`The Ontological and Epistemological Underpinnings of Galileo’s Mathematical Realism`

CARLA RITA PALMERINO

ON MAY 7, 1610, Galileo Galilei wrote to Belisario Vinta, Secretary of State of the Grand Duchy of Tuscany, about the terms of his future position as a court mathematician. In his letter Galileo expressed the wish that “His Majesty add the name of Philosopher to that of Mathematician,” motivating his request with the fact that he had “spent more years studying philosophy than months studying pure mathematics” (Galilei 1890–1909, 10:353).

The fact that Galileo spoke of “pure mathematics,” and not just of “mathematics,” is highly significant. In his view, to be a philosopher meant to be a mathematician, but one who was interested in discovering the real constitution of the physical world. In the Third Day of the Two New Sciences (1638) Galileo describes his approach to the study of accelerated motion, and explains in which sense the philosopher’s mathematical method differs from that of a pure mathematician:

Not that there is anything wrong with inventing at pleasure some kind of motion and theorizing about its consequent properties, in the way that some men have derived spiral and conchoidal lines from certain motions, though nature makes no use of these paths.… But since nature does employ a certain kind of acceleration for descending heavy things, we decided to look into their properties so that we might be sure that the definition of accelerated motion which we are about to adduce agrees with the essence of naturally accelerated motion (Galilei 1974, 153; 1890–1909, 8:197).1

Similarly, in his Dialogue (1632), Galileo contrasts the ambition of the mathematical astronomer, who contents himself with saving the phenomena, to that of the “astronomer philosopher,” who seeks “to investigate the true constitution of the universe—the most admirable problem that there is” (Galilei 1967, 341). For although whatever we read in the book of nature—a book that, as Galileo famously argues in the Assayer (1623), is written in the language of mathematics—“is the creation of the omnipotent Craftsman,… nevertheless that part is most suitable and most worthy which makes His works and His craftsmanship most evident to our view” (Galilei 1957, 3).2

But how is it possible for the natural philosopher to get access to the mathematical language of the book of nature and to find out which geometrical line or which mathematical formula corresponds to the essence of a particular physical phenomenon? As I shall try to show in this chapter, Galileo’s answer to this question was much more sophisticated than scholars generally assume. In addressing the question concerning the relation between mathematical and physical truths, Galileo carefully kept ontological considerations distinct from epistemological considerations. In his view, the fact that physical phenomena cannot always be translated into simple mathematical laws was not an argument against mathematical realism, but was only a sign that the mathematical order of nature is often too complex to be grasped by the human mind.

`IS THE BOOK OF NATURE REALLY WRITTEN IN THE LANGUAGE OF MATHEMATICS?`

In The Crisis of European Sciences and Transcendental Phenomenology (1970), Edmund Husserl devoted considerable attention to Galileo’s mathematization of nature. According to Husserl, Galileo took it for granted that geometry, which “produces a self-sufficient, absolute truth,” could be applied to nature “without further ado,” without reflecting “on how the free, imaginative variation of this world and its shapes results only in possible empirically intuitable shapes and not in exact shapes” (Husserl 1970, 49). In other words, Galileo overlooked the fact that concrete bodies are not “the geometrically pure shapes which can be drawn in ideal space,” but “are thinkable only in gradations: the more or less straight, flat, circular, etc.” (Husserl 1970, 25).

Whereas in Husserl’s eyes Galileo was a Pythagorean Platonist mathematician who performed a “surreptitious substitution” of the ideal objects of mathematics for the real objects of the physical world (see Soffer 1990; De Gandt 2004; Moran 2012), many scholars believe that Galileo cannot be considered a Platonist precisely because he regarded mathematical laws as idealizations that do not have a counterpart in the actual world. In a passage of the Two New Sciences, for example, Galileo explicitly admits that: “No firm science can be given of such accidents of heaviness, speed, and shape, which are variable in infinitely many ways. Hence to deal with such matters scientifically, it is necessary to abstract from them. We must find and demonstrate conclusions abstracted from the impediments, in order to make use of them in practice under those limitations that experience will teach us” (1974, 225).

According to Noretta Koertge’s influential interpretation, Galileo’s concern with the “problem of accidents”—a problem for which he developed increasingly sophisticated solutions in his writings—shows that the Assayer’s image of the universe as a book written in the language of geometry should not be “taken as a significant or careful statement of Galileo’s philosophical views” (Koertge 1977, 402). In her view, Galileo endorsed neither a Pythagorean ontology nor a Platonist epistemology. Likewise, Robert Butts perceived an inconsistency between Galileo’s practice of science and his professed mathematical realism, arguing that, “Galileo’s argument that mathematics applies to the world was more a metaphysical faith than a philosophically established conclusion. He seems to have concluded that if the world does not conform to the truths of mathematics, so much the worse for the world” (Butts 1978, 81). Similarly, in an article published in 1985, Ernan McMullin argued that Galileo “is not entirely single-minded” in maintaining the view that the effects of the physical impediments can be calculated. “He sometimes lapses back into a Platonic pessimism about the ‘imperfections of matter, which is subject to many variations and defects.’ … But if this were the case, the Book of Nature would not really be written in the language of mathematics, or would, at least, be poorly written” (McMullin 1985, 251). Finally, Maurice Finocchiaro has recently claimed that Galileo’s remark in the Assayer that the book of nature is written in mathematical language was “more a plea for independent-mindedness” than a statement of mathematical realism. Moreover, “if and to the extent that the remark on the book of nature … can be taken as an expression of mathematical realism or Platonism, it should be noted that the remark is an epistemological reflection, not an instance of concrete scientific practice, and one can raise the question whether Galileo’s words and deeds correspond” (Finocchiaro 2010, 115–16). In Finocchiaro’s view, they often don’t correspond.

