Brouwer criticises Cantor e.g. in Intuitionistiche Mengenlehre. Is there a link or reference to some streamlined modern account of Brouwer's ideas?
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$\begingroup$ It is a common idea that early intuitionism propounds the point of view that there are only denumerably many objects. It would be nice to have an accessible reference for this. $\endgroup$– Frode Alfson BjørdalCommented Jan 11, 2015 at 1:31
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$\begingroup$ When you say "streamlined modern account", do you mean a mathematicized account: one written in modern-day mathematical language? Or do you mean more of a retrospective that comments on what Brouwer said and published? $\endgroup$– Todd TrimbleCommented Jan 11, 2015 at 15:02
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$\begingroup$ If you want something streamlined and modern, I wouldn't look at Brouwer's intuitionist ideas. $\endgroup$– user44143Commented Jan 11, 2015 at 17:24
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$\begingroup$ @Todd I am interested in in what you take to be a mathematicized account. $\endgroup$– Frode Alfson BjørdalCommented Jan 11, 2015 at 22:36
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$\begingroup$ @Matt I am interested in a streamlined modern account of Brouwer, and not an account of Brouwer as streamlined or modern. $\endgroup$– Frode Alfson BjørdalCommented Jan 11, 2015 at 22:38
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3 Answers
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The life story and the scientific biography of Brouwer are the subject of the book L.E.J. Brouwer – Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life by Dirk van Dalen.
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$\begingroup$ Does the book treat the topic? I would prefer something available online at least by means of JSTOR. $\endgroup$ Commented Jan 11, 2015 at 12:15
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Troelstra is a well-known exponent of intuitionism. Here are two online articles that contain philosophical and historical material that may be useful to you:
You might check out particularly section 6 (of the latter article) on Brouwer's intuitionistic analysis and account of the continuum, and some of the subsequent history.
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$\begingroup$ Thanks for these. I have been aware of Troelstra and others as famous exponents of intuitionism. But your reference here does not to me seem to locate something that vindicates that Brouwer had ideas which somehow successfully (justifies by its own philosophy, as it were) evaded Cantor's idea that there are uncountable sets. $\endgroup$ Commented Jan 11, 2015 at 23:11
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1$\begingroup$ Yes, I think the story must be complicated (and unfortunately I'm no scholar here). The second reference touches on Brouwer's notion of choice sequences which were apparently crucial to his conception of the continuum. Thus (and I think this gets closer to your concerns), my rough reading is that Brouwer could not accept the idea of the continuum as a "bag of points" which could be considered in isolation, but considered that the continuum had to be given as a whole (and so he rejected the classical constructions of the continuum). $\endgroup$ Commented Jan 11, 2015 at 23:48
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$\begingroup$ What's a little puzzling to me about this discussion is that, to the best of my knowledge, contemporary mathematicians accept Cantor's argument that a set cannot map onto its power set as completely consistent with intuitionistic principles (I am not considering objections coming from predicative mathematics, where power sets are rejected). $\endgroup$ Commented Jan 11, 2015 at 23:52
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$\begingroup$ I am puzzled for the same reason. $\endgroup$ Commented Jan 12, 2015 at 0:05
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Laura Crosilla has a rather comprehensive bibliography for constructive and intuitionistic set theory on her website. Including her SEP entry on the subject. Some of the papers, like Grayson's on Heyting-valued models, actually develop some mathematics inside the set theory they define.
This might be relevant re what intuitionism has to propose in the stead of Cantor's development.
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$\begingroup$ Theanks. It seems to me that Laura Crosilla deals with modern intuitionistic set theories which either (1) do not have the power set or (2) accept Cantor's theorem that there are uncountable sets. I am looking for something with Brouwer that was geared to avoid uncountable sets. $\endgroup$ Commented Jan 11, 2015 at 23:06