By Reuben Hersh
This startling new choice of essays edited through Reuben Hersh includes frank proof and critiques from top mathematicians, philosophers, sociologists, cognitive scientists, or even an anthropologist. each one essay offers a difficult and thought-provoking examine fresh advances within the philosophy of arithmetic, demonstrating the chances of pondering clean, sticking just about real perform, and fearlessly letting pass of ordinary shibboleths.
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Extra info for 18 Unconventional Essays on the Nature of Mathematics
The reflected image cannot be touched; if I put my hand on it, I would touch only the cool water. As a matter of fact, the reflected image does not really exist; it is illusion, nothing else. SOCRATES image? Is there nothing in common between the rock and its reflected HIPPOCRATES Well, in a certain sense, the reflected image is a faithful picture of the rock. The contour of the rock, even its small abutments, are clearly visible in the reflected image. But what of it? Do you want to say that the world of mathematics is a reflected image of the real world in the mirror of our thinking?
In this book I do not consider all philosophical questions concerning mathematics, even less all philosophical questions concerning knowledge, 80 See, for example, Giaquinto 2002, Shapiro 2000. See Benacerraf-Putnam 1983, Ewald 1996, Hart 1996, Jacquette 2002, Mancosu 1998, van Heijenoort 1977. 82 See, for example, Barbin-Caveing 1996. 83 A basic choice can be found in Baum 1973. 84 See, for example, Tymoczko 1998. 85 Pólya 1948, p. 158. 86 Pólya 1954, I, p. vi. , I, p. v. , I, p. vi. 89 See, for example, Cellucci 1998a, 1998b, 2000, 2002b.
While physical hypotheses come and go, none is definitive, and so “in physics nothing is completely certain”, mathematics, as based on the axiomatic method, “lasts an eternity”71. The view expressed in this book is instead that mathematics is a body of knowledge but contains no truths. Speaking of truth is not necessary in mathematics, just as it is not necessary in the natural sciences, and is not necessary anywhere except perhaps in theology and in lovers’ quarrels. Assuming that mathematics is a body of truths leads to an inextricable muddle, which results in self-defeating statements such as: It is legitimate “to argue from ‘this theory has properties we like’ to ‘this theory is true’” 72.
18 Unconventional Essays on the Nature of Mathematics by Reuben Hersh