In this chapter I shall try to show that Galileo’s claim that nature is written in the language of mathematics, far from being a rhetorical statement or an unwarranted metaphysical conviction, is grounded in coherent ontological and epistemological arguments. In his works Galileo repeatedly argues that mathematical entities are ontologically independent from us and that the physical world has a mathematical structure. This structure is, however, too complex to be fully grasped by our finite intellect, which is why we need to simplify physical phenomena in order to be able to deal with them mathematically. What scholars have regarded as an opposition between the abstract and the concrete, the mathematical and the physical, was intended by Galileo as a distinction between what is mathematically simple, and hence easy for our intellect to grasp, and what is mathematically complex and hence unknowable.

In the following pages I shall first focus on Galileo’s use of the metaphor of the book of nature, paying particular attention to his views concerning the different properties of verbal and mathematical language. Then I shall turn my attention to Galileo’s reflections on the relation between mathematical and physical truths which, as I shall try to demonstrate, are fully in accordance with his scientific practice.

`GALILEO ON VERBAL AND MATHEMATICAL LANGUAGE`

In his book on Nominalism and Constructivism in Seventeenth-Century Mathematical Philosophy, David Sepkoski observes that “the epistemology of mathematization is fundamentally linked to the epistemology of language” (Sepkoski 2007, 2). Indeed, early modern authors such as Gassendi, Hobbes, Locke, and Berkeley adopted a nominalistic theory of language that also influenced their views concerning the relation between mathematics and the physical world. While Kepler and Galileo conceived of mathematics as the “language of nature,” these authors regarded it “as a ‘language’ for describing nature that was subject to the same epistemological conventions that govern the structure, objects, and claims to knowledge of natural languages” (125).

As I shall try to document in the following pages, Galileo’s works contain very important, albeit unsystematic, considerations concerning the conventional nature of verbal language, which display interesting analogies with those of his nominalist contemporaries. Contrary to the latter, however, Galileo was not willing to extend his conclusions to mathematical language. Rather, the chief function of Galileo’s use of the metaphor of the book of nature is precisely that of contrasting the exact and “obligatory” character of mathematical language to the imprecise and arbitrary character of verbal language.3

In the famous passage of the Assayer to which we have already referred, Galileo reminds his Jesuit opponent that philosophy is not

a book of fiction created by some man, like the Iliad or Orlando Furioso—books in which the least important thing is whether what is written in them is true. Well, 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 geometrical figures, without which it is humanly impossible to comprehend a single word of it. (1957, 3)

In other places Galileo uses the same topos to convey the originality of his approach to philosophy. While his Scholastic opponents spend their time commenting on Aristotle’s books and disputing ad utranque partem, he prefers to study “the book of nature, where things are written in one way only” (1890–1909, 248). Finally in the Copernican Letters, which represent the manifesto of his ideas concerning the relationship between revealed and physical truths, Galileo compares the book of nature, which God wrote at the moment of creation, to the book of Scripture, which he dictated to evangelists and prophets:

Holy Scripture and nature derive equally from the Godhead, the former as the dictation of the Holy Spirit, and the latter as the most obedient executrix of God’s orders; moreover, to accommodate the understanding of the common people it is appropriate for Scripture to say many things that are different (in appearance and in regard to the literal meaning of the words) from the absolute truth; on the other hand, nature is inexorable and immutable, never violates the terms of the laws imposed upon her, and does not care whether or not her recondite reasons and ways of operating are disclosed to human understanding. (Galilei 1989, 93)

As has been observed by Giorgio Stabile, in these lines Galileo relies on the medieval image of God’s two books, but attributes a greater binding force to the natural law (lex naturae) than to the divine law (lex divina). While the Bible must follow the logic of ordinary language, which is conventional and hence negotiable, the book of nature, being the reification of God’s word, is unmediated by human conventions, fixed and inviolable (Stabile 1994, 55–56). It is in this regard interesting to see that Galileo’s distinction between the respective status of verbal and mathematical languages also influences his judgment concerning the accessibility of God’s two books. Augustine, an author Galileo often quotes, claimed in his Enarrationes in Psalmos (XLV, 7), that while the pages of the Bible could only be enjoyed by those who know how to read, the book of the universe is accessible to everyone. Galileo believes, on the contrary, that the book of nature is more difficult to decipher than Scripture, because the latter is adjusted to the intellectual capacities of common people, whereas the former is not. As he explains in a letter to Fortunio Liceti in January 1641: “the book of philosophy is that which stands perpetually open before our eyes. But being written in characters different from those of our alphabet, it cannot be read by everybody; and the characters of this book are triangles, squares, circles, spheres, cones, pyramids and other mathematical figures, the most suited for this sort of reading” (1890–1909, 18:295).

