Investigations into the Applicability of Geometry

Douglas Bertrand Marshall, Investigations into the Applicability of Geometry, Ph.D. Thesis, Harvard University (2011)

Below pages correspond to those in PDF.

Beginning of Abstract

“Philosophical reflection about the sciences has persistently given rise to worries that mathematics, while true of its own special objects, is inapplicable to nature or to the physical world. Focusing on the case of geometry, and drawing on the histories of philosophy and science, I articulate a series of challenges to the applicability of geometry based on the general idea that geometry fails to fit (or correspond to) nature. This series of challenges then plays two major roles in the dissertation: it clarifies the ways in which the applicability of geometry poses a problem for two major 17th century natural philosophers, viz., Galileo and Leibniz, and it allows for the investigation of the relationship between geometric structures and nature by means of an investigation of the applicability of geometry.”

“I begin with the challenge pressed by some thinkers in the Aristotelian tradition that the results which geometry proves about its objects are false when interpreted as assertions about objects in nature. Despite the durable influence of this challenge and the Aristotelian theory of science which inspires it, I argue that Aristotle himself did not oppose the use of geometry in empirical inquiry, but rather offered an account of it. I then examine how Galileo takes on the objection that geometric results are false if understood as claims about nature in his Dialogue Concerning the Two Chief World Systems. On my interpretation, Galileo argues the objection should be recast as the claim that there are no geometric points, lines, or surfaces in nature. This is an objection both Galileo and Leibniz take seriously in developing their new mathematical physics, although I argue that Galileo and Leibniz react to the objection very differently: Galileo rejects the objection as false and grounded on a misconception of the relationship between geometry and nature, whereas Leibniz grants the truth of the objection and tries to show that it is not damaging for the project of mathematical physics.”

Challenging the Applicability of Geometry

p. 21 “[Protagorean Challenge] For all phi in Gamma, phi is a theorem of geometry, and when it comes to physical and material things,  not phi.”

p. 25 “[No-Shapes Challenge] There are no geometric objects in nature. That is, there are in nature no points, lines, or surfaces which satisfy the axioms of geometry.”

p. 37 “[No-Structure Challenge] Nothing in nature is isomorphic either to Euclidean space, or to any Euclidean curve, or to any Euclidean surface.”

p. 41 “[No-Discrepancies Challenge] Given any natural item N and any geometric item G, there is no determinate or well-defined discrepancy between N and G.”

p. 41 ‘An intended consequence of the No-Discrepancies Challenge is that geometric objects do not even approximate things in nature. This already implies the truth of the No-Structure Challenge, but it implies a good deal more. An apt motto for the No-Discrepancies Challenge would be: “Nature is blurry.”‘


p. 75 “Although AnPst has encouraged many readers to regard Aristotle as rejecting mathematical methods in empirical inquiry, I have argued that this is a misinterpretation. On Aristotle’s overall picture of the workings of science, it is true that the practitioners of a science consider a limited range of objects and a correspondingly limited range of attributes. Moreover, in most cases the subject genera of distinct sciences do not bear a close enough relationship to each other to support one science’s making an application of the other. Nonetheless, in light of the stunning successes achieved by mathematical pursuits in optics, harmonics, astronomy, etc., Aristotle repeatedly makes explicit exceptions for areas of natural science to apply branches of pure mathematics. Good interpretations of AnPst must not render these applications unintelligible. In this chapter, I have suggested how Aristotle accounts for these applications given both the explanatory resources and the technical constraints of AnPst.”


p. 105-106 “To claim that geometric objects do not exist in nature is at least prima facie to challenge the descriptive applicability of geometry. As I interpret Galileo’s response to Simplicio, the bulk of what follows the discussion of the mathematical proof that a sphere touches a plane tangent to it at a point is an attempt to meet that challenge. We saw in §3.2 – 3.3 that Galileo offers two strategies for defending the descriptive applicability of geometry against such a challenge. There is the inflationary strategy of getting his opponent to grant that there is more to geometry and to the physical world than the opponent has so far recognized. There is also the idealizing strategy which tries to identify conditions under which one may legitimately use geometry to approximate some natural phenomena even when the phenomena do not correspond precisely to the geometric approximation. A common theme of the two strategies is that they involve coordinating our conceptions of geometry and physics.”


