Weyl's tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called "the mixed account," according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To motivate this account as a genuine solution to (...) Weyl's tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest's (1995) account in response to Weyl's tile argument, which can be considered as a restricted version of the mixed account. (shrink)

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what (...) mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity. (shrink)

Van Bendegem has recently offered an argument to the effect that, if time is discrete, then there should exist a correspondence between the motions of massive bodies and a discrete geometry. On this basis, he concludes that, even if space is continuous, it should nonetheless appear discrete. This paper examines the two possible ways of making sense of that correspondence, and shows that in neither case van Bendegem’s conclusion logically follows.

My topic in this paper will be Hume’s claim that we have no idea of a vacuum. I offer a novel interpretation of Hume’s account of our ideas of extension that makes it clear why those ideas cannot include any ideas of vacuums, and I distinguish my interpretation from prominent readings offered by other Hume scholars. An upshot of Hume’s account, I will argue, is his commitment to a remarkable and distinctly Humean view I call “perceptual relationism.” Perceptual relationism is (...) a fundamental characteristic of Hume’s “universe of the imagination,” and a manifestation of just how “loose and separate” the constituents of that inner universe are. Once we understand perceptual relationism and its entailments, we are in a better position to understand the rest of Hume’s sometimes puzzling remarks on space and the vacuum. (shrink)

We propose a new interpretation of objective deterministic chances in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set--up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the 'distance' (in terms of physical resources) from a physical state to another, and (3) the size of the set of time--complexity functions that are compatible with the physical (...) resources required to reach a physical state from another. (shrink)

Weyl’s tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called the mixed account, according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To motivate this account as a genuine solution to (...) Weyl’s tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest’s (Synthese 103:327–354, 1995) account in response to Weyl’s tile argument, which can be considered as a restricted version of the mixed account. (shrink)

Strict finitism is a minority view in the philosophy of mathematics. In this paper, we develop a strict finite axiomatic system for geometric constructions in which only constructions that are executable by simple tools in a small number of steps are permitted. We aim to demonstrate that as far as the applications of synthetic geometry to real-world constructions are concerned, there are viable strict finite alternatives to classical geometry where by one can prove analogs to fundamental results in classical geometry. (...) We consider this as one of many early steps investigating the extent to which strict finite foundations can be developed for the application of mathematics to the real-world. (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)