As we have seen, Galileo contrasts the book of nature with works of fiction, which do not pretend to tell the truth, with Scholastic books, which tell questionable truths, and with scriptural books, which do tell the truth, but a truth that is often merely “adumbrated.”4 What these books have in common is the fact that they are written in a language that is ambiguous by its very nature.

Galileo’s considerations concerning natural language, which are scattered throughout his works, touch upon three main themes, notably the arbitrary character of names, the inconstancy of meanings, and the link between words and appearances. In fact, Galileo’s ideas on these subjects show an interesting resemblance with those contained in Locke’s Essay Concerning Human Understanding.

In the Third Book of the Essay, where he explains that words are made “arbitrarily the mark of an idea” by a “voluntary imposition,” Locke (1690) criticizes Scholastic philosophers for coining names such as “saxietas, metallietas, lignietas and the like …, which should pretend to signify the real essences of those substances whereof they knew they had no ideas” (3.2.1, 187; 3.8.2, 230).

The conventional character of verbal language was already emphasized by Galileo, often in the context of a critique of the essentialist definitions put forward by his Aristotelian opponents. In answering Ludovico delle Colombe, Galileo notices, for example, that “the explications of terms are free” (1890–1909, 4:632) and that the attribution of a name can hence never be mistaken. However, precisely because of their arbitrary character, words cannot reveal the essences of things. A similar point is made in the Letters on the Sunspots and in the Assayer, where Galileo takes issue with the scientific nomenclature used by his Aristotelian opponents. Welser is free to call the sunspots “stars” and Grassi may well refer to a comet as a “planet” provided they don’t pretend that their word choice can solve a dispute concerning the nature of celestial bodies. In the Third Letter on the Sunspots to Mark Welser, we read: “In truth, I am not insisting on nomenclature, for I know that everyone is free to adopt it as he sees fit. As long as people did not believe that this name conferred on them certain intrinsic and essential conditions … one might also call solar spots ‘stars,’ but they have, in essence, characteristics that differ considerably from those of actual stars” (Galilei and Scheiner 2010, 289).

And in the Assayer Galileo observes: “I am not so sure that in order to make a comet a quasi-planet, and as such to deck it out in the attributes of other planets, it is sufficient for Sarsi and his teacher to regard it as one and so name it. If their opinions and their choices have the power of calling into existence the things they name, then I beg them to do me the favor of naming a lot of old hardware I have about my house, ‘gold’ ” (1957, 253).

In the Dialogue Galileo also points out that verbal language is full of misleading synonymies, which can sometimes hinder the process of knowledge (1967, 403). When for example Simplicio voices his skepticism concerning Gilbert’s geomagnetic theory, Salviati asks himself whether it is not only because of “a single and arbitrary name” that his Aristotelian interlocutor is reluctant to accept the idea that the earth is a big lodestone. If our planet had not been called “earth,” a term which also signifies “that material which we plow and sow,” but rather “stone,” then “saying that its primary substance was stone would surely not have met resistance or contradiction from anybody.”

In the Essay Concerning Human Understanding, Locke was to regard the unsteady application of names as one of the great abuses of language. For although words are “intended for signs of my ideas, to make them known to others, not by any natural signification, but by a voluntary imposition, it is plain cheat and abuse, when I make them stand sometimes for one thing and sometimes for another” (1690, 3.10, 240–51). Galileo made a somewhat similar point when he observed that misunderstandings and errors do not originate from the “first definition” of a name, which being conventional can never be mistaken, but from the fact that “one doesn’t stick to the terms originally included in the definition, or forms different concepts of the defined thing” (1890–1909, 4:632). Galileo repeatedly accuses Scholastic authors of being incoherent in the application of terms that they themselves have coined. If, following Aristotle, one defines the term “place” as the “surface of the surrounding body,” then it makes no sense to inquire, as Ludovico delle Colombe does, whether the outermost heaven is in a place. Similarly, if one agrees with Aristotle that humid bodies are those that are not confined within limits of their own, but adapt to the form of their container (De gen. 2.2, 329b), then one must reach the conclusion that fire is humid (Galilei 1890–1909, 4:632–33). Disputes concerning the imposition of names are the business of grammarians, not of philosophers. The latter must however make sure that terms are not “first defined in one way, and then applied to scientific demonstrations in another” (Galilei 1890–1909, 4:698–700).

