18 Unconventional Essays on the Nature of Mathematics by Reuben Hersh

By Reuben Hersh

This startling new number of essays edited by way of Reuben Hersh includes frank proof and reviews from prime mathematicians, philosophers, sociologists, cognitive scientists, or even an anthropologist. every one essay presents a not easy and thought-provoking examine contemporary advances within the philosophy of arithmetic, demonstrating the chances of considering clean, sticking just about genuine perform, and fearlessly letting pass of ordinary shibboleths.

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Perhaps you can help me. HIPPOCRATES I would do so with pleasure, but I am afraid you are mocking me again. Do not make me ashamed by asking my help, but tell me frankly the question which I overlooked. SOCRATES You will see it yourself if you try to summarize the results of our discussion. HIPPOCRATES Well, when it became clear why mathematics is able to give certain knowledge about a world different from the world in which we live, about the world of human thought, the question remained as to the use of this knowledge.

According to the dominant view, the logic of mathematics is deductive logic. For theorems “are justified by deductive inference”51. In fact, “deductive inference patently plays a salient part in mathematics. The correct observation that the discovery of a theorem does not usually proceed in accordance with the strict rules of deduction has no force: a proof has to be set out in sufficient detail to convince readers, and, indeed, its author, of its deductive cogency”52. Admittedly, “deduction is only one component in mathematical reasoning understood in the broad sense of all the intellectual work that goes on when solving a mathematical problem.

Uncertainty and doubt have replaced the self-complacent certainty of the past. As some supporters of the dominant view, like Leary, also acknowledge, by Gödel’s incompleteness theorems and related results, “mathematics, which had reigned for centuries as the embodiment of certainty, had lost that role”73. 12. According to the dominant view, the question of the applicability of mathematics to the physical sciences is inessential for the philosophy of mathematics. Mathematics is “a unified undertaking which we have reason to study as it is, and the study of the actual methods of mathematics, which includes pure mathematics, quickly reveals that modern mathematics also has goals of its own, apart from its role in science”74.

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