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As of today, most important results in mathematics are conjectured long before they are proven.

Are there any examples of (important) mathematical discoveries that were proven by chance rather than from the authors conducting a targeted research on a hypothesis? I'm interested especially in results obtained in the last 100 years or so.

imanoob
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    Someone with more historical background can expand on this but I think many of the results behind the theory of Monstrous Moonshine were accidental observations coming from a wide variety of fields: https://en.wikipedia.org/wiki/Monstrous_moonshine – Alex R. Mar 16 '16 at 22:27

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A fun instance in proof-theory. The Curry-Howard correspondence links computational calculi to logical systems. In particular, there are (old) isomorphisms between a Hilbert-style deduction system and combinators, and between natural deduction and lambda-calculus, both in the case of intuitionist logic. It was widely believed that there wasn't anything similar for classical logic, or rather logicians all knew it was an impossible task, because classical logic didn't have any computational meaning. (Note: classical logic is intuitionist logic with the law of excluded middle, $A\vee \neg A$, added as an inference rule.)

But in 1989, a computer scientist, who didn't know it was impossible, did it: he gave a type system for Scheme, with the $\texttt{call/cc}$ primitive, which was typed by Peirce's law, a rule equivalent to the law of excluded middle. From that the Curry-Howard isomorphism was generalized to classical logic and then many more complex calculi, which greatly influenced the field of proof theory, influencing proof assistants and programming language design.

Mikhail Katz
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Graffitics
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In 1979-80 when Benoit Mandelbrot was a Visiting Professor of Mathematics at Harvard University he had the chance to use the brand new Vax computer. On the $1$st March $1980$ he had a first detailed picture of an island molecule in the Mandelbrot set for \begin{align*} z\rightarrow z^2-c \end{align*}

B. Mandelbrot wrote in his contribution Fractals and the Rebirth of Iteration Theory in The Beauty of Fractals by Heinz-Otto Peitgen and Peter Richter:

... The beauty of many fractals is the more extraordinary for its having been wholly unexpected ...

Markus Scheuer
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I would like to expand a bit on the comment made by @AlexR, citing the so-called Monstrous Moonshine.

The Monstrous Moonshine is a link - that stayed elusive for a long time, and was worth Richard Borcherds a fields medal for making it explicit - between two apparently unrelated domains: irreducible representations of the Monster group (whence the name) and modular forms. The serendipitous remark was the following:

The Fourier expansion of the $j$-invariant (some function on the upper half complex plane with some interesting properties) is given by $$j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2+\ldots$$ where $q=e^{2\pi i\tau}$. At the same time, we have the Monster group, i.e. the biggest of the sporadic finite simple groups. We look at its smallest irreducible representations and notice they have dimension $r_1 = 1$ (the trivial representation), $r_2 = 196883$, $r_3 = 21296876$,... Those two sequences of numbers are eerily similar, and in fact one notices that $196884 = r_1 + r_2$ and $r_1+r_2+r_3$ (and similar identities also hold for higher coefficients of the Fourier expansion and higher dimensional irreducible representations). This was remarked by McKay in 1978, and the natural question that arose is: Is there some hidden structure that explains those apparently nonsensical relations?

This question stimulated a great deal of research, and an answer was reached with the work of Borcherds in 1992 via vertex operator algebras.

Daniel Robert-Nicoud
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Not sure if I can call this a mathematical result, as this phenomenon is far from understood, but I think that the notion of Ulam spiral is famous enough and was discovered by chance.

Peter Franek
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Until the summer of 1960 most historians and philosophers (and arguably many mathematicians) believed (following Bishop Berkeley, Moigno, Cantor, Russell, and others) that infinitesimals (as a possible foundation for analysis) had been proven inconsistent and consigned to the dustbin of history. Cantor actually published a purported "proof" of inconsistency of infinitesimals that influenced numerous scholars like Russell, as analyzed by Philip Ehrlich in this 2006 article.

Walking toward Fine Hall at Princeton University in the fall of 1960 Abraham Robinson realized that a way can be found to make infinitesimals (as a possible foundation for analysis) rigorous. For additional details see Dauben's biography of Robinson:

Dauben, J. Abraham Robinson, The Creation of Nonstandard Analysis: A Personal and Mathematical Odyssey (Princeton, N.J.: Princeton University Press, 1995).

This serendipitous discovery explained why the historical mathematicians like Leibniz, Euler, and Cauchy made so few mistakes in manipulating infinitesimals, and set the ground for a fruitful area of research.