Another issue that is dear to Galileo and Locke alike is the relation between names and appearances. In the third book of his Essay, Locke introduces a famous distinction between the real essence and the nominal essence of things, which he anchors in the distinction, made in the second book, between the primary qualities of bodies (i.e., solidity, extension, motion or rest, number or figure) and their secondary qualities, which are the sensations produced in us by the primary qualities. Due to our “ignorance of the primary qualities of the insensible parts of bodies,” the real essence of a substance, which Locke identifies with its internal constitution, remains unknown to us, and we have “no name that is the sign of it” (1690, 4.3.12, 271; 3.3.18, 196). The abstract general ideas we form of substances “with names annexed to them, as patterns, or forms,” refer to their nominal essences, which is the collection of particular qualities that one observes together in a substance (3.3.13, 193). As Locke explicitly acknowledges, “the ideas that our complex ones of substances are made up of, and about which our knowledge concerning substances is most employed, are those of their secondary qualities.” (4.3.11, 271). Given “that there is no discoverable connexion between any secondary quality and those primary qualities,” the gap between the real and the nominal essence of substances remains unbridgeable (4.3.12, 271). According to Locke, the only category of objects for which nominal and real essences coincide is constituted by geometrical figures, the definition of which is “not only the abstract idea to which the general name is annexed, but the very essentia or being of the thing itself; that foundation from which all its properties flow, and to which they are all inseparably annexed” (3.3.18, 195).

The relation between names and things, attributes and essences is also addressed, though in an unsystematic way, in Galileo’s writings. In the Dialogue, Simplicio states with conviction that the cause of free fall is called “gravity.” Salviati reminds him that to know the name of a thing is not the same as to know its essence:

You are wrong, Simplicio; what you ought to say is that everyone knows that it is called “gravity.” What I am asking you for is not the name of the thing, but its essence, of which essence you know a bit more than you know about the essence of whatever moves the stars around. I except the name which has been attached to it and which has been made a familiar household word by the continual experience that we have of it daily. But we do not really understand what principle or what force it is that moves stones downward, any more than we understand what moves them upward after they leave the thrower’s hand, or what moves the moon around. We have merely, as I said, assigned to the first the more specific and definite name “gravity,” whereas to the second we assign the more general term “impressed force,” and to the last-named we give “spirits,” either “assisting” or “abiding”; and as the cause of infinite other motions we give “nature.” (1967, 234–35)

In the Letters on Sunspots (Galilei and Scheiner 2010, 91) Galileo claims that “names and attributes must accommodate themselves to the essence of the things, and not the essence to the names, because things come first and names afterwards.” It is however vain “to try and penetrate the true and intrinsic essence of natural substances,” which means that we have to content ourselves with definitions dependent on the perceived qualities of bodies:

If upon inquiring into the substance of clouds, I am told that it is a moist vapor, I will then wish to know what vapor is. Perhaps I will be informed that it is water, attenuated by virtue of warmth and thus dissolved into vapor, but being equally uncertain of what water is, I will in asking about this finally hear that it is that fluid body flowing in rivers that we constantly handle and use. But such information about water is merely closer and dependent on more [of our] senses, but not more intrinsic than [the information] I had earlier about clouds. (Galilei and Scheiner 2010, 254)

That information derived from our senses is not able to reveal the intrinsic essences of things, is something Galileo also claims in a famous passage of the Assayer that anticipates Locke’s distinction between primary and secondary qualities. When explaining that tastes, odors, and colors do not reside in the perceived objects, but only in the perceiving subject, Galileo declares that they are “mere names.” The fact that we have imposed upon sensory qualities “special names, distinct from those of the other and real qualities mentioned previously” (i.e., size, figure, quantity, motion, and rest) makes us wrongly believe that “they really exist as actually different from those” (1957, 274). What in fact happens is that our senses transliterate the mathematical language of the book of nature into the language of experience, which is riddled by synonymies and homonymies. The famous fable of sounds, told in the Assayer, is nothing other than a way of proving that “the bounty of nature in producing her effects” is such that “our senses and experience” sometimes judge as identical phenomena that are in fact produced by utterly different causes (258–59).

Although our definitions of natural substances are inevitably dependent on the senses, it is not a vain enterprise to try and provide an accurate analysis of some of their properties. In the case of sunspots, Galileo notices, for example, that attributes such as their “location, motion, shape, size, opacity, mutability, appearance, and disappearance” can “be learned by us and then serve as our means better to speculate upon other more controversial conditions of natural substances” (Galilei and Scheiner 2010, 255). Natural philosophers must hence draw their attention to the quantitative properties of bodies, which contrary to sensorial qualities are not “mere names,” but have an independent ontological status. While names have been arbitrarily imposed by men onto things, numbers and geometrical figures have been inscribed by God into things, and hence have the power of disclosing their essences.

