Download 18 Unconventional Essays on the Nature of Mathematics by Reuben Hersh PDF

By Reuben Hersh

This startling new choice of essays edited via Reuben Hersh includes frank evidence and reviews from top mathematicians, philosophers, sociologists, cognitive scientists, or even an anthropologist. every one essay offers a difficult and thought-provoking examine contemporary advances within the philosophy of arithmetic, demonstrating the probabilities of considering clean, sticking just about real perform, and fearlessly letting cross of ordinary shibboleths.

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Extra resources for 18 Unconventional Essays on the Nature of Mathematics

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It is the only discipline in which 46 47 48 49 50 51 52 53 54 Giusti 1999, p. 20. Hersh 1997, p. 6. Mayberry 1994, p. 34. , p. 35. Ibid. Maddy 1984, p. 49. Dummett 1991, p. 305. Prawitz 1998, p. 332. Ibid. “Introduction” to Filosofia e matematica 27 deductive logic is the sole arbiter of truth. As a result mathematical truths established at the time of Euclid are still held valid today and are still taught. No other science can make this claim”55. ) inferences in addition to deductive inferences.

That mathematics is not a body of truths does not mean that it has no objective content. , statements compatible with existing knowledge. The objectivity of mathematics does not depend on its being a body of truths but on its being a body of plausible statements. Moreover, the idea that mathematics is a body of absolutely certain and hence irrefutable truths, overlooks the fact that we cannot be sure of the current proofs of our theorems. For, by Gödel’s incompleteness theorems and related results, we cannot be sure of the hypotheses on which they are based.

The activity “of a creating brain has never had any rational explanation, neither in mathematics nor in other fields”59. In particular, the discovery of axioms has nothing to do with logic, because “there is no hope, there is, as it were, a leap in the dark, a bet at any new axiom”, so “we are no longer in the domain of science but in that of poetry”60. Generally, “the creative and intuitive aspects of mathematical work evade logical encapsulation”61. The “mathematician at work relies on surprisingly vague intuitions and proceeds by fumbling fits and starts with all too frequent reversals.

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