus) describes the

structure of nature as being inherently geometrical, and therefore only subject to explanation

through mathematical concepts. According to Koyré (followed by Crombie and others), Galileo is

essentially a ‘Platonist’ insofar as he attributes a crucial importance to the use of mathematics in

1 See e.g. Wallace 1974: 79 (with references); McMullin 1978: 210. 2 Koyré's views concerning the predominance of natural philosophy (in the Aristotelian sense) over the application of mathematical reasoning to the physical world constitute one the main underpinnings of his well-known denial that the ancients ever developed any ‘technology’ in the modern sense of the term (cf. especially From the Closed World to the Infinite Universe, Baltimore - London 1957).

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natural science (ubi mathematica, ibi Plato).3 C. Dollo (1989), however, has rightly pointed out

that Galileo’s Platonism is constituted by well-defined and circumscribed elements of method and

cosmological content, rather than by a mere set of ‘Platonic ideas’, vaguely related to the

mathematization of the world, but devoid of factual significance.4

Other scholars, most notably T.P. McTighe (1968) and E. McMullin (1978), have

emphasized the ‘Aristotelian’ character of Galileo's epistemology by drawing attention to his

conception of physics - and of mechanics in particular - as a demonstrative science, or a ‘science

of the necessary’. Demonstration, for Aristotle provides the gnoseological foil against which the

explanatory power of any other science is to be measured: an extremely influential idea, if one

thinks - for example - of Veblen's claim that the formulation of a science endowed with “necessity

and sufficiency” is the goal of any investigations of the foundations of geometry.5 Moreover,

building upon an intuition originally suggested by Cassirer,6 A.C. Crombie (1953) and N. Jardine

(1976) have shown striking affinities between Galileo's account of scientific demonstration and

the Aristotelian theories of Paduan Renaissance intellectuals such as Agostino Nifo and Giacomo

3 It is now known that the principal experiments described in Galileo's Discorsi, and thought by Koyré to be pure ‘thought experiments’, were in fact performed (see e.g. Dubarle 1968: 305). 4 Similarly, J. Hankins (2000) rightly warns against the tendnecy of much Galilean scholarship to approach the Platonic problem from an unhistorical perspective: rather, we should understand Platonism in terms of what it meant for Galileo in his own time, rather than in terms of what it means for us now. For example, Ficino’s view on how the planets’ motion is a composite of rectilinear and circular motion is much closer than Plato’s Timaeus to Galileo’s own cosmogonic account (cf. Hankins 2004: 159). 5 Veblen 1903: 309. 6 Cassirer 1906: 134-141. A fundamental innovation of Paduan Aristotelianism consists, according to Cassirer, in bringing together the compositional and the analytical method, in order to establish procedures for acquiring knowledge that move from effects to causes and viceversa. Such a method is not so much a way of ‘resolving’ phenomena into fundamental principles as it is about shedding light on the hidden causes of those phenomena: in fact, it is an essentially causal-explanatory procedure. Cassirer sees in the role of mathematics the main, decisive difference between Zabarella’s regressus and Galileo’s own scientific method. Aristotelian syllogistics cannot, however, escape from the issue of whether and how the first principles can be proven: hence, for the new science, a need for hypotheses.


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Zabarella, whose doctrines most probably acted as ‘mediators’ between ancient (or medieval)

Aristotelianism and Galileo's epistemology.7

This paper focuses on a particular aspect of Galileo’s methodology, i.e. his “metodo

resolutivo”, which I intend to relate to the Platonic (and neo-Platonic) concept of hypothetical

method. In order to do so, I will examine Galileo’s conception of hypothesis and hypothetical

reasoning, showing that his use of the term “ipotesi” is by no means unequivocal. However, at

least one of his ways of employing the notion is, I will argue, eminently Platonic.

Galileo the realist?

The main piece of evidence for mathematical ‘Platonism’ in Galileo's method of science is

usually taken to be a well-known and oft-quoted passage from The Assayer (1623), in which

Galileo polemizes against Orazio Grassi's failure to understand the ‘mathematical language’ which

constitutes the basis of the scientific intelligibility of the world:

Il Saggiatore, 6.34-37 Parmi, oltre a ciò, di scorgere nel Sarsi ferma credenza, che nel filosofare sia necessario

appoggiarsi all'opinioni di qualche celebre autore, sì che la mente nostra, quando non si maritasse col discorso d'un altro, ne dovesse in tutto rimanere sterile ed infeconda; e forse stima che la filosofia sia un libro e una fantasia d'un uomo, come l'Iliade e l'Orlando Furioso, libri ne’ quali la meno importante cosa è che quello che vi è scritto sia vero. Signor Sarsi, la cosa non istà così. La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l'universo), ma non si può intendere se prima non s'impara a intender la lingua, e conoscer i caratteri, ne’ quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto. (Galilei 2005: 99)

It seems to me that I discern in Sarsi a firm belief that in practising philosophy it is

essential to support oneself upon the opinion of some celebrated author, as if when our minds are not wedded to the reasoning of some other person they ought to remain completely barren and sterile. Possibly he thinks that philosophy is 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 is true. Well, Sig. Sarsi, that is not the way matters stand. Philosophy is written in this grand book - I mean the universe - which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other

7 For the contemporary debate on ‘Aristotelianism’ and ‘Platonism’ in Galileo, cf. notably Cellucci 2012, De Caro 2012, Finocchiaro 2010, Hatfield 2004.

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geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is vainly wandering about in a dark labyrinth. (Drake 1960: 183-184, modified)

Reading this passage as an endorsement of (Platonic) mathematical realism would be

reading too much into it and reading it out of context, as L. Geymonat already pointed out in his

pioneering monograph.8 But it is not simply “a plea for [...] independent-mindedness” against

established authority:9 it is Galileo's own version of the time-honored idea of the Book of Nature,

which for Galileo is encoded in mathematical language, and therefore understandable to a (well-

educated) human mind.

Whether or not Galileo had Plato's Timaeus (53c-55c) in mind, according to which the

structure of the world can be explained in terms of five (Platonic) solids that can in turn be

reduced to triangles, Galileo's criticism of Sarsi (= Grassi) is clearly aimed at highlighting the

latter's underestimation of the role of mathematics in natural science.10 In fact, the Assayer as a

whole has been read as “a heroic poem in prose, a ‘Sarsiad’, a protracted tale of right against

wrong, good against evil, innocence against deceit”.11 Even though Galileo's polemical statement

is more concerned with his epistemology than with his actual scientific practice, it does reveal that

Galileo conceives of nature as being ultimately readable.12

Similar thoughts are expressed by Galileo in a famous programmatic document, the letter

to the Grand Duchess Christina of Tuscany (1615): the argument proceeds ex negativo and with a

mainly apologetic tone, since the central purpose of the letter is not to prove that the Bible

8 Geymonat 1957: 199-202. 9 Finocchiaro 2010: 116. 10 Cf. also this passage from the Second Day of the Two New Sciences: “Sagr. Che diremo, Sig. Simplicio? non convien egli confessare, la virtù della geometria esser il più potente strumento d’ogni altro per acuir l’ingegno e disporlo al perfettamente discorrere e specolare? e che con gran ragione voleva Platone i suoi scolari prima ben fondati nelle matematiche? Io benissimo avevo compreso la facultà della leva, e come crescendo o sciemando la sua lunghezza, cresceva o calava il momento della forza e della resistenza; con tutto ciò nella determinazione del presente problema m’ingannavo, e non di poco, ma d’infinito. Simp. Veramente comincio a comprendere che la logica, benché strumento prestantissimo per regolare il nostro discorso, non arriva, quanto al destar la mente all’invenzione, all’acutezza della geometria.” 11 Heilbron 2010: 245; see also Blumenberg 19862: 72. 12 See Blumenberg 19862: 74-75; Finocchiaro 1980: 121.


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supports the Copernican theory, but rather to show that it does not refute it. Galileo, in other

words, argues that there are no scriptural objections to the heliocentric model; that theology is not

binding in physical investigation; that neither scriptural consensus nor the unanimity of the

Church fathers nor the authority of the Church itself are sufficient to force a literal interpretation

of the biblical text.13

In arguing for the inescapable value of sensory experience and necessary demonstration,

whose authority - not that of the Scriptures - is the starting point of science, Galileo states that the

course of nature is “inexorable and immutable”. Hence the irrevocability of physical laws, once

they have been conclusively proved by means of experience and demonstration. The mysteries of

nature are to be investigated by reading the “open book of the sky”, which can obviously never be

contradicted by the other divine book (= the Scriptures).14

Here, however, a puzzle arises: if the workings of the natural world are readable and

intelligible, why would the astronomer need to rely on hypothetical reasonings? In other words,

how does Galileo conceive of scientific hypotheses and of their role in natural philosophy? As a

matter of fact, his usage and treatment of the ‘technical’ language concerning the semantic sphere

of ‘hypothesis’ still has not been thoroughly investigated, and such will be one of the main goals of

this paper.

In attested Italian, the noun ‘ipotesi’ (a quasi-transliteration from the Greek ὑπόθεσις) is

not common before the seventeenth century,15 the idea being more frequently expressed with the

13 See Finocchiaro 2010: 84. 14 “Il proibir tutta la scienza, che altro sarebbe che un reprovar cento luoghi delle Sacre Lettere, i quali ci insegnano come la gloria e la grandezza del sommo Iddio mirabilmente si scorge in tutte le sue fatture, e divinamente si legge nell'aperto libro del cielo? Né sia chi creda che la lettura degli altissimi concetti, che sono scritti in quelle carte, finisca nel solo veder lo splendor del Sole e delle stelle e 'l lor nascere ed ascondersi, che è il termine sin dove penetrano gli occhi dei bruti e del vulgo; ma vi son dentro misteri tantro profondi e concetti tanto sublimi, che le vigilie, le fatiche e gli studi di cento e cento acutissimi ingegni non gli hanno ancora interamente penetrati con l'investigazioni continuate per migliaia e migliaia d'anni” (from the Letter to Christina). 15 See e.g. F. Sabatini and V. Coletti (eds.), Dizionario della lingua italiana (Milan 2008), s.v. ipotesi.

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term ‘supposizione’ (from Latin suppositio, which literally has the analogous meaning of ‘putting

under’). This paper will argue that, in both his early and his mature works, Galileo not only

oscillates is in his use of the Italian words “ipotesi” and “supposizione” (with its Latin equivalent

suppositio), but that - at the same time - he also operates with (at least) three distinct concepts of

scientific hypothesis. One of his methodological notions of hypothesis is, as I intend to argue,

strikingly akin to the one employed in the ‘analytic method’ dear to the Platonic (and neo-

Platonic) tradition. Such different conceptualizations, as I will show, contribute to highlighting

different aspects of Galileo's methodology in ways that have not hitherto been adequately

explored.

