Balestra and the pythagorean metholology of Galileo and Descartes

Vena, MERTZD,M. laa a NA BALESTRA AND THE PYTHAGOREAN METHODOLOGY OF GALILEO AND DESCARTES Y way of introduction, | am pleased to say that, as a basis on which to make the following comments, E am the fortunate beneficiary of the relevant work of Richard Blackwell in the form of both the content of his graduate seminars and in his published research. Prof, Blackwell was my graduate advisor and much admired teacher for a number of courses. My first published papers, papers on Peirce, Kuhn. Piaget. and — especially relevant here — papers on Galileo's method! came directly from seminars taught by Prof. Blackwell. | have said facetiously in moments of frustration that the drive to philosophize — to seek to know and answer the fundamental questions — is a ‘divine madness’ from which we would all do better to be cured. Et is my experience of the living scholarship and teaching of a few paradigm philosophers such as Richard Blackwell that re-inspire my own efforts. | join the participants of this conference in honoring a man whose scholarship, teaching, and genuine concern for students ts exemplary and a model for all of us who would be philosophers and teachers. There are a number of interesting points raised by Prof. Balestra? that are worthy of comment, though for brevity | will focus on only the main issue of how to interpret the significance of the ‘Pythagorean method’ of Galileo and Descartes. The advocacy and adoption of this methodology. rightly understood, is central, ] propose. to the rationality of the seventeent century scientific revolution, and to the core of what should be included under Balestra's term of demythologized Pythagorcanism, This core is, in brief, the promotion of siructaral explanations and the integral and natural role of mathematics therein. For, what ought to be implied by the assertion that Galileo's and Descartes's “Pythagoreanism is a philosophy of nature is, in regard to its lasting legacy to the history of science, not an ontology of mathematics-incarnate (mathematical entilies — numbers, geometric figures — as actual constituents of physical reality). but rather a methodology for generating the distinct type of explanation we call structural, A paradigm example of such an explanation is Galileo's Copernican argument from the tides. the example Balestra focuses on." Such explanations are inherently impossible for the Aristotelian essentialism of monadic pronerties and The Modera School, LXXVI demas? Mench 1999

concomitanti syllogistie logic, de then-canonical rival to the mathematical would propose that this is how we best understand Galileo's conception of the physics of Galileo and Descartes. { propose that it was not Descartes’s argument in the Meditations that in either a theoretical or practical sense dislodged the grip role of mathematics in physics. When he says that nature is written in the lanof an Aristotelian theory of nature in favor of mathematical physics, as Balestra understand this in a straight forward manner as saying that these geomelric fipguage of mathematics where its characters are triangles, circles, etc. we should argues, but rather it was the practical explanatory and discovery power of strucures are the characters — the alphabet — of the language describing nature, but tural explanations, aided by the gradual acceptance of the practical impossibility of establishing demonstrative certainty in physical matters (acknowledged. for not as such part of the nature described.* Galileo did not subscribe to mathematiexample, by Christiaan Huygens). In the polemics with their rivals, both Galileo and Descartes maintained the cal essentialism.? In contrast to Galileo, Descartes did think it necessary to have a metaphysics of mathematical physics. one that identified the essence of material nature with prerequisite for getting a hearing within the then orthodox scientific/philosophic extensión. This was mathematics — or specifically. geometry-incarnate — and thus Descartes held to a geometric essentialism a “demythologized community. Yet, in practice both Galileo and Descartes accepted as established Pythagoreanism.” Conforming to the prevailing Aristotelian canons for proper many scientific explanations that were less than certain, explanations which are physical explanation that required this be in terms of statements of essences of physical subjects, Descartes’s essentialism made it possible for him to pose physideal of demonstrative certainty for