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    • CommentAuthortrb456
    • CommentTimeJun 13th 2013
     
    This is my first time on Meta, so please be gentle. I am mostly a lurker on MO and somewhat active on SE.

    I want to make my query specific. This question, http://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-without-axiom-of-power-set, is only the latest in a series of many questions (it seems to me) questioning the validity of transfinitism and nonconstructivism. Any trained mathematician knows that these controversies have been around for a long time. So my question is: why now? Has something happened in the world of mathematics or math education that is causing what seems to me a rethinking on this?

    I saw one comment that jokingly suggested it was Wiles' fault for solving FLT, with the cranks moving on to Cantor and the like. More seriously, though:

    1) Is the rise of computer science and coding in importance driving any of this? Specifically, does anyone actually involved in CS see examples of educators opening suggesting/calling for finitist teaching universally? I know of some examples, but how widespread? I realize that theoretical CS deals with essentially finite objects, but why now the urge to shut down the rest of the mathematical world?

    2) It strikes me that finitism and the like is a claim that we should restrict knowledge, since finitism and constructivism are perfectly fine subsets of classical mathematics. If you a writing a computer program, what you are doing is necessarily computable. But why restrict the exploration into the non-computable? Again, I'm looking for examples from the practicing mathematicians here of examples they see where this idea of pedagogy is taking hold. I can understand how much of this is non-intuitive at first, but a key part of math education is learning to follow the logic wherever it leads. Why now the urge to suppress this?

    I ask this because I am not a practicing mathematician or an educator (I have an MS degree, and use math in an applied field; but I am thoroughly classical in outlook). Perhaps I'm just seeing things that aren't there. Perhaps increasing interconnectedness just amplifies some voices beyond their actual influence. But I sense that it may be more than this, and since I don't actively deal with the research and educational community, I'm interested in their views. I'm wondering if some more grassroots in education is happening.

    Speaking only for myself, calls to restrict exploration into acquiring knowledge is profoundly anti-scientific and anti-mathematical. As I saw one wise contributor suggest, if some area doesn't interest you, study something else. But why the desire to shut off other discussion? I'm talking about non-crank motives--we'll never reach them. In particular, why teach impressionable students to not be curious and chase knowledge wherever it leads? Again, not here to discuss philosophy, but is anyone here seeing this urge in some area of teaching, and if so, what are their stated motives?

    Thanks in advance!
  1.  

    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.

  2.  

    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.

  3.  

    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.)

  4.  
    I consider it perfectly reasonable to ask what can be proved in various limited axiom systems, especially if those systems admit reasonably natural models. In particular, the *title* of the question about power set looks quite reasonable to me. For a full-fledged question, one should, of course, say which axiom system one intends to remove the power set axiom from, and, as Joel Hamkins explains in his answer, there is interesting mathematics there. My objection to the body of the question is that it presupposes that "set" ought to mean "definable set". That is a plausible idea, and it was debated at considerable length by prominent mathematicians early in the 20th century, mostly in connection with the axiom of choice. Eventually, most mathematicians came to accept that definability should not be required, partly because the axiom of choice leads to nice results, but mostly because of the difficulties that arise when one tries to make notion of definability precise. Especially since Tarski's work on undefinability of truth, but to some extent already earlier in paradoxes like "the smallest natural number not definable in fewer than twenty English words", people realized that definability is a subtle issue, probably too subtle to be built into the very foundations of mathematics. So when a question comes along that ignores the subtleties and presupposes (without supporting argument) the opposite of the generally accepted answer to "must sets be definable", people like me react with "Oh, yuck."

    Although the question about power set seems to be based on a form of "countabilism" or "definabilism" rather than "finitism", I regard finitism and even ultrafinitism as respectable topics for mathematical investigation. (In the case of ultrafinitism, the main technical problem seems to be to provide an axiomatization that captures the intended ideas. In the case of finitism, the system PRA of "primitive recursive arithmetic" seems to be widely accepted as an appropriate foundation, but I don't know how solid the philosophical support for this acceptance is.)

    The bottom line, for me, is that I have no problem with people saying "we should study such-and-such notion of set" and asking "what can we prove with this notion", provided they give a clear indication of what their notion is, and provided they don't preach "this is the right notion of set and everybody who uses a different notion is crazy."
    • CommentAuthorgrp
    • CommentTimeJun 13th 2013
     
    I'd say postings on MathOverflow are too small a sample space to use to draw
    conclusions on the questions you are asking. Likely as not, a single person or
    small team of people are posting the questions. I think you need to gather
    more data before presupposing a shift in a larger system of education.

    Regarding the subject matter, I think well behaved and cogent questions
    within the intended demesne of MathOverflow are allowed even if they
    are contrary to some conventional understanding of how things are.
    The current question is close enough to it, and allows for specific answers,
    and does not suggest that its point of view should be universal, just that it
    should be considered. This is a slightly different presentation of Andreas
    Blass's bottom line, with which in the main I agree.

    Gerhard "Always Watch For The Bottom Line" Paseman, 2013.06.13
    • CommentAuthortrb456
    • CommentTimeJun 13th 2013
     
    Thanks very much for the great comments. I think Andreas' bottom line also reflects my view, and darijgrinberg's comment on CS is helpful to hear. I guess my only remaining question is whether anyone is aware of finitist ideas being pursued not as good research or bad crankdom, but perhaps at the level of undergraduate pedagogy; e.g. finitist methods are being viewed by some educators as a more helpful/useful way to teach math. MO might not be the right forum to ask this, as I assume you are all mostly practicing researchers, and don't necessarily know about what's happening in undergraduate education. Thanks again!
  5.  
    Concerning undergraduate pedagogy, Norm Wildberger has some videos on the web teaching undergraduate mathematical topics, and some of his videos express his finitist views on set theory. See for example.
    • CommentAuthorDrewS
    • CommentTimeJun 14th 2013
     
    I, too, am primarily a lurker. I just signed up for an account, and wanted to let you know that I am pretty sure, no almost completely sure, that Albino is William Mueckenheim. I can tell, as he presents himself as a "Classical Mathematician", and by the way he reference Cantor as being the culprit for the power set in Set Theory.