‘It's only a hypothesis’

In the introductory essay to Copernicus' De revolutionibus, Osiander wrote that his “new

hypotheses” are no less probable than the ancient (i.e. Ptolemaic) ones, and that nobody should

“expect anything certain from astronomical models, which cannot furnish it, lest he accept as the

truth ideas conceived for another purpose, and depart from this study a greater fool than when he

entered it”. 16 Copernicus' reasoning, in other words, is presented by Osiander as purely

hypothetical: the heliocentric - or, more precisely, geokinetic - theory is based on the argument

that, if we assume the earth's motion around the sun, then the observed phenomena are explained;

to infer from this that the earth does indeed move would amount to a logical fallacy, known as

affirmatio consequentis.17 Galileo's preface to the Dialogue Concerning the Two Chief World

Systems (1632) describes the author's Copernican option along similar lines:

16 Trans. Rosen 1978: xx. 17 Galileo's first discussion of Copernican astronomy is found in a letter to Jacopo Mazzoni, a senior colleague at the University of Pisa, dated 30 May 1597. Mazzoni himself had just published a book critically discussing and comparing Plato and Aristotle (and taking an anti-Aristotelian view on the motion and speed of falling bodies), but also containing anti-Copernican arguments: Galileo congratulates his friend, then embarks on a lengthy refutation of Mazzoni's anti-Copernicanism (Finocchiaro 2010: 47). For


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Dialogo dei massimi sistemi: al lettore A questo fine ho presa nel discorso la parte Copernicana, procedendo in pura ipotesi

matematica, cercando per ogni strada artifiziosa di rappresentarla superiore, non a quella della fermezza della Terra assolutamente, ma secondo che si difende da alcuni che, di professione Peripatetici, ne ritengono solo il nome, contenti, senza passeggio, di adorar l'ombre, non filosofando con l'avvertenza propria, ma con solo la memoria di quattro principii mal intesi. (Galilei 1963: 8)

To this end I have taken the Copernican side in the discourse, proceeding as with a pure mathematical hypothesis and striving by every artifice to represent it as superior to supposing the earth motionless - not, indeed, absolutely, but as against the arguments of some professed Peripatetics. These men, indeed, only deserve the bare name, although they do not even walk about; they are content to adore the shadows, philosophizing not with due circumspection but merely from having memorized a few ill-understood principles. (Drake 2001: 5-6, modified)

Galileo's allegiance to ‘the Copernican side’ is stated as clearly as his hostility to the

‘Aristotelians’: the latter, instead of ‘walking about’ (= περιπατεῖν) in a spirit of genuine research,

remain content with mere ‘shadows’ (the image is perhaps reminiscent of Plato's cave). At the

same time, however, Galileo calls his line of inquiry a ‘pure mathematical hypothesis’, which he is

going to defend in all sorts of ‘artificial ways’. What exactly are these ‘artifices’? They are never

explicitly defined, but they do not seem to involve any ‘empirical proof’ in particular: a few lines

later, in fact, Galileo acknowledges the empirical equivalence of the Copernican and the Ptolemaic

hypothesis.

Dialogo dei massimi sistemi: al lettore Prima cercherò di mostrare, tutte le esperienze fattibili nella Terra essere mezi

insufficienti a concluder la sua mobilità, ma indifferentemente potersi adattare così alla Terra mobile, come anco quiescente; e spero che in questo caso si paleseranno molte osservazioni ignote all'antichità. Secondariamente si esamineranno li fenomeni celesti, rinforzando l'ipotesi copernicana come se assolutamente dovesse rimaner vittoriosa, aggiungendo nuove speculazioni, le quali però servano per facilità d'astronomia, non per necessità di natura.18

First, I shall try to show that all experiments practicable upon the earth are insufficient

measures for proving its mobility, since they are indifferently adaptable to an earth in motion or at rest. I hope in so doing to reveal many observations unknown to the ancients. Secondly, the celestial phenomena will be examined, strengthening the Copernican hypothesis until it might

Galileo's early interest in Plato and Aristotle (partly mediated by Mazzoni) and his juvenile commentaries on Aristotle's Physics, cf. also Heilbron 2010: 46-47. 18 Cf. also the following passage from the Second Day: “[Salv.] Il vedere se l'una e l'altra posizione [scil. il moto diurno esser della Terra sola <o> dell'universo, trattone la Terra] sodisfaccia egualmente bene [= aeque bene], si comprenderà da gli esami particolari dell'apparenze alle quali si ha da sodisfare, perché sin ora si è discorso, e si discorrerà, ex hypothesi, supponendo che quanto al sodisfare all'apparenze amendue le posizioni sieno egualmente accomodate” (Galilei 1963: 159).

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seem that this must triumph absolutely. Here new reflections are adjoined which might be used in order to simplify astronomy, though not because of any necessity imposed by nature. (Drake 2001: 6) Galileo stresses the importance of ‘simplicity’ in a scientific theory, which merely

strengthens its plausibility, but certainly does not derive from natural necessity: the aim of an

astronomical hypothesis is to ‘save the phenomena’. The main idea is, of course, an ancient one.

Ptolemy himself, in fact, makes important remarks on the empirical equivalence of competing

hypotheses in the first book of his main treatise, the Almagest.19 Even though some of the

phenomena recently observed on the Earth, Galileo claims, are ‘unknown to antiquity’, they are

still not sufficient to disprove one hypothesis in favor of the other.

He argues, however, that the importance of ‘celestial’ phenomena can be taken as far as to

‘reinforce’ the Copernican hypothesis as if it were to be conclusively demonstrated. I emphasized

the words as if (“come se”) because they can shed light on the concept of “ipotesi” that Galileo

employs in his preface. In his seminal book Philosophie des Als Ob, H. Vaihinger argued that

ancient Greek thinkers had a multifarious concept of hypothesis: for the Greeks, the word

ὑπόθεσις could mean either ‘foundational proposition’ (supporting something else), or

‘assumption’ (Lat. suppositio, as opposed to demonstratio or affirmatio), or even stylistic-

rhetorical ‘fiction’ (Lat. fictio).

Whereas Plato and Aristotle - according to Vaihinger - mainly employed various versions

of the first two types of ὑπόθεσις, the sceptics later brought the identification of hypothesis with

fiction to its extreme consequences.20 These issues need not be further explored here: let it suffice

for now to say that, for Vaihinger, when Plato and Aristotle employ the term ὑπόθεσις in the

sense of ‘fiction’, they only do so in order to introduce a reductio ad impossibile, in the course of

19 Cf. Ptol., Almag. 1.7 (H21-H24): “One can show by the same arguments as the preceding that the earth cannot have any motion in the aforementioned directions, or indeed ever move at all from its position at the centre. For the same phenomena would result as would if it had any position other than that the central one” (trans. Toomer). 20 Vaihinger 1911: 248.


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which a certain proposition is ‘fictionally’ (or even ‘rhetorically’) assumed to be true, in order to

show the absurd consequences that follow therefrom.21

In Galileo's preface to the Dialogue, the idea of ‘hypothesis’ as ‘fiction’ appears to play a

major role, insofar as the alleged ‘fictionality’ of the Copernican model enables him not to defend

it in terms of reality, which would have explicitly violated the Diktat established by the

Congregation of the Index in 1616. Bellarmine, for example, had already formulated the principle

of ‘purely hypothetical’ reasoning in his (in)famous letter to Foscarini, dated April 12, 1615:

Primo, dico che V. P. et il Sig.r Galileo facciano prudentemente a contentarsi di parlare ex suppositione e non assolutamente, come io ho sempre creduto che habbia parlato il Copernico. Perché il dire, che supposto che la Terra si muova e il Sole sia fermo si salvano tutte le apparenze meglio che con porre gli eccentrici et epicicli, è benissimo detto, e non ha pericolo nessuno; e questo basta al mathematico: ma volere affermare che realmente il Sole stia nel centro del mondo e solo si rivolti in sé stesso senza correre dall'oriente all'occidente, e che la Terra stia nel terzo cielo e giri con somma velocità intorno al Sole, è cosa molto pericolosa non solo d'irritare i filosofi e theologi scolastici, ma anco di nuocere alla Santa Fede con rendere false le Scritture Sante [...].

First, I say that it seems to me that Your Paternity and Mr. Galileo are proceeding

prudently by limiting yourselves to speaking suppositionally and not absolutely, as I have always believed that Copernicus spoke. For there is no danger in saying that, by assuming the earth moves and the sun stands still, one saves all the appearances better than by postulating eccentrics and epicycles; and that is sufficient for the mathematician. However, it is different to want to affirm that in reality the sun is at the center of the world and only turns on itself without moving from east to west, and the earth is in the third heaven and revolves with great speed around the sun; this is a very dangerous thing, likely not only to irritate all scholastic philosophers and theologians, but also to harm the Holy Faith by rendering Holy Scripture false [...]. (cit. and trans. Finocchiaro 2008: 146) Bellarmine appears to attribute Osiander's position to Copernicus himself. As long as the

Earth is hypothetically supposed to be in motion, Bellarmine says, ‘nobody is in danger’ and the

‘mathematician’ is satisfied; however, to assert that the Earth really moves around the Sun would

amount to a dangerous falsification of the Scriptures. This argument seems to imply that, should

observational experience and necessary demonstration ever prove conclusively that the Earth

moves and the Sun is at rest, the Catholic Church itself would have to rethink more cautiously the

21 Vaihinger 1911: 242-243.

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traditional interpretation of the seemingly opposed passages of the Scriptures. 22 In fact,

Bellarmine will later thaw to admit that such a possibility needs to be allowed for, should a

conclusive and necessary demonstration of the geokinetic theory arise.23 Does Galileo reach such a

binding demonstration of the Copernican hypothesis?