scientific results, and this would have been characterized at places by Galileo as “very little lower than mathematical proof and by Descartes as “morally certain” (as opposed to metaphysically certain). ical problems mathematically and to treat results as physics and not irrelevant The physico-mathematical explanations offered by Galileo and Descaries (and pure mathematics. So contra Balestra, in so construing geometry Descartes indeed those of modern science generally) do not need a ‘realistically’ construed remains under the grip of Aristotelian meta-theory. mathematics, realistic in the sense of actual placing mathematical entities in Descartes’s physics was continuous with his metaphysics which remained in material nature, in order to provide vertdical explanations making entological essential ways Aristotelian, and this was a source of considerable weakness, A claims. Structural explanations make ontological claims about the posited phystparticular weakness is the thesis that all attributes of material entities — motion, rest. plurality, weight, resistance, shape, figure, etc. — are modes of extension. cal structures. Le. the network of posited relations (cansalltemporal/spatiat). that given physical and/or mathematical structures model, a modeling warranted by However, to carry out his reduction, Descartes needs motion as à principle coobservational verification. Internal essences of isolated single substances and equal with extension, and indeed he must rely upon motion more than extension their properties are not the focus here. but rather the polyadic relations that otherin order to generate the observed primary qualities of things. As Blackwell observes the unequal burden on motion over extension, and the dual status of wise unknown subjects enter into with other such subjects in the thus explanatory network. Based in part upon his rejection of explanations in terms of essences, | motion as both a mode of extended substance and as a coequal principle for generating the universe, are indictments of Cartesian metaphysical/geometrical Donald Mertz, “On Galileo's Method of Causal Proportionality.” Stidies in she History and Philosophy ef Science VI (1980): 229-242, Cottingham. R. Stoothoff. & D. (Cambridge: Cambridge Murdoch University Press, and “The Concept of Structure in Galileo: ths 19851. Vol, MI trans. Cottingham, Stwotholl. Murdoch, and A, Kenny (Cambridge: Role in the Methods of Proportionality and Fix Cambridge Untversity Press. 10911 12898. Suppositione as Applied to the Tides.” did. 13 Hereafter abbreviated PW. (F982): 11-131. Dominic Balestra, “At the Origins of Galilea, trans. Stillman Drake (New York: Modern Science: Doubleday, 1957), 237-238, Demythologizing Pytha- For an analysis of Galtleo’s use of structural especially for Discoverica amt Opinions of For the clan that Galileo did not hold a porcanisny” ta this volume. arguments, “Galileo, Ihe Copernican nithematical ontology see Gary Hatfield. “Metaphysics and the New Science.” in motion of the earth, see Mertz. "On Galileo's Reappraisats of the Scientific Revolution, ed. Method” an “The Concept of Structure.” ‘Galileo, Dialogue Concerning the Five D. Lindberg and R. Westman (Cambridge: physics. Despite the fact that Descartes rejects explanations in terms of substantial forms and in their place promotes structural theories. he nevertheless retains an Aristotehan-like ontology of substance, attribute, and mode, as laid out in the Principles of Philosophy (1644). It is strange from the perspective ofthis schema that, as Blackwell observes, Descartes treated extension as a monadie property. Not only would one expect extension to be treated as an underlying subject substance rather than as a predicate of what would then apparently be an incoherent “bare particular” but there is also the problem of how specific multi-subject spalial relations can be generated from the generic monadic property of extension. Pace Descartes. extension ts a dependent, secondary attribute of an entity. one whose predication is based upon the logically prior obtaining of specific spatial Cambridge University Press, 1990). 94-166 Chief World Systems. trans, Silman Drake "Richard. Blackwell. “Descartes” Concept (Berkeley: University of California Press. al Matter” in the Concept of Matter in Modern 1967), 408. Philosophy, ed. Ernan MeMullin (Notre Dame; Truth and Judement in Aquinas University of Note Dame Press, 1978), 50-75, John Peterson Rene Descartes, The Philosophical Writings of Descartes, vol. | & Ih trans. 1.

