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I don't think the questions on MO are representative of what is considered mainstream mathematics. Fads on MO come and go. When it started, MO was mostly algebraic geometry discussion. A few other groups have been prominent in the mean time. Presumably other groups will appear more common in the future.
I see nothing at all wrong with asking mathematical questions about predicative mathematics, which is certainly alive and well and intensively studied. However, the question that you linked to is not at a professional level, and should under reasonable standards be closed (perhaps as 'not a real question').
Questions that have an ideological axe to grind, so to say, would also generally be viewed askance.
Speaking only for myself, calls to restrict exploration into acquiring knowledge is profoundly anti-scientific and anti-mathematical.
Maybe I'm not understanding what trb456 has in mind here, but I can safely say that many, probably most professionals who pursue predicative mathematics, constructive mathematics, etc. do not do so because of some ingrained philosophical prejudice, but for pragmatic reasons, and such pursuits are inevitably all about expansion of knowledge, not suppression. In the case of intuitionist or constructive mathematics, a major point is that by weakening the logic, one can dramatically expand the worlds or semantics in which the mathematics will still be valid -- quite a powerful tool. For example, categorical logicians are frequently interested in intuitionist mathematics because the results therein are valid in toposes much more general than the category of sets. Another case study is intensional dependent type theory, which is exceedingly active these days.
From my experience as both a constructivist and a mathematician working quite close to computer science, I think trb456's point number 1 is the main reason for the ideological shift towards constructivism, along with possibly the fact that ZFC, too, was found insufficient for what homological and homotopical algebraists nowadays want and seems to require a lot of patching (of course, constructivism in its current form isn't a panacea to this; if it was, it probably would be the leading foundation of mathematics by now). Since constructive proofs are stronger (in the meaning of conveying more information) than classical ones, I don't believe that this shift is anti-scientific, at least when it leads to rewriting and re-proving results in a constructive way rather than just throwing them away because their usual formulation is not computational.
An annoying side effect of this particular shift is, of course, that cranks have quickly caught up to it because it is quite visible (even MO had its share of legit constructivism discussions already) and everybody, except probably mathematicians, seems to believe he is perfectly capable of understanding any issue on mathematical logic. The latter reason seems to be the prevalent one -- I've seen a lot of logic cranks without any finitist/constructivist agenda. (Cantor is still the most popular subject: http://scientopia.org/blogs/goodmath/tag/cantor-crank/ . And this one works just as well in constructive logic, even if "uncountable" isn't the same as "bigger than countable" there.)
I came to meta because I didn't agree with closing the question that trb456 mentions and I wanted to see if there was a meta discussion about it. I have voted to reopen. The question is not ideally written but it is a real question and a legitimate one for MO in my opinion. (Should I start a separate meta thread about this?)
Regarding trb456's question, I'm not closely in touch with pedagogical practice nowadays, but my impression is that there are very few if any people "officially" trying to indoctrinate the next generation with finitist philosophical presuppositions. What I see happening is similar to what trb456 mentioned: the influence of computer science has caused increasing numbers of people to develop a feeling that reality is finite and discrete and anything else is just so much metaphysical nonsense. People with this kind of attitude may not consciously try to promote it as an agenda, but it has a tendency to spill out whether they intend to or not.
In some ways, I prefer the zealots who are open about their agenda to the "silent majority" who don't state their assumptions explicitly, because the former tend to have thought through their position more carefully and are less likely to exude pure prejudice.
Perhaps http://mathoverflow.net/questions/133789/standard-natural-numbers-do-not-form-a-set-closed should be in this thread too.
In full it says:
The standard natural numbers do not form a set. Why is that?
IMO, this is not unrelated.
I really wished standards regarding "foundational question" were somewhat more in line with those of the rest of the site.
Now, that my main point got confirmed so quickly and so amply is quite amazing. :-)
Option 2.
Another vote for Option 2.
And as a compromise solution: AFAIK, but I might be missing something, in such cases you could delete the accounts of OP of question, preserving whatever of the content might be perceived as valuable but keeping the danger from such accounts (in particular as soon as they have some points) under control.
Re Scott's query, I'd like to mention that cranks sometimes leave destructive comments on posts other than their own. So while I interpret Option 2 as "Delete all of the crank's questions and answers", my own choice would be Option 2-prime: "Delete all of the crank's questions, answers, and comments." (I'm sure Scott will realize that I have a specific crank in mind, but I'd favor Option 2-prime quite generally.)
Option 2. The number of inappropriate posts seems to be increasing and is a distraction.
WM on main 'on this topic' http://mathoverflow.net/questions/133867/is-subcountable-a-valid-substitute-for-uncountable [Added: now, deleted, so not generally visible anymore]
I vote for option 2. Also, could their comments on meta also be removed?
This discussion has veered off topic.
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