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    This question was closed but is nevertheless gathering many upvotes. I agree that the question as posed is borderline contentious, not to mention the fact that it's a statement and not a question; however I believe that there is a good question in there somewhere.

    First, it should not be a question about physicists. There are mathematical results by established mathematicians (Euler, Hodge and Yamabe come immediately to mind, but there are probably many others) which have stood the test of time and have eventually been proved rigourously, but whose original "proofs" had gaps or used questionable manipulations not unlike the ones you might find in the Physics literature and to which the original question alludes.

    The question could be about results which were "proved" but then turned out to be false, regardless the discipline (Physics, Maths,...) at the origin of the "proof". I seem to recall another question along these lines, though -- but I cannot find it. (Search in MO could be improved, I think.)

    • CommentAuthorCSiegel
    • CommentTimeJul 1st 2010
    You might be thinking about this one: about the Italian school of algebraic geometry and their mistakes?

    I notice the question has been reopened.

    Harald, the question was closed by 2 and reopened by 5. And it sounds a nice wiki question, at least if it is understood in a sufficiently broad sense: physics and physicists should not be offended.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 1st 2010 edited

    I reopened it because I hope to use the information in my ongoing fight against physicists.

    Another plus is that it contains the words "Physicists" and "wrong".

    Harry, at least a few folks on MO and in most math departments could nominally be described as physicists. Trolling them is probably not a great idea.
    • CommentAuthorMariano
    • CommentTimeJul 1st 2010

    Well, it is a safe bet that at least some of them, apart from being physicists, have a sense of humour, so it is also not a terribly ungreat idea!

    Regarding physicists and humor, I submit the following well known tale: An engineer, a physicist, and a mathematician were traveling through Scotland when they came across a black sheep. The engineer said, "this sheep is black, so I suppose all sheep in Scotland are black." The physicist said, "don't be ridiculous. All you know is that this particular sheep is black." The mathematician said, "Another completely unmotivated assumption. All you really know is that one side of this particular sheep is black."
    • CommentAuthorWM
    • CommentTimeJul 1st 2010
    It seems to me that mathematicians do not like to be reminded that some of their guild have been wrong. My recent answer, mentioning some of those cases, was promptly deleted. I am new in MO. Is that kind of behaviour customary here?

    Regards, WM
    • CommentAuthorMariano
    • CommentTimeJul 1st 2010

    It was deleted? Is it not the same answer that's there with the 3 downvotes?

    I really think it is quite off-topic, in a question quite explicitely about physicists!


    I certainly think that there is merit in some version of this question -- wasn't there a good question a while back which asked for specific examples, recounted without crowing or sneering, of where the old Italian school of algebraic geometry were led to erroneous statements/predictions by being sloppy about "general position" arguments?


    WM: You have just made a rather broad claim about mathematicians in your comment, and it doesn't seem to be backed by much in the way of evidence. If you look around MathOverflow, you will find a lot of mistakes made by active mathematicians (myself included). The wording of your first sentence suggests a disconnect with the reality of the mathematical community.

    I deleted your answer because its relation to the question is quite tenuous, and because it was flagged as offensive by another user. You never mentioned anything involving physical heuristics in any of your examples, and you wrote the answer in an inflammatory tone. My guess is that you were voted down because you were lowering the signal-to-noise ratio, not because you were challenging the orthodoxy.

    The answer reappeared after deletion because I made the mistake of leaving it unlocked when I deleted it. This is due to my own unfamiliarity with the moderator interface.

    • CommentAuthorWM
    • CommentTimeJul 1st 2010
    Agreed. So I will give another answer. Now concerning physics.

    Regards, WM
    • CommentAuthorWM
    • CommentTimeJul 2nd 2010
    Now I am sure that some mathematicians do not like to be reminded that some of their guild have been wrong. The physical misbeliefs of Cantor were deleted. Is there any reason to extinguish historical facts that fit the question?

    Regards, WM
    @WM: The question is about physical reasoning leading to mathematical mistakes. Cantor's physical opinions are not relevant to this: what was the mathematical mistake?

    Let's not be disingenuous: you are notorious on the internet for your writings about set theory and especially Cantor's uncountability arguments. But Cantor's work on set theory has been explored and vetted with extreme care by mathematicians for more than a hundred years. Nowadays our attitude to allegations of flaws in Cantor's work is similar to that of many biologists when presented with attacks to evolution from "creation scientists": it is not a debate we are eager to have, and we feel that we are at least entitled to restrict ourselves to discussants who show an understanding and technical mastery of the relevant material (which is, for mathematics, not that technical: for instance, many bright high school students know it well). There's certainly room for philosophical doubts about uncountable (or even countably infinite) sets, but this is not the appropriate forum for that.

