Do Mathematicians Quarrel?

Abstract

Mathematics is often taught as an “absolute science” where there is a clear distinction between true and false or right and wrong, which should be uni- versally accepted by all professional mathematicians and every enlightened layman. This is true to a large extent, but there are important aspects of mathematics where agreement has been lacking and still is lacking. The development of mathematics in fact includes as fierce quarrels as any other science. In the beginning of the 20th century, the very foundations of mathematics were under intense discussion. In parallel, a split between “pure” and “applied” mathematics developed, which had never existed before. Traditionally, mathematicians were generalists combining theoretical mathematical work with applications of mathematics and even work in mechanics, physics and other disciplines. Leibniz, Lagrange, Gauss, Poincaré and von Neumann all worked with concrete problems from mechanics, physics and a variety of applications, as well as with theoretical mathematical questions.

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)