Galileo’s dictum that the book of nature “is written in the language of mathematics” is emblematic of the accepted view that the scientific revolution hinged on the conceptual and methodological integration of mathematics and natural philosophy. Although the mathematization of nature is a distinctive and crucial feature of the emergence of modern science in the seventeenth century, this volume shows that it was a far more complex, contested, and context-dependent phenomenon than the received historiography has indicated, and that philosophical controversies (...) about the implications of mathematization cannot be understood in isolation from broader social developments related to the status and practice of mathematics in various commercial, political, and academic institutions. (shrink)

Newton frequently characterized his methodology as distinctive and capable of achieving greater evidential support than that of his contemporaries, due to its mathematical character. Newton's pronouncements reflect a striking position regarding the role of mathematics in natural philosophy. We can give an initial characterization of his position by considering two questions central to seventeenth century debates about the applicability of mathematics. First, how are we to understand the distinctive universality and necessity of mathematical reasoning? One common way to preserve the (...) demonstrative character of mathematics was to restrict its domain, as far as possible, to pure abstractions. The subject matter of mathematics is then taken to be abstracted from the changeable natural world, consisting of quantity and magnitude themselves rather than the objects bearing quantifiable properties. Yet restricting the domain in this way leaves it difficult to see how mathematics relates to natural phenomena. How could the book of nature be written in a language of pure abstractions? Second, what is the proper role of mathematical reasoning in natural philosophy? Many followed Aristotle in consigning mathematics to a subordinate role. Mathematics could not fulfill the main aim of natural philosophy, namely to provide demonstrations reflecting the essences of things and nature's causal order. A related concern was also pressing for the mechanical philosophers: a merely mathematical demonstration fails to provide an intelligible mechanical explanation. The very title of Newton's masterpiece, Philosophiae Naturalis Principia Mathematica, would have been extremely perplexing: how could natural philosophy be based on mathematical principles? My aim is to articulate Newton's position regarding the mathematization of nature in response to these two concerns. (shrink)

In the Treatise of Human Nature, David Hume mounts a spirited assault on the doctrine of the infinite divisibility of extension, and he defends in its place the contrary claim that extension is everywhere only finitely divisible. Despite this major departure from the more conventional conceptions of space embodied in traditional geometry, Hume does not endorse any radical reform of geometry. Instead Hume espouses a more conservative approach, claiming that geometry fails only “in this single point” – in its purported (...) proofs of infinite divisibility – while “all of its other arguments” remain intact. -/- In this paper, after laying out the prima facie case for Hume’s radical challenge to traditional geometry, I consider five strategies for blocking the arguments for infinite divisibility while conserving most of geometry. I show that each of these interpretive strategies suffers from serious substantive problems, and so none of them delivers an interpretation of Hume’s account that provides him with a way of blocking the geometric arguments for infinite divisibility while sustaining his geometric conservatism. (shrink)

For Aristotle, the shape of a physical body is perceptible per se (DA II.6, 418a8-9). As I read his position, shape is thus a causal power, as a physical body can affect our sense organs simply in virtue of possessing it. But this invites a challenge. If shape is an intrinsically powerful property, and indeed an intrinsically perceptible one, then why are the objects of geometrical reasoning, as such, inert and imperceptible? I here address Aristotle’s answer to that problem, focusing (...) on the version of it that he presents in De caelo III.8. I argue that if we grant that Aristotle conceived of the shape of a sensible body as some kind of causal power, then the satisfactory resolution of that challenge pushes us to interpret him as having conceived of it as being, more specifically, an impure power—that is, as a property that is not only intrinsically powerful but also, in some way, intrinsically non-powerful as well. This is a notable result not only insofar as it illuminates Aristotle’s conception of shape but also insofar as it contributes to our knowledge of Aristotle’s theory of dunameis and his ontology more broadly. (shrink)

We advance a theory of the representational role of Euclidean diagrams according to which they are samples of co-exact features. We contrast our theory with two other conceptions, the instantial conception and Macbeth’s iconic view, with respect to how well they accommodate three fundamental constraints on theories of the Euclidean diagrammatic practice— that Euclidean diagrams are used in proofs whose results are wholly general, that Euclidean diagrams indicate the co-exact features that the geometer is allowed to infer from them (...) and that Euclidean diagrams play the same role in both direct proofs and indirect proofs by reductio—and argue that our view is the one best suited to account for them. We conclude by illustrating the virtues of our conception of Euclidean diagrams as samples by means of an analysis of Saccheri’s quadrilateral. (shrink)