M. Finocchiaro (2010) divides the development of Galileo's defense of Copernicanism

into three periods: a pre-telescopic stage, marked by indirect or implicit pursuit (in which his

judgment is based on the theory's progressiveness, problem solving success in dynamics, and

explanatory coherence in astronomy); then a full-blown middle period (1609-1616), characterized

by explicit or qualified acceptance on mainly tentative or practical grounds (such as empirical

accuracy, e.g. in the case of the ‘Medicean stars’); finally, a problematic post-1616 stage, during

which the issue is complicated by the intervention of the Catholic Church, and which is

correspondingly dominated by the relationship of astronomy with religious beliefs

(Copernicanism must be proved with a conclusive, necessary demonstration, not just on

hypothetical grounds: otherwise, the Church will continue to ban it).24

Now, the post-1616 ecclesiastical caveat provides precisely the context in which the

preface of Galileo's Dialogue has to be read and understood. In its last page, the merely ‘fictional’

status of mathematical hypothesis is affirmed with a ‘self-deprecatory’ tone that is simply too

forceful not to sound ironical:

Dialogo dei massimi sistemi: al lettore

22 Cf. the same Letter to Foscarini, a few lines below: “Third, I say that if there were a true demonstration that the sun is at the center of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary, and say rather that we do not understand them than that what is demonstrated is false. But I will not believe that there is such a demonstration, until it is shown to me. Nor is it the same to demonstrate that by assuming the sun to be at the center and the earth in heaven one can save the appearances, and to demonstrate that in truth the sun is at the center and the earth in heaven” (trans. Finocchiaro 2008: 147). 23 Heilbron 2010: 213. Cf. similarly Barberini's somewhat clumsy effort to persuade Galileo of the purely hypothetical status of astronomical claims to natural knowledge (Heilbron 2010: 222). 24 Finocchiaro 2010: 63.


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Spero che da queste considerazioni il mondo conoscerà, che se le altre nazioni hanno navigato più, noi non abbiamo speculato meno, e che il rimettersi ad asserir la fermezza della Terra, e prender il contrario solamente per capriccio matematico, non nasce da non aver contezza di quant’altri ci abbia pensato, ma, quando altro non fusse, da quelle ragioni che la pietà, la religione, il conoscimento della divina onnipotenza, e la coscienza della debolezza dell'ingegno umano, ci somministrano.

I hope that from these considerations the world will come to know that if other nations

have navigated more, we have not theorized less. It is not from failing to take count of what others have thought that we have yielded to asserting that the earth is motionless, and holding the contrary to be a mere mathematical caprice, but (if for nothing else) for those reasons that are supplied by piety, religion, the knowledge of Divine Omnipotence, and a consciousness of the limitations of the human mind. (Drake 2001: 6)

Passages such as this one are evidently not to be taken at face value, especially considering

Galileo's use of colloquial and quasi-comic language (“per capriccio matematico”).25 The use of

“ipotesi” in the introductory letter of the Dialogue is ironically (and cautiously) derogatory, in

that it implies the arbitrary or ‘capricious’ ficticiousness of the Copernican theory or model. A

hypothesis, in this case, is a ‘fiction’ insofar as its explanatory value is confined to the realm of

abstract mathematical reasoning. As Osiander already did, Galileo states that Copernicus' theory

is a mere mathematical hypothesis, which religious and theological reasons do not allow to take

too seriously. This claim, nevertheless, is precisely what the Dialogue is ultimately meant to refute:

Bellarmine and his colleagues should instead read the preface, and be content with it.

Aristotelian assumptions

Why does science need hypotheses? The Peripatetic tradition grounded the substantial

distinction between ‘demonstrative’ and ‘non-demonstrative’ knowledge on the fact that, whereas

the former deals with ‘what cannot be otherwise’ (τὸ μὴ ἐνδεχόμενον ἄλλως ἔχειν), the latter is

concerned with what is subject to change: that is, with ‘what is capable of being otherwise’ (τὸ

25 For a similar style, cf. the finale of the Fourth Day: “[Salv.] Credo veramente che l'imaginazion vostra, più che la nostra tardanza, abbia allungato il tempo; e per non lo prolungar più, sarà bene che, senza interporre altre parole, venghiamo al fatto, e mostriamo come la natura ha permesso (o sia che la cosa in rei veritate stia così, o pur per ischerzo e quasi per pigliarsi giuoco de’ nostri ghiribizzi), ha, dico, permesso, che i movimenti, per ogni altro rispetto che per soddisfare al flusso e reflusso del mare, attribuiti gran tempo fa alla Terra, si trovino ora tanto aggiustamente servire alla causa di quello [...]” (Galilei 1963: 511).

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ἐνδεχόμενον ἄλλως ἔχειν). Consider, for instance, this assessment of scientific knowledge and its

principles offered by Aristotle in the Nicomachean Ethics:26

Aristot., EN 6.1140b31-1141a8 Since scientific knowledge is a mode of conception concerning the universal and what

exists by necessity, and demonstrated conclusions, like all scientific knowledge, proceed from first principles -for science involves reasoning-, it follows that there is no science, nor craft, nor prudence of the first principles of what is scientifically known. In fact, objects of scientific knowledge are demonstrable, while craft and prudence concern things that allow for change. [The knowledge] of first principles is not a matter of wisdom, either: for the wise man has to reach some conclusions by way of demonstration. Thus, if the faculties whereby we arrive at truth and never slip into falsehood, be it about things that allow for change or about invariable things, are scientific knowledge [episteme], prudence [phronesis], wisdom [sophia], and intellect [nous], and if none of these three - I mean prudence, scientific knowledge, and wisdom - allows us to apprehend the first principles, it remains that first principles must be apprehended by intellect [nous].27

In order to qualify as ‘demonstrative science’, knowledge (ἐπιστήμη) has to fulfil all three

senses of ἀπόδειξις, i.e. to prove, to explain, and to teach.28 This is why science, in the Aristotelian

epistemology, is defined as ‘necessary knowledge through causes’ (cognitio certa per causas): even

when its reasonings are phrased in a conditional form, they are still apodeictic (or

‘demonstrative’) insofar as their logical conclusions are based upon necessary or evident

premisses.29 Is it possible to have necessary, demonstrative, and therefore scientific knowledge of

natural phenomena, which are contingent and variable by definition? Far from being radically

anti-Aristotelian, Galileo's scientific methodology is largely based on an ideal of science (scientia)

as demonstration, even though it is complicated by oscillating views on how to define this

demonstrative character itself.30

The main source for Aristotle's conception of demonstrative science is his Organon. In

particular, the Prior Analytics contain an elaborate theorization of hypothetical argument per

impossibile, which aims at demonstrating the contradictory of the initial supposition by showing

the absurd consequences deriving from the assumption of the initial proposition itself as true.

26 Cf. also De an. 433a30, Met. 1015a34, EN 1140b22. 27 Translation mine. 28 Cf. e.g. Aristot., An. Post. 71b17 ff.; 99b15-19. 29 On knowledge ex condicione in the later Aristotelian tradition, see notably Crombie 1953: 268. 30 See notably Wallace 1974: 89 and McMullin 1978: 211.


13

Aristot., An. Pr. 1.23, 41a21-38 It is evident, then, that the ostensive syllogisms come to their conclusion in the

aforementioned figures. That this is so also for the arguments that lead to the impossible will be clear from the following. All those who reach a conclusion through the impossible deduce the falsehood by a syllogism, but prove the initial thesis from a hypothesis, when something impossible results from the assumption of the contradictory. For example, one proves that the diagonal [of the square] is incommensurable [with the side] because odd numbers turn out to be equal to even ones if one assumes that it is commensurable. Now that odd numbers turn out to be equal to even ones is deduced by syllogism, but that the diagonal is incommensurable is proved from a hypothesis, since a falsehood results because of its contradictory. For this is what was meant by ‘deducing through the impossible’, namely showing that something impossible follows because of the initial hypothesis. Thus, since there is an ostensive syllogism for the falsehood in arguments that lead to the impossible while the initial thesis is proved from a hypothesis, and since we said before that ostensive syllogisms come to a conclusion through those figures, it is evident that syllogisms through the impossible will also be in those figures. And the same holds for all other arguments from a hypothesis, for in all of them the syllogism is for the substituted proposition, while the initial thesis is reached through an agreement or some other kind of hypothesis. And if this is true, it is necessary that every demonstration and every syllogism come about through the three aforementioned figures. But once this has been proved, it is clear that every syllogism is perfected through the first figure and is reduced to the universal syllogisms in this figure. (trans. Striker 2009: 38, modified) Which proposition is the hypothesis that explains Aristotle's classification of this type of

reasoning as ‘arguments from a hypothesis’? Ancient and modern commentators are divided: is it

the contradictory of the demonstrandum (Alexander of Aphrodisias), or the denial of the

impossible conclusion (Mignucci), or the logical rule used in the step from the impossibility of the

first conclusion to the assertion of the demonstrandum (Ross)? In the latter case, the hypothesis is

a logical law that is evidently valid.

More generally, however, Aristotle conceives of arguments ‘from a hypothesis’ as based

on a proposition that is accepted as true by an explicit agreement (ὁμολογία) between the

διαλεγόμενοι.31 Why does Aristotle choose not to deal with hypothetical arguments as a whole,

and just focuses on reductio ad impossibile instead? Perhaps this is due to the fact that his readers

would have been already familiar with Plato's hypothetical arguments (cf. e.g. Meno 86e ff.),

whereas the classification of reductio ad impossibile under the same category is Aristotle's own

innovation.32 Let it suffice, for our purposes, to say that Galileo's conception of ex suppositione

31 Striker 2009: 177. 32 Striker 2009: 174; cf. similarly An. Pr. 1.44, 50a35 ff.

Platonic Hypotheses

14

arguments is ostensibly influenced by Aristotle's application of hypothetical reasoning to

arguments per impossibile alone.

