I'd like to read some well-vetted and modern pieces arguing for intuitionism (the "intuitionist stance"?), which I take to be:

As the intuitionist sees it, the rules of logic used by mathematicians have an empirical character. Certain methods of proof came to be commonly used by mathematicians, and, over the years, were codified into a body of rules. These rules were observably correct in their original context, but—after they were codified—they came to be used uncritically in totally different contexts in which they no longer applied...it is only when they[axioms developed constructively] are transposed to problems involving an infinite domain of objects, or in which the objects are not given by an explicit construction, that the rules are incorrect.

A question I have asked on the main site on this point is doing quite poorly, and I do not want to argue for it, but instead mainly understand why it is not a research level question.

I was hoping that I would get recommendations for articles that argue for intuitionism in way that makes, for example someone like Paul Halmos, understandable in his criticism of non-standard analysis, since he was of the "intuitionist constructivist" group founded by Brouwer. I left this sort of detail out, but perhaps that was a mistake?

What is it exactly, that's so convincing for him regarding this stance? I can barely get my mind around why intuitionism is even a thing, since in some sense, I view mathematics as the "careful manipulation of symbols, based on an arbitrary set of rules".

A link to the question in question: Powerful arguments for the intuitionist stance

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    $\begingroup$ Have you considered the possibility that other people do not view mathematics as a bare exercise in symbol manipulation? $\endgroup$ – S. Carnahan Jan 6 '18 at 21:16
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    $\begingroup$ Putting aside the biased attitude, it sounds like what research you've done here has been minimal and superficial, showing no real acquaintance with intuitionist reasoning, hence "not research level" (in the sense of professionalized mathematics). But one reason for being interested in intuitionistic reasoning is that it applies much more generally; e.g., when working internally in a topos, classical modes of reasoning are no longer valid. You might start with Andrej Bauer's fine article: ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/… $\endgroup$ – Todd Trimble Jan 6 '18 at 21:32
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    $\begingroup$ As an aside: this is the first time I've heard of Halmos being one of the intuitionist constructivist group founded by Brouwer. News to me! (Hint: irony.) $\endgroup$ – Todd Trimble Jan 6 '18 at 21:34
  • $\begingroup$ @S.Carnahan my question clearly showed my bias, but i didn't want to simply "write away" intuitionist thought as "useless" --- instead, I wanted to understand why it is that much smarter people than myself chose to be intuitionists, when on the surface, it seems "crazy" to me. In other words, I do not share my bias because I intend to denigrate, but rather I share my bias to show how difficult/far removed intuitionist concepts are to me (likely because of the style of my education?). $\endgroup$ – user89 Jan 7 '18 at 0:12
  • $\begingroup$ @ToddTrimble I was reading about criticism for non-standard analysis, where it is discussed that Halmos chose a student of his, Bishop, to write one of the most scathing critiques of the subject. Bishop seems very familiar with Brouwer's writing in his critiques: "Our program is simple: To give numerical meaning to as much as possible of classical abstract analysis. Our motivation is the well-known scandal, exposed by Brouwer (and others) in great detail, that classical mathematics is deficient in numerical meaning." [cont.] $\endgroup$ – user89 Jan 7 '18 at 0:19
  • $\begingroup$ @ToddTrimble [cont. from prev.] Since students are often share views with their advisors, and Halmos and Bishop seemed to be on the same page regarding their criticism of n-s-a, I assumed (incorrectly?) that Halmos too must share Brouwer's views to some extent. Could you clarify further (i.e. instead of ironic statements, may I hear what you know plainly?). $\endgroup$ – user89 Jan 7 '18 at 0:19
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    $\begingroup$ Bishop got his PhD under Halmos as an analyst (his brilliance was plainly on display there). His work on constructive mathematics came rather later. As for Halmos: he freely used excluded middle and the axiom of choice in his mathematics without any qualms, and did not identify as a constructivist. Having read his automathography, I come away believing that he wasn't philosophically opposed to nonstandard analysis, but thought it was a language that he didn't speak easily and probably didn't much need. IIRC he (affectionately) said Bishop was eventually "lost" to the cause of constructivism. $\endgroup$ – Todd Trimble Jan 7 '18 at 0:38
  • $\begingroup$ @ToddTrimble Thank you! $\endgroup$ – user89 Jan 7 '18 at 0:57
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    $\begingroup$ Here is an extract of Halmos on Bishop: "He was always a strong-minded man, and I cannot claim to have influenced him much or taught him -- he made himself..." (followed by words of admiration that he wrote a fine thesis, then discovered he had been anticipated, and so scrapped it to write an even greater thesis). "He discovered many of the fundamental concepts about function algebras and the relations among those concepts. Then, almost discontinuously, he "got religion", went into constructive mathematics, wrote the book that made the phrase famous, and started the sect (continued) $\endgroup$ – Todd Trimble Jan 7 '18 at 1:01
  • $\begingroup$ of which he was the leading but somewhat reluctant guru till the day he died. Functional analysis misses him, and so does constructive mathematics, and so, most of all, do we, his friends." (pp. 161-162 of Halmos's I Want To Be A Mathematician). $\endgroup$ – Todd Trimble Jan 7 '18 at 1:05
  • $\begingroup$ @ToddTrimble But, no one just "gets religion" randomly, right? There must be something there that he saw. Do you have any insight on what it is that Bishop saw? Also, I am very much anjoying the AMS article you linked to (it is pleasantly readable). $\endgroup$ – user89 Jan 7 '18 at 1:25
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    $\begingroup$ Well, it may be impossible to say how Bishop got involved in constructivism (but I think we can confidently assert that Halmos had nothing to do with it). What we do know is that, by all accounts, Bishop was a terrifyingly powerful mathematician (see for instance this review of his selected papers: projecteuclid.org/euclid.bams/1183554746). Maybe he had heard that doing analysis constructively was considered to be hopeless, and not being one to shrink from a challenge, looked into the matter and then got hooked. Perhaps that's not a likely explanation, but ... who knows? :-) $\endgroup$ – Todd Trimble Jan 7 '18 at 1:38
  • $\begingroup$ The question is deleted now, but if it's the one that I'm thinking about, then you shouldn't expect people to help you when your question is laden thick with insults of them and their work from the beginning to the end. And yes, the question shows no research effort at all, plenty of explanations of intuitionism is given both in general and on this site. Unless you show that your question is somehow not answered there, there's nothing to discuss. $\endgroup$ – Anton Fetisov Jan 7 '18 at 18:55
  • $\begingroup$ @AntonFetisov There were no insults in that post, because I am not the sort of person who insults people. There was flowery language, which was misconstrued, but there were no insults. I simply do not have a high enough opinion of myself to be able to imagine insulting others. In fact, I hoped to convey that while intuitionist ideas were very alien to me, there are people who clearly think in those terms, and its very likely I am missing something big. $\endgroup$ – user89 Jan 8 '18 at 2:26
  • $\begingroup$ @AntonFetisov It's difficult for me to understand what sort of research you expected me to conduct, given that the question was a reference for powerful arguments for intuitionism (in particular, I requested a comment from the answerer regarding how much of a role the article played in convincing them). The question on the other hand, is research level because it tackles foundational issues that I at least, had not come across in my undergraduate education. Regardless, I would appreciate it if you helped me answer questions I raise for Gerhard Paseman in the comments to his answer below. $\endgroup$ – user89 Jan 8 '18 at 2:28