On the basis of what is just said, it might appear strange to see Galileo argue, in the Two New Sciences, that mathematical definitions are “the mere imposition of names, or we might say abbreviations of speech,” which is exactly what many years before he had declared about verbal language.5 The arbitrary character of mathematical language resides, however, only in the choice of the signifier, not in that of the signified. As Galileo repeatedly explains, the language of mathematics is strict and unambiguous. Each sign carries one and only one meaning, and the propositions we can construct out of these signs can but be true or false. While verbal language is the language of persuasion, mathematical language is the language of certainty: “Geometrical things cannot be affected by cavils and paralogisms, as they are true in one way only, can be explained in one way only and intended in one way only” (1890–1909, 458).

The extent to which Galileo thinks that the language of the book of nature is accessible to the mathematical philosopher is a question that is addressed in the following section.

`DECODING THE BOOK OF NATURE`

Galileo explicitly acknowledges that mathematical laws are idealizations that do not exactly correspond to the behavior of physical bodies. This has led some scholars to regard the claim that the book of nature is written in the language of mathematics as a rhetorical statement, which is neither representative of Galileo’s philosophical views nor in accordance with his scientific practice.

As I have argued elsewhere (Palmerino 2006, 39), the fault with this interpretation is that it attributes an ontological meaning to considerations that are in fact epistemological. When Galileo claims, for example, that in the study of motion one must abstract from the “accidents of heaviness, speed, and shape, which are variable in infinitely many ways,” he just means that their random variations are too complex to be translated into a simple mathematical law. As Galileo explains in the Assayer

Regular lines are called those that, having a single, firm and determinate description, can be defined and whose accidents and properties can be demonstrated … But the irregular lines are those that, not having any determination whatsoever, are infinite and casual, and thus indefinable, and of which therefore no property can be demonstrated and nothing, in sum, can be known. To say that “this accident happens according to an irregular line” is for that reason the same as to say “I do not know why it happens.” (1890–1909, 458)

In Galileo’s eyes the problem is hence not that irregular lines (and physical accidents) are not mathematical, but rather that their mathematical structure is beyond the reach of our intellectual skills.

The issue of the relation between physical and mathematical truths is explicitly addressed by Galileo in the second day of the Dialogue (1967, 203). Having patiently listened to a long mathematical demonstration proposed by Salviati, the Aristotelian Simplicio objects that mathematical truths lose their validity when they are applied to physical matters. “A sphere touches a plane in one point” is, for example, a typical case of a proposition that is true in the abstract, but not in the concrete. Salviati offers a very articulated answer to this objection, the first step of which consists in providing a mathematical proof of the validity of the proposition at stake.6 Such a proof can of course not satisfy Simplicio, who regards it as valid “for abstract spheres, but not for material ones” (206). Challenged by Salviati to explain why what is conclusive for immaterial and abstract spheres should not apply to material ones, Simplicio observes that “material spheres are subject to many accidents,” like for example porosity and weight, which make it impossible to “achieve concretely what one imagines of them in the abstract” (206–7). Salviati is however not willing to accept the equation between “abstract” and “perfect,” on the one hand, and “concrete” and “imperfect” on the other hand. “Even in the abstract, an immaterial sphere which is not a perfect sphere can touch an immaterial plane which is not perfectly flat in not one point, but over a part of its surface, so that what happens in the concrete up to this point happens the same way in the abstract” (207). Contrary to Simplicio, Salviati uses the attributes “perfect” and “imperfect” not to draw a boundary between mathematics and physics, but to distinguish what is regular, and hence mathematically simple, from what is irregular, and hence complex.7 Galileo’s spokesman is ready to admit that the behavior of physical bodies is not always translatable into a simple mathematical law, but he insists that “the philosopher geometer, when he wants to recognize in the concrete the effects which he has proved in the abstract, must deduct the material hindrances, and if he is able to do so, I assure you that things are in no less agreement than arithmetical computations. The errors, then, lie not in the abstractness or concreteness, nor in geometry or physics, but in a calculator who does not know how to make a true accounting” (207–8). In order to be able to “make a true accounting,” the philosopher geometer must hence abstract from those material hindrances that render physical phenomena too complex to be grasped. As Salviati explicitly states in the first day of the Dialogue (1967, 103), the human intellect can in fact only understand a limited number of mathematical propositions, but “with regard to those few which the human intellect does understand … its knowledge equals the divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness.” When Galileo argues in the Two New Sciences that it is not possible for the natural philosopher to reach a “firm science … of such accidents of heaviness, speed, and shape, which are variable in infinitely many ways” (1974, 225) he does not mean that these accidents are not mathematical, but just that we are not able “to deal with them scientifically,” due to their complex mathematical character.

