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By Reuben Hersh

Collection of the main fascinating fresh writings at the philosophy of arithmetic written via hugely revered researchers from philosophy, arithmetic, physics, and chemistry

Interdisciplinary ebook that would be beneficial in different fields—with a cross-disciplinary topic region, and contributions from researchers of varied disciplines

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First published 1964]. Beth, Evert Willem, 1957. La crise de la raison et la logique, Gauthier-Villars, Paris, and Nauwelaerts, Louvain. Cellucci, Carlo, 1998a. Le ragioni della logica, Laterza, Bari. [Reviewed by Donald Gillies, Philosophia Mathematica, vol. 7 (1999), pp. 213-222]. Cellucci, Carlo, 1998b. ‘The scope of logic: deduction, abduction, analogy’, Theoria, vol. 64, pp. 217-242. Cellucci, Carlo, 2000. ‘The growth of mathematical knowledge: an open world view’, in Emily R. ), The growth of mathematical knowledge, Kluwer, Dordrecht, pp.

Is he a discoverer as the sailor or an inventor as the painter? HIPPOCRATES It seems to me that the mathematician is more like a discoverer. He is a bold sailor who sails on the unknown sea of thought and explores its coasts, islands and whirlpools. SOCRATES Well said, and I agree with you completely. I would add only that to a lesser extent the mathematician is an inventor too, especially when he invents new concepts. But every discoverer has to be, to a certain extent, an inventor too. For instance, if a sailor wants to get to places which other A Socratic Dialogue on Mathematics 11 sailors before him were unable to reach, he has to build a ship that is better than the ships other sailors used.

The view that mathematics is in essence derivations from axioms is backward. In fact, it’s wrong”47. The limitations of the axiomatic method are also acknowledged by some supporters of the dominant view, like Mayberry, who recognizes that “no axiomatic theory, formal or informal, of first or of higher order can logically play a foundational role in mathematics”48. For “it is obvious that you cannot use the axiomatic method to explain what the axiomatic method is”49. Since any theory put forward “as the foundation of mathematics must supply a convincing account of axiomatic definition, it cannot, on pain of circularity, itself be presented by means of an axiomatic definition”50.

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