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I would like to ask about the classification of finite simple groups, in particular, how we can be confident that it is correct. This post asks essentially the same question, so if I were to ask, it would presumably be flagged as a duplicate.

However, the accepted answer does nothing more than state 'yes, we can be confident', and links a short article which overviews the status of the proof, and outlines it a bit. As far as I can tell this does nothing to answer the question besides that the author also states that they are confident the CFSG is correct, and that the 2nd and 3rd generation proofs are being done for clarity, not because of doubts over their proof.

The comments also provide links to two MathOverflow questions. This one gives updates on the status of the second generation CFSG. This one asks if there is an intuitive reason to expect something like CFSG to be true, and the answer is broadly 'no'. So neither do anything to suggest we should have more confidence in CFSG.

What can I do to get an answer to this question?

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    $\begingroup$ I do not think you get a better answer about classification than the one already posted. The truth is, it depends on who "we" refers to. Everybody decides for themselves. You can try Mathoverflow, but I think it will be the same as here. $\endgroup$ Commented Mar 9, 2023 at 14:21
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    $\begingroup$ Generally, you have two standard options for the general situation (not the specific question you ultimately want answered): one is to put a bounty on the question, and explain in the bounty notice why you think the given answers are unsatisfactory. A second option is to post the question, linking to the old question and making clear what it is about it that you find unsatisfactory and what you would like an answer to address, which is not already addressed. If done properly, this ought to prevent closure as duplicate. That said, I doubt you'll get a different answer in this particular case. $\endgroup$ Commented Mar 9, 2023 at 19:49
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    $\begingroup$ i think the only solution is to try to ask your question on a different site. If you reask the question on this site, somebody will surely mark it as duplicate and closed, no matter how you may pray not to do it. I consider this as the site limitation. $\endgroup$
    – kludg
    Commented Mar 14, 2023 at 14:56
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    $\begingroup$ The answer of @hardmath says what I want to say, especially the first paragraph. But I'll add a thought or two. If you want your question to not be a duplicate of another post, ask something unique not covered by that post. Also, ask something with actual mathematical content. Learn a little about the proof; yes, it's a humingoenormousgigantic proof, but you can still learn a little bit about it. Formulate a reasonable mathematical doubt, and use that to formulate a question. $\endgroup$
    – Lee Mosher
    Commented Mar 19, 2023 at 18:33
  • $\begingroup$ Would this be a reasonable question on Math.OF? $\endgroup$
    – Brian Tung
    Commented Mar 22, 2023 at 2:42

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That was originally a comment, later posted as an answer by request. It's definitely not optimal.

I don't know whether you'll get a better answer, though, because as currently phrased, it's really more of an epistomology question. How can we know that any sufficiently detailed proof is valid? Well, either you read it and verify it yourself, or you defer to consensus of experts in the field.

Getting deeper into it than that quickly becomes too broad. After all, "how do we know CFSG is correct" is essentially an entire field of research in and of itself-- that's why an overview of it exists to be linked to.

If you want more, I'd encourage you to write your own question that focuses in on a specific part.

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    $\begingroup$ I'm not sure what I'm asking is just epistemology, as I think there is a middle ground between deferring to consensus and checking everything yourself. I would like to know why experts are so confident the proof is correct- not in the sense of understanding the theorem itself, but in understanding the scrutiny that it has been subject to- and intuition for why 'gaps' found in it can be expected to be easily fixed. So I think there is an answerable question here. Also, sorry if I came off as harsh. I was just annoyed I didn't find what I was looking for. $\endgroup$
    – Zoe Allen
    Commented Mar 8, 2023 at 5:23
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    $\begingroup$ @ZoeAllen Oh yeah, it's no problem-- I'm still unsatisfied, too, years later. It's a difficult proof to get even a high level understanding of. $\endgroup$
    – Alexander Gruber Mod
    Commented Mar 8, 2023 at 5:34
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I recommend posing a Question with a less ambitious scope. Perhaps your concern about how much scrutiny the proof of the classification of finite simple groups has received should be informed by some basic context. Give the statement of the theorem and a brief account of what you know about its history.

The specific case you ask about involves a 16 step plan/proof program outlined by Gorenstein (1972) and completed over "several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004". How much exposition you will inspire your Readers to devote to the topic is going to depend on how much effort you put in to delineating your own understanding of the math and the history.

The large extent of the reasoning involved makes an account of the submission and review history of these articles too big for a typical Math.SE Answer. What might be more tractable and of significant interest to other Readers is a treatment of the Feit-Thompson (odd order) theorem (1962,1963).

Note that this earlier result has "been totally checked in Coq", a formal proof assistant (2012). You might tell us how much this affects your "confidence" in a theorem.

Update: The Question Why do we trust the Classification of Finite Simple Groups? has now been asked on main Math.SE by the same user. It asks about "confidence" in the result without giving a statement of the theorem. A number of votes have been cast to close as opinion-based. While I agree with that, I did not vote to close myself as I have already had my say here.

One could not really have confidence in any result when one is not familiar with its exact statement, and the post there eschews "asking about the structure of the proof itself." The main post goes so far as to say, "This might not be anything specific to CFSG."

The basic rule for me is whether a Question can be resolved by reasoned mathematical argument. Without specifics there can be no mathematical reasoning.

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MathSE reviewers will sometimes be too quick to label a posting as a duplicate, when the linked posting does not fully address the question(s)/issues/uncertainty of the new posting. This is human nature, and (to some extent) is caused by the frustration that MathSE reviewers feel at being inundated with postings that are either of low quality, because no work was shown, or postings that actually are duplicates, where the poster did not bother to search for duplicates before posting.

Then, once a MathSE reviewer labels the new posting as a duplicate, you have a difficulty. If you attempt to reason with reviewers, after the labeling, explaining why you think that the old posting does not fully answer your new posting, you run the risk of MathSE reviewers emotionally digging their heels in.


Going forward, with your future MathSE postings, the way to avoid this is to be pro-active. That is, try to anticipate which postings that MathSE reviewers might cite as duplicates, that do not actually answer your question. This pro-active searching should include searching on MathSE, as well as searching via approach0.

If (for example), you cite two or three such MathSE/Approach0 references that appear to make your posting a duplicate, and you very carefully explain, in your posting, at the time that you first post the question, why your posting is not a duplicate of these postings, your behavior will create a positive impression among the MathSE reviewers.

Such pro-active behavior by a poster is rare. If you couple that with an otherwise high quality posting, such as is discussed in this article, this will positively influence reviewers right from the start. This will prevent them from digging their heels in.

You may not then get the answer that you need, because reviewers might feel that it is too much trouble to completely grapple with your posting. However, on MathSE, this pro-active approach is clearly your best chance.

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