In commenting on Galileo’s reflections on physical-mathematical reasoning, Maurice Finocchiaro has recently argued that “we have neither separation nor identification of the two domains, but rather correspondence. We can never know in advance that a particular mathematical truth corresponds to physical reality, or that a particular physical situation is representable by a particular mathematical entity, but we can claim in advance as a matter of methodological prescription that every physical situation is representable by some mathematical entity” (2010, 119). While I believe that these lines perfectly catch the meaning of Galileo’s ideas concerning the relation between physical and mathematical truths, I don’t fully understand why Finocchiaro claims that “when so interpreted,” the Assayer’s remark on the book of nature being written in the language of mathematics, “is a long way from the extreme mathematical realism or Platonism sometimes attributed to Galileo” (119). If “mathematical realism or Platonism” means—as Finocchiaro claims—the “identification or conflation of mathematical and physical truth,” then Galileo is indeed not a Platonist. As I have argued elsewhere (Palmerino 2006, 41–42), Galileo in fact believes that while what is true in physics must necessarily be true in mathematics, not all mathematical propositions must necessarily find an instantiation in the physical reality. However, if by “mathematical realism or Platonism” one intends the view that mathematical entities are ontologically independent from our mind, then Galileo is certainly a Platonist. In his writings he not only asserts that the structure of reality is intrinsically mathematical, but also claims that mathematical propositions are true independently of whether they are known by us. While God knows all mathematical propositions, we only understand a limited amount of them.

In his book, Finocchiaro takes issue not only with those scholars who explicitly talk of Galileo’s Platonism, but also with those who implicitly attribute to him a conflation between mathematical and physical truths. In his view, “the most common instance of such implicit conflation is the interpretation that Galileo was certain about the truth of Copernicanism because of its mathematical simplicity” (Finocchiaro 2010, 115n21). Also in this case, I only partially agree with Finocchiaro’s point of view. To be sure, Galileo did not think that the criterion of mathematical simplicity could yield a demonstration of the validity of the Copernican system, for otherwise he would not have put forward the physical proof of the earth’s motion based on the phenomenon of the tides. By arguing that the flux and reflux of the sea could not be brought about by any other cause than the double motion of the earth, Galileo tried to transform what was just a probable hypothesis into a demonstrated scientific truth, the only truth capable of outrivaling the authority of the book of Scripture. If one looks at Galileo’s scientific practice, one sees however that he did attribute an important heuristic function to the principle of simplicity of nature. Besides being invoked in the Dialogue to argue that the Copernican system is more probable than the Ptolemaic one (1967, 123–24), that principle is presented in the Two New Sciences as the guiding assumption in Galileo’s search for the true law of free fall.

Further, it is as though we have been led by the hand to the investigation of naturally accelerated motion by consideration of the custom and procedure of nature herself in all her other works, in the performance of which she habitually employs the first, simplest, and easiest means. And indeed, no one of judgment believes that swimming or flying can be accomplished in a simpler or easier way than that which fish and birds employ by natural instinct. Thus when I consider that a stone, falling from rest at some height, successively acquires new increments of speed, why should I not believe that those additions are made by the simplest and most evident rule? … And we can perceive the increase of swiftness to be made simply, conceiving mentally that this motion is uniformly and continually accelerated in the same way whenever, in any equal times, equal additions of swiftness are added on. (1974, 153–54)

Also in this case, however, Galileo regards mathematical simplicity as an indication not of the truth, but of the plausibility of his definition of natural accelerated motion. In order to conclusively demonstrate that the law of free fall that can be mathematically derived from this demonstration corresponds to the “acceleration employed by nature in the motion of her falling heavy bodies,” he invokes the result of the famous experiment with a bronze ball rolled down a groove in an inclined plane (1974, 169–70). Mathematical reasoning can help the natural philosopher to draw the boundaries of what is physically possible, but only sensory experiences can establish whether a specific physical phenomenon obeys a specific mathematical law (see also Palmerino 2006, 43).

Galileo’s repeated appeals to the principle of simplicity of nature seem to be at odds with other passages of his works, where he claims that nature does not conform to the human criteria of perfection and simplicity. In a letter of July 1611 to Gallanzone Gallanzoni, Galileo observes for example that if a man had been allotted the task of defining the relation among the respective movements of the celestial spheres, he would have chosen the “first and most rational proportions” (1890–1909, 149). God, however, “without consideration for our sense of symmetry, arranged those spheres according to proportions which are not only incommensurable and irrational, but totally inaccessible to our intellect.” Galileo illustrates this point with an interesting mathematical example. A person having an insufficient understanding of geometry might wonder why the circumference of the circle “has not been made exactly three times as long as the diameter or corresponding to it in some better known proportion.” The answer is that, if this had been the case, many “admirable” properties of the circle would have been lost: “the surface of a sphere would not have been four times as big as the maximum circle, nor would the volume of a cylinder have been 3/2 of that of a sphere, and in sum no other geometrical property would have been true as it is now” (1890–1909, 149–50). In order to appreciate the perfection and simplicity of the sphere, one should hence know all of its mathematical properties, which is something our finite intellect cannot achieve.

Two important conclusions can be drawn from the letter to Gallanzone. First, Galileo regards mathematical entities as divinely created objects that are ontologically independent from our mind. Second, he believes that God always operates in the most simple and rational manner, even when his acts look irrational to us.