Let us now turn our attention to the discussion of natural motion at the beginning of the

First Day of the Dialogue Concerning the Two Chief World Systems:

Dialogo dei massimi sistemi: giornata prima [Salv.] Se i corpi integrali del mondo devono esser di lor natura mobili, è impossibile che i

movimenti loro siano retti, o altri che circolari: e la ragione è assai facile e manifesta. Imperocché quello che si muove di moto retto, muta luogo; e continuando di muoversi, si va più e più sempre allontanando dal termine ond’ei si partì e da tutti i luoghi per i quali successivamente ei va passando; e se tal moto naturalmente se gli conviene, adunque egli da principio non era nel luogo suo naturale, e però non erano le parti del mondo con ordine perfetto disposte: ma noi supponghiamo, quelle esser perfettamente ordinate: adunque, come tali, è impossibile che abbiano da natura di mutar luogo, ed in conseguenza di muoversi di moto retto. (Galilei 1963: 25)

If all integral bodies in the world are by nature movable, it is impossible that their motions should be straight, or anything else but circular; and the reason is very plain and obvious. For whatever moves straight changes place and, continuing to move, goes ever farther from its starting point and from every place through which it successively passes. If that were the motion which naturally suited it, then at the beginning it was not in its natural place. So then the parts of the world were not disposed in perfect order. But we are assuming them to be perfectly in order; and in that case, it is impossible that it should be their nature to change place, and consequently to move in a straight line. (Drake 2001: 21, modified)

Filippo Salviati, Galileo's spokesman, denies that integral bodies are endowed by nature

with rectilinear motion: in so doing, he employs the Aristotelian language of reductio ad

impossibile. The proposition ‘all parts of the universe are disposed in perfect order’ (or the like) is

assumed to be true as a preliminary ‘supposition’, and it thereby contributes to prove the

‘impossibility’ (that is, the falsehood) of the original hypothesis - namely of the proposition ‘the

motions of integral bodies is rectilinear’, or the like - once the consequences of the latter are

shown to contradict the supposition concerning the perfect order of the universe. The argument,

in other words, aims at proving that integral bodies do not move in a straight line, i.e. at

demonstrating the contradictory of the hypothesis. Appearing at the outset of the First Day, this


15

argument could be read as a rhetorical move on Salviati's part, striving to beat the Peripatetics at

their own game.33

It is worth noting, in fact, that the language of ex suppositione reasoning is strikingly

shared by both the ‘Galilean’ character of the Dialogue (Salviati) and the ‘Aristotelian’ character

(Simplicio). Let us consider, for instance, this passage from the Fourth Day:

Dialogo dei massimi sistemi: giornata quarta [Simpl.] Non mi par che si possa negare che il discorso fatto da voi proceda molto

probabilmente, argumentando, come noi dichiamo, ex suppositione, cioè posto che la Terra si muova de i due movimenti attribuitigli dal Copernico: ma quando si escludano tali movimenti, il tutto resta vano ed invalido; l'esclusion poi di tale ipotesi ci viene dall'istesso vostro discorso assai manifestamente additata. Voi con la supposizion de i due movimenti terrestri rendete ragione del flusso e reflusso, ed all'incontro, circolarmente discorrendo, dal flusso e reflusso traete l'indizio e la confermazione di quei medesimi movimenti: e passando a più specifico discorso, dite che l'acqua per esser corpo fluido, e non tenacemente annesso alla Terra, non è costretta ad ubbidir puntualmente ad ogni suo movimento, dal che inducete poi il suo flusso e reflusso. (Galilei 1963: 536)

I do not think it can be denied that your argument goes along very plausibly, the reasoning being ex suppositione, as we say; that is, assuming that the earth does move in the two motions assigned to it by Copernicus. But if we exclude these movements, all the rest is vain and invalid; and the exclusion of this hypothesis is very clearly pointed out to us by your own reasoning. Under the assumption of the two terrestrial movements, you give reasons for the ebbing and flowing; and vice versa, arguing circularly, you draw from the ebbing and flowing the sign and confirmation of those same two movements. Passing to a more specific argument, you say that on account of the water being a fluid body and not firmly attached to the earth, it is not rigorously constrained to obey all the earth's movement. From this you deduce its ebbing and flowing. (Drake 2001: 506-507)

Simplicio starts by approvingly classifying Salviati's discourse as a type of ex suppositione

argument. What does he mean by ex suppositione in this context? If we follow Simplicio in

representing the argument in a propositional form, ‘if p then q’, we can see that p stands for a

mathematical hypothesis (= the earth's movement as posited by Copernicus) and q stands for

observable phenomena (= the tides). Simplicio, in other words, construes Salviati's argument as a

form of ‘hypothetico-deductive’ reasoning, which he then labels as ‘circular’ by charging it with

affirmatio consequentis.

33 As for this dialectical strategy, we shall see that it is employed by Galileo in other parts of the Dialogue, too.

Platonic Hypotheses

16

More importantly, Simplicio's attempt at categorizing Salviati's proof as hypothetico-

deductive reasoning implicitly places it outside of the domain of science, which needs to be - in an

Aristotelian perspective - strictly demonstrative, in order to be qualified as such. On the contrary,

hypothetico-deductive arguments are eminently non-demonstrative, since they are based on non-

evident (and therefore non-necessary) hypotheses: in fact, any hypothetico-deductive process

proceeding from a hypothesized beginning to a necessarily verified conclusion cannot ipso facto

transform its hypotheses into non-hypothetical truths.34

Now, the traditional procedure of ex suppositione reasoning - as it was understood in the

Aristotelian epistemology of Galileo's contemporaries - is a conditional argument of the form ‘if p

then q’, where, however, p stands for an inductive (i.e. non-necessary) generalization deriving

from the observation of nature, and q is a (non-observable) antecedent cause or condition

necessary to produce it, as in the modus ponendo ponens of scholastic syllogistics.35 This way of

reasoning is also exemplified in Galileo's writings; yet Galileo gives it a particular turn which is

extremely influential in his scientific methodology. Let us consider, for example, the following

exchange between Simplicio and Salviati in the First Day of the Dialogue:

Dialogo dei massimi sistemi: giornata prima [Simplicio] Aristotile fece il principal suo fondamento sul discorso a priori, mostrando la

necessità dell'inalterabilità del cielo per i suoi principii naturali, manifesti e chiari; e la medesima stabilì doppo a posteriori, per il senso e per le tradizioni degli antichi.

[Salviati] Cotesto, che voi dite, è il metodo col quale egli ha scritta la sua dottrina, ma non credo già che e’ sia quello col quale egli la investigò, perché io tengo per fermo ch’e’ proccurasse prima, per via de’ sensi, dell’esperienze e delle osservazioni, di assicurarsi quanto fusse possibile della conclusione, e che doppo andasse ricercando i mezi da poterla dimostrare, perché così si fa per lo più nelle scienze dimostrative: e questo avviene perché, quando la conclusione è vera, servendosi del metodo resolutivo, agevolmente si incontra qualche proposizione già dimostrata, o si arriva a qualche principio per sé noto; ma se la conclusione sia falsa, si può procedere in infinito senza incontrar mai verità alcuna conosciuta, se già altri non incontrasse alcun impossibile o assurdo manifesto. (Galilei 1963: 64-65)

“[Simplicio] Aristotle first laid the basis of his argument a priori, showing the necessity of

the inalterability of heaven by means of natural, evident, and clear principles. He afterward supported the same a posteriori, by the senses and by the traditions of the ancients.

34 See e.g. McTighe 1968: 371. 35 Wallace 1974: 95.


17

[Salviati] What you refer to is the method he uses in writing his doctrine, but I do not believe it to be that with which he investigated it. Rather, I think it certain that he first obtained it by means of the senses, experiments, and observations, to assure himself as much as possible of his conclusions. Afterward he sought means to make them demonstrable. That is what is done for the most part in the demonstrative sciences; this comes about because when the conclusion is true, one may by making use of the analytic method hit upon some proposition which is already demonstrated, or arrive at some principle known in itself; but if the conclusion is false, one can go on forever without ever finding any known truth - if indeed one does not encounter some impossibility or manifest absurdity. (Drake 2001: 57-58, modified) Even though Salviati might prima facie appear to advocate a form of hypothetico-

deductive method, the type of reasoning proposed here is eminently hypothetico-conditional: a

conclusion, in fact, is first ‘hypothesized’ on the grounds of observational experience, then

demonstrated by way of successive logical implications leading to an already known or

demonstrated proposition. The latter part is what Salviati calls ‘analytic method’ (= “metodo

resolutivo”). In this respect, Galileo's debt to his Renaissance predecessors and contemporaries

has been thoroughly investigated by historians of early modern science.36

Yet the hallmark of Galileo's original approach to Renaissance Aristotelianism lies, in my

view, precisely in the careful effort to distinguish the way in which Aristotle wrote and ‘presented’

his doctrine from the way in which he actually went about ‘investigating’ it. In the Dialogue,

moreover, he seems to attribute to Aristotle what is in fact Salviati's (= his own) method of

scientific inquiry. The language of the passage here considered well illustrates how Galileo

remains consistent in adopting Aristotelian ‘idioms’ (cf. for instance “scienze dimostrative”,

“verità alcuna conosciuta”) in order to ‘Aristotelianize’ his own epistemological position, whose

Platonic elements I am about to investigate.

Platonic analysis

To be sure, an earlier version of the ‘analytic method’ mentioned by Salviati had already

been outlined, before Aristotle, by Plato's Socrates in the Phaedo and elsewhere. The Phaedo, in

36 See McTighe 1968 and Jardine 1976: 306; cf. also Morrison 1997: 16-22.

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particular, has long been acknowledged to offer the best and most complete account of the

method of hypothesis in Plato.37 At Phaed. 100a3-8, for instance, Socrates sets out to tackle the

problem of the immortality of the soul, and he explains that all his inquiries concerning

philosophical matters proceed from an initial assumption, or hypothesis, of whose truth he is

already assured. In fact, he says, the starting hypothesis must be a proposition or a principle

deemed to be ‘the strongest’:

καὶ ὑποθέμενος ἑκάστοτε λόγον ὃν ἂν κρίνω ἐρρωμενέστατον εἶναι, ἃ μὲν ἄν μοι δοκῇ τούτῳ συμφωνεῖν τίθημι ὡς ἀληθῆ ὄντα, καὶ περὶ αἰτίας καὶ περὶ τῶν ἄλλων ἁπάντων ὄντων, ἃ δ’ ἂν μή, ὡς οὐκ ἀληθῆ. βούλομαι δέ σοι σαφέστερον εἰπεῖν ἃ λέγω· οἶμαι γάρ σε νῦν οὐ μανθάνειν.

