By Reuben Hersh
Collection of the main attention-grabbing fresh writings at the philosophy of arithmetic written through hugely revered researchers from philosophy, arithmetic, physics, and chemistry
Interdisciplinary booklet that may be important in numerous fields—with a cross-disciplinary topic region, and contributions from researchers of varied disciplines
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Extra info for 18 Unconventional Essays on the Nature of Mathematics
For mathematics “is the purest product of conceptual thought, which is a feature of human life that both pervasively structures it and sets it apart from all else”76. Mathematics is “unconstrained by experience”, enters “the world touched only by the hand of reflection”, and is “justified by pure ratiocination, perceptual observation via any of our five sensory modalities being neither necessary nor even relevant”77. 73 74 75 76 77 Leary 2000, p. 3. Maddy 1997, p. 205, footnote 15. Franks 1989b, p.
The axiomatic method is available for the whole of mathematics, because “all mathematical theories, when sufficiently developed, are capable of axiomatization”39. Moreover, the axiomatic method provides “a strategy both for finding and remembering proofs”, because “relatively few properties, Bourbaki’s few, so-called basic structures, have been found adequate for similar strategies in a very broad domain of mathemathics”, although “the use of axiomatic analysis as a proof strategy does not seem to be well known to people writing on heuristics, like Polya”40.
The sailor also can build any sort of ship at his discretion, but, of course, he would be mad to build a ship which would be crushed to pieces by the first storm. Now I think that everything is clear. SOCRATES If you see everything clearly, try again to answer the question: what is the object of mathematics? HIPPOCRATES We came to the conclusion that besides the world in which we live, there exists another world, the world of human thought, and the mathematician is the fearless sailor who explores this world, not shrinking back from the troubles, dangers and adventures which await him.