Errett Bishop - Quotes

Quotes

• (A) "Mathematics is common sense";
• (B) "Do not ask whether a statement is true until you know what it means";
• (C) "A proof is any completely convincing argument";
• (D) "Meaningful distinctions deserve to be preserved".

(Items A through D are principles of constructivism from his 1973 Schizophrenia in contemporary mathematics, reprinted in Rosenblatt 1985)

• "The primary concern of mathematics is number, and this means the positive integers. . . . In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself." (Bishop 1967, Chapter 1, A Constructivist Manifesto, page 2)
• "We are not contending that idealistic mathematics is worthless from the constructive point of view. This would be as silly as contending that unrigorous mathematics is worthless from the classical point of view. Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof." (Bishop 1967, Preface, page x)
• "Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable. The proof is essentially Cantor's 'diagonal' proof. Both Cantor's theorem and his method of proof are of great importance." (Bishop 1967, Chapter 2, Calculus and the Real Numbers, page 25)
• "The real numbers, for certain purposes, are too thin. Many beautiful phenomena become fully visible only when the complex numbers are brought to the fore." (Bishop 1967, Chapter 5, Complex Analysis, page 113)
• "It is clear that many of the results in this book could be programmed for a computer, by some such procedure as that indicated above. In particular, it is likely that most of the results of Chaps. 2, 4, 5, 9, 10, and 11 could be presented as computer programs. As an example, a complete separable metric space X can be described by a sequence of real numbers, and therefore by a sequence of integers, simply by listing the distances between each pair of elements of a given countable dense set. . . . As written, this book is person-oriented rather than computer-oriented. It would be of great interest to have a computer-oriented version." (Bishop 1967, Appendix B, Aspects of Constructive Truth, pages 356 and 357)
• "Very possibly classical mathematics will cease to exist as an independent discipline” (Bishop, 1970, p. 54)
• “Brouwer’s criticisms of classical mathematics were concerned with what I shall refer to as ‘the debasement of meaning’ ” (Bishop in Rosenblatt, 1985, p. 1)