    One of his favorite rants on google sci.logic/math is that since we can't have uncountable names for things, we can't have uncountably many real numbers. He attempts to formulate "alternative theories", but nothing ever pans out.
    He presents unfounded proofs of internal contradictions, and would accept his logic errors. He only writes statements in plain English and then uses ambiguities in the language to show that he is right.

    I sure you know what I'm talking about.
    Just a heads up.

    Thanx, Drew
    • CommentAuthortrb456
    • CommentTimeJun 14th 2013
     
    Gerry Myerson, I definitely know about Wildberger, and I was wondering if approaches like his were becoming more widespread. As discussed above, I agree it's not a problem discussing finitism within the context that it is a subfield of classical, certainly legitimate for research, but not universally accepted foundationally. But if you tell students at the undergraduate level that perfectly acceptable classical mathematics is a sham, this just creates problems getting students interested in research mathematics. This is what concerns me.
  6.  
    DrewS: I think Mueckenheim's first name is Wolfgang, not William.

    trb456: I agree that teaching undergraduate students that ordinary classical mathematics is wrong will create difficulties, and not only for their potential interest in research mathematics. It will create difficulties for their understanding of the next semester's course if that's taught in the normal way. Even undergraduate mathematics, especially analysis, becomes considerably more difficult to develop if one restricts to subsystems rather than using the full power of classical mathematics. Bishop has shown how to carry out the development of a good deal of analysis in a constructive system, and I believe there were earlier efforts along such lines by Lorenzen, but the weaker logical and set-theoretic framework must be compensated by more work and more careful statements of theorems. (Some would say "weakenings" or "circumlocutions" instead of "more careful statements".) Reducing the framework further, to a form of finitism, we have even more difficult (to the best of my knowledge) work of Nelson. I don't think anything like this belongs in the undergraduate curriculum, except perhaps in a specialized and rather advanced course in foundations, for students who already know the standard approach.
    • CommentAuthortrb456
    • CommentTimeJun 14th 2013 edited
     
    Andreas Blass: Could not have said it better myself. Everything you said is exactly what I think, and what I potentially fear!
  7.  

    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.

    • CommentAuthorquid
    • CommentTimeJun 14th 2013 edited
     

    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.

    • CommentAuthorquid
    • CommentTimeJun 14th 2013
     

    Now, that my main point got confirmed so quickly and so amply is quite amazing. :-)

    • CommentAuthortrb456
    • CommentTimeJun 15th 2013
     
    My reason for starting this thread was not about closing questions, but I don't mind if they are discussed (as long as that is OK under Meta's rules).

    @Timothy Chow: Well said. The problem with many of these discussions is unstated or unclear assumptions.
  8.  
    Perhaps it is time to open a general policy question to the community. We essentially have the following choices:

    1. keep a crank's question open when there are interesting answers.

    2. make everything written by said crank disappear, even if it is a reasonable-looking question with good answers.

    We've tried option 1 in the recent past. It seems to lead to a general increase in pointless arguments and some unhappy users. We've also tried Option 2, and it seems to lead to some collateral damage from deleted answers in the form of unhappy users.

    Timothy Chow seems to advocate for option 1, but perhaps without full recognition of the context.
  9.  
    My vote is for option 2. There seem to have been a lot of cranks here recently, and I want them to feel as unwelcome as is possible within the bounds of remaining professional.
  10.  

    Option 2.

    • CommentAuthorquid
    • CommentTimeJun 15th 2013
     

    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.

    • CommentAuthorvoloch
    • CommentTimeJun 15th 2013
     
    Option 2.
  11.  

    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.)

  12.  
    Also, and with due hesitation about suggesting anything that makes more work for the moderators, it might be good to maintain an official repository of good answers to deleted questions.
  13.  

    Option 2. The number of inappropriate posts seems to be increasing and is a distraction.

    • CommentAuthorMax1
    • CommentTimeJun 16th 2013 edited
     
    Timothy Chow, thanks. I'm on your side.

    " make everything written by said crank disappear..."

    In this way it is possible to classify(blaze) of "not convenient", for someone, by some reasons, mainly subjective(we are people) ,questions/answers/comments as "crank", "spam" and so on.

    In some cases this may leads to excessive usage of administrative resourse(reputation),
    unfair competition with "punitive psychiatry" as main method.
    I wish to MO do not slide towards banal censorship, possible in sheep's clothing.

    The number of closed posts seems to be increasing and is a distraction.
    • CommentAuthorAngelo
    • CommentTimeJun 16th 2013
     
    Another vote for option 2.
    • CommentAuthorquid
    • CommentTimeJun 16th 2013 edited
     

    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]

  14.  
    Option 2 sounds great. I'm usually not a big fan of slippery slope arguments, but this time it seems to have proved itself.

    We can begin with the post quoted by quid.
  15.  
    Can't we combine 1 with discouragement: Suspending the crank and leaving the question open with a moderator note that is was left because of the ineresting answers.
  16.  
    Michael Greinecker: we have tried that, and what often happens is that various arguments are joined by new users with suspiciously similar opinions.
  17.  

    I vote for option 2. Also, could their comments on meta also be removed?

  18.  

    This discussion has veered off topic.