Abstract
My text this morning comes from a highly regarded German algebraist of the nineteenth century, Leopold Kronecker. He was engaged in a vigorous debate with other mathematicians on exactly this question, what are the foundations of mathematics? What set of axioms can be used as a basis for deriving everything else? Most mathematicians were following directions of thought which invoked the concept of infinity, infinite sequences, infinite sets, infinite classes, and reciprocally, the notion of infinitesimal. By consistent application of logic, their premise inexorably led to further elaborations such as trans-finite numbers which are esoteric in the experience of most of us. Kronecker vigorously asserted that following this path was an error. He claimed that one must begin, and not only begin but remain content with the integers and those things which can be described algorithmically in terms of them. “God made the integers, and all the rest is the work of man.”
- 1 See, for example, the very readable book entitled G&ohuml;del's Proof, Nagel and Newman, New York University Press, 1958.Google Scholar
- 2 For a brief but lucid account of the issues and history, see the article on Foundations of Mathematics by S. C. Kleene in the Encyclopedia Britannica, 14th Ed., 1973 printing.Google Scholar
- 3 Washington Post December 16, 1974, p. A-1 and A-8.Google Scholar
Index Terms
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