I assume [hypothémenos] in each case a principle that I deem to be the strongest [erromenéstaton], and I regard as true whatever seems to me to agree with it, whether concerning the cause or anything else, and whatever disagrees with it, I consider as untrue. But I want to explain you more clearly what I mean: in fact, I think that you do not understand now.38

supponensque rationem sempre, quam esse judico validissimam, quaecunque huic consonare videantur, pono equidem tanquam vera; idque ago et circa rerum causas et circa reliqua omnia: quae vero dissonant, vera esse nego. Volo equidem, quae dico, tibi apertius explanare. puto enim te nondum intelligere. (Ficino)

A few lines below, at 101c9-102a1, Socrates makes a further statement concerning the

hypothetical method, urging Cebes not to answer objections raised against the hypothesis itself

“until you have examined the [logical] consequences of the hypotheses, to see if they agree or

disagree with each other” (some hypotheses are indeed stronger than others, as Simmias points

out at 92d2-e2).39 Socrates also asserts that a hypothesis may be corroborated by being deduced

from a “higher” proposition (Phaedo 101d5-e3). Thus, the initial hypothesis acts both as a premiss

37 Cf. e.g. Cellucci 2012: 57. 38 Translation mine. 39 Whether the meaning of ‘accord and disaccord’ in the whole section refers to logical deducibility or to internal consistency is not relevant for my present purposes: for the problem cf. Robinson 19532; Rowe 1993; Newton Byrd 2007; Benson 2015: 136. Dancy 2004: 297 offers a good attempt at overcoming the antinomy between ‘logical entailment’ and ‘logical consistency’ by simply acknowledging that Plato had never done any formal logic: rather, according to Dancy (2004: 298), one should construe symphonein as referring to a loose form of ‘enthymematic entailment’, where hypothesized premises are needed to let the inference proceed, as well as a general notion of consistency whereby the further claims to which the hypothesis gives rise are jointly defensible. Thus, Socrates’ ‘concordant with’ means neither ‘entailed by’ nor ‘consistent with’, but something much vaguer.


19

for causal-explanatory reasoning and as a conclusion that must itself be proved - or refuted - on

the basis of higher principles (cf. Phaedo 101e, 107b).

Whether Galileo read it in Sebastiano Erizzo’s 1574 Italian edition or in Marsilio Ficino’s

1484 Latin version (more probably the latter),40 there is no reason to doubt that he was familiar

with Plato’s Phaedo. Galileo's analytic method can, on the one hand, be regarded as consonant

with Plato's insofar as both specify that the researcher must have a ‘preliminary’ grasp of the

validity and soundness of the hypothesis, before being able to build upon it and to define the

truth-value of further propositions as depending on whether they agree or disagree with the initial

hypothesis. On the other hand, Plato's Socrates - unlike Galileo's Salviati - never makes clear in

the Phaedo what the hypothesis itself must eventually receive logical confirmation from.

Some light might be shed on this issue by a passage in the well-known ‘Divided Line’

section of Plato's Republic.

Plato, Resp. 510b2-511a1 Σκόπει δὴ αὖ καὶ τὴν τοῦ νοητοῦ τομὴν ᾗ τμητέον. Πῇ; Ἧι τὸ μὲν αὐτοῦ τοῖς τότε μιμηθεῖσιν ὡς εἰκόσιν χρωμένη ψυχὴ ζητεῖν ἀναγκάζεται ἐξ ὑποθέσεων, οὐκ ἐπ’ ἀρχὴν πορευομένη ἀλλ’ ἐπὶ τελευτήν, τὸ δ’ αὖ ἕτερον—τὸ ἐπ’ ἀρχὴν ἀνυπόθετον—ἐξ ὑποθέσεως ἰοῦσα καὶ ἄνευ τῶν περὶ ἐκεῖνο εἰκόνων, αὐτοῖς εἴδεσι δι’ αὐτῶν τὴν μέθοδον ποιουμένη. Ταῦτ’, ἔφη, ἃ λέγεις, οὐχ ἱκανῶς ἔμαθον. Ἀλλ’ αὖθις, ἦν δ’ ἐγώ· ῥᾷον γὰρ τούτων προειρημένων μαθήσῃ. οἶμαι γάρ σε εἰδέναι ὅτι οἱ περὶ τὰς γεωμετρίας τε καὶ λογισμοὺς καὶ τὰ τοιαῦτα πραγματευόμενοι, ὑποθέμενοι τό τε περιττὸν καὶ τὸ ἄρτιον καὶ τὰ σχήματα καὶ γωνιῶν τριττὰ εἴδη καὶ ἄλλα τούτων ἀδελφὰ καθ’ ἑκάστην μέθοδον ταῦτα μὲν ὡς εἰδότες, ποιησάμενοι ὑποθέσεις αὐτά, οὐδένα λόγον οὔτε αὑτοῖς οὔτε ἄλλοις ἔτι ἀξιοῦσι περὶ αὐτῶν διδόναι ὡς παντὶ φανερῶν, ἐκ τούτων δ’ ἀρχόμενοι τὰ λοιπὰ ἤδη διεξιόντες τελευτῶσιν ὁμολογουμένως ἐπὶ τοῦτο οὗ ἂν ἐπὶ σκέψιν ὁρμήσωσι. Πάνυ μὲν οὖν, ἔφη, τοῦτό γε οἶδα. Οὐκοῦν καὶ ὅτι τοῖς ὁρωμένοις εἴδεσι προσχρῶνται καὶ τοὺς λόγους περὶ αὐτῶν ποιοῦνται, οὐ περὶ τούτων διανοούμενοι, ἀλλ’ ἐκείνων πέρι οἷς ταῦτα ἔοικε, τοῦ τετραγώνου αὐτοῦ ἕνεκα τοὺς λόγους ποιούμενοι καὶ διαμέτρου αὐτῆς, ἀλλ’ οὐ ταύτης ἣν γράφουσιν, καὶ τἆλλα οὕτως, αὐτὰ μὲν ταῦτα ἃ πλάττουσίν τε καὶ γράφουσιν, ὧν καὶ σκιαὶ καὶ ἐν ὕδασιν εἰκόνες εἰσίν, τούτοις μὲν ὡς εἰκόσιν αὖ χρώμενοι, ζητοῦντες δὲ αὐτὰ ἐκεῖνα ἰδεῖν ἃ οὐκ ἂν ἄλλως ἴδοι τις ἢ τῇ διανοίᾳ.

Now consider again in what way the section of the intelligible is to be divided. What way is that? In the one section, the soul, using as images what was imitated at previous stages, is

forced to investigate from hypotheses, not proceeding toward a first principle but toward a

40 See Favaro 1886: 244-245.

Platonic Hypotheses

20

conclusion; but in the other section, the soul moves from the hypothesis toward an unhypothesized beginning which transcends hypotheses, without making any use of the images employed in the previous section, relying solely on forms and progressing systematically through forms.

I do not really understand what you mean, he said, but tell me again. I will, said I. I think you will understand more readily after a few preliminary remarks. I

think you know that those who study geometry and arithmetic and similar subjects postulate the odd and the even, geometrical figures and the three kinds of angles, and other relationships of this sort according to each system of inquiry. So, taking these things as known, they make them their hypotheses and do not think it worth their while to provide any justification for them to themselves or others, on the grounds that they are evident to everyone. And starting from these, they go on through the remaining steps and end up in agreement at the point they set out to reach in their investigation.

Yes, of course, I know all that! So you'll also know that they make use of the visible forms as well and make their

arguments about them, although considering not the actual things, but those they resemble, making their arguments on the basis of the square itself and the diagonal itself, but not the line they are drawing, and similarly with everything else. These very things they are forming and drawing, of which shadows and reflections in waters are images, they now in turn use as their images and aiming to see those very things which they could not otherwise see except in thought.41

Plato's Socrates distinguishes between two types of hypothetical reasoning. The first one,

in which the soul makes use of images belonging to the visible world and proceeds from

hypothesis to a conclusion, is assimilated to geometrical reasoning; the second one, in which the

mind grasps an unhypothesized first principle by virtue of reason itself, is equated with dialectic.

Geometers, in fact, use postulates, axioms, and hypotheses as cornerstones for their arguments,

just assuming their validity without proof, since they hold them to be self-evident. Mathematical

reasoning, in other words, is unable to escape from the non-demonstrability of its hypotheses,

which it cannot transcend.

The dialectician, on the other hand, makes use of ‘genuine’ hypotheses as ‘steps and

starting points’ (511b3-4), only to transcend them and to reach an unhypothetical first principle

(ἀρχὴ ἀνυπόθετος): once such a principle has been grasped, the argument can move backwards

again, eventually comes down to a conclusion that is consistent with the first principle. In

performing this upward-then-downward movement, the dialectician does not need to rely on

sense perception or visible images at all.

41 Trans. Emlyn-Jones and Preddy 2013: 2.99-100 (modified).


21

If we now compare Galileo's ‘analytic method’, as outlined by Salviati in the First Day of

the Dialogue, with Plato's ‘dialectic method’, as outlined by Socrates in the sixth book of the

Republic, we can notice that both aim at reaching a non-hypothetical proposition, which can

contribute (by way of successive logical implications) to demonstrate the conclusion (hypothesis)

from which the reasoning started.42 This ‘unhypothesized’ proposition, however, need certainly

not be ‘self-evident’ for Galileo, let alone an extra-dianoetic principle ‘transcending’ the

hypotheses themselves.43 Salviati, in fact, makes it clear that the first principle reached by the

analytic method can be an ‘already demonstrated’ proposition, or a truth independently known.

It has been suggested44 that one likely source of Galileo's association of the best method of

research with geometrical demonstration is Pappus, Collectio Mathematica.45 Galileo probably

knew Commandino's translation of Pappus,46 where resolutio is systematically used for the Greek

term ἀνάλυσις. In antiquity, Plato himself was often credited with the invention of the geometric

method of analysis.47 The latter, however, is now largely considered to have been common

currency among Greek mathematicians of Plato’s own time,48 and it has been convincingly shown

42 In his juvenile writings (especially the Additamenta), Galileo calls ‘resolution ex suppositione’ a type of reasoning in which “a conclusion is resolved to principles that have been supposed, but which need not have taken the form of suppositions because they are capable of proof; then the resolution continues until any further suppositions that might be required for their proof are uncovered, and each of these is proved in turn, until one comes finally to principles that are most easily grasped and that require no supposition whatever” (Wallace 1984: 120). 43 Cf. e.g. McTighe 1968: 371. 44 Jardine 1976. 45 Ed. F. Hultsch, Berlin 1876-78, vol. II, 635-36. 46 Pesaro, 1588: 157r-v. 47 Cf. e.g. Diog. Laert. 3.24; see further Sayre 1969: 23. 48 Cf. especially Meno 86d-87c. A scholion to the fifth book of Euclid’s Elements attributes the content of the book itself, which mainly concerns the axiomatic construction of a theory of proportions, to Eudoxus of Cnidus, whose work is said to have been contemporaneous with Plato’s (D32 Lasserre). Cf. also Procl., In Eucl. El. 67.2-8 Friedlein, where Eudoxus is credited with an increase in the number of ‘general’ (καθόλου) theorems. At any rate, it is more plausible that Plato was referring to a ‘general tendency’ in Greek mathematics rather than to a specific author, such as Eudoxus himself.

