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I read in a PDF document where the author made a comment that it is “dangerous” to use indirect proof method/contradiction proof method (as far as I can remember, and of course I am paraphrasing) as it may lead to false result. He gave an example of John G. Thompson’s famous proof of finite group theory saying that the first proof Thompson submitted to journal, had to be retracted immediately due to the gap caused by the indirect proof method.

I have forgotten where I read that, I can’t remember the name of author, and can’t find the PDF file either.

Can anyone confirm what I stated above? If possible please provide reference.

Also if anyone read the PDF file I am referring to please comment. Also if you have any suggestion how can I find that document please share.

Update:

  1. The document I am referring to was about proof methods.
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    $\begingroup$ This is something that is addressed in many philosophical discussions of mathematics (going back for several hundred years, at least). You may as well be asking for the .pdf file where the discussion of why self-driving cars will likely cause economic disruptions. That said, the most compelling argument I've seen for direct proofs is the following from p. 187 of Blumberg's On the change of form: "An analogous idea, stressed by G. Hessenberg, (continued) $\endgroup$ – Dave L Renfro Aug 25 at 9:06
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    $\begingroup$ is that the reductio ad absurdum should be used most sparingly, inasmuch as it rests only on deductive logic, and proves only one proposition, while a direct proof deals with actualities and representations, and yields factual results all along the way." $\endgroup$ – Dave L Renfro Aug 25 at 9:06
  • $\begingroup$ Thomas Hales has a paper "Developments in Formal Proofs" that discusses the formal proof by Georges Gonthier and a large team at Microsoft Research-Inria of Feit and Thompson's odd order theorem. Hales gives a lengthy discussion of the constructive nature of the formal proof, and its avoidance of proof by contradiction. I did not find anything about one of Thompson's proofs having to be retracted, however. The Gonthier et al. achievement received a lot of press, so the story you are looking for may be in one of the many articles published at the time. $\endgroup$ – Will Orrick Aug 25 at 15:18
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    $\begingroup$ Page 33 of "An Introduction to Pure Mathematics" by Peter M. Neumann discusses potential pitfalls of proof by contradiction, and mentions the Feit-Thompson theorem as a famous example of a theorem whose proof uses the technique. But, again, no mention of any incorrect proof or retraction. Interestingly, Neumann suggests that we don't know any way to avoid the use of contradiction in the proof of the Feit-Thompson theorem. $\endgroup$ – Will Orrick Aug 26 at 4:04
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    $\begingroup$ If you remember enough keywords from the article, even ones unrelated to the story that interests you, it shouldn't be hard to find. On thing that might help is if you could clarify what theorem your phrase "famous proof of finite group theory" refers to. Do you mean the odd order theorem, or one of the many other important results Thompson proved? Also, retraction of a paper is very serious, and would have left a permanent record had it occurred. Is it possible that something milder occurred, e.g. the manuscript was modified prior to publication, or an erratum was issued after publication? $\endgroup$ – Will Orrick Aug 26 at 5:26

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