relations between distinguishable parts of the entity. Descartes inverted this are transitional figures who retained as an ideal of scientific explanation the residdependency in an attempt at what is a failed essentialism, a physics that was ual Aristotelian standard of demonstrative certainty, and this no doubt remforced excessively geometric at the expense of physical concepts. the appropriateness of adopting mathematics as an explanatory tool. a thesis debated at the time as non-Aristotelian. Yet, what equally appealed to both Galileo and Descartes was the power of applied mathematics as an instrument of Whether it was perceived at the time as failed or not, Descartes” extensional metaphysics did not succeed in dislodging the grip of an Aristotelian theory of structural explanation, Conforming to Balestra’s apt injunction, I shall below discovery. And it is, I contend, that aspect of the nature of applied mathematics which empowers it as an instrument for extending our explanatory knowledge briefly “recontextualize” the use of structural explanations in the works of Galileo that is the lasting legacy of Galileo’s and Descäartes’s contribution to the sevenand Descartes. icenth century scientific revolution. nature, as Balestra holds, but rather it was the practical success of the method of Central to the concept of rationality in natural science is the achievement of Now, what is it about the nature of applied mathematics that makes it a tool an effective methodology for providing explanations of physical phenomena. for gaining new insights and explanations of natural phenomenon? To answer this. it must be understood that applied mathematics is a component of Ute broad- Characterizing the “modern” methods of Galileo and Descartes, they have been described as proper mixtures of mathematics and physical observation/experimentation. Both Galileo and Descartes saw themselves as rejecting the traditional er category of structural explanations, specifically, explanations in terms of strie tural isomorphisms or modelings. Stated precisely, a structural explanation method of unaided/uncontrived observation plus syllogistic logic for a more consists in a partial isomorphism between two parallel structures or complexes, potent method synthesizing experimentation and mathematics in a broad sense of one a given physical and/or mathematical structure or model and the other a what we would call structural modeling. Both men reject appeal to forms and hypothesized structure held to effect some given phenomena. In causal explanaqualities of the Aristotelian/scholastic tradition as non-explanatory. Galileo describes the attempt to know of such properties as an “impossible undertaktions, the connecting mapping that is the isomorphism rests on the principle of similar effects having similar causes. The promotion of structural explanations ts ing," and Descartes holds that “no one has ever succeeded in deriving any praca pivotal element of the seventeenth century scientific revolution. as fundamental, tical benefit from ‘prime matter”, ‘substantial forms’. “occult qualities’, and the if not more so, than the increased emphasis on active experimentation. To use an like! Syllogistic logic is also rejected because, as Galileo puts it, “though much preached [logic] is not very powerful." Descartes agrees, saying “with regard to analogy from modern logic, while experimentation provides the semantical content for scientific explanations, the structural form provides the formation and logic that syllogisms and most of its other techniques are of less use for learning transformation rules, and because of the latter the potential for the retroductive things than for explaining to others the things one already knows." For both introduction of conceptual novelty. What makes partial isomorphisms between Galileo and Descartes, syllogistie logic is to be rejected because it is not a tool for paralleling structures a ‘logic of discovery” is that properties and relations gaining new knowledge — if is not a “logic of discovery”. The supposed intellecrevealed by the examination of the given structure suggest analogous properties tual power of “quickness of wit’ (achinoia), the gift Aristotle said humans have for and relations in the hypothesized structure. N may be possible that these new hitting upon causal middle terms in demoustrative syllogisms. is here rejected as hypothesized attributes are verified directly and singly by experiments. or they ineffectual. The true and efficacious logic of discovery for physies is applied may be warranted only holistically on how well they hold together with other mathematics; geometry for Galileo, geometry (specifically, the geometric diaattributes that are more directly verified. In the Principles, alter using the example gram)" assisted by algebra for Descartes, Galileo has Simplicio say “Truly I of a structural explanation of the decoding of an encrypted letter, Descartes begin to understand that although logie is a very excellent instrument to govern asserts of his theories of “magnetism, fire and the fabric of the entire world” that our reasoning, Il does not compare with the sharpness of geometry in awakening “even if they think that my assumptions of these principles was arbitrary and the mind to discovery (invenzione)! And Descartes in a letter of 1648 asserts, groundless, they will still perhaps acknowledge that it would hardly have been “A study of mathematics, then, is a prerequisite for making new discoveries, both possible for so many items to fit into a coherent pattern if the original principles in mathematics itself and in philosophy." In short, both Galileo and Descartes had been false" Here we have coherence as the warrant lor a retroductive form of the hypothetico-deductive method. "Quoted in Stillman Drake, Galileo at Work (Chicago: University of Chicago Press, 1978). 199, OPW 11.224. IPAM ESL & O7IT. "Gahleo, Vive Mew Sciences, trans, Süllman Drake (Madison: Press. 19741. 133, UCGatilea, Dietogne, 191. "PWHEAST. NPW,EIIN. "PW.1:290 214 University of Wisconsin Balestra and the Pythagorean Methadatogy

