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Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu(x)$ and $Y_\mu(x)$ are the Bessel function of the first and second kind. It is supported by numerical calculation with hundreds of digits of precision for many different values of $\mu, \nu$. The question is open for several days with +500 bounty on it and is not resolved yet. But my question here is not about if this conjecture true or false.

Obviously, several possible closed forms matched my numeric calculations, for example: $$\left(\frac{\pi}{2}+7^{-7^{7^{7^{7^{7^{\sqrt{5}+\sin \mu\nu}}}}}}\right)(\mu^2-\nu^2).$$ But for some reason that I cannot clearly explain (or even understand) I selected the simpler one, and I am strongly inclined to search for its proof rather than a disproof. I believe most people would feel and behave exactly the same way.

(1) Are there any mathematical or philosophical reasons supporting this position?

Why when we calculate some sum or integral (which do not contain explicit tiny quantities like $10^{-10^{10^{.^{.^{10}}}}}$) with thousands of digits of precision and it matches some simple closed-form expression, we inclined to believe this is the exact equality rather than an accidental very close value?

(2) Are there known cases when such intuition turned out to be wrong?

And one more question:

(3) Do you believe there can be exact closed forms for some infinite sums or integrals, that cannot be proved in $ZFC$ or any its reasonable extension (like adding some large cardinal axioms) - so to speak, equalities that hold without any reason.

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Imagine an a priori probability distribution over all closed form expressions. For any $n$, there are only finitely many closed form expressions with at most $n$ characters. Hence, the probability of a closed form expression goes to zero as its length increases, thus, it makes at least some sense to weight closed forms according to their simplicity. If you are curious about the philosophical questions involved, you should look up "Occam's razor" and "Kolmogorov prior". –  David Cohen May 22 '13 at 23:21
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@DavinCohen It is difficult for me to think in terms of probabilities, because I do not see any random process here. –  Vladimir Reshetnikov May 22 '13 at 23:51
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Your conjecture is very interesting and attractive by itself! I spent several hours on it and still have no clue how to resolve it. –  Oksana Gimmel May 23 '13 at 2:42
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@Vladimir Reshetnikov "Evidential probability, also called Bayesian probability (or subjectivist probability), can be assigned to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility, or the degree to which the statement is supported by the available evidence." Quote from en.wikipedia.org/wiki/Probability_interpretations –  Waldemar May 23 '13 at 9:13
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xkcd.com/217 –  Noam D. Elkies May 25 '13 at 18:04

7 Answers 7

up vote 46 down vote accepted

Part of what makes this question subtle is that what's intuitive depends on your background knowledge. In particular, the question of what counts as an "explicit tiny quantity" is hard to pin down. For example, Bailey, Borwein, and Borwein pointed out the example $$ \left(\frac{1}{10^5} \sum_{n=-\infty}^\infty e^{-n^2/10^{10}}\right)^2 \approx \pi, $$ which holds to over 42 billion digits. If you know something about Poisson summation, it's pretty obvious that this is a fraud: Poisson summation converts this series into an incredibly rapidly converging series with first term $\pi$, after which everything else is tiny. Even if you don't know about Poisson summation, the $10^5$ and $10^{10}$ should raise some suspicions. However, it's a pretty amazing example if you haven't seen this sort of thing before.

As for the third question, the truth is more dramatic than that. There are finite identities that are independent of ZFC, or any reasonable axiom system, so you don't even need anything fancy like infinite sums or integrals. This follows from a theorem of Richardson (D. Richardson, Some Undecidable Problems Involving Elementary Functions of a Real Variable, Journal of Symbolic Logic 33 (1968), 514-520, http://www.jstor.org/stable/2271358).

Specifically, consider the expressions obtainable by addition, subtraction, multiplication, and composition from the initial functions $\log 2$, $\pi$, $e^x$, $\sin x$, and $|x|$. Richardson proved that there is no algorithm to decide whether such an expression defines the zero function. Now, if the function is not identically zero then it can be proved not to be zero (find a point at which it doesn't vanish and compute it numerically with enough accuracy to verify that it is nonzero). If all the identically zero cases could be proved in ZFC too, then that would give a decision procedure: do a brute force search through all possible proofs until you find one that resolves the question. Thus, there exists an expression that actually is identically zero but where there is no proof in ZFC. In fact, if you go through Richardson's proof you can explicitly construct such an expression, although it will be terrifically complicated. (To be precise, you'll be able to prove in ZFC that if ZFC is consistent, then this identity is unprovable, but by Gödel's second incompleteness theorem you won't be able to prove that ZFC is consistent in ZFC.)

Being independent of ZFC doesn't mean these identities are true for no reason, and in fact Richardson's proof gives a reason. However, his paper shows that there is no systematic way to test identities. You can indeed end up stuck in a situation where a proposed identity really looks true numerically but you just can't find a proof.

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The near-identity $e^\pi - \pi = 19.999099979\ldots \approx 20$ from xkcd.com/217 is actually related with this Poisson fraud: differentiating the $t \leftrightarrow 1/t$ identity for $\sum_{n=-\infty}^\infty e^{-n^2\pi t}$ at $t=1$ yields $e^\pi \approx 8\pi - 2$; now compare with $7 \pi \approx 22$, etc. –  Noam D. Elkies May 25 '13 at 18:08

This is an old and fascinating question but it is very difficult to make concrete progress on it. For example, Harvey Friedman made a brief start on a closely related problem but as far as I know did not do any serious follow-up work. In the comments above, David Cohen mentions the names Occam and Kolmogorov. The trouble is that Kolmogorov complexity and related theories, as currently formulated, are asymptotic, and there's an arbitrary constant hanging around that can absorb any finite amount of "slush." This means that any specific finite expression can be declared to be simpler than any other specific finite expression, and that's not what we want.