    I am sorry if you feel that your views are being excluded by some sort of clubbish or defensive attitude on the part of professional mathematicians. I do think it is fair to say that mathematicians bring a particular point of view to these issues of infinity. From our point of view, your criticisms are simply not valid. Other than different people willing to explain to you why your ideas and arguments are incorrect, I'm not sure what is to be gained by bringing the discussion here.

    WM: The question was not about bad physical theories cooked up by mathematicians (of which there are plenty), but cases where physical heuristic reasoning about mathematics yielded wrong mathematical answers. Your responses have not fit the question at all. If you can find a documented example of Cantor applying physical heuristic reasoning to conclude a false mathematical statement, you are welcome to point it out.

    The fact that you are citing Cantor's private communications in your answers leads me to suspect your motivations. People do not make claims in private letters with the same confidence that they do in their published papers, and letters are traditionally a place where people can exchange speculative, incomplete thoughts. For that reason, when you claim that someone was wrong in a letter, it does not carry the sort of judgmental weight that you seem to be seeking. Of course, such false claims could be of historical interest, since we often like to know what was going on in someone's head when a correct theory later came out of something wrong. Such a discussion might be on-topic at MathOverflow, but only if prompted by a concrete question, e.g., about the historical development of a particular theory.

    • CommentAuthorWM
    • CommentTimeJul 2nd 2010
    @ Pete.
    The question is about physical reasoning leading to mathematical mistakes. Cantor's physical opinions are not relevant to this: what was the mathematical mistake?

    Transfinite set theory. The greatest error in intellectual history of mankind (may I say so in this hidden place? or will it cost me a further suspension?)
    You can be sure that I understand the technical items of set theory. But I do not believe in actual infinity because it leads to contradictions. But most set theorists do not even know what actual infinity means and that it is completely different from potential infinity (the latter can never yield transfinite numbers). The "care" you talk about is no argument. Compare Schroeder's proof of the equivalence theroem which stood up for 30 years. And compare the handling of counter-arguments here and elsewhere.

    The interesting thing is that my first argument posted here, the binary tree, has been "refuted" by many, but with always different arguments the easily can be recognized as invalid. Same ist with the list of all words. There is no refusal but only dismission. I am interested to see how long it will be readable here.

    Regards, WM
    • CommentAuthorWM
    • CommentTimeJul 2nd 2010
    @ Scott
    I don't know whether you understand German. Here is a part of a paper that Cantor intended to publish in Acta Mathematica. It was rejected by Mittag-Leffler in 1884. There he talks about the reasons for his theory of order types. He devised set theory for application to physics and chemistry and even biology:

    Die mathematische Physik wird von der Typentheorie gleichfalls betroffen, weil sich
    letztere als ein mächtiges und tief einschneidendes Werkzeug zur Ergründung und zur
    begrifflichen Construction der sogenannten Materie ausweist.
    Damit hängt auch die Anwendbarkeit der Typentheorie in der Chemie zusammen; es ist aber die hier gemeinte Typentheorie nicht zu verwechseln mit der ebenso benannten Theorie von GERHARDT, welcher die Chemie wesentlich ihre gegenwärtige Gestaltung verdankt,obgleich die GERHARDT'sche Theorie längst nicht mehr in ihrer ursprünglichen Form anerkannt wird, sondern sich erheblichen Umgestaltungen hat unterziehen müssen, welches Schicksal sie, meines Erachtens, nothwendig mit allen vergangenen oder noch kommenden Theorien theilen wird, die auf der chemischen Atomistik ihr Gebäude errichten. Mit dieser Typentheorie hat die meinige nichts als den Namen gemein.
    Von ganz besonderem Interesse scheinen mir aber die Anwendungen der mathematischen Typentheorie auf das Studium und die Forschung im Gebiete des Organischen zu sein.

    An unpublished paper by Georg Cantor: Principien einer Theorie der Ordnungstypen. Erste Mittheilung. Ivor Grattan-Guinness, Acta Mathematica 124 (1970) 65 - 107

    Regards, WM
    Is there any reason to make a show from the answer? This discussion was started with a different purpose.

    WM: If you can prove that transfinite set theory leads to contradictions, then there is a very straightforward way to convince all set theorists: write a complete proof in formal language. Start with the axioms of Zermelo-Frankel set theory (or your favorite variant), and derive a contradiction, justifying each step with rules of inference. Nowadays, the validity of such proofs can be checked by computer (look up Coq, for example), so if you write your proof in the correct format, you don't even have to worry about imposing on someone else's time. If you do this, we will welcome your arguments with open arms. In the absence of such a proof, we have no reason to pay attention to your claims.

