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Related:

The questions above use the term 'lost' to refer to theorems that exist in the literature, but may be so obscure and/or not commonly useful nowadays that few people actually bother to learn them. Are there any theorems that were truly lost - that is, they were reasonably well-received by the mathematical community, were used for some time, and then were truly lost to history, unavailable in any extant literature, textbook, or other resource. All available copies might have been lost due to fire, war, or an intentional purge, or available copies might remain but only in an undeciphered code. Examples would include:

  • Theorems that have been truly forgotten and literally cannot be learned today from any source
  • Theorems that were truly lost but were rediscovered many years later, after the original author and all of their students were deceased

I am not talking about theorems that are commonly classified as lost simply due to obscurity or uselessness, but can be learned by a student determined enough to see out applicable resources, instructors, etc. These are what the two questions at the top are speaking of.

To some extent, Fermat's Last Theorem seems to match this, but I understand that there is quite a bit of controversy whether Fermat ever even had a proof to begin with.

To be clear, I am obviously not asking about theorems that nobody has ever heard of, but theorems that we are fairly certain existed (e.g. there are citations to them), but for which all copies have apparently been lost. For example, something like this could answer the question:

We currently have proof that this expression converges when N > 3. A proof that it converges for all N (Smith and Higgins, 1724) has been cited by Jones (1745) and Hooper (1763), but the only known copy of the full proof was reported lost in a fire at the Bodelian Library in 1789. Searches were undertaken in 1812 and 1826 to find a copy in another library, but these searches were unfruitful.

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  • $\begingroup$ I've seen a lot of papers like this one --- Ivor Grattan-Guinness, An unpublished paper by Georg Cantor: Principien einer Theorie der Ordnungstypen: Erste Mitteilung [Principles of the theory of order types: First report], Acta Mathematica 124 (July 1970), 65-107 --- but to what extent the theorems Cantor proved in this manuscript could be considered truly unknown today, I don't know. Keep in mind that there are plenty of Ph.D. dissertations all of whose results have not been published (I personally know about several), (continued) $\endgroup$ Dec 26 '19 at 16:12
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    $\begingroup$ I think your constraints prevent such an example being known, because if we knew that someone proved/did something that was "lost", our knowledge of this fact in itself would provide the existence of a record of its existence surviving to us. $\endgroup$ Dec 26 '19 at 16:24
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    $\begingroup$ If they "were truly lost to history, unavailable in any extant literature, textbook, or other resource", " truly forgotten and literally cannot be learned today from any source" we wouldn't know anything about them, would we? $\endgroup$
    – Conifold
    Dec 27 '19 at 3:33
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    $\begingroup$ @Conifold well, we could know they existed, but not their content. I'm thinking along the lines of certain ancient books that we can find contemporary citations to but no extant copies. $\endgroup$ Dec 27 '19 at 3:38
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    $\begingroup$ In that case, is Zenodorus's work on the isoperimetric problem an example? "His work was lost. We know of it mainly through Pappus and Theon of Alexandria", Blasjo, The Isoperimetric Problem. $\endgroup$
    – Conifold
    Dec 27 '19 at 3:47
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Archimedes' books, Stomachion and The Method of Mechanical Theorems were lost until rediscovered in 2006. The only known copy is the Archimedes Palimpsest. These two texts comprise many theorems. The Method describes Archimedes' very early use of Riemann sums to compute areas and a variation of Dedekind cuts (via a pair, one a strictly monotonically increasing sequence of rationals and the other strictly monotonically decreasing, where the difference between sequence member of the same index decreases below any specified bound) to identify real numbers.

A few other books of Archimedes are still lost, titles known only through indices and reference by other authors: Catoptrica, a work on optics; Principles, explaining the number system used in The Sand Reckoner; On Balances and Levers; On Centers of Gravity; and On the Calendar. We do not know how many of these theorems we do not know.

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    $\begingroup$ Also On Polyhedra, identifying what are now known as Archimedean solids. $\endgroup$
    – Mark S
    Dec 27 '19 at 13:58
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    $\begingroup$ Wikipedia says it was rediscovered in 1906 rather than 2006 - a bit of a discrepancy! $\endgroup$
    – Nathaniel
    Dec 27 '19 at 15:21
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    $\begingroup$ @Nathaniel : You seem to be conflating "was spotted in 1906 by a Danish classics scholar who made unverifiable claims about the contents of the palimpsest before it was lost again" and "had new texts actually extracted from the document by hyperspectral imaging, announced in 2006". $\endgroup$ Dec 27 '19 at 20:33
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    $\begingroup$ @EricTowers I know nothing of any of those events. I was just noticing that one source gave a date 100 years later than another source and thought it worth mentioning. If you'd like it to be clearer to others in future, the best strategy is probably to edit the Wikipedia article. $\endgroup$
    – Nathaniel
    Dec 27 '19 at 20:59
  • $\begingroup$ @EricTowers : I read a translation of that work of Archimedes, with some lacunae, in 1982. The translation was published in the early 20th century. New things were announced in 2006, but most of the work was in the translation I read. And remember that the book, the sole surviving copy, was sold at auction in New York in the late 1990s after a lawsuit. Hardly possible if it was not rediscovered until 2006. $\endgroup$ Dec 29 '19 at 2:01
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The Rogers-Ramanujan identities?

The formulas have a curious history, having been proved by Rogers (1894) in a paper that was completely ignored, then rediscovered (without proof) by Ramanujan sometime before 1913. The formulas were communicated to MacMahon, who published them in his famous text, still without proof. Then, in 1917, Ramanujan accidentally found Roger's 1894 paper while leafing through a journal. In the meantime, Schur (1917) independently rediscovered and published proofs for the identities (Hardy 1999, p. 91).
LINK

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  • $\begingroup$ "independently rediscovered and published proofs for the identities" Were the proofs the same? $\endgroup$ Dec 28 '19 at 13:45
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    $\begingroup$ I don't know, but perhaps that is a good question for you to ask. $\endgroup$ Dec 28 '19 at 15:29
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    $\begingroup$ @MaxBarraclough: Proofs by Rogers, Ramanujan and Schur were all different. Schur's proof is combinatorial in nature. None of the proofs are simple by the way. The manipulations in Ramanujan's proof are easy but there is no insight about why some particular expressions were chosen to be manipulated that way. $\endgroup$ Feb 18 at 3:31
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    $\begingroup$ @MaxBarraclough: you may have a look at Ramanujan's proof. $\endgroup$ Feb 18 at 5:25
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I did once write an email to Robert Geroch, because an unpublished paper of his was listed as a bibliographical reference in a paper of Hajicek, which contained a few theorems regarding non-Hausdorff spacetimes. He apparently had no recollection of this paper, and it was never published. I assume there are many other such papers, the myriad of unpublished papers, private communications and unwritten conference presentations.

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  • $\begingroup$ How do we know this is actually a lost theorem that was received by the mathematical community? (Isn't it possible that he didn't publish it because it was found to be irreparably flawed?) $\endgroup$
    – user21820
    Dec 27 '19 at 14:06
  • $\begingroup$ At least one of the theorem of this paper found its way in Hajicek's paper. This topic was also apparently approached by Geroch in another venue, his presentation at the 1968 Battelle Rencontres (as referenced in Penrose's article in the Einstein centenary survey). So the various ideas thrown around were known by at least two people. What theorems that have been lost and which ones were simply reused in other papers of course, hard to say! $\endgroup$
    – Slereah
    Dec 27 '19 at 14:12
  • $\begingroup$ If you have evidence that any one of those theorems were presented publicly or subsequently (re)proven, you should include that information into your post and it would be a very good example. $\endgroup$
    – user21820
    Dec 27 '19 at 14:14
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    $\begingroup$ Along these lines, see the MathOverflow question, Results that are widely accepted but no proof has appeared. This is mathematics that has not been lost yet, but if the authors die before writing up their results, it could become lost. So far, I think that the mathematical community has been lucky, in the sense that I don't know of a major result that everyone accepted based on the mathematician's reputation, but whose proof died with the author and has resisted all attempts at reconstruction. But I think it's bound to happen sooner or later. $\endgroup$ Jul 14 at 13:43
  • $\begingroup$ (Continued) I don't count Fermat's Last Theorem, by the way, since the consensus is that Fermat did not have a valid proof. $\endgroup$ Jul 14 at 13:44
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I'd be willing to bet that quite a number of theorems were lost simply due to the Library of Alexandria getting destroyed.

Take a look at this History Stack Exchange question and some of the answers. Particularly where it mentions that all mathematics before Euclid was lost, and that the mathematics of that age wasn't really surpassed until the 19th century. Given that level of loss and backtracking, it's pretty safe to say that there were some theories that ancient greek mathematic scholars would have received well, but were lost over the years.

(The post in question is referencing Lucio Russo's 'Forgotten Revolution', which covers the knowledge of the Hellenistic world that was lost through history; the author is both a historian and a mathematician, which is pretty on-point for the question.)

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Richard Stanley's article Hipparchus, Plutarch, Schröder, and Hough (Amer. Math. Monthly 104 (1997), 344–350) describes a possible example. Among Plutarch's writings appears the following enigmatic statement.

Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952.)

These numbers mystified generations of historians, until in 1994, David Hough noticed that 103,049 is the tenth (little) Schroeder number. Moreover, although Stanley's article does not offer an explanation of the number 310,952, it was noticed soon afterward that the number 310,954 (off by 2) is the tenth element of a closely related sequence. Thus the circumstantial evidence is overwhelming that these numbers are what Plutarch and Hipparchus were referring to.

Along the lines of Conifold's remark, now that we know this, the mathematics is no longer lost, right? Well, it's still not clear exactly what is meant by a "compound proposition," nor is it clear why 310,952 is off by 2 from 310,954. Perhaps more importantly, the method by which Hipparchus computed these numbers has been lost. We do know of recurrences that Hipparchus could have used to calculate these numbers, and it is a plausible conjecture that he used such a recurrence, but unless someone discovers further corroborating documents from that time period, this conjecture remains unconfirmed. In any case, computing these numbers requires considerable sophistication in combinatorics; as far as I know, we have no other evidence from that far back of any other combinatorial calculation of comparable difficulty.

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