This can help explain the presence in the Dialogue of two apparently contradictory statements. Galileo claims, on the one hand, that we should not “make human abilities the measure of what nature can do,” as there is no single physical phenomenon of which we can achieve “a complete understanding” (1967, 101). At the same time, however, he insists that the mathematical philosopher, in his attempt to unveil the real constitution of the universe, should let himself be guided by the assumption that nature “does not act by means of many things when it can do so by means of few.”8

The principle of simplicity of nature finds an ontological justification precisely in the passage of the Dialogue in which Galileo addresses the question of the relation between mathematical and physical truths. It might hence be useful to resume our analysis of this crucial text.

After having made the point that “abstract” and “concrete” are not necessarily synonymous with “perfect” and “imperfect,” Salviati observes that “meeting in a single point is not at all a special privilege of the perfect sphere and a perfect plane.” So, even if Simplicio were right in claiming that neither a perfect sphere nor a perfect plane can be found in nature, there are still good reasons to think that an imperfect material sphere and an imperfect material plane would touch each other at a single point. For in order to touch each other “with parts of their surfaces,” these parts must either be both “exactly flat, or if one is convex, the other must be concave with a curvature which exactly corresponds to the convexity of the other.” Such conditions are, however, “much more difficult to find, because of their too strict determinacy, than those others in which their random shapes are infinite in number” (1967, 208). Salviati’s remark seems to have a double function. The first is to guarantee right of existence to the mathematical point, an entity that plays a crucial role in Galileo’s physics. In the Two New Sciences, in fact, Galileo advocates the composition of space, time, and matter out of nonextended physical atoms.9 Second, by observing that random shapes, being infinite in number, are more likely to be found in nature than regular shapes, Salviati provides an explanation of the fact that material bodies, with their variable accidents, don’t behave according to simple mathematical laws. Salviati’s remark, however, puzzles Sagredo, who does not see why it should be more difficult to obtain from a block of marble a perfect sphere or pyramid rather than a perfect horse or grasshopper. Salviati’s reply reads as follows:

I say that if any shape can be given to a solid, the spherical is the easiest of all, as it is the simplest, and holds that place among all solid figures which the circle holds among surfaces—… The formation of a sphere is so easy that if a circular hole is bored in a flat metal plate and a very roughly rounded solid is rotated at random within it, it will without any other artifice reduce itself to as perfect as spherical sphere as possible … But when it comes to forming a horse or, as you say, a grasshopper, I leave it to you to judge, for you know that few sculptors in the world are equipped to do that. (209)

Sagredo agrees with Salviati that “the great ease of forming a sphere stems from its absolute simplicity,” whereas the production of complex figures is rendered difficult by their “extreme irregularity.” This is for him a reason to wonder whether objects with a regular shape are really as rare as many believe: “If of the shapes which are irregular, and hence hard to obtain, there is an infinity which are nevertheless perfectly obtained, how can it be right to say that the simplest and therefore the easiest of all is impossible to obtain?” (210). Sagredo’s curiosity remains unsatisfied as Salviati suddenly puts an end to the discussion to go back to “serious and important things.” The right answer to Sagredo would of course be that, in purely probabilistic terms, it is equally difficult to find an object that exactly corresponds to any given irregular figure than to any given regular figure. But this is not the conclusion that Galileo wants to suggest. In his view, the fact that regular figures are “the simplest and easiest of all” increases the probability of their occurrence in nature. And, in turn, the fact that “nature … does not act by means of many things when it can do so by means of few” (117) increases the chance that the mathematical philosopher will be able to decode some chapters of the book of nature.

`CONCLUSION`

At this point we can return to the question raised in the introduction, as to whether Galileo’s remark that the book of nature is written in the language of mathematics may be taken as a literal expression of his philosophical views. As I have tried to show in this chapter, this question must be answered in the affirmative, as there is no conflict between Galileo’s professed mathematical realism and his scientific practice. When Galileo claims that the natural philosophers must abstract from accidents and impediments in order to be able to mathematize physical phenomena, he does not mean that these phenomena are intrinsically nonmathematical, but just that their mathematical setup is too complex to be grasped by our finite intellect. Moreover, the very fact that the mathematical philosopher is able, simply by “eliminating the material hindrances,” to translate physical phenomena into exact mathematical laws, is in Galileo’s eyes a sign of the fact that nature is organized according to the criteria of mathematical order and simplicity.

In his above-mentioned book on seventeenth-century mathematical philosophy, David Sepkoski (2007, 83) analyzes Isaac Barrow’s considerations on the ontological and epistemological basis of mathematics, which he regards as being “reminiscent of Gassendi’s.” In Sepkoski’s view, Barrow’s constructivist view of mathematics, according to which “mathematical objects are created by the mathematician and do not necessarily represent real objects in physical nature,” finds “a broad and general epistemological justification” in Gassendi’s nominalism (124–25). By claiming that “a mathematical number has no existence proper to itself,” Barrow destabilizes the ontological foundations of arithmetic (Barrow 1734, 41, 103).10 Sepkoski admits, however, that Barrow “is more hesitant when it comes to geometry, since he believes that on some level geometrical demonstrations do correspond with physical realities” (Sepkoski 2007, 103).