(i.e.. historical development), and structural. with the latter receiving surprisingly non power of explanation. In a 1638 letter to Morin he states, “I nuuntain, thereforo. that analogies of this sort are the most appropriate means available to the human mind for laying bare the truth in problems of physics. | would go so far as to say that. when someone makes an assertion concerning nature which cannot be explained by any such analogy, | think I have demonstrative knowledge that the point is false" In this regard, Desmond Clarke comments that it is no exaggerittion to say that Descartes's entire scientific project is one of imaginatively conlittle attention.!* Yet, MeMullin observes that structural explanations are more structing models descriptive of motions intended to explain natural phenomena." powerful than the other two types. because they warrant ontological claims about Descartes's strategy is to construct large-scale models that duplicate the hypothesized underlying micro-structure. The assumption is that the general structure and Belore touching further upon the role of structural explanations in the works of Galileo and Descartes, itis important to observe that the role and power of structural explanations in the history of science have not been sufficiently appreciated. Two exceptions to this can be found in works of Ernan MeMullin and Richard Blackwell. MeMullin has pointed out that there are three broad categories of explanations found within the natural sciences: nomological, genetic unobserved processes, and they make for correct prediction and technical control. A good structural model provides resources for imaginative extension for some time after its introduction, Now, these same three categories of explanation can be seen at work at the meta-level of philosophy of science — in the theories. respectively, of logical empiricism. the Kuhnian theory of revolutionary paradigm shifts, and in a structuralist-Piagetian account. Blackwell holds that the direct object of epistemological analysis is structure, where the essential constituents are the interconnecting relations, the “transformation.” These transformations can be dynamic (an event structure such as a complex system of cause and effect) or static (a formal logical or mathematical structure). Blackwell argues that scientilic theories grow out of the cognitive activity of problem solving and that they are dynamic structures in PiageUs sense — self-regulated structures with the functional invariants of organization, equilibration, assimilation, accommodation. and re-equilibration. IF we understand this properly, we have the possibility of a rational logic of discovery, something that neither the logical empiricist nor Kubnian theortes can provide. These theses bear on the issue of rationality of set entific change raised by Balestra in his introduction. Blackwell's adaptation theory provides a non-° Whiggish'” account of the rationality of scientific change. Returning to Galileo and Descartes, Galileo did not acknowledge structural explanations as an explicitly recognized category. though his practice is one of using "proportional reasoning’ in the extended sense of isomorphic correlation between structures. This was for him a standard mode of explanation, including in his argument for the tides From the Copernican motion of the earth. Descartes, on the other hand. was explicit in attributing to models and ‘analogies a sine qua »Ernan MeMullin. Structural Explanitions. American Philosophical Quarterly 15 1978: 139 147. Richard I Scrence (University Park Pennsylvania State University Press. 1982), 124, "Galileo. five Mew Setences. 169 Blackwell, "N Stucteralisi SA Rive, "Galilée el La Lin D'Inerte n Account of Scientilie Theories” dtesstational Endes Galiléennes (Paris: Hermann, 1539). Philosophical Quarterty 10 (19760) 203-274. IT. Io2lt. Also see “Scientific Discovery and the Linvs of Concept al Matter” in The Concept of Matter: Lovie.” New Schotasticisn 5041970) ed. Finan MeMultin (Notre Dame: University 333-344, OPW AEA 22. "Desmond Clarke, Descartes’ Phidoyopiiv of 216 Also see Blackwell. al Notre Dame Press. 19789, 69, laws of matter apply to all levels of matter, macro- and microscopic, For Galileo and Descartes, structural explanations are sometimes in concrete terms of material and efficient causes described mechanistically. as in Galileo's sloshing-water/moving-barge model for the Mediterranean-tides/Copernicanmotion of the earth, or Descartes's tennis ball model for deducing Snell's law of refraction, Descartes’s general theory of vortices is a structural explanation. Other explanations are in the more abstract terms of comparisons between geometrie continua and paralleling continua of spatial and temporal points of composing posited trajectories of hypothesized moving bodies, Examples of this include Galileo's derivation of the law of falling bodies in the Tiro New SciencesE or, specifically in the case of Descartes. the structural parallel between the solution sets of algebraic equations and geometric figures. and the latter with the ultimate subject matter of physics — extended substance. Appropriate here ts an observation made by Koyré, and re-emphasized by Blackwell, that Descartes's conception of motion is a static geometrical idealization consisting in the trajectory ofa moving body. not the dynamic change of place itself. This static conception of motion is exactly what one would expect when motion is defined as a mode of extension as with Descartes. A tinal but crucial point needs to be made concerning why mathematics is congenial to structural explanations, A structure is by definition a network or system of interrelated elements where the dra-connecting relations are the essential elements, the natures of the connected nodes serving the role of bases for these relations. Indeed. the natures of the connected elements may be in themselves unknown: itis sufficient fora structural explanation that we know only their relations to other nodes within the system. In this regard, there ts an interesting asser: tion by Galileo. “In our speculating we either seck to penetrate the true and internal essence of natural substances or content ourselves with a knowledge of “Descartes” Balestra and the Pyihagercan Methodology

