1 What is Intuitionism? Intuitionism
Abstract:
, who developed, in the years following 1905, a completely new philosophy of mathematics. I'll try to outline two basic points here, without attempting to go too deeply into the quite sophisticated philosophical discussion which ensued [the curious reader is referred to the books and papers of Troelstra, Beeson, Bishop & Bridges and Smorynski in the references]. In order to understand these points it is perhaps useful to recall some developments in late 19th century mathematics which to some extent rocked the average mathematician's working intuition. These were the creation of set theory by Cantor, and the development of mathematical logic by Frege and Russell. Both were very successful attempts to codify mathematics and mathematical thinking, and Russell went so far as to propose that mathematics was not more than a deductive system, completely subsumed by logic. This view was also taken by his pupil Wittgenstein who taught that mathematical truths are vacuous tautologies. Brouwer opposed this view vehemently. Against it he raised two related philosophical issues [this is an extremely simplified picture]: solipsism and the role of language. ffl Solipsism states, in a crude form, that man is basically alone and has no way of firmly establishing the existence of an outside world, let alone facts about such a world. Mathematics can therefore not be concerned with the discovery of absolute truths which are, in some way or another, hidden `out there'. The
Citations
| 248 | Vulnerabilities Analysis – Bishop - 1999 |
| 206 | 1985], Foundations of Constructive Mathematics – Beeson |
| 34 | Intuitionistic propositional logic is polynomial-space complete – Statman - 1979 |
| 4 | Logical number theory I: An introduction – Smoryński - 1991 |
| 3 | Logic in the writings of Brouwer and Heyting, in: Atti del Convegno Internazionale di Storia della Logica, San Gimignano, 4-8 dicembre 1982, edited by – Troelstra - 1983 |
| 1 | On the early history of intuitionistic logic – Troelstra |
| 1 | of constructivism in the twentieth century – Troelstra |