David Cohen also alluded to statistical concepts, and those are probably the best tools currently available. The idea would be something like this. You have two expressions that agree up to $n$ decimal digits. You now hypothesize that this is a "coincidence." I won't try to define "coincidence" but it should mean that if you compute the $(n+1)$st decimal digit then the probability of equality will be 1/10. If you compute, say, three more digits, then the probability of equality under the coincidence hypothesis will be 1/1000, so if you do get equality then you could argue that you are "99.9% confident" that it is not a coincidence. (I've formulated this in terms of hypothesis testing but you could tell a similar story in a Bayesian framework.)

The trouble with this naive statistical approach is that in real life there aren't just two alternatives, "pure coincidence" and "true." There can be underlying structure that causes numerical agreement for a long time without causing literal equality. Typically, it is impossible to quantify the "probability of unknown underlying structure," so we're stuck with waving our hands. In this regard, let me mention that Andrew Odlyzko does not consider the existing numerical computations of the zeros of the zeta function to be overwhelming evidence for the truth of the Riemann Hypothesis. In one talk that I heard him give many years ago, he vaguely sketched one possibility for how it could be false and yet for there to be some kind of underlying structure that would not allow a zero off the critical line until far beyond the realm of current computations. He wasn't saying that he actively believed that this was what was going on, but gave it as an example of why he was unwilling to accept the numerical evidence as overwhelming.

Of course there might be limited domains in which we feel we understand what kinds of structure could possibly be in play, and then we might feel more assurance in writing down probabilities. But this has to be approached on a case-by-case basis.

As for cautionary tales, I recommend the section "Coincidence and Fraud" in Bailey and Borwein's AMS article on experimental mathematics. Here's one example that they give: the integral $$\int_0^\infty \cos(2x) \prod_{n=1}^\infty \cos\biggl({x\over n}\biggr)\ dx$$ agrees with $\pi/8$ to over 40 decimal places. They also give a related example where someone was led astray by numerical evidence to conclude that a computer algebra package had a bug when it didn't.

Finally, regarding your third question, see this MO question for a discussion of "truth by accident" and its relationship (or lack thereof) to formal unprovability.

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I see now that the computer algebra bug has already been mentioned in Michael Biro's answer and the attached comments. –  Timothy Chow May 23 '13 at 15:31

For $(1)$, I would say aesthetics is good enough for me even if it doesn't always turn out the way I'd like.

For $(2)$, the two (small) examples that come to mind are:

$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925$

and

$\int_0^\infty \text{sinc}(x)\text{sinc}(\frac{x}{3}) \dots \text{sinc}(\frac{x}{15})dx = π(\frac{1}{2} - \frac{6879714958723010531}{935615849440640907310521750000})$

$= 0.49999999999265 \pi$

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I want to add that the last integral is called a Borwein integral, if you want to look it up. Someone here was working on writing computer algebra software and this example made him think he had a bug... if you omit the last $sinc$ term, it is in fact $\pi/2$, and similarly if you omit several of the last $sinc$ terms. –  Eric Tressler May 23 '13 at 5:01
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Yes, the paper is "Some Remarkable Properties of Sinc and Related Integrals" by David and Jonathan Borwein. –  Michael Biro May 23 '13 at 5:03

For addressing (2) here are links to relevant (sorted in decreasing order) MO questions.

  1. The phenomena of eventual counterexamples, which contains a wealth of a list of examples akin to those mentioned by H. Cohn, and others above.
  2. This famous MO question on false beliefs, which contains a wealth of examples of either misunderstandings, or of false intuition due to either insufficient knowledge, misleading definitions, obstinacy, or sundry other reasons.
  3. Examples of interesting false proofs

For (1), "Occam's razor" does suggest itself, but perhaps recursively so, because an appeal to Occam's razor is itself perhaps the simplest explanation for the phenomenon that you raise.

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Let me try to give a tentative speculation to question (1): Bessel functions are solutions of the simplest singular ODEs; the coefficients are lowest degree polynomials with integer coefficients. Your formula respects this simple built. That $\pi$ appears is expected becuase these ODEs behave not too badly with respect to Fourier transform. You did not do unnatural operations with which would be against the spririt of Bessel functions. In particular, there is no source for the 7-ish small constant that you mention.

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There is an aspect that has not been mentioned so far, namely that of usability. I do not believe that there is a context in which the OP's additional term of $7^{-7^{7^{7^{7^{7^{\sqrt{5}+\sin \mu\nu}}}}}}$ could actually have been used in a further calculation - that is, in its exact form, not just for an estimate.

Indeed, when there was suspicion that the value of $\frac\pi2 (\mu^2 - \nu^2)$ were no correct, the next best conjecture for the value of the integral would probably be something like $(\frac\pi2 + \epsilon) (\mu^2 - \nu^2)$ with suitable bounds on $\epsilon$: Many conjectures of this form have been made in other contexts.

Could it be that usability is actually the main criterium that is employed when making conjections? I believe that this is entirely possible.

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You might like the Wikipedia article

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