By Reuben Hersh

Collection of the main attention-grabbing contemporary writings at the philosophy of arithmetic written by way of hugely revered researchers from philosophy, arithmetic, physics, and chemistry

Interdisciplinary publication that may be worthwhile in different fields—with a cross-disciplinary topic region, and contributions from researchers of varied disciplines

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The view expressed in this book is instead that the reflection on mathematics does in fact require detailed knowledge of mathematics. Neglecting this has led the philosophy of mathematics to deal with marginal issues, deliberately excluding the broader ones. The philosophy of mathematics has done so on the assumption that, although the broader questions are “more interesting, more pressing, more significant than the narrower logical questions that are properly foundational”, the latter are “amenable to solution, whereas solutions to the broader questions may depend upon further advances in mathematics itself, advances which we cannot as yet foresee”21.

SOCRATES Let us go back to the point where we established that the mathematician is not dealing with the number of sheep, ships or other existing things, but with the numbers themselves. Don’t you think, however, that 12 Alfréd Rényi what the mathematicians discover to be true for pure numbers is true for the number of existing things too? For instance, the mathematician finds that 17 is a prime number. Therefore, is it not true that you cannot distribute 17 living sheep to a group of people, giving each the same number, unless there are 17 people?

Inferences in addition to deductive inferences. It is by non-deductive inferences that one finds the hypotheses by which mathematical problems are solved. The logic of mathematics is not, therefore, that studied by mathematical logic, which is simply a branch of mathematics, but consists of a set of non-deductive methods and techniques in addition to deductive methods and techniques, and hence is not a theory but a set of tools. To claim that the logic of mathematics is deductive logic because theorems are justified by deductive inference, restricts mathematical experience to ways of reasoning found only in textbooks of mathematical logic, and neglects those that are really used in mathematical activity.