    • CommentAuthorWM
    • CommentTimeJul 2nd 2010
    Ah, I knew it. It has already happened. KGB could not more eagerly and efficiently throw out its putative enemies.

    Ben Webster?

    You must be very anxious that people may recognize how miserable set theory is. Unless brain-washed by at least two years of due study, most students understand extremely easily how the binary tree contradicts uncountability.

    Here you coulkd learn it if you could:,34,Folie 34
    And here you can see the opinions of some less dense mathematicians and philosophers:,53,Folie 53
    I would spend some dollars to see you studying it.

    Regards, WM
    @WM: So you're saying that the part of your answer which you claim makes it on topic was something you had to intentionally omit because of its known unpopularity to mathematicians? Wow.

    I am not sure that you understand the technical items of set theory. Your arxiv papers contain no arguments which are written in the language of modern set theory, i.e., formalized via ideas and techniques from mathematical logic. Rather, you confine yourself to a sort of literary analysis of very old papers, mostly from the 19th or early 20th centuries. You use terms like completed versus potential infinity, which are not part of the modern vernacular.

    Your argument about the infinite binary tree is not written clearly enough to be easily refuted. The burden of writing clearly enough for others to understand you must lie on your side, otherwise discourse is impossible. I don't understand at all what the contradiction between the set of nodes being countably infinite and the set of paths being uncountably infinite is supposed to be. If you want to try to be understood, you could try out your argument on the following simpler case: consider the infinite graph on the integers where for all n, n is adjacent precisely to n-1 and to n+1. There are countably many nodes on this graph but there are uncountably many random walks: they again correspond to infinite sequences from a two-element set. This is disturbing to you because...?
    • CommentAuthorWM
    • CommentTimeJul 2nd 2010
    @ Scott: I can prove that one result of set theory is wrong, namely that there are uncountably many real numbers. This proof is very easily done with the binary tree.,34,Folie 34

    Everybody able and interested to do it, may it recast in ZFC + FOPL. I am not interested in ZFC + FOPL. And I need not produce an egg in order to judge whether it is bad.

    Concerning AC: It is impossible to prove well-ordering of the real numbers or even larger sets without AC or an equivalent assumption.

    Regards, WM
    @Scott, Pete: perhaps we should take a few moments to think about what Archimedes Plutonium did on Usenet, and what if anything should be applied from such past lessons to MO.
    • CommentAuthorMariano
    • CommentTimeJul 2nd 2010

    @Steve, or just browse the sci.math archives in googlegroups to see the terabytes already spent in trying to explaining to WM, hmm well, something.


    @Scott, @Pete: I admire your patience and civility in dealing with WM. I am not as nice as you guys, so I will stay out of the discussion. Enough said.

    @Steve, Harald: your point is well-taken. I think I've just about had my fill of this discussion as well.
    1) are replies that receive a certain number of downvotes automatically deleted?

    2) can users on meta be suspended?

    Robin, the best practice when dealing with spam/offensive posts is to flag the post as spam/offensive. When a post gets enough spam/offensive flags, it's locked and deleted by the community user and the owner is penalized 100 reputation. See this meta.stackoverflow post.

    Thanks, Francois, that is very useful information. I see from the FAQ that
    users with 15+ reputation points can flag posts.

    May I take this opportunity to apologise for starting this thread? It has degenerated into something very different than what I had intended.


    @figueroa: No. If the thread degenerated, that is not your fault for starting it. You have nothing to apologize for, so don't.

    • CommentAuthorWM
    • CommentTimeJul 3rd 2010
    You have recognized that none of the initial segments of the Binary Tree contains an infinite path. But you believe that the union of these segments contains uncountably many infinite paths.
    I think you are not only lacking niceness.

    Regards, WM
    I thought this was a forum for discussing the running of MO, not for mathematical questions.

    @Robin: Indeed.

    It appears that WM is using another account "skeptic" while he's suspended. Probably Anton would not consider blocking certain IPs as an acceptable practice (it would certainly be easy to get around such blocking), but I thought somebody should bring it up anyway, since this situation has the clear potential for becoming a long-term nuisance.
    Steve, he is also using a/c 7282 "Henri Poincare" (would you believe!).
    By my reckoning that's three sock puppets he's used now.

    Steve, Robin, perhaps you should email Anton or one of the mods if you haven't done so already?


    Also, to be fair, I think that on at least one of the accounts he signs his comments/posts with "regards, WM", so this is not sock-puppetry per se (just against the norms of MO).

    @ Yemon--Done.
    • CommentAuthorWM
    • CommentTimeJul 3rd 2010
    Poincaré is the correct spelling.
    Regards, HP (WM)