some of their properties. The former | hold to be as impossible an undercation of relations must now be explained away. Recall Leibniz’s assertion that we would not want “to posit an accident which would inhere simultaneously in taking with regard to the closest elemental substances as with the more remote celestial things... But if what we wish to fix in our minds is the two subjects — one which, so to speak, has one leg in one and another leg in the apprehension of some properties of things, then it seems to me that we other" Hence, we have the torturous and spurious attempts at reducing relations to properties of their subjects. a tack found repeatedly throughout Western philosneed not despair of our ability to acquire this respecting distant bodies Just as well as those close at hand. . . [These properties are] location. Note that each one of the latter are relational/structural properties. What makes ophy up to the turn of this century, and even now sporadically.” As it were, Ihe monadic tail is wagging the polyadic dog. To borrow from Peter Geach a characterization of a closely related issue, the containment model of predication was, mathematics congenial to structural explanations is the fact that the underlying and in some quarters residually continues to be, a disaster for philosophy compalogic of mathematical systems is a relational logic — a logic of polyadic predirable to the Fall of Adam.Y For in that model, relations lose any ontological, and hence any explanatory standing; they become, as Aristotle stated, “least of all real. Under this conception of relations the Aristotelian tradition was right to give mathematics. which embodies an intrinsic relational logic, only a purely descriptive role of ‘saving the appearances. Though unintended, what Galileo motion, shape, size, opacity, mutability, generation. and dissolution"? cates. This is in contrast to the monadic, subject/predicate logic of Aristotle. Moreover, as we now know, it is impossible to reduce the logic of relations to the logie of properties. It is interesting to note that scholars of the sixteenth and seventeenth centuries (including Christopher Clavius) attempted to demonstrate that the proofs of Euclid’s Elements where reducible to syllogistie form." This is an and Descartes helped to initiate in regard to the foundations of modern science area worthy of more scholarship, where a central question is whether these scholwas the loosening of the grip of two erroneous models — demonstratively certain ars themselves thought they were successful in demonstrating this syllogistic reduction, Whatever the answer, lan Mueller has shown that not even the first axiomatic mathematics as a model for the form and character of scientifically established truths, and at a deeper level. the containment model of predication proposition of the Elements is reducible to syHogistic form, the proposition that robbed structural explanations of ontological legitimacy. depending ultimately on the relations between three straight lines and not on properties of them taken as pairs which a syllogistic analysis would require.” In sum, Aristotelian scientific methodology as well as its underlying ontolpredication. The Aristotelian model of material predication is comtainment - - an ogy were incapable of providing structural explanations. but a ‘Pythagorean’ methodology incorporating applied mathematics could. And though structural explanation is inherently hypothetical, it was its relative power and its facility asa ‘logic of discovery” that impressed Galileo and Descartes and provided the sucattribute is ‘present in, ‘inheres in. "is immanent in’ its subject. The model is one cess that eventually dislodged the grip of the Aristotelian theory of nature. But there is a deeper ontological issue here of which the logical difficulties are only a symptom. Tt has to do with the underlying theory of entic or ‘material’ of part to subsuming whole. This model of necessity restricts all predication ultimately to single subjects, Le., all predicates are monadic predicates -properties. The related inference engine is, of course. syllogistic logic. Leibniz was perhaps the most consistent philosopher in explicitly enforcing the containment model in his doctrine of praedicatum inest subjecto. The simultaneous multi-subject predi- Quote in Drake. Galileo at Work, 199, Reidel. 1974), 35-74 "On the irreducibility of polyadic relations Gottfried W, Leibniz, Phdosephical Papers to monadic properties and a history of such and Letters, Ind. ed. ed. Ley Loemker (Dordrecht: Reidel. 1969), 704. attempted reductions, see my Moderate Realisen and tix Logic (New Haven: Yale University Press, 1906), "References tor these attempted reductions of proofs tn Buclid’s Elements lo syllogistic “For example, Keith Campbell in bis Abstract Particulars (Oxford: Basil Bluckwell. 1990) presses lor a monadic reduction of rela tens. See my Atoderate Reatisnr, VOL. Tor a form are found in Neal Gilbert, Renaissance Concepts of Method (New York: Columbia rebuttal. University Press, 1960), 89-90, University of California Press, 1980). 47 and Man Mucller, “Greek Mathematics and Greek Logie.” in Ancient Logie and tts Modern 240, Interpretations, ed. John Corcoran (Dordrecht: 218 “Peter Geach. Logic Matters (Berkeley: “Aristotle, Aetaphvsics 1088120), Balestra and the Pythagorean Methodology