Platonic Hypotheses

22

that Plato’s use of geometrical analysis as a heuristic method is primarily meant to provide a foil

or model for philosophical inquiry.49

The philosopher, for Plato, is meant to follow the geometers’ example in proceeding from

logically posterior to logically prior propositions, so as to discover the principles suitable to

answering conclusively a certain question. In philosophical terms, Galileo's reminiscence of

Plato’s analytic method in the Dialogue might be part of a larger strategy, aiming at highlighting

the superiority of geometrical demonstration over syllogistic, Aristotelian regressus.50 Galileo, in

fact, seems to resist the syllogistic methodology of Zabarella and his contemporaries, while

revitalizing a distinctively Platonic tradition of analytic reasoning, according to which the

hypothetical method is endowed with a scientifically heuristic value (rather than just being part of

a system of logic or a theory of causation).

To sum up, for Salviati, the natural scientist is required to have a preliminary grasp of the

soundness of a hypothesis, before being able to embark on a full-fledged demonstration and

define the truth-value of further propositions based on whether they logically agree or disagree

with the hypothesis itself. The latter, in turn, must be supported by already demonstrated

propositions, self-evident general principles, or statements independently known to be true.

Unlike Plato, Salviati does not hint at the notion of a universal, all-encompassing, ἀρχή that

transcends the dianoetic level of reasoning and ultimately serves to unify all sciences:51 in this

respect, Salviati's conception of ‘first principles’ distances itself from the Platonic view, but his

notion of hypothesis is remarkably indebted to the Platonic tradition.

Even before the Dialogue, Galileo had already approached the issue of hypotheses and

conclusions in the brief, witty treatise The Assayer (1623). Among the numerous objections to his

scientific procedures that Galileo has to confront, one point of ‘concern’ is constituted by his use

49 See e.g. Menn 2002: 216. For late-antique notions of analysis and method, cf. notably Morrison 1997. 50 Cf. Jardine 1976: 307. 51 Cf. e.g. An. Post. 76a16-18.


23

of new technological instruments. In fact, some opponents argued, Galileo would never have

achieved his most important astronomical discoveries without the help of the telescope:

Il Saggiatore, 13.29-30 Ma forse alcuno mi potrebbe dire, che di non piccolo aiuto è al ritrovamento e risoluzion

d'alcun problema l'esser prima in qualche modo reso consapevole della verità della conclusione, e sicuro di non cercar l'impossibile, e che perciò l'avviso e la certezza che l'occhiale era di già stato fatto mi fusse d'aiuto tale, che per avventura senza quello non l'avrei ritrovato. A questo io rispondo distinguendo, e dico che l'aiuto recatomi dall'avviso svegliò la volontà ad applicarvi il pensiero, che senza quello può esser ch’io mai non v’avessi pensato; ma che, oltre a questo, tale avviso possa agevolar l'invenzione, io non lo credo: e dico di più, che il ritrovar la risoluzione d’un problema segnato e nominato, è opera di maggiore ingegno assai che ’l ritrovarne uno non pensato né nominato, perché in questo può aver grandissima parte il caso, ma quello è tutto opera del discorso. (Galilei 2005: 156)

Perhaps someone will say, however, that in the discovery and solution of a problem it is of

no little assistance first to be conscious in some way that the conclusion is true and to be certain that one is not attempting the impossible; and hence that my knowledge and certainty that a telescope had already been made were of so much help to me that without this I should perhaps not have made the discovery. To this I shall reply by making a distinction. I say that the aid afforded me by the news awoke in me the will to apply my mind to it, and without this I might never have thought about it; but beyond that I do not believe that such news could facilitate the invention. I say, moreover, that to discover the solution of a known and designated problem is a labor of much greater ingenuity than to solve a problem which has not been thought of and defined, for luck may play a large role in the latter while the former is entirely a work of reasoning. (trans. Drake 1960: 212)

Albeit highly polemical and rhetorical, Galileo's self-defense tackles a crucial issue of

epistemology, namely the role played by technological progress in a scientist's work. His

argument draws a clear distinction between the role of the telescope as an incentive, or ‘stimulus’,

to apply his thought to an astronomical problem (which he acknowledges), and the centrality of

the telescope in actually bringing about the new astronomical discoveries themselves (which he

altogether denies).

In fact, for Galileo, even before starting to look for a solution to any given scientific

problem, the researcher must be ‘aware, in some way,’ of the validity of the conclusion that still

needs to be proved. To be sure, new technologies can greatly help the scientist to find solutions to

already known and familiar problems: but it is much more difficult to solve a time-honored

Platonic Hypotheses

24

problem than to ‘invent’ another one from scratch, since only the former can be adequately

described as being a purely ‘discursive’ matter.52

It ought to be noted that, about ten years before the Dialogue, Galileo is already

convinced that, in any matter of scientific reasoning, the validity of the conclusions must at least

be preliminarly shown to be plausible before they can start being demonstrated: this is what

Galileo considers, as we have seen, the hallmark of ‘demonstrative science’. In casting his

epistemic procedures as ‘demonstrative science’, Galileo continues to ‘Aristotelianize’ his

language: a terminological and conceptual palimpsest which was perhaps facilitated, in the case of

his conception of hypothesis, by Ficino’s and Erizzo’s translation of Plato's ὑπόθεσις as suppositio

and “supposizione” respectively. A comparison between the above passage from The Assayer and

Salviati's reply to Simplicio in the First Day of the Dialogue (quoted above) can readily show how

this idea is a constant element in Galileo's epistemological thought, at least in its ‘mature’ phase.

More geometrico

Both Plato's dialectics and Aristotle's theory of first principles are heavily influenced by

mathematical - and specifically geometrical - ideas, as H.D.P. Lee has shown.53 For Aristotle, in

particular, first principles are true, indemonstrable, necessary, causal, and prior to the conclusions

drawn from them. Principles can be of three kinds: axioms, definitions, hypotheses. Each science

thus has a minimum of preliminary, indemonstrable assumptions that allow to deduce its

propositions and conclusions logically. Axioms (or ‘common notions’) are shared by more than

one science, and are presupposed before the reasoning process starts. Definitions, which are never

ontological, are meant to answer the question τί ἐστι about any object of a particular science,

whereas hypotheses assume the existence of the object in question (and are therefore ontological).

52 Or an inquiry μετὰ λόγου, in Aristotelian terms? 53 Lee 1935.


25

Aristotle's tripartite system can be mapped onto Euclid's three types of first principle (which are

all self-evident and assumed without proof): namely, common notions (κοιναὶ ἔννοιαι),

definitions (ὅροι), and postulates (αἰτήματα).54

In the Posterior Analytics, Aristotle draws an important distinction between hypotheses,

which are assumed without proof despite their being provable, and postulates, which he considers

as illegitimate insofar as they are assumed without the preliminary consent of both the

participants to the discussion. In fact, the very existence of hypotheses and postulates presupposes

a dialogue between a teacher and a learner:

Aristot., An. Post. 1.10, 76b23-34 That which necessarily is per se and which must necessary be held as existing is neither a

hypothesis nor a[n illegitimate] postulate. In fact, demonstration is not directed to external reasoning, but to the [internal] one that is in the soul, since this is true of any syllogism. For it is always possible to raise objections to external reasoning, whereas this is not always true of the internal one. On the one hand, those propositions are hypothesized which, despite being provable, are assumed [by the teacher] without proof, if the learner believes them; and they are not hypotheses simpliciter, but only in relation to the learner. On the other hand, those propositions are [illegitimate] postulates which [the teacher] assumes without the learner having an opinion on them, or even if the learner has a contrary opinion. And this is the difference between hypothesis and [illegitimate] postulate: the latter is the opposite of the learner's opinion, [whereas the former is] demonstrable, but assumed and used without proof.55

Aristotle's conception of postulates, as outlined in this passage, is very different from that

of Euclidean mathematics: this might be due to the fact that what we call ‘Euclid's geometry’ had

not yet been systematized into a single chain of deduction from first principles at the time in

which the Posterior Analytics were written. 56 On the other hand, the Aristotelian idea of

hypothetical reasoning, in the Posterior Analytics, is fundamentally comparable with the outline

of geometrical arguments offered by Plato in the Republic, except for the fact that Aristotle

distinguished between hypotheses simpliciter and those that - despite being provable - are

54 In fact, Aristotle’s views on demonstration show the familiarity with geometric analysis in early Academic circles: see Menn 2002: 209. 55 Translation mine. 56 Lee 1935: 117.

Platonic Hypotheses

26

assumed without proof, whereas Plato conceives of geometrical hypotheses as unproven

propositions par excellence (cf. Aristotle's ‘postulates’).

In fact, a major point of ‘agreement’ between the two thinkers - at least in the domain of

mathematics - lies in their similarly ‘sceptical’ judgment concerning preliminary assumptions

made without proof:57 a substantial hint at the fact that both Plato and Aristotle share a common

image of geometrical reasoning. 58 At any rate, as already observed, they both consider

mathematics as a blueprint for the main characteristics of any science claiming to be

‘demonstrative’: an idea that Galileo, who doubtlessly conceives of mathematics as a ‘Euclidean’ -

i.e. systematic - discipline, is hardly willing to abandon. Consider, for instance, the following

passage from the Two New Sciences (1638):

Discorsi e dimostrazioni matematiche intorno a due nuove scienze, 3.197 Le proprietà del moto equabile sono state considerate nel libro precedente: ora dobbiamo

trattare del moto accelerato. E in primo luogo conviene investigare e spiegare la definizione che corrisponde esattamente al moto accelerato di cui si serve la natura. Infatti, sebbene sia lecito immaginare arbitrariamente qualche forma di moto e contemplare le proprietà che ne conseguono (così, infatti, coloro che si immaginarono linee spirali o concoidi, originate da certi movimenti, ne hanno lodevolmente dimostrate le proprietà argomentando ex suppositione, anche se di tali movimenti non usi la natura), tuttavia, dal momento che la natura si serve di una certa forma di accelerazione nei gravi discendenti, abbiamo stabilito di studiarne le proprietà, posto che la definizione che daremo del nostro moto accelerato abbia a corrispondere con l'essenza del moto naturalmente accelerato.