Abraham Fraenkel wrote in the 1960s that "my student Robinson saved the honor of infinitesimals."

Robinson's framework has been recently championed by Terry Tao who argues for the conceptual advantages of using the hyperreals and hyperreal-related structures.

Mikhail Katz
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    I don't think Robinson discovered this "by chance," which is what the question is asking for. Also, I'd dispute the claim that "most mathematicians believed that infinitesimals had been proven inconsistent" - I think that pre-1930s mathematicians hadn't thought about their consistency (as independent from their "true" existence) at all, and that post-1930s mathematicians who thought about consistency knew that they were consistent via the compactness theorem. I strongly doubt Robinson was the first to notice this - just the first to emphasize it as a serious way to approach calculus. – Noah Schweber Mar 14 '16 at 16:52
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    Re: the consistency of infinitesimals, also note that non-Archimedean ordered fields (i.e. ordered fields with infinitesimals) had been explicitly studied at least as early as 1900 - see https://en.wikipedia.org/wiki/Non-Archimedean_ordered_field. – Noah Schweber Mar 14 '16 at 16:54
  • That's not relevant to analysis (see my answer). – Mikhail Katz Mar 14 '16 at 16:55
  • @NoahSchweber, there is of course Skolem's work in arithmetic, but do you have any source for your claim that mathematicians before Robinson thought infinitesimals were consistent (with the possible exception of Schmieden and Laugwitz in 1958 which was little known). – Mikhail Katz Mar 14 '16 at 16:59
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    @user72694 Well, non-Archimedean ordered fields such as the Dehn field were well known long before Robinson, so it was well-known that the existence of infinitesimals is consistent with the ordered field axioms (although that would have been a very strange thing to say pre-1930s). I don't know if anyone had thought of the question "Is the existence of infinitesimals consistent with the theory of the reals as a field?" Do you have a source of anyone claiming that infinitesimals are not consistent with the theory of the reals? – Noah Schweber Mar 14 '16 at 17:02
  • @NoahSchweber, The Dehn field provided a counterexample proving the independence of axioms of geometry that Hilbert was looking for. I don't think they were thought of as having implications for analysis. Note that the point I made is fairly well known and in particular emphasized by Dauben, so you are arguing against a number of historians. – Mikhail Katz Mar 14 '16 at 17:03
  • @user72694 But "having implications for analysis" is very different from "being consistent" - you claimed that Robinson was the first to notice that infinitesimals were consistent! Yes, Robinson was (to the best of my knowledge) the first to realize that you could rigorously ground infinitesimals as a foundation for analysis, but the first sentence of your answer makes a different claim. – Noah Schweber Mar 14 '16 at 17:04
  • @NoahSchweber, in retrospect it is easy to say that people should have seen this coming, but Russell for one was convinced by Cantor that infinitesimals are inconsistent, and popularized this view, as documented by P. Ehrlich in his detailed study from 2006 that you might want to consult. – Mikhail Katz Mar 14 '16 at 17:06
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    @user72694 From page 3 the Ehrlich paper you mention (http://www.ohio.edu/people/ehrlich/AHES.pdf): "whereas most late nineteenth- and pre-Robinsonian twentieth-century mathematicians banished infinitesimals from the calculus, they by no means banished them from mathematics." Again, I'm not disputing that Robinson was the first to use them to ground calculus, just that the claim at the beginning of your answer that most pre-Robinson mathematicians believed infinitesimals were inconsistent is wildly off-base, or at least a bad oversimplification. – Noah Schweber Mar 14 '16 at 17:09
  • Regardless, there's still the question of whether this was "serendipitous" in the sense of the question. There was a lot of philosophical work pre-Robinson around whether infinitesimals could make sense in calculus, and Robinson was certainly aware of this work; he may have discovered nonstandard analysis by accident, but I'd be surprised if that were the case. – Noah Schweber Mar 14 '16 at 17:11
  • @NoahSchweber, Ehrlich details work by mathematicians like Stolz, Levi-Civita, Veronese, du Bois-Raymond and a handful of others that continued to work on infinitesimals but the dominant view was as I described and Ehrlich is not disputing that; see his comments on Russell. Besides, none of these people attained anything like a system that can serve as a full-fledged foundation for analysis, and I am not aware of anyone before Robinson who thought this possible. – Mikhail Katz Mar 14 '16 at 17:12
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    @user72694 For the last time, I'm not disputing that Robinson was the first to rigorously ground analysis in infinitesimals! I'm disputing the claim in your opening sentence that most mathematicians thought that infinitesimals were inconsistent, which is a much stronger statement (and pretty clearly false). Also, there's still the issue of whether this was in fact serendipitous in the sense of the OP. – Noah Schweber Mar 14 '16 at 17:13
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    Abraham Fraenkel wrote in his book that Robinson "saved the honor of infinitesimals". This may have been slightly exaggerated but if this was known before Robinson, what was there to save? – Mikhail Katz Mar 14 '16 at 17:14
  • Their mathematical (and foundational) relevance - something being consistent is a far cry from it being useful, or interesting, or etc. Again, I'm not disputing the novelty or importance of Robinson's discovery, just the first sentence of your answer, which is just false. EDIT: Rather, the first sentence of your answer, pre-recent edit. – Noah Schweber Mar 14 '16 at 17:27
  • Dauben seems to imply that the discovery was "serendipitous". I can give you his email address if you would like to lodge a complaint. – Mikhail Katz Mar 14 '16 at 18:43
  • Dauben wrote only that "Robinson had been thinking about nonstandard models of arithmetic, particularly Skolem's work, and one day, as he walked into Fine Hall, the idea of nonstandard analysis suddenly flashed into his mind!". That does not sound at all like the discovery in late 1960 was serendipitous. Robinson at his January 1961 plenary lecture on nonstandard analysis described it as an answer to the question of whether what Skolem did for N could also be done for R. That's a natural question, though Dauben doesn't say if it was the path to discovery. – zyx Mar 19 '16 at 15:42
  • @zyx, note that Dauben's prose is often well... prosaic which is normal and perhaps even to be expected for a professional historian. The exclamation point at the end of his sentence (which is apparently in the original though I haven't checked) as well as the term suddenly do seem to suggest a great measure of surprise, on the part of Robinson's fellow mathematicians and perhaps even on the part of Robinson himself. If you like replace "serendipitous" by "unexpected" but beyond that we are quibbling about semantics. – Mikhail Katz Mar 20 '16 at 10:21
  • Not that I want to belittle this discovery or anything, but "Made so few mistakes"... It sounds more like the discovery or invention was designed to find a beautiful way to confirm something that had been known to be useful in various applied fields for a long time. Like confirmation bias. – mathreadler Mar 20 '16 at 18:15
  • @mathreadler, Terry Tao and others have recently produced applications of Robinson's framework to a variety of fields. See for example Terry Tao and Van Vu, "Sum-avoiding sets in groups," here As far as historians like Dauben are concerned, they are primarily interested in issues of historical interpretation, and here also Robinson's framework is illuminating; see for example this recent post – Mikhail Katz Mar 21 '16 at 08:25
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Does Srinivasa Ramanujan count as a mathematical discovery?

This is admittedly a 'cutesy' sort of answer for this type of question, but my tongue is just barely in my cheek here. G. H. Hardy, a hugely productive and influential mathematician in his own right*, was asked once in an interview what he thought his greatest contribution to mathematics was. He immediately replied with Ramanujan. It's scary to think how easily Ramanujan could have remained an anonymous man in India who filled up notebooks with jibberish in his spare time.

*Bohr famously remarked, "Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy–Littlewood." Presumably he hadn't been introduced to Ramanujan. :)

David H
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What is "important" and what is "mathematical"?
The first time I encountered the word "serendipity" was in an engineering context, this one:

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Are Finite Elements important? Sure they are in e.g. Structural Mechanics.
Are Finite Elements mathematical? Oh well, they are quite a bit practical,
but that doesn't mean they aren't interesting for theoretical mathematics.

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Han de Bruijn
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Candes's and Romberg's perfect reconstruction of the Logan-Shepp phantom, leading to the pioneering work in Compressed Sensing, was unexpected. Terence Tao tells the story here:

https://mathoverflow.net/questions/178139/examples-of-unexpected-mathematical-images/178210#178210

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littleO
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  • Yes almost looks like that phantom happens to be perfect for that method. – mathreadler Mar 20 '16 at 19:40
  • @mathreadler It's not just that the phantom is an easy example; that would be less interesting. Something much deeper is going on. Tao writes, "Much to my surprise, I found instead that random matrix theory could be used to guarantee exact reconstruction from a remarkably small number of measurements." Compressed sensing is useful for reconstructing real MRI images using fewer measurements than had previously been required. – littleO Mar 20 '16 at 21:27
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The accidental discovery of chaos by Edward Lorenz in 1963 (see here), but have in mind also the work of Boris Chirikov (see here).

corey979
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