p. 109 “Unlike Galileo, Leibniz accepts the view that nothing in nature corresponds strictly to any geometric object. Writing in 1686, Leibniz claims that ‘no determinate shape can be assigned to any body, nor is a precisely straight line, or circle or any other assignable shape of any body, found in the nature of things’ (RA, p. 315). Later, in 1702, Leibniz reaffirms the view: ‘It is true that perfectly uniform change, such as the mathematical idea of motion, is never found in nature any more than are actual figures which possess in full rigour the properties which we learn in geometry’ (G4.568, L, p. 583).3 Despite his frequent changes of mind on other topics, Leibniz’s rejection of what he calls ‘precise’ or ‘definite’ shapes in nature seems to be a stable part of his view from the 1680’s until the end of his life.”

p. 110 “When Leibniz is confronted with this difficulty, the justification he tends to offer is that even if nothing in nature corresponds exactly to any geometric shape, there can be things in nature which approximate geometric shapes to within any specified margin of error. For example, Leibniz writes in 1679 that ‘[E]ven if straight lines and circles do not and cannot possibly exist in nature, it suffices nonetheless that there can exist figures which differ so little from straight lines and circles that the error be less than any given’ (A6.4.159).”

p. 133 “To get some further purchase on his view of the existence and legitimacy of geometric approximations in particular instances, I will examine a case from Leibniz’s own scientific practice. In ‘An Essay on the Causes of Celestial Motions’, published in the Acta Eruditorum in 1689, Leibniz presents an argument which is meant to explain why the planets in our solar system move in ellipses with the sun at one focus. Leibniz’s chief assumption is the existence of a fluid vortex circulating around the sun; the key property of the vortex is that it circulates harmonically, which is to say that ‘the velocities of circulation round the centre decrease proportionally as the distances from the centre increase’ (GM6.149-150, Tentamen, pp. 129-130). Leibniz gives a demonstration of the elliptical trajectories of the planets from the assumption of the fluid vortex [30]. The demonstration is offered as an explanation, assuming the vortex theory, of why the planets move as they do: they are being pushed by the fluid around them.”

p. 141 “All that science requires is for the error between the real shape and the approximating geometric curve to be small enough so as not to cause errors significant in light of the aim at hand. To return to the example I discussed at some length, Leibniz’s soundest justification for approximating the trajectory of Mars with a particular ellipse would be to say that the error between the trajectory and the ellipse is small enough so as not to cause significant errors given the aims of astronomy. This justification stands in contrast with another possible justification which I argued provides us with a poorer interpretation of Leibniz: namely, that the error between the trajectory and the ellipse is less than any given quantity.”

p. 139-140 “When bodies and their motions appear to us, their shapes appear to us as mathematical surfaces and their trajectories appear as mathematical curves. But the appearances are misleading: nothing in nature or in the works of God corresponds precisely to mathematical curves and surfaces. The boundaries and trajectories of bodies are in finitely complex, they are beyond our comprehension, and they are not part of our sensory experience of the world. So described, it is hard to see how these boundaries could be ‘phenomenal’ or the contributions of our sensory faculties. Rather, the physical boundaries exist in nature and are part of what God does, though they are hidden from us by the workings of our minds.”

Geometry and Nature

p. 154 “In this final chapter, I will argue that the applicability of a given geometry to nature does impose a non-trivial constraint on the relationship between nature and the corresponding geometric structure: in particular, when one represents some natural objects or processes by a geometric structure, there must be determinate discrepancies between features of the natural objects or processes and the geometric structure.”