The properties belonging to uniform motion have been discussed in the preceding

section; but accelerated motion remains to be considered. And first of all it seems desirable to find and explain a definition best fitting natural phenomena. For anyone may invent an arbitrary type of motion and discuss its properties; thus, for instance, some have imagined helices and conchoids as described by certain motions which are not met with in nature, and have very commendably established the properties which these curves possess, arguing on the basis of their hypothesis; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. (trans. Crew 1991: 160, modified)

Strikingly enough, the ‘first principles’ (cf. “in primo luogo”) of Galileo's treatment of

uniformly accelerated motion are not hypotheses, but ‘definitions’. We have already seen that,

57 For Plato's ‘scepticism’ concerning hypothetical propositions assumed without any test of their validity, see e.g. Robinson 1980: 147. 58 It is also important to note that Aristotle's conception of mathematics is largely based on the (quasi-Platonic) idea of geometry as the study of universals, which are separable in thought from tangible matter, but result from the combination of geometric properties and intelligible matter (see Mueller 1979: 105).


27

besides hypotheses and axioms, Aristotle similarly classifies ‘definitions’ as one of the possible

‘first principles’ of scientific reasoning: in fact, at An. Post. 75b31, he asserts that a definition

(ὁρισμός) can be the ἀρχή of a demonstration.59 Yet Galileo's account includes a further proviso:

in order for a demonstration to be valid (in the domain of mechanics), the preliminary definitions

must ‘correspond exactly’ to the phenomena observed in nature.60

A mathematician is always allowed, in fact, to ‘hypothesize’ arbitrary definitions (which

do not correspond to any phenomenon empirically observable in nature) and logically deduce

certain properties therefrom:61 but this type of ex suppositione argument cannot, of course, lead to

any scientific theory capable of ‘agreeing’ with the phenomena. The idea that definitions must

‘save the phenomena’ might prima facie seem to be consistent with the Aristotelian theory (An.

Post. 100b2 ff.) whereby induction, based on experience deriving from the accumulation of

perceptions and memories, can lead to the formation of first principles.62 Now, Galileo is no

59 Galileo's knowledge of Aristotle's Posterior Analytics is well documented: as a young man, Galileo wrote a commentary on Aristotle's Posterior Analytics, entitled Disputationes de praecognitione et demonstratione (cf. McMullin 1978: 217). 60 The need for a careful assessment of the validity of the intial hypothesis is expressed by Plato's Socrates at Crat. 436c8-d7: “if the giver [of names] made a mistake in the first place and then distorted the rest to meet it and compelled them to accord with him, it would not be at all surprising, just as in diagrams sometimes, when a slight and inconspicuous mistake is made in the first place, all the huge mass of consequences agree with each other. It is about the beginning of every matter that every man must make his big discussion and his big inquiry, to see whether it is rightly laid down (ὑπόκειται) or not; and only when that has been adequately examined should he see whether the rest appear to follow from it” (trans. Robinson 1980: 147). 61 For Galileo's language, cf. the well-known chapter 15 of Machiavelli's Principe (referring to Plato's ‘imaginary’ Republic): “sendo l'intento mio scrivere cosa utile a chi la intende, mi è parso più conveniente andare drieto alla verità effettuale della cosa, che alla immaginazione di essa. E molti si sono immaginati repubbliche e principati che non si sono mai visti né conosciuti essere in vero; perché elli è tanto discosto da come si vive a come si doverrebbe vivere, che colui che lascia quello che si fa per quello che si doverrebbe fare, impara più tosto la ruina che la perservazione sua [...]”. It is likely that Galileo has Archimedes in mind when referring to those who “have imagined helices and conchoids”, in the same way as Machiavelli alludes to Plato. 62 Cf. also Ptol., Almag. 9.2 (H212): “[...] and we know too that assumptions made without proof, provided only that they are found to be in agreement with the phenomena, could not have been found without some careful methodological procedure, even if it is difficult to explain how one came to conceive them (for, in general, the cause of first principles is, by nature, either non-existent or hard to describe); we know, finally, that some variety in the type of hypotheses associated with the circles [of the planets] cannot be plausibly considered strange or contrary to reason (especially since the phenomena exhibited by the actual planets are not alike [for all]); for, when uniform circular motion is preserved for all without exception, the individual

Platonic Hypotheses

28

inductivist: for him, the hypothesized principles of a demonstrative science must have the status

of necessary premisses. 63 But how can a definition, or any hypothesized proposition, be

‘necessarily’ valid? This is left unexplained by Galileo. And even assuming that the initial

definition does necessarily ‘match’ nature's phenomena, are the conclusions logically drawn from

it not in need of empirical verification?

An important page written by Galileo shortly after the publication of his Two New

Sciences, namely the letter to Baliani, can shed some light on this issue. In the letter, Galileo seems

to argue that, once a correct definition of accelerated motion (i.e. one corresponding to the

motion actually observed in nature) is established, the mathematical properties of motion

deduced from it need not be empirically verified. He claims to be reasoning ex suppositione about

accelerated motion defined in such a way that, “even though the consequences might not

correspond to the properties of natural motion of falling heavy bodies, it would little matter to

me, just as the inability to find in nature any body that moves along a spiral line would take

nothing away from Archimedes' demonstration”.64

In fact, he goes on to assert that he has been ‘lucky’ (“avventurato”) in observing that the

properties of falling heavy bodies correspond “puntualmente” to the properties demonstrated on

the basis of his definition of uniformly accelerated motion.65 For Galileo, it seems, nature's

phenomena are demonstrated in accordance with a principle which is more basic and more generally applicable than that of similarity of the hypotheses [for all planets]” (trans. Toomer). 63 See McMullin 1978: 235. 64 Cit. and trans. in Wallace 1974: 94. The main ‘Archimedean’ aspect of Galileo's method lies, according to Wallace, in his proclamation of the power of mathematics in physical explanation: Wallace sees the notion of ex suppositione as straddling the boundaries between physics and mathematics (Wallace 1974: 99; see contra McMullin 1978: 234). 65 If natural phenomena ‘fail’ to correspond to the expected properties, a ‘Platonist’ scientist may well make appeal to the ‘recalcitrant’ and ‘imperfect’ behavior of matter (see e.g. Molland 1976: 37; McMullin 1978: 230; Wallace 1984: 304). Cf. also Ptol., Almag. 13.2, H2.532: “For once each of the phenomena is preserved in accordance with the hypotheses, why should anyone think it strange that such complications can come about in the motions of heavenly things when they do not have a nature that produces hindrance, but one that is adapted to yield and give way to the natural motions of each, even if they are opposed to one another? Thus, simply, all the masses can easily pass through and be seen through all others, and this ease of transit applies not only to the individual circles, but to the spheres themselves and their axes of revolution.


29

intrinsic intelligibility is consequent upon its essentially geometrical structure, which makes

experimental verification almost entirely superfluous (according to the letter to Baliani).66 To sum

up, in Galileo's method, the resolutive or ‘analytic’ moment - which is based on the analysis of an

experiential datum into its geometrical structure in order to reach a self-evident or per se notum

principle - must be followed by composition or synthesis, which involves the mathematical

deduction of further theorems from such a principle, without the need of empirical verification

thereof.

Several interpreters of the letter to Baliani, and most notably W. Wallace (1974) have

suggested the possibility of a different interpretation: in denying the need for ‘saving the

phenomena’ (if not by a ‘stroke of luck’), Galileo might be rhetorically adopting the persona

loquens of the Archimedean mathematician, just as in the preface to the Dialogue he defended, for

rhetorical purposes, the ‘Osianderian’ conception of the Copernican theory as a ‘mere

mathematical hypothesis’. Nevertheless, in earlier works Galileo similarly argues that geometrical

demonstrations proceeding from well-grounded definitions need not be empirically confirmed.

In fact, experience and experiment as regulative elements are almost entirely absent from Galileo’s

early writings,67 which are dominated by a marked confidence in the scientist’s ability to reach

solid first principles legitimizing a physical theory.

In order to defend the intellectual priority of his Two New Sciences over Baliani's treatise

De motu naturali gravium solidorum, which independently reached the same conclusions,68

We see that in the models (εἰκόσιν) constructed on earth the intertwining and combination of these same [elements] in the different motions is laborious, and difficult to achieve in such a way that the motions do not hinder each other, while in the heavens no obstruction whatever is caused by such combination [...]” (trans. Toomer, modified). 66 See McTighe 1968: 374-375. For a similar point, cf. the letter sent by Evangelista Torricelli (one of Galileo’s pupils) to M. Ricci on Feb. 10, 1646: “[...] che i principi della dottrina de motu siano veri o falsi a me importa pochissimo. Poiché se non sono veri, fingasi che sian veri, conforme habbiamo supposto, e poi prendansi tutte le altre specolazioni derivate da essi principi, non come cose miste, ma pure geometriche” (cit. in Torrini 1993: 241). 67 See Shea 1977: 7. 68 See E. Grillo in Dizionario Biografico degli Italiani (vol. 5, Rome 1963), s.v. Baliani.

Platonic Hypotheses

30

Galileo advocates for the autonomous validity of mathematical reasoning with respect to

empirical data. While the letter to Baliani certainly does not prove that Galileo regarded

experimental confirmation as being completely extraneous to his ex suppositione arguments, these

still remain solidly founded in logical demonstration proceeding from hypothesized ‘definitions’

(“posto che...”), and can therefore be construed as hypothetical arguments in conditional form.69

In guise of a conclusion

Galileo's use of the words (and concepts of) “ipotesi” and suppositio does not authorize us

to interpret his epistemological statements in terms of the modern hypothetico-deductive

method. Nor does he appear to have ‘one’ consistent view on the role of hypotheses and first

principles in natural science: indeed, the very range of variation examined in this paper does not

justify any particular ‘label’ (e.g. ‘Platonic’, ‘Aristotelian’, ‘Archimedean’, etc.) for his conception

of science as a whole. For rhetorical reasons, or for the purposes of self-fashioning, Galileo

sometimes adopts Osiander's ‘reductive’ idea of astronomical models as mere mathematical

hypotheses, by no means aimed at explaining the phenomena and at laying bare the language in

which the ‘book of the universe’ is written.

Elsewhere, in a narrower sense, he refers to the ‘analytic’ method whereby an ‘intuitive

grasp’ (based on observation) of the validity of the hypothesized conclusion leads, by way of

subsequent propositions logically implying one another, to its ultimate confirmation. This

strengthens the view that Galileo strove to attribute to physics (and mechanics in particular) the

69 In fact, Galileo assimilates definitions to “supposizioni” (= hypotheses) in an early treatise, Le Meccaniche: “Quello che in tutte le scienze demostrative è necessario di osservarsi, doviamo noi ancora in questo trattato seguitare: che è di proporre le deffinizioni dei termini proprii di questa facultà e le prime supposizioni dalle quali, come da fecondissimi semi, pullulano e scaturiscono consequentemente le cause e le vere demostrazioni delle proprietà di tutti gl’instrumenti mecanici, i quali servono per lo più intorno ai moti delle cose gravi. Però determineremo primamente quello che sia gravità” (Galilei 2002: 48).


