From the Editors

In this chapter, Mark Steiner first describes what makes the applicability of mathematics a thorny philosophical problem—and explains that it is really two problems, one that the philosophers have been discussing and a quite different one that physicists have been asking. He then concentrates on one aspect of the problem, which at first seems quite unproblematic, the question of addition. It leads to a better understanding of what Wigner might have meant when he asked, many years ago, about the “Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

Mark Steiner is a Professor of Philosophy in the Faculty of Humanities at the Hebrew University of Jerusalem (socrates.huji.ac.il/Prof–Mark–Steiner.htm). His interests include philosophy of mathematics, philosophy of science, Wittgenstein, and Hume. He has worked on philosophical issues in contemporary mathematics education.

In the first paragraph of this chapter, he has written “to the extent one can talk about a ‘consensus’ of the philosophical community, I'm not in it.” This is because he has consistently worked, independently of the trends in the philosophy of mathematics, on issues involving how mathematics is actually practiced. His work has several times been far ahead of that of his contemporaries. His book, Mathematical Knowledge, published in 1975, was one of the first works to say that none of the traditional foundational schools—logicism, formalism, intuitionism—describe how mathematicians come to have mathematical knowledge. Mathematical knowledge cannot be identified with formal proofs, as mathematicians very rarely give formal proofs. Also, some informal arguments have actually produced mathematical certainty, or near certainty, as in some of the work of Euler.

Philosophers identify Frege with the doctrine that arithmetic truths are “analytic.” But Frege's main goal, as many now realize, was mathematical: to show that the truths of arithmetic are provable, against many who denied this, each for a different reason. Mathematicians had by Frege's time shown that analysis could be grounded in what we would today call second-order arithmetic: arithmetic plus “logic” Frege's goal was to remove the dependence on arithmetic by beefing up the “logic.”

This observation explains what otherwise seems a puzzling digression in the Grundlagen. At Grundlagen §10, Frege suddenly considers the question of whether the laws of arithmetic might not be “inductive truths,” that is, established nondeductively by verifying particular cases. This question is not reducible to the question of whether mathematics is “analytic” in Kant's (or even Frege's) sense. Nor is Frege concerned here, as he is elsewhere, with the more specific claim of John Stuart Mill that arithmetic is empirical. Even if arithmetic is non-empirical, even if it is analytic, it might still be inductive – a possibility that few great philosophers had considered, but that could not escape a mathematician.

Frege's reply is: