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The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, as the result the mathematics itself had to be thrown out. Old results might need a new assumption or two. Certainly the rewritten assumptions often allow wonderful new results, but have we actually lost anything?

Note I would like to rule out the case where an area has been rendered unimportant by the development of different techniques. In that case the results still hold, but are no longer as interesting.

I wrote up a longer version of this question with a look at a little of the history: http://maxwelldemon.com/2012/05/09/have-we-ever-lost-mathematics/

Edit in response to comments

My thinking was about results that have been undermined from below. @J.J Green's example in the comments of Italian algebraic geometry seems like the best example I have seen. The trisection and individually wrong results do not seem to grow into areas, but certainly I would find interesting any example where a flawed result had built a small industry before it was found to be wrong. I am fascinated by mathematics that has been overlooked and rediscovered (ancient and modern) but that is perhaps a different question.

Yemon Choi
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Edmund Harriss
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    If we had truly lost it, can you expect us to know enough about it to tell you? Gerhard "Still Looking For A Proof" Paseman, 2012.05.09 – Gerhard Paseman May 09 '12 at 22:03
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    What counts as "anything"? Certainly incorrect theorems have been published. Sometimes they were unimportant, but occasionally they have been genuinely interesting and important results that could not be salvaged once the mistake was identified. Does that count as lost mathematics? I think there's a whole continuum here, from isolated errors to fundamental flaws in large parts of mathematics. I don't know of any really large-scale examples. – Henry Cohn May 09 '12 at 22:23
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    I had some mathematics in my pocket the other day, but I seemed to have lost it. Perhaps it is just buried in the mess of my desk... – Asaf Karagila May 09 '12 at 22:34
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    I think the OP might have meant a case when a whole field has been invalidated. I can't recall such a case. – Felix Goldberg May 09 '12 at 22:47
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    I take the question to be, is there a mathematical equivalent of the phlogiston theory of combustion? – Gerry Myerson May 09 '12 at 22:50
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    http://en.wikipedia.org/wiki/Obsolete_scientific_theory makes no mention of mathematics... – Felix Goldberg May 09 '12 at 23:00
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    In which case, there are several instances of such theories that continually reappear, e.g. trisecting the angle using compass and straightedge alone. They aren't lost, unfortunately. Gerhard "Ask Me About System Design" Paseman, 2012.05.09 – Gerhard Paseman May 09 '12 at 23:02
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    But trisection have never been really accepted as valid, have they? Maybe a better example would be proofs of Euclid's fifth postulate. As I understand, they were being "improved" throughout the ages until the whole enterprise imploded. – Felix Goldberg May 09 '12 at 23:05
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    Didn't something along these lines happen to Italian algebraic geometry in the 1930s? see http://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry for example – J.J. Green May 09 '12 at 23:13
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    I would phrase the question 'Have we had to lose any mathematics?' (or "mathematics", or "theorems") There is probably some ancient Greek stuff that we know about but don't have (we have actually actually lost it, and not on purpose), but this is not what you are asking about. – David Roberts May 09 '12 at 23:59
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    This sounds like a different phenomenon from the one that you are refering to (is it?), but Indian Mathematics, Chinese Mathematics, Babylonian Mathematics, etc. were effectively "lost" (at least in large part), and have only recently been partially "rediscovered" as "archeology". More recently, 19th century invariant theory. It wasn't that they were false; it was that numerical methods became less valuable for these problems, because general methods were discovered, or else the calculations didn't draw enough attention because of lack of practical applications. – Daniel Moskovich May 10 '12 at 00:06