31

fully demonstrative status of a cognitio certa per causas.70 Galileo's effort to turn mechanics into a

demonstrative science is further shown by his - occasionally extreme - application of the language

of mathematics (geometry) to hypothesized ‘definitions’ and ‘demonstrations’ concerning the

properties of accelerated motion, even though Galileo himself could have purposefully

‘exaggerated’ his rhetorical denial of the need for empirical verification at the end of the

demonstrative process.

In this paper, I have aimed to show the deep consonance between Plato’s hypothetical

method (especially as it is described in the Phaedo) and the “metodo resolutivo” of Galileo’s

Dialogue. To that end, I have argued that, in both cases, the hypothesis is simultaneously a

premiss, of whose force the researcher is convinced, and a demonstrandum awaiting proof on the

basis of higher, i.e. already known or proven, principles. In fact, I identify this conception of the

analytic method, whereby the scientist is required to have a preliminary grasp of the soundness of

the hypotheses themselves, as one of the main Platonic elements in Galileo’s scientific

methodology. Furthermore, both Plato’s and Galileo’s hypotheses are causal-explanatory in

nature: as such, they are the fundamental component of a heuristic process. Finally, the Platonic

character of Galileo’s “metodo resolutivo” is deliberately and rhetorically clothed in ‘Aristotelian’

terms, familiar to the epistemology of Galileo’s contemporaries: on several occasions, his use of

‘Platonic’ hypotheses confirms the priority he assigns to geometrical reasoning over empirical

proof.

However, even restricting our focus to any one of Galileo's ‘mature’ works, we notice that

none of his conceptions of hypothetical reasoning is employed consistently throughout. This can

be probably explained by considering the tension, constant in Galileo's epistemology, between the

demonstrative ideal that he inherited from the Greek tradition and the ‘retroductive’ or ‘analytic’

method, exemplified in discussions of phenomena whose causes are remote, enigmatic, or 70 See McMullin 1978: 250.

Platonic Hypotheses

32

invisible. Galileo was required - not just by the Congregation of the Index, but by his own

fundamental conception of ‘firm demonstration’ - to produce a necessary and demonstrative

account of his astronomical tenets, based on the reduction of the complexity of observable natural

phenomena to evident or certifiable first principles through the use of mathematical reasoning. In

the (ultimately impossible) attempt to do so, he contributed to founding an entirely different, and

eminently modern, methodology of science.

Cambridge, Massachusetts

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FINOCCHIARO 2010: M.A. Finocchiaro, Defending Copernicus and Galileo, Dordrecht GALILEI 1963: G. Galilei, Opere, vol. 2, ed. Franz Brunetti, Turin GALILEI 2002: G. Galilei, Le meccaniche, ed. R. Gatto, Florence GALILEI 2005: G. Galilei, Il Saggiatore, ed. O. Besomi and M. Helbing, Rome GEYMONAT 1957: L. Geymonat, Galileo Galilei, Turin (trans. S. Drake, New York 1965) HANKINS 2000: J. Hankins, Galileo, Ficino and Renaissance Platonism, in J. Kraye and M.W.F. Stone (eds.), Humanism and Early Modern Philosophy, London, 209-237 HANKINS 2004: J. Hankins, Humanism and Platonism in the Italian Renaissance, vol. 2, Rome HATFIELD 2004: G. Hatfield, Metaphysics and the New Science, Cambridge HEILBRON 2010: J.L. Heilbron, Galileo, Oxford JARDINE 1976: N. Jardine, Galileo's Road to Truth and the Demonstrative Regress, «Studies in the History and Philosophy of Science» 7, 277-318 KOYRÉ 1943: A. Koyré, Galileo and Plato, «Journal of the History of Ideas» 5, 400-428 LEE 1935: H.D.P. Lee, Geometrical Method and Aristotle's Account of First Principles, «CQ» 29, 113-124 MCMULLIN 1978: E. McMullin, The Conception of Science in Galileo's Work, in R.E. Butts and J.C. Pitt (eds.), New Perspectives on Galileo, Dordrecht, 209-257 MCTIGHE 1968: T.P. McTighe, Galileo's Platonism: A Reconsideration, in Galileo. Man of Science (cit.), 365-387 MENN 2002: S. Menn, Plato and the Method of Analysis, «Phronesis» 47, 193-223 MORRISON 1997: D.R. Morrison, Philoponus and Simplicius on Tekmeriodic Proof, in E. Kessler et al. (ed.), Method and Order in Renaissance Philosophy of Nature. The Aristotle Commentary Tradition, Farnham, 1-22 MUELLER 1979: I. Mueller, Aristotle on Geometrical Objects, in J. Barnes, M. Schofield, and R. Sorabji (eds.), Articles on Aristotle, London, 96-107 NEWTON BYRD 2007: M. Newton Byrd, Dialectic and Plato’s Method of Hypothesis, «Apeiron» 40, 141-158 ROBINSON 19802: R. Robinson, Plato's Earlier Dialectic, New York ROSEN 1978: E. Rosen (ed. and trans.), Nicholas Copernicus. On the Revolutions, Baltimore ROWE 1993: C.J. Rowe, Explanation in Phaedo 99c6-102a8, «OSAPh» 11, 49-69 SAYRE 1969: K.M. Sayre, Plato’s Analytic Method, Chicago SHEA 1977: W.R. Shea, Galileo’s Intellectual Revolution, New York STRIKER 2009: G. Striker (ed.), Aristotle. Prior Analytics, Book I, Oxford TORRINI 1993: M. Torrini, “Galileo, Platone e la filosofia,” in P. Prini (ed.), Il Neoplatonismo nel Rinascimento, Rome, 233-245 VAIHINGER 1911: H. Vaihinger, Philosophie des Als Ob, Berlin VEBLEN 1903: O. Veblen, Hilbert's Foundations of Geometry, «The Monist» 13, 303-309 WALLACE 1974: W.A. Wallace, Galileo and Reasoning ex suppositione: the Methodology of the Two New Sciences, «Boston Studies in the Philosophy of Science» 32, 79-104 WALLACE 1984: W.A. Wallace, Galileo and His Sources: Heritage of the Collegio Romano in Galileo's Science, Princeton

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IN BREVE. La questione delle origini antiche dell’epistemologia di Galileo continua ad

interessare filosofi e storici della scienza (cfr. Cellucci 2012, De Caro 2012, Finocchiaro 2010, Hatfield 2004). In particolare, il contrasto tra quanti considerano Galileo un ‘platonico’ (cfr. Koyré 1943) e quanti lo ritengono un ‘aristotelico’ (cfr. McMullin 1978, Wallace 1984) sembra dominare la storia degli studi almeno dalla metà del Novecento. Lo scopo della mia comunicazione è quello di mostrare importanti, seppur finora trascurate, correlazioni tra il “metodo resolutivo” adottato da Galileo nelle sue opere maggiori e il metodo ipotetico delineato da Platone nel Fedone.

È da tempo riconosciuto che il Fedone offre la versione più compiuta e dettagliata del metodo ipotetico nell’opera di Platone. In 100a3-8, Socrate introduce la dimostrazione dell’immortalità dell’anima con una discussione metodologica volta a chiarire come ogni indagine filosofica debba basarsi su di una ὑπόθεσις iniziale, considerata come la più cogente (ἐρρωμενέστατον), dalla quale si parte per considerare “come vero tutto ciò che si accorda con essa, e come falso tutto ciò che non le si accorda.”

Poco dopo, in 101c9-102a1, Socrate precisa che un’eventuale obiezione mossa contro l’ipotesi stessa non va confutata prima di aver esaminato a fondo “le conseguenze [logiche] dell’ipotesi, per verificare se si accordano o meno fra loro.” Socrate asserisce, inoltre, che un’ipotesi viene corroborata qualora la si deduca da una proposizione “più alta” (101d5-e3). In tal modo, l’ipotesi costituisce sia la premessa del ragionamento sia una conclusione da dimostrare - o refutare - sulla base di principi superiori (101e, 107b).

Analogamente, Filippo Salviati (il portavoce di Galileo nel Dialogo sopra i due massimi sistemi del mondo, 1632) afferma, nella Prima Giornata del Dialogo, che il fisico deve possedere una comprensione preliminare della validità delle proprie ipotesi, prima di poterne dare una dimostrazione mediante l’analisi delle conseguenze logiche dell’ipotesi stessa. Quest’ultima, inoltre, va sostenuta attraverso il ricorso a proposizioni già dimostrate, principi auto-evidenti, o asserzioni indipendentemente note come vere (Galilei 2001: 57-8).

Questo procedimento, definito da Salviati “metodo resolutivo” (o analitico), era già diffuso nella trattatistica scientifica antica e tardo-antica, come testimonia soprattutto Pappo nella sua Collectio Mathematica (2.635-6 Hultsch). Non poche corrispondenze concettuali, tuttavia, permettono di far risalire il metodo analitico di Pappo, almeno in parte, alla matematica greca del tempo di Platone (cfr. Menone 86d-87b).

L’interesse galileiano verso la questione delle ipotesi e dei principi primi è già evidente in opere scientifiche precedenti al Dialogo, come ad esempio il Saggiatore (1623). In entrambi i contesti, il carattere ‘platonico’ del metodo ipotetico delineato da Galileo è confermato dal tono polemico con cui lo scienziato contrappone le proprie linee di ricerca alle argomentazioni ex suppositione della sillogistica neo-aristotelica, adottando peraltro, con fini retorici, stilemi del linguaggio aristotelico in molteplici occasioni.

Il mio contributo mira dunque a mostrare la profonda consonanza epistemologica tra il metodo ipotetico del Fedone e il “metodo resolutivo” del Dialogo galileiano. A tale scopo, mi propongo di esaminare i seguenti punti fondamentali: 1) l’ipotesi, per entrambi gli autori, è tanto una premessa quanto un demonstrandum da confermare sulla base di proposizioni già note; 2) per entrambi, l’ipotesi svolge una funzione causale-esplicativa; 3) la natura platonica del metodo analitico di Galileo è deliberatamente e polemicamente contrapposta all’epistemologia aristotelica dei suoi contemporanei.

platonic hypotheses: galileo’s “analytic method” and the...

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