Mark Steiner

(1942 - 2020)

In lieu of an abstract, here is a brief excerpt of the content:400 KANT'S MISREPRESENTATIONS OF HUME'S PHILOSOPHY OF MATHEMATICS IN THE PROLEGOMENA In 1783, Immanuel Kant published the following reflections upon the philosophy of mathematics of David Hume, words which have colored all subsequent interpretations of the letter's work: Hume being prompted to cast his eye over the whole field of a priori cognitions in which human understanding claims such mighty possessions (a calling he felt worthy of a philosopher) heedlessly severed from it a whole, and indeed its most valuable, province, namely, pure mathematics ; for he imagined its nature or, so to speak, the state constitution of this empire depended on totally different principles, namely, on the law of contradiction alone; and although he did not divide judgments in this manner formally and universally as I have done here, what he said was equivalent to this: that mathematics contains only analytical, but metaphysics synthetical, a priori propositions. In this he was mistaken, and the mistake had a decidedly injurious effect upon his whole conception. But for this, he would have extended his question concerning the origin of our synthetical judgments far beyond the metaphysical concept of causality and included in it the possibility of mathematics a priori also, for this latter he must have assumed to be equally synthetical. And then he could not have based his metaphysical propositions on mere experience without subjecting the axioms of mathematics equally to experience, a thing which he was far too acute to do. The good company in which metaphysics would thus have been brought would have saved it from the danger of a contemptuous illtreatment, or the thrust intended for it must have reached mathematics, which was and could not have been Hume's intention. Thus that acute 401 man could have been led into considerations which must needs be similar to those that now occupy us, but would have gained inestimably by his inimitably elegant style. In other words, Hume failed to notice that mathematics, despite its a priori character, is synthetic. It was thus easy for him to conclude that all a priori judgments are analytic, and that therefore metaphysics (conceived of as synthetic ¿ priori judgments) is impossible. Now what considerations led Kant himself to the opposite conclusion, that mathematics is synthetic despite being a priori? Let's consider Kant's own words in a typical passage: Just as little is any principle of geometry analytical. That a straight line is the shortest path between two points is a synthetical proposition. For my concept of straight contains nothing of quantity, but only a quality. The concept 'shortest' is therefore altogether additional and cannot be obtained by any analysis of the concept 'straight line.' Here, too, intuition must come to aid us. It alone makes the synthesis possible. I should like the reader now to compare the following passage from Hume's Treatise of Human 2 Nature, Book I, Part II, Section IV: 'Tis true, mathematicians pretend they give an exact definition of a right line, when they say, it is the shortest way betwixt two points. But in the first place, I observe, that this is more properly the discovery of one of the properties of a right line, than a just definition of it. For I ask any one, if upon mention of a right line he thinks not immediately on such a particular appearance, and if 'tis not by accident only that he considers this property? A right line can be comprehended alone; but this definition is unintelligible 402 without a comparison with other lines, which we conceive to be more extended. In common life 'tis establish 'd as a maxim, that the streightest way is always the shortest; which wou'd be as absurd as to say, the shortest way is always the shortest, if our idea of a right line was not different from that of the shortest way betwixt two points. Style and terminology apart, I defy the reader to distinguish between these two arguments and their conclusions. Hume argues that 'The shortest distance between two points is a straight line' is no definition; furthermore he says almost explicitly that 'There is a longer distance between two points than a straight line...

Discussions of the applicability of mathematics in the natural sciences have been flawed by failure to realize that there are multiple senses in which mathematics can be ‘applied’ and, correspondingly, multiple problems that stem from the applicability of mathematics. I discuss semantic, metaphysical, descriptive, and and epistemological problems of mathematical applicability, dwelling on Frege's contribution to the solution of the first two types. As for the remaining problems, I discuss the contributions of Hartry Field and Eugene Wigner. Finally, I argue that there are epistemological problems concerning the applicability of mathematics that nobody in the philosophical community has yet confronted, though the problems are well known to physicists.

During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are ‘internally’ connected to their canonical applications, i.e. , that mathematical theorems are ‘hardened’ empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications for all of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay.

Remarks on the Foundations of Mathematics, Wittgenstein, despite his official 'mathematical nonrevisionism', slips into attempting to refute Gödel's theorem. Actually, Wittgenstein could have used Gödel's theorem to good effect, to support his view that proof, and even truth, are 'family resemblance' concepts. The reason that Wittgenstein did not see all this is that Gödel's theorem had become an icon of mathematical realism, and he was blinded by his own ideology. The essay is a reply to Juliet Floyd's work on Gödel: what she says Wittgenstein said, I say he should have said, but didn't (couldn't).