16

Has anyone published in print or on a web site or elsewhere a compilation of successful illogical formal arguments? By those I mean arguments that follow a form in disregard of the legality of its applicability in the circumstances to which it is applied, and get a right answer.

One notable example occurred when a formula was discovered for finding solutions of third-degree algebraic equations with real coefficients, when it was known that a real root exists. It was necessary to take square roots of negative numbers. These "imaginary" numbers canceled out and then (the punch line): when the resulting values were substituted into the equation, it checked.

Many things that Euler did could be considered examples. It has been noted here in stackexchange that Euler's derivation of the product formula for the sine function considered the infinitely large leading coefficient of the polyonomial that was the product (which is an infinite product and so has no leading coefficient).

Paul Dirac's delta function was seen to lead to demonstrably correct results at a time when it was not known to to have any logically rigorous justification.

Here's an example from Paul Halmos' article Does Mathematics Have Elements:

In the general ring theory question there are no numbers, no absolute values, no inequalities, and no limits - those concepts are totally inappropriate and cannot be brought to bear. Nevertheless an impressive-sounding classical phrase, "the principle of permanence of functional form", comes to the rescue and yields an analytically inspired proof in pure algebra. The idea is to pretend that $1/(1-ba)$ can be expanded in a geometric series (which is utter nonsense), so that $$(1-ba)^{-1} = 1 + ba + baba + bababa + \cdots.$$ It follows (it doesn't really, but it's fun to keep pretending) that $$(1-ba)^{-1} = 1 + b(1 + ab + abab + ababab + \cdots )a,$$ and, after one more application of the geometric series pretense, this yields $$(1-ba)^{-1} = 1 + b(1-ab)^{-1}a.$$ Now stop the pretense and verify that, despite its unlawful derivation, the formula works. If, that is, $c = (1-ab)^{-1}$, so that $(1-ab)c = c(1-ab) = 1$, then $1 + bca$ is the inverse of $1 - ba$. Once the statement is put this way, its proof becomes a matter of (perfectly legal) mechanical computation. Why does it all this work? What goes on here? Why does it seem that the formula for the sum of an infinite geometric series is true even for an abstract ring in which convergence is meaningless? What general truth does the formula embody? I don't know the answer, but I note that the formula is applicable in other situations where it ought not to be,[ . . . ]

Question: Has anyone published in print or on a web site or elsewhere a compilation of examples of this phenomenon? (Maybe annotated with explanations of the resolution of the seeming paradox in cases where it is known.)

Michael Hardy
  • 1
  • 7
    Umbral calculus is another example of this. Also 17th-century use of infinitesimals, which weren't put on a rigorous footing for 300 years. – MJD Jul 07 '12 at 23:42
  • @MarkDominus Exactly the example I thought of when I saw the question. – Alex Becker Jul 07 '12 at 23:49
  • 1
    There are various ways to "explain" why the "proof" quoted by Halmos works - see my MO question: How would you solve this tantalizing Halmos problem? – Bill Dubuque Jul 07 '12 at 23:52
  • 1
    There are also some arguments about regular languages from infinite series. The Kleene closure $L^$ of a language $L$ is defined to be $\bigcup_{i=0}^\infty L^i$, or $1 + L + LL + LLL +\ldots$ where $1$ is the language containing only the empty string, $+$ is set union and (invisible) multiplication is regular language concatenation. Then you can say $L^ = (1-L)^{-1}$ as if this makes sense, although subtraction of languages is risible. I forget how it goes from there, but you can actually get a useful conclusion out of this nonsense, so there must be something to it. – MJD Jul 07 '12 at 23:57
  • 8
    I think the MathOverflow thread on "jokes in the sense of Littlewood" is in the same spirit. –  Jul 08 '12 at 00:57
  • @MarkDominus: I believe I saw similar expressions in Concrete Mathematics with some combinatorial objects to resolve some recurrences related to them, for which "multiplication" was actually concatenation of sorts, even expanding them into power series, kind of like writing $(1-Lx)^{-1}$ to denote $\sum L^i x^i$ (whatever that's even supposed to mean) in your example. That did make me chuckle a little when I've first seen it. – tomasz Jul 08 '12 at 02:00
  • @tomasz I only remember Concrete Mathematics doing this in connection with generating functions, which is well-known and quite justifiable. Doing it with regular languages seems much less plausible. I'm sure I read a paper about this in the past couple of years, but I can't remember what it was. – MJD Jul 08 '12 at 02:28
  • 4
    Sounds like a good candidate for a big-list question... – Nate Eldredge Jul 08 '12 at 03:15
  • @MarkDominus : Well, I mentioned the 18th-century use of infinitesimals. – Michael Hardy Jul 08 '12 at 03:41
  • You mean proofs where the end result is correct while the proof is not correct? Does it matter if they are fixable mistakes? Or are you looking for results where the author of the proof knew it is not a correct proof? If I understand the question correctly, you are looking for application of mathematical results in places where the assumptions about the result do not hold, yet the result of the application of the theorem in that case is correct. ps: I think "logic" is not the issue here, all occurrences of the word "logic" in the question can be changed to "mathematically rigorous/correct". – Kaveh Aug 02 '12 at 06:05
  • @Kaveh : In $16/64$, if one cancels the $6$, one gets $1/4$, and there the result is correct although the method is not. That is NOT the sort of thing I have in mind. I have in mind arguments not strictly justified by any logic known or mentioned in the proof, but that may become justified by other discoveries. I think that's what you see in my list of examples. – Michael Hardy Aug 02 '12 at 18:17
  • How can we know if the argument for 16/64=4 may not become justified? Are you looking for proofs that were unjustified at some point but become justified by later developments? Or are you looking for arguments which are not justified but there is some (non-rigorous) reasoning about why they should be true (which would support why they may become justified)? Or are you looking for new mathematical axioms? (The examples you gave are not about logic, so I get a strange feeling every time you use the word. They are not illogical, they are non-logical.) – Kaveh Aug 02 '12 at 22:44
  • For example, would results supported by heuristic physical arguments but without rigorous mathematical support qualify? (e.g. physicists' predictions about random k-sat) I think I have seen a question along those line either on MO or cstheory. – Kaveh Aug 02 '12 at 22:48
  • @RahulNarain : Maybe you should post your comment as an answer. – Michael Hardy Dec 08 '12 at 19:42
  • http://math.stackexchange.com/questions/123633/pseudo-proofs-that-are-intuitively-reasonable/ seems relevant. Is http://math.stackexchange.com/questions/123633/pseudo-proofs-that-are-intuitively-reasonable/123895 the kind of thing you're looking for? – Matthew Towers Dec 08 '12 at 20:35
  • @mt : Those look more like proofs with gaps than like the sort of thing described above. – Michael Hardy Dec 08 '12 at 20:47
  • This may be related: https://en.wikipedia.org/wiki/Mathematical_fallacy – Erel Segal-Halevi Nov 23 '13 at 22:12
  • @ErelSegalHalevi : It may be related, but of course mathematical fallacies were not what I was looking for. – Michael Hardy Nov 24 '13 at 18:21

1 Answers1

2

I think the MathOverflow thread on "jokes in the sense of Littlewood" is in the same spirit.

Community
  • 1