I think it is a research level question. However, it is not in mathematics.

The question strikes me as being not just foundational in nature, but bordering on philosophy. While it is reasonable to ask it here, this forum is actually intended less for philosophy and more for the technical questions that are part of the realm of the community. I am unsurprised at your question's reception by this community.

There are better fora for your question than MathOverflow. Go seek them.

Gerhard "There Is No PhilosophyOverflow. Yet." Paseman, 2018.01.06.

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    $\begingroup$ Yes, it struck me more as a philosophy question as well. $\endgroup$ – Todd Trimble Jan 7 '18 at 1:05
  • $\begingroup$ Doesn't it influence how people do mathematics though? For example, how they prove things, and which proofs they accept? It seems like a bit of philosophy that has very strong impact in terms of application, rather than the usual philosophy which is a question you consider briefly, before forgetting? $\endgroup$ – user89 Jan 7 '18 at 1:27
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    $\begingroup$ It might even affect which mathematics gets outside funding. I don't know. However, my point here is that the philosophical nature of the post makes MathOverflow a poor fit for it. If you want to make a similar question that is a good fit for this forum, you need to rewrite it. I am turned off by the usage of "garbage heap" and "me". The question comes across like you expect it's our job to convince you of the merits of intuitionism. Also, what you call the intuitionistic stance seems a poor relation to what is actually studied. Gerhard "I Say Start Completely Over" Paseman, 2018.01.06. $\endgroup$ – Gerhard Paseman Jan 7 '18 at 1:37
  • $\begingroup$ Why would you be turned off by "throw into my mental garbage heap"? It was simply a flowery way of saying "before I abandon it"? The "me", was actually referring to the rating the answerer would give to the article they are about to share: e.g. SomeAnswerer writes "I am going to share an article that convinced me because...". I am not sure how two people misconstrued that to somehow mean that I am grandiose enough to declare "convince me, my minions!"---but if two people did, I am probably the one in the wrong. In any case, I have deleted the question. $\endgroup$ – user89 Jan 7 '18 at 3:40
  • $\begingroup$ Also, I have been thinking further about your description of taking a intuitionist/non-intuitionist stance, and I find myself increasingly unsure why it is philosphical, rather than technical. Wouldn't describing it as a philosophical problem mean that one is essentially waving it off as a matter of "it's a simply how you look at things", which is in stark contrast to intuitionist/constructivist opinion that something is "very seriously wrong" with non-constructive proofs? $\endgroup$ – user89 Jan 7 '18 at 3:44
  • $\begingroup$ Also, why is what I call the intuitionist stance in poor relation to what is actually studied? (sorry for the flurry of questions, I certainly appreciate any time you spend answering them) $\endgroup$ – user89 Jan 7 '18 at 3:49
  • $\begingroup$ "There are better fora for your question than MathOverflow." and "There Is No PhilosophyOverflow. Yet." - aren't these claims contradictory to each other? @user89 Would maybe Philosophy.SE be worth considering (they have philosophy-of-mathematics tag.) Or perhaps Mathematics site (which has both constructive-mathematics and philosophy tags.) $\endgroup$ – Martin Sleziak Jan 7 '18 at 7:09

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