Do Mathematicians Quarrel?

Johan Hoffman (), Claes Johnson () and Anders Logg ()
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Johan Hoffman: New York University, Courant Institute of Mathematical Sciences
Claes Johnson: Chalmers University of Technology, Department of Computational Mathematics
Anders Logg: Chalmers University of Technology, Department of Computational Mathematics

Chapter 18 in Dreams of Calculus, 2004, pp 121-139 from Springer

Abstract: Abstract The proofs of Bolzano’s and Weierstrass theorems have a decidedly non-constructive character. They do not provide a method for actually finding the location of a zero or the greatest or smallest value of a function with a prescribed degree of precision in a finite number of steps. Only the mere existence, or rather the absurdity of the nonexistence, of the desired value is proved. This is another important instance where the ”intuitionists” have raised objections; some have even insisted that such theorems be eliminated from mathematics. The student of mathematics should take this no more seriously than did most of the critics. (Courant) I know that the great Hilbert said “We will not be driven out from the paradise Cantor has created for us”, and I reply “I see no reason to walking in”. (R. Hamming) There is a concept which corrupts and upsets all others. I refer not to the Evil, whose limited realm is that of ethics; I refer to the infinite. (Borges). Either mathematics is too big for the human mind or the human mind is more than a machine. (Gödel)

Keywords: Real Number; Natural Number; Mathematics Education; Rational Number; Turing Machine (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-3-642-18586-1_18

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