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I've seen internet jokes (at least more than 1) between mathematicians like this one here about someone studying a set with interesting properties. And then, after a lot of research (presumably after some years of work), find out such set couldn't be other than the empty set, making the work of years useless (or at least disappointing), I guess.

Is this something that happens commonly? Do you know any real examples of this?

EDIT: I like how someone interpreted this question in the comments as "are there verifiably true examples of this well-known 'urban legend template'?"

FeedbackLooper
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    For example, Reinhardt cardinals? (They do not exist due to Kunen inconsistency theorem.) – Hanul Jeon Nov 10 '20 at 08:32
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    " making the work of years useless, I guess." Not necessarily. Knowing that something is the empty set might be disappointing, but it is in any case better that having a wrong idea about it. – Francesco Polizzi Nov 10 '20 at 08:51
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    The set of all solutions in integers to $a^n + b^n = c^n$, with $n \geq 3$ and $abc \neq 0$, is one example that comes to mind.... – Rivers McForge Nov 10 '20 at 08:54
  • Thanks Francesco Polizzi! You are totally right. I guess I wanted to say (as you pointed out) that the result might be disappointing. Someone could "feel" like the work was useless at first glance. – FeedbackLooper Nov 10 '20 at 08:56
  • Those sound like nice examples , thanks – FeedbackLooper Nov 10 '20 at 08:59
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    Isn’t every proof by counter example investigating properties of the empty set? – Gordon Royle Nov 10 '20 at 10:38
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    @GordonRoyle for sure. But In a proof by counter example you usually expected the set to be empty right from the beginning. I was looking for examples in which this was unexpected, and than finding that out took presumably a lot of work and time. – FeedbackLooper Nov 10 '20 at 10:41
  • I bet some people are working on consequences of having non-trivial zeros of the zeta function outside of the critical line ! Though we don't know if this is empty of course :) – Adrien Nov 10 '20 at 12:54
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    If you work for a while in the belief that an object with certain properties should exist, and eventually learn otherwise, then you will have discovered that the set of such objects is empty. I just now gave an MO answer of this type! https://mathoverflow.net/a/376107/2926 – Todd Trimble Nov 10 '20 at 13:26
  • Somehow I think that many proofs by counterexample started life as a..... proof. And then, surprise surprise, a contradiction arises. – Lee Mosher Nov 10 '20 at 21:25
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    @LeeMosher, but, nowadays, we don't get to end with such a flourish as "this is repugnant to the nature of the straight line". – LSpice Nov 10 '20 at 22:21
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    Does this count? https://mathoverflow.net/a/53172/27343 – Ari Brodsky Nov 11 '20 at 09:21
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    As a principle every proof by contradiction applies. Still the question can be a bit narrowed to the specific thorough study of objects that are expected (or not) not to exist. For instance, the smallest finite non-abelian simple group of odd order (or satisfying some given properties). It could happen that such results appear way before they're shown not to exist. – YCor Nov 11 '20 at 14:33
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    Echoing a preceding comment: analytic number theorists pour significant effort into zero density theorems for the Riemann zeta function (and its relatives), which amount to elaborate upper bounds on the cardinality of sets that are widely expected to be empty. – Terry Tao Nov 11 '20 at 16:48
  • A very pleasant example and in fact a story regarding non existence of points of increase of one dimensional Brownian motion can be found here https://imstat.org/2019/11/15/the-curious-incident-of-the-car-in-the-night-time/ – Aditya Guha Roy Nov 11 '20 at 19:09
  • It's of course pretty common that if one thinks there may not exist any foofnagles, then one might try to prove that every foofnagle satisfies certain conditions, e.g. is green and smelly, and then eventually determine that these conditions are so strong that they cannot be satisfied. But maybe the spirit of the question would be more about proving properties of the set of all foofnagles - e.g. it is compact in some topology, it is a group, it has cardinality less than whatever - instead of proving properties satisfied by foofnagles individually. – Nate Eldredge Nov 11 '20 at 21:24
  • I once attended a talk by Donald Knuth where he said he discovered a polyhedron with nine faces, which he later realized does not exist. (But I don't think he did much with it.) As follows: three regular pentagonal faces share a vertex, and the three sides opposite that vertex are in a common plane, and you take another copy of that structure and glue those three edges to the corresponding three edges of that other copy, and then the remaining three faces are rhombuses or rhombi or whatever you call them. But then he realized that the four edges of such a rhombus are not coplanar. – Michael Hardy Nov 12 '20 at 17:01
  • Somewhat related question: https://mathoverflow.net/questions/160265/analogues-of-p-vs-np-in-the-history-of-mathematics – Sam Hopkins Nov 16 '20 at 17:34
  • I think you can probably cook up an example from the early results of Whitney in manifold theory: e.g. the fact that there is no difference between a $C^{\infty}$ vs. $C^{\omega}$ manifold (unless I'm misremembering this fact...). – Sam Hopkins Nov 16 '20 at 17:37

11 Answers11

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Jonathan Borwein, page 10 of Generalisations, Examples and Counter-examples in Analysis and Optimisation, wrote,

Thirty years ago I was the external examiner for a PhD thesis on Pareto optimization by a student in a well-known Business school. It studied infinite dimensional Banach space partial orders with five properties that allowed most finite-dimensional results to be extended. This surprised me and two days later I had proven that those five properties forced the space to have a norm compact unit ball – and so to be finite-dimensional. This discovery gave me an even bigger headache as one chapter was devoted to an infinite dimensional model in portfolio management.

The seeming impass took me longer to disentangle. The error was in the first sentence which started “Clearly the infimum is ...”. So many errors are buried in “clearly, obviously” or “it is easy to see”. Many years ago my then colleague Juan Schäffer told me “if it really is easy to see, it is easy to give the reason.” If a routine but not immediate calculation is needed then provide an outline. Authors tend to labour the points they personally had difficulty with; these are often neither the same nor the only places where the reader needs detail!

My written report started “There are no objects such as are studied in this thesis.” Failure to find a second, even contrived example, might have avoided what was a truly embarrassing thesis defence.

Glorfindel
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Gerry Myerson
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At the beginning of XX century, Hilbert and his students were actively investigating the properties that a consistent, complete and effective axiomatization of arithmetic should have.

As we all know, this line of research was unexpectedly wiped out (at least, in his initial formulation) by Gödel's First Incompleteness Theorem (1931), saying that no such axiomatization can exist.

Francesco Polizzi
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    Might want to add "recursively enumerable" somewhere in there. – Pace Nielsen Nov 10 '20 at 23:53
  • @PaceNielsen: you are of course right, I was being sloppy. I added "effective" in the assumptions about axiomatization (a formal system is said to be effectively axiomatized if its set of theorems is a recursively enumerable set). – Francesco Polizzi Nov 11 '20 at 09:15
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What about:

There are no infinite order polynomially complete lattices, after all by Goldstern and Shelah.

Mohammad Golshani
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Not exactly the empty set, and not years of work, but Milne told the following story about some research he and a colleague were doing in ring theory. They proved a few theorems; then, they made some assumptions on the ring, and proved some stronger theorems; then, they made some more assumptions on the ring, and proved some even stronger theorems; then, they made a few more assumptions, and were amazed at the strength of the results they were getting – until they realized that any ring satisfying all those assumptions had to be a field.

Gerry Myerson
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    This answer would really benefit from some details or a reference - though understandably I imagine you don't have any. – Wojowu Nov 10 '20 at 11:09
  • This is the kind of situations I refereed. As @Wojowu said, it would be great if you had a reference for this. Thanks anyway! – FeedbackLooper Nov 10 '20 at 11:14
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    @Wojowu correct on both counts. Milne told the story during a lecture to a class in which I was enrolled as a graduate student nearly 45 years ago. I don't know whether he ever put it i writing. I suppose if you really want details, you could write to him to ask for them. – Gerry Myerson Nov 10 '20 at 11:15
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    My interpretation of the question was, are there verifiably true examples of this well-known 'urban legend template'? So the details are important. Otherwise, it's easy to find examples, e.g., this one or this one. – Timothy Chow Nov 10 '20 at 17:04
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    @Timothy I chose not to post an answer about the Holder-or-Lipschitz continuous function story precisely because I couldn't find any retelling with checkable details. The Milne story is not something I heard from a friend of a friend; I was there, in the classroom, at the time. As I wrote before, if you want details, you could try writing to Milne. – Gerry Myerson Nov 10 '20 at 21:55
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"The next two chapters [Chapters 9 and 10] show more recent technology which was developed to replace the unproven Riemann hypothesis in applications to the distribution of prime numbers. We are talking about [zero density] estimates for the number of zeros of $L$-functions in vertical strips which are positively distanced from the critical line. Hopefully in a future one will say we were wasting time on studying the empty set."

Henryk Iwaniec and Emmanuel Kowalski, Analytic Number Theory, page 2

2734364041
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The following two paragraphs are the last footnote on p. 69 of [1]. I found this such good advice that I began the first chapter of my 1993 Ph.D. dissertation, on p. 6, with this quote.

[1] William Henry Young, On the distinction of right and left at points of discontinuity, Quarterly Journal of Pure and Applied Mathematics 39 (1908), pp. 67−83. (Also here.)

Mark the importance of testing not only the accuracy but also the scope of one's results by constructing examples. To quote an instance which has come under my notice in the course of my present work, Dini (p. 307) states that if a left-hand derivate and a right-hand derivate both exist and are finite and different at every point of an interval $\ldots$ $\ldots$ certain results follow.

The reader might well imagine not only that such a case could occur, but that Dini knew of a case where it did occur. As a matter of fact, however, the hypthesis [sic] is an impossible one. In default of an example it could, in such a case, only stimulate research to state that an example had not been found.

Incidentally, I don’t know whether “p. 307” is for the 1878 Italian original of his real functions book or for the 1892 German translation of his real functions book. Young’s previous footnote appears to cite the 1878 Italian original, but p. 307 of the German translation seems more likely (based on math symbols appearing; I can’t read German or Italian).

For some more context about the fact that no such function exists, see B. S. Thomson’s answer to If $f$ is bounded and