This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word coined for (...) this lecture. To repeat, a subregular polyhedron has congruent faces and congruent [polyhedral] angles. A subregular polyhedron whose faces are all regular polygons is regular—using standard terminology. -/- Geometers before Euclid showed that there are “essentially” only five regular polyhedra: every regular polyhedron is a tetrahedron (4 faces), a hexahedron or cube (6 faces), an octahedron (8 faces), a dodecahedron (12 faces), or an icosahedron (20 faces). -/- The first question is whether there are subregular polyhedra that are not regular. For example, are there tetrahedra having congruent angles and congruent triangular faces but whose faces are not equilateral triangles? -/- Another question is the classification of subregular polyhedra if they exist. For example, considering the fact that the regular tetrahedra all have equilateral triangles as faces, we ask which triangles other than equilaterals are faces of subregular tetrahedra. Similarly, considering the fact that the regular hexahedra all have squares as faces, we ask which quadrangles other than squares are faces of subregular hexahedra. -/- After introductory remarks that include historical and philosophical points, we concentrate on tetrahedra. A triangle that is congruent to each of the four faces of a tetrahedron is called a generator of the tetrahedron. The main result proved is that every acute triangle is a generator of a subregular tetrahedron. The proof includes an algorithm –implementable with scissors and paper –that constructs from any given acute triangle a subregular tetrahedron whose faces are congruent to the given triangle. -/- Algorithm: Given any acute triangle. Construct a similar triangle whose sides are double the sides of the given triangle. Draw the three lines connecting the three midpoints of the sides (making four triangles congruent to the given triangle—a central triangle surrounded by three peripheral triangles). Make three “hinges” along the lines connecting the midpoints. “Fold” the peripheral triangles together (into a tetrahedron). [LIGHTLY EDITED VERSION OF PRINTED ABSTRACT] Acknowledgements: William Lawvere, Colin McLarty, Irvin Miller, Frango Nabrasa, Lawrence Spector, Roberto Torretti, and Richard Vesley. -/- . (shrink)

Spinoza's Ethics self-consciously follows the example of Euclid and other geometers in its use of axioms and definitions as the basis for derivations of hundreds of propositions of philosophical si...

The status of the laws of nature in Hobbes’s Leviathan has been a continual point of disagreement among scholars. Many agree that since Hobbes claims that civil philosophy is a science, the answer lies in an understanding of the nature of Hobbesian science more generally. In this paper, I argue that Hobbes’s view of the construction of geometrical figures sheds light upon the status of the laws of nature. In short, I claim that the laws play the same role as (...) the component parts – what Hobbes calls the “cause” – of geometrical figures. To make this argument, I show that in both geometry and civil philosophy, Hobbes proceeds by a method of synthetic demonstration as follows: 1) offering a thought experiment by privation; 2) providing definitions by explication of “simple conceptions” within the thought experiment; and 3) formulating generative definitions by making use of those definitions by explication. In just the same way that Hobbes says that the geometer should “put together” the parts of a square to learn its cause, I argue that the laws of nature are the cause of peace. (shrink)

In his doctoral dissertation On the Principle of Sufficient Reason, Arthur Schopenhauer there outlines a critique of Euclidean geometry on the basis of the changing nature of mathematics, and hence of demonstration, as a result of Kantian idealism. According to Schopenhauer, Euclid treats geometry synthetically, proceeding from the simple to the complex, from the known to the unknown, “synthesizing” later proofs on the basis of earlier ones. Such a method, although proving the case logically, nevertheless fails to attain the raison (...) d’être of the entity. In order to obtain this, a separate method is required, which Schopenhauer refers to as “analysis,” thus echoing a method already in practice among the early Greek geometers, with however some significant differences. In this essay, I here discuss Schopenhauer’s criticism of synthesis in Euclid’s Elements, and the nature and relevance of his own method of analysis. (shrink)

Berkeley and Kant are known for having developed philosophical critiques of materialism, critiques leading them to propose instead an epistemology based on the coherence of our mental representations. For all that the two had in common, however, Kant was adamant in distinguishing his own " empirical realism " from the immaterialist consequences entailed by Berkeley's attack on abstract ideas. Kant focused his most explicit criticisms on Berkeley's account of space, and commentators have for the most part decided that Kant either (...) misunderstood or was simply unfamiliar with the Bishop's actual position. Rather than demonstrate that Kant understood Berkeley perfectly well—an argument that has already been forcefully made by Colin Turbayne—I want to take a different tack altogether. For it is by paying attention to Berkeley's actual account of space, an account oriented by his rejection of spatial " geometers " like Descartes, and spatial " absolutists " like Newton, that we discover an account of embodied cognition, of spatial distance and size that can only be known by way of the body's motion and touch. Perhaps even more striking, I will want to suggest, is the manner in which Kant's approach to the problem of incongruent counterparts will equally need to rely on a proprioceptive cognition, one requiring a different geometry of position altogether. My discussion proceeds in three stages, with stage one focused on Kant's efforts to distinguish his philosophical project from Berkeley's own idealist system, and stages two and three describing the manner in which their approach to spatial orientation both challenges and extends the traditional narrative of their differences as laid out in stage one. (shrink)

Although their contributions to the history of economic thought and their scholarly reputations are firmly established, relatively little is known about the relationship between Carl Menger, founder of the Austrian School of economics, and his son, Karl Menger, the mathematician, geometer, logician, and philosopher of science, whose famous Mathematical Colloquium at the University of Vienna was central to the early literature on the existence of general equilibrium and the concomitant development of mathematical economics. The present paper begins to fill (...) this gap. Karl Menger’s diaries, held in the David M. Rubenstein Rare Book and Manuscript Library at Duke University, offer insight into the intimate relationships within the Menger clan, Karl’s work and study habits, and the development of his uncommonly broad intellect, as well as on life in a vanquished city, Vienna, in the immediate wake of the humiliating defeat of the First World War and the disintegration of the Habsburg Empire. (shrink)