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Descartes, IIRC, somewhere says something about the vagaries of memory influencing our justification for believing in our memory, and thence for believing in proofs involving many steps that we have to mnemonically track as we go along. Not that our justification is entirely negated, but just weakened modulo hyperbolic doubt, say.

Now, with respect to the preface paradox directly, does this paradox indicate that a properly modest mathematician should not completely accept a proof that goes beyond a certain (arguably vague) boundary of length? I've read that some proofs go over 200 pages, for example. Granted, typesetting will be a factor in this kind of situation to the extent that irregular typesetting can inflate a text's size beyond necessity, but I rarely see full academic writers performing that kind of inflation operation (I can remember one and only one dissertation I read where the font was Courier and the spacing was egregious, so what probably could've been a 300-page document ballooned to ~600 pages). But so I doubt that most long mathematical proofs appear to be so long mainly on account of how they were physically jotted down.

But at any rate, then, perhaps more intriguingly, does the viability of long mathematical proofs undermine the preface paradox instead?

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  • There's a subjective element to proofs which kinda messes up yer project, assuming it is one. Fermat's Last Theorem (Google).
    – Agent Smith
    Mar 12 at 12:09
  • @AgentSmith that's the one I was thinking of for the "some proofs are 200 pages long" remark. I will admit that most any question I ask here is for the sake of research to be used, potentially, in other writing, but on the other hand, there are too many commitments to try to balance for me to also be too categorical by the by. So, then, does e.g. the proof of FLT fall prey to the preface paradox, or is it a counterexample to that paradox? Or does the requirement that the mathematical community evaluate the proof (they found an error in Round 1 of judging it or something) safeguard the result?
    – Kristian Berry
    Mar 12 at 13:35
  • The lattermost option (not the last option, period, maybe; I don't know that all the options have at all been stated!) given corresponds to the possibility of social intuitionism, or social constructivism in mathematics (where "constructivism" has to be taken, at the same time, in both the mathematical/logical sense and the critical/paramodernist one). How does the preface paradox intersect claims of large bodies of socialized knowledge (e.g. the notion of "common knowledge")?
    – Kristian Berry
    Mar 12 at 13:38
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    I was going to answer your question, but by the time I got to the bottom I forgot the first paragraph.
    – Boba Fit
    Mar 12 at 16:36
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    If you think 200 pages is bad, take a look at the Classification Theorem of Finite Simple Groups. The proof runs to thousands of pages in hundreds of different papers by dozens of different authors. It is unlikely that any one person has ever read all of it, let alone checked all of it. I think Russell said something similar to your Descarted quote, about trying to keep a proof in his mind in order to understand all of it, and the most he could manage was about a minute.
    – Bumble
    Mar 14 at 4:47

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