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    A lot of classical Greek mathematics is only known to us via Arabic translations (so it was "lost" to Europeans for centuries) and some classical mathematical works referred to in others have been lost altogether. Of course it is rather unlikely that any original mathematics in those works has not since been rediscovered. – Brendan McKay May 10 '12 at 03:23
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    Would gematria count? – Terry Tao May 10 '12 at 03:35
  • @Terry: I still see numerology around, especially in California. Then again, I don't go out of my way to avoid it; I really enjoyed Underwood Dudley's "Mathematical Cranks". I think (hope) it doesn't count, though. – Eric Tressler May 10 '12 at 03:56
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    According to http://www.cecm.sfu.ca/organics/covering/html/node4.html, "We have reached the point of decay in some areas. Richard Askey has observed that Gregory Chudnovsky knows things about hypergeometric functions that no one has understood since Riemann and that, with Chudnovsky's eventual passing, no one is likely to understand again." I've wondered what this refers to, but I've never asked Askey whether this quote is accurate or what he meant. – Henry Cohn May 10 '12 at 04:25
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    @Terry and @Eric Gematria is a neat example, certainly built on foundations we now feel are unfounded. The actual mathematics built on top of it, however, some combinatorics is still valid. So perhaps this is actually evidence the other way that good mathematics CAN be built on the weakest of foundations. – Edmund Harriss May 10 '12 at 04:42
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    @Henry Sadly the decay is not a new thing. For example the role that Coxeter played in protecting many results in Geometry, or the nineteeth century interest in the geometry of linkages that is now being rediscovered. A nice example of this is Kempe's result that any bounded region of an algebraic curve can be made by a linkage. Though in this case the decay is being reversed as O'Rourke and Demaine give (the first correct) proof in http://www.amazon.com/Geometric-Folding-Algorithms-Linkages-Polyhedra/dp/0521857570 – Edmund Harriss May 10 '12 at 04:57
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    @Daniel: I don't understand what you mean by "it was that numerical...practical applications". 19th century invariant theory was not about doing numerical calculations. Also, it was an endeavor in pure mathematics. Practical applications were not at the top of the agenda. – Abdelmalek Abdesselam May 10 '12 at 13:02
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    @Edmund: Actually, the first correct proof appeared in M. Kapovich, J. Millson, Universality theorem for configuration spaces of planar linkages, Topology, Vol. 41 (2002), no. 6, p. 1051--1107. – Misha May 13 '12 at 21:25
  • @Misha Thanks for that. Will look up that paper. – Edmund Harriss May 14 '12 at 01:19
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    @Henry: In a similar vein, I have heard the fear expressed that the last generation of experts on the proof of the Classification of Finite Simple Groups will die out and posterity will never recover the current level of insight into CFSG, because it will be too much effort for too little reward for newcomers to reach this level of expertise. – Colin Reid May 14 '12 at 05:34
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    Since the issue seems to come up repeatedly in answers (despite @YemonChoi's heroic efforts), would you mind making a further edit to clarify that you mean 'lost' not in the sense of "unable to be found" ("the lost work of …"), but rather in the sense of "realised not to be true" (something more like "lost innocence")—unless that isn't what you mean, in which case a clarification to that effect would probably be appropriate? – LSpice May 11 '16 at 01:55
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    @LSpice The OP http://mathoverflow.net/users/15516/edmund-harriss is, at time of writing, "last seen Sep 6 '14" – Yemon Choi May 11 '16 at 02:03