Barrow’s considerations concerning the ontological status of geometrical objects show, in my opinion, that his philosophy of mathematics is far closer to Galileo’s than to Gassendi’s. Like the former, and contrary to the latter, Barrow seems in fact willing to admit that geometrical figures have a real existence outside of our intellect.11

In a passage of The Usefulness of Mathematical Learning, Barrow (1734) explicitly takes issue with the Jesuit Giuseppe Biancani, according to whom mathematical figures have “no other existence in the nature of things than in the mind alone.” In reaction to this claim, Barrow observes that,

if the Hand of an Angel (at least the Power of God) should think fit to polish any Particle of Matter without Vacuity, a Spherical Superfice would appear to the Eyes of a figure exactly round; not as created anew, but as unveiled and laid open from the Disguises and Covers of its circumjacent Matter. Nay I will go farther and affirm that whatsoever we perceive with any sense is really a mathematical Figure, though for the most part irregular; for there is no reason why irregular figures should exist everywhere, and regular ones can exist nowhere. Moreover if it be supposed that Mathematical things cannot exist, there will also be an end of those ideas or types formed in the mind, which will be no more than mere Dreams or the Idols of Things no where existing. (77)

The similarity between Barrow’s (1683–86) and Galileo’s (1890–1909, 4:52) views concerning the relation between mathematical and physical truths is truly striking. In the lines just quoted, Barrow claims, exactly like Galileo in the Dialogue, that all physical objects possess a geometrical shape, and although these are “for the most part irregular,” there is no reason to exclude a priori that regular figures can be found in nature. Moreover, even if a mathematical entity does not find instantiation in the physical world, it still exists in the mind of God, who, as both Galileo and Barrow claim, quoting the Platonizing Book of Wisdom, “arranged all things by number, weight and measure.”

`NOTES`

I wish to thank the participants of the Work-in-Progress Seminar of the Center for the History of Philosophy and Science of Radboud University Nijmegen (Delphine Bellis, Hiro Hirai, Klaas Landsman, Christoph Lüthy, Elena Nicoli, Kuni Sakamoto, Francesca Vidotto, and Rienk Vermij) for their comments on an earlier draft of this paper, as well as Antonio Cimino for his elucidation of Husserl’s interpretation of Galileo.

1. Galilei 1974, 153 (`=` 1890–1909, 8:197). As Ofer Gal and Raz Chen-Morris have recently observed there is a striking resemblance between this passage and a statement found in the dedicatory letter of Kepler’s Ad Vitellionem Paralipomena: “And I have not satisfied my soul with speculations of abstract Geometry, namely with pictures, ‘of what there is and what is not’ to which the most famous geometers of today devote almost their entire time. But I have investigated the geometry that, by itself, expresses the body of the world following the traces of the Creator with sweat and heavy breath” (Dedication to the Emperor, in Kepler [1937], quoted in Gal and Chen-Morris [2012]).

2. A similar point is made by Galileo in the “First Letter on Sunspots”: “The latter [`=` the philosophical astronomers], besides the task of saving the appearances in whatever way necessary, try to investigate, as the greatest and most marvelous problem, the true constitution of the universe, because this constitution exists, and it exists in a way that is unique, true, real and impossible to be otherwise” (Galilei and Scheiner 2010, 95).

3. I have dealt with Galileo’s use of the topos of the book of nature in Palmerino (2006). On the same topic see, among others, Stabile (1994), Howell (2002), and Biagioli (2003).

4. For Galileo’s theory of “adumbratio,” see Stabile (1994, 54–56).

5. Compare Galilei (1974, 36) with Galileo’s answer to Di Grazia: “Everybody has free authority in the imposition of names and in the definition of terms, as similar definitions are nothing else than abbreviations of speech” (1890–1909, 4:697).

6. I cannot agree with Rivka Feldhay (1998), according to whom Salviati’s answer to Simplicio is “surprisingly poor” if analyzed against the background of the Renaissance debate de certitudine mathematicarum.

7. For a critique of the Aristotelian notion of perfection, see also Galilei (1890–1909, 4:446; 6:319–20; 7:35; 11:149–50).

8. Galilei (1967, 117). See also page 60 (“natura nihil frustra facit”), page 117 (“nature … does not act by means of many things when it can do so by means of few”), and page 123 (“frustra fit per plura quod potest fieri per pauciora”).

9. For Galileo’s views on the composition of space, time, and matter see Palmerino (2011).

10. Barrow’s claim that “a mathematical number has no existence proper to itself” is quoted in Sepkoski (2007, 100).

11. This is what Gassendi denies in his Disquisitio metaphysica, seu dubitationes et instantiae adversus Renati Cartesii metaphysicam: “at non est dicendum … triangulum esse reale quid, veramque naturam praeter intellectum” (Gassendi 1658). This passage is quoted and discussed in Osler (1995).

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