3 ms·

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

by chalst 15y ago

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) predicativist. I don't get the point about the cavalier attitude to truth: constructivist rejection of PEM comes from worries that some propositions are not sharp enough to be determinately true or false, the classical Brouwerian examples coming from topology. This stance more deserves to be called pernickety than cavalier.

pash 15y ago
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 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.
- chalst 15y ago
  OK, that is clearer. 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.

by chalst 15y ago

pash 15y ago

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 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.

chalst 15y ago
OK, that is clearer. 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.