11–14 of 14 posts
How (if at all) do you do the same thing with dz's?
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> i'm a lapsed physicist/astronomer that's been interested in intuitionist and constructivist ideas for a long time. yet whenever i ask mathematicians about them i am treated much the same way as i treat people who ask me about new age hippy crap. Somehow I suspect the problem is not "the mathematical community doesn't like constructivism", because e.g. watching MathOverflow it seems to come up a lot, and it apparent…
One wonders whether most mathematicians and physicists reject the author's point of view because to accept it requires either (1) a deep understanding of set theory and its limitations or (2) an attitude towards truth that's rather more cavalier than you'll usually find in the faculty lounge. I suspect that the author himself fails (1) and so must believe what he does due to (2). I will not pretend to understand the…
The condescending tone towards Andrej Bauer is unwarranted, who is from the Dana Scott PhD stable and who has a well-thought out attitude towards constructivism and realist foundations of mathematics. Nelson is respected but not what I would call very influential. I took this to be the point of Bauer's jibe about ultrafinistists. Note that, AFAICS, Nelson does not call himself an ultrafinitist, but rather a (radical)…
Writing 'cavalier attitude to truth' was a poor choice of words, but let me explain what I meant. I agree that intuitionistic math is logically more rigorous, and it sets a higher bar for what can be considered true. But because it rejects the validity of tools like proof by contradiction and the law of the excluded middle—tools that classical mathematicians use to disprove many concepts—intuitionistic math contains a much wider universe of possible mathematical objects. It sets a lower bar for what should be considered untrue.
So when you use mathematical objects that only potentially exist—that is, objects whose existence you can neither prove nor disprove—it seems to most classical mathematicians as more than a bit dishonest, because they work only with objects that must exist. (I brought up Nelson's axiomatic approach to infinitesimals because it is analogous in this regard.) Thus Russell's likening postulates to theft.
Intutionists get to play with toys that other mathematicians don't because the former are, in a sense, more broad-minded about what toys are (could be) out there. That's all I meant by 'cavalier attitude to truth'.
Like I said, I admire this approach, and have tried to pick up some axiomatic approaches myself when they're helpful, but I still do think that Russell's bit of pith is quite apt.
Earlier quoted context omitted.
The condescending tone towards Andrej Bauer is unwarranted, who is from the Dana Scott PhD stable and who has a well-thought out attitude towards constructivism and realist foundations of mathematics. Nelson is respected but not what I would call very influential. I took this to be the point of Bauer's jibe about ultrafinistists. Note that, AFAICS, Nelson does not call himself an ultrafinitist, but rather a (radical)…
In my defense, that's a written-at-five-in-the-morning tone, not a condescending tone. I'm not sure what I was trying to write with the "I imagine the author himself fails (1)" bit, as I am sure Bauer is far beyond me on the subject, as I admitted in my comment. Apologies to Bauer and to all for coming off wrong. Writing 'cavalier attitude to truth' was a poor choice of words, but let me explain what I meant. I agree…
I guess we could call the revised argument the "cavalier attitude to refutation". I don't agree about the "much wider universe": Kripke semantics draws a line around what is possible, and tells us that intuitionistic structures can be embedded in families of partial classical structures that evolve towards (but need not reach) a classical structure. Structures inconsistent with this view can be intuitionistically refuted.