The new intuitionism
I confer the title ‘intuitionism’ and its cognates, here and throughout, upon intuitionistic mathematics solely, the mathematics inaugurated officially by L.E.J. Brouwer but prefigured in the writings of Paul du Bois-Reymond and Émile Borel. I do not use that word to designate any philosophy or philosophies of mathematics that might, at one time or another, have danced attendance upon the mathematics. Nor do I use ‘intuitionist’ as Felix Klein once did, in his Evanston lectures of 1893 (Klein 1911), to denote a mentality prompting certain mathematicians to favor geometric visualization over algorithmic calculation or logical analysis. Even with that delimitation, I freely admit that the definite article in my title is inappropriate. In the proclamation of another ‘new’ intuitionism, there is little new. Since the day Klein bestowed the appellation, some mathematicians and philosophers, among them Brouwer, have trumpeted the presumptive advantages of the very latest, the very ‘newest’ intuitionism. Surveying the foundational trends of the past forty years, one espies a number of developments, each modish in its day, each then meriting the designation ‘new intuitionism’: the proof-theoretic intuitionism of Georg Kreisel, the category-theoretic intuitionism of William Lawvere, the intuitionistic type theories of Per Martin-Löf, the constructivism of Errett Bishop and that of Paul Lorenzen.
KeywordsLogical Sign Classical Mathematician Choice Sequence Conventional Logic Continuity Theorem
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