9 Answers9

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Hilbert's $16^{\rm th}$ problem.

In 1923 Dulac "proved" that every polynomial vector field in the plane has finitely many cycles [D]. In 1955-57 Petrovskii and Landis "gave" bounds for the number of such cycles depending only on the degree of the polynomial [PL1], [PL2].

Coming from Hilbert, and being so central to Dynamical Systems developments, this work certainly "built a small industry". However, Novikov and Ilyashenko disproved [PL1] in the 60's, and later, in 1982, Ilyashenko found a serious gap in [D]. Thus, after 60 years the stat-of-the-art in that area was back almost to zero (except of course, people now had new tools and conjectures, and a better understanding of the problem!).

See Centennial History of Hilbert's 16th Problem (citations above are from there) which gives an excellent overview of the problem, its history, and what is currently known. In particular, the diagram in page 303 summarizes very well the ups and downs described above, and is a good candidate for a great mathematical figure.

Community
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Rodrigo A. Pérez
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I was once told of a paper in homological algebra where a new class of functors was introduced, generalizing Ext and Tor. For some years they were studied, and various properties were proved. Finally someone managed to give a complete description of the entire class. It consisted of two elements, Ext and Tor. (Sorry, I don't have more details.)

Per Manne
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    This is interesting, anyone have more details? – Edmund Harriss May 10 '12 at 09:44
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    This reminds me of a colloquium talk I heard at Harvard some 40 years ago, in which a famous speaker generalized some results of Bott to an abstract setting. At the end, Bott, who was sitting in front, asked the speaker if he knew any examples of his theoretical objects other than, as I recall, sections of vector bundles over (possibly compact?) manifolds. The answer was "No, I don't." – roy smith May 10 '12 at 16:04
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    Edmund, this may be related to your question.

    http://mathoverflow.net/questions/93716/cov-right-exact-additive-functors-that-dont-commute-with-direct-sums

     – roy smith May 10 '12 at 16:14
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    I heard the same story about a PhD thesis, where someone in the audience announced during the defense that the class of objects having the amazing properties described by the student was actually empty. I assume all such stories are apocryphal. – JeffE May 13 '12 at 08:47
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    That's not a loss, at least not a complete loss... it's a classification theorem. – Rex Butler Feb 28 '17 at 22:36
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Volume II of Frege's Grundgesetze der Arithmetik (Basic Laws of Arithmetic) had already been sent to the press when Bertrand Russell informed him that what we now call "Russell's paradox" could be derived from one of his basic laws. I do not know to what extent Frege's work was known and publicly accepted (volume I was published 10 years before volume II), but this seems a clear case where a major body of work was undermined "from below", to use the words of the OP.

Upon learning of Russell's observation, Frege quickly wrote up an appendix to volume II, where he writes, "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." (This translation appears in the Wikipedia article.)

Todd Trimble
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    There has been an attempt to recover the ideas of Frege's approach while avoiding the paradox: https://plato.stanford.edu/entries/frege-theorem/ – arsmath Jun 26 '20 at 20:46
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There are "Lectures on Lost Mathematics" by B. Grünbaum. They were given at the University of Washington in 1975. The notes are available here

Rodrigo A. Pérez
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Kristal Cantwell
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    Not to forget the book "A la recherche de la topologie perdue" edited by Guillot and Marin. – anonymous May 10 '12 at 06:27
  • @anonymous: why do you mention that book? Is it about lost math? – hopflink May 13 '12 at 07:29
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    Given that the title translates roughly as "Remembering Lost Topology," I'd assume so. – Daniel McLaury May 14 '12 at 07:19
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    This does not seem to address the question being asked: the OP apparently wanted examples of mathematics that had to be discarded because of errors – Yemon Choi May 10 '16 at 19:39
  • It addresses the original question not the one that is now there after today's revision. I think the original question asked about lost mathematics. – Kristal Cantwell May 11 '16 at 16:28
  • Kristal, I have read the original question and it had the meaning specified here. All I did was to expand the title. The revision history http://mathoverflow.net/posts/96510/revisions shows that at the time when you wrote your answer, the question sought examples where maths was "lost" through the discovery of errors. The link in the question makes this very clear, I think – Yemon Choi May 11 '16 at 17:58
  • In section 3 of the notes I published it is noted that there are problems with the results on stiffness namely mainly many of the results are wrong due to problems with foundations such as bad definitions. In applied mathematics with some work being done by people who are not mathematicians there are probably going to be problems with the foundations that become apparent with inspection. – Kristal Cantwell May 11 '16 at 19:41
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I feel the answer is obviously "yes", and indeed that much of 19th century mathematics was lost, in a serious sense, for much of the 20th century. I was struck recently by discovering that Henry Fox Talbot, the photographic pioneer, had written on what is clearly the area around Abel's theorem for curves, and that probably it is a long time since anyone reconstructed what he was doing. Also that George Boole's main work, as far as his contemporaries were concerned, dropped out of sight within a couple of decades.

The fact is that mathematics now is (a) axiomatic and (b) dominated by a canon. I'm reminded of Bertrand Russell's nightmare - where, a century after his death, the last copy of the Russell-Whitehead Principia Mathematica is in danger of being thrown out by an ignorant librarian. It actually isn't obvious that even such a pioneering work makes it into the mathematical logic "canon" around later developments. (I hear protests!) Maybe it is worth pointing out Hilbert's interest in Anschauliche Geometrie, in other words non-axiomatic, intuitive geometry. And the canon should be "porous", as has been argued by some of the Moscow school. It seems quite an illuminating take on mathematics as a living tradition that simple accretion of "known results" is misleading.

Charles Matthews
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    I always understood Russell's nightmare as reflecting insecurity in the lasting importance of his work, rather than the intelligence of its judges. – Pablo Zadunaisky Dec 16 '12 at 22:34
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I don't know if this is an example of what you're asking. In mathematical logic, the Hilbert Program of the 1920's intended to come up with a finitary consistency proof and a decision procedure for analysis and set theory. Many luminaries including Hilbert himself, Bernays, Ackermann, von Neumann, etc. gathered in Göttingen for this purpose. Ackermann in 1925 published a consistency proof for analysis (that turned out to be incorrect) and many other promising results emerged. Then in 1931, Gödel's incompleteness theorem shut the whole thing down. Some valid theorems came out of it, but the program as a whole had to be (in some interpretations) completely abandoned.

http://en.wikipedia.org/wiki/Hilbert_program

none
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    This is a great example, but it shows more that mathematics itself is not lost. In fact this, as part of the quest for the foundations is perhaps the canonical example. Doubt was cast on the foundations of the whole subject. Yet we only lost research directions rather than worlds of results. – Edmund Harriss May 12 '12 at 22:47
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I'm not sure that Drach's 1898 thesis on differential Galois theory "built a small industry", but it was certainly accepted and praised by his examiners before Vessiot pointed out a very serious flaw. However, there was no public acknowledgement of this at the time (or later) by any of the parties involved.

It wasn't until the 1983 publication of Pommaret's Differential Galois Theory that the story came to light. In his 1988 book Lie Pseudogroups and Mechanics, Pommaret reproduced and translated into English the original examiners' reports, and the key correspondence describing the error.

For more context and details about Drach's work, see T. Archibald, "Differential equations and algebraic transcendents: French efforts at the creation of a Galois theory of differential equations 1880 - 1910", Revue d'histoire des mathématiques, 17 (2011) 373- 401.

Phil Harmsworth
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A exposition along this vein about Arabic mathematics.

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html

Eugene
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There are some results in coding theory (explicit constructions of codes, published in 1990) that are thought to be lost. See the heading Lost Codes in https://www.win.tue.nl/~aeb/codes/Andw.html - page maintained by one of the authors of the 1990 paper.

Dima Pasechnik
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    Isn't the question about maths that had to be "thrown out" because of flaws, rather than maths which was "mislaid"? – Yemon Choi May 10 '16 at 19:35
  • well, results from a paper that one cannot reconstruct fall under lost, no? – Dima Pasechnik May 10 '16 at 19:39
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    I still don't think this is what the OP intended - see https://maxwelldemon.com/2012/05/09/have-we-ever-lost-mathematics/ – Yemon Choi May 10 '16 at 19:40
  • the author cannot reconstruct the results in question; nor anyone else could. It's not "mislaid". – Dima Pasechnik May 10 '16 at 20:20
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    Dima, this is an interesting example, but I really don't think it meets the OP's aim as implied in the first paragraph: "The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, as the result the mathematics itself had to be thrown out. Old results might need a new assumption or two. Certainly the rewritten assumptions often allow wonderful new results, but have we actually lost anything?" – Yemon Choi May 10 '16 at 20:47
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    See also this part near the end: "My thinking was about results that have been undermined from below. @J.J Green's example in the comments of Italian algebraic geometry seems like the best example I have seen. The trisection and individually wrong results do not seem to grow into areas, but certainly I would find interesting any example where a flawed result had built a small industry before it was found to be wrong." – Yemon Choi May 10 '16 at 20:47
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    we found some apparently wrong published constructions (thought of as "classical") of Hadamard matrices: https://arxiv.org/abs/1601.00181 – Dima Pasechnik May 10 '16 at 21:05
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    and I found some wrong published constructions of abstract regular polytopes: https://arxiv.org/abs/1603.01710

    (these are of course not as spectacular as the whole Italian school story :))

     – Dima Pasechnik May 10 '16 at 21:07