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In math/physics, or generally in science, there are many moments where the success and the triumph come from the accidental, unplanned attempts. Moreover, there are some cases that originally having attempts for one specific question or a goal, but solve another seemly unrelated problems, or reach seemly the opposite goals. There are these kinds of moments leading to breakthrough of math or science.

For example,

Calabi conjecture: S.T. Yau firstly attempted to disprove the theorem, but in the end he figure out his disproof's logical gap pointed out by Calabi can be corrected, so he prove the theorem instead.

[Question]: Can any of the readers here list more and others in the subject of math? To give us some inspiration and high motivations to be subconsciously aware of those random accidental moments.

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closed as too broad by Andy Putman, Nate Eldredge, Stefan Kohl, David White, Andrey Rekalo Dec 20 '13 at 22:23

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs. If this question can be reworded to fit the rules in the help center, please edit the question.

This is far too broad. I mean, if you consider the Calabi conjecture to be an example, then probably half of all math would count (it's very common to be trying to prove something and then find a counterexample, for instance). –  Andy Putman Dec 20 '13 at 20:31
I have no idea where you got the idea of a "typical homework problem" from what I said. The work "breakthrough" is totally nebulous; however, there are many, many strong results proved every year, and I suspect that many of them are "accidents" according to your overly broad definition. –  Andy Putman Dec 20 '13 at 20:54
Voted to close as "too broad". –  Stefan Kohl Dec 20 '13 at 21:13
Thanks, I do not mind to be discredited. And I may agree it is broad. But one could not feel satisfied if explicit examples is provided, say an order of 10 or a hundred so. I believe such accidental breakthroughs do not occur to Gauss, Euler, Riemann, etc and some other great minds. So those accidental breakthrough moments are not universal. It is hard to imagine some mathematicians define something is {\it too broad} but without thinking about explicit examples. –  Idear Dec 20 '13 at 21:31
As far as I know, it is common to investigate statements we don't know whether we will prove, disprove, or can disprove but can salvage by adding a condition. It is normal, not exceptional, to work one moment on constructing a counterexample, and the next turn a difficulty in the construction into an attempt at a proof. You are suggesting that this is noteworthy and you want historical documentation. You are trying to impose your idea of how mathematics is done. I think this is misguided. Mathematics is not shooting pool. It is not necessary to call the pocket before making an attempt. –  Douglas Zare Dec 24 '13 at 1:21

2 Answers 2

I believe Killing discovered the exceptional Lie algebras when trying to prove that such things could not exist.

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For motivating this question, let me initiate and borrow the physics facts from Phys.SE. Accidental, unplanned breakthroughs in physics, relevant to mathematical physics of QFT:

(2) Asymptotic freedom (QCD running couplings to small at high energy): from Wilczek's book ``Longing for the Harmonies,'' at that time David Gross originally aimed to prove that QFT cannot explain the Bjorken scaling, i.e. prove that (non-Abelian) QFT always lead to large coupling at high energy, simply that QFT cannot explain the couplings run to small at high energy. But the final result is the opposite. And they won the Nobel Prize.

(3) Yang-Mills theory original paper has attempt to explain a theory of nucleon, such as neutron and proton interactions with isospin symmetry, but it turns out that Yang-Mills theory as a non-Abelian gauge theory suitable to describe a more fundamental subject, the gauge fields coupled to quarks and leptons in the standard model (EM, and especially the weak interaction and the strong interaction.)

(4) String theory: originally aimed for explaining strong interaction, but not as successful as QCD. Later string theory turned out to be a candidate for a theory of everything.

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-1 I think it is inaccurate to claim that string theory was abandoned. –  S. Carnahan Dec 21 '13 at 5:19
Hi, I remove that "String had then been abandoned for a long while." As you suggest. –  Idear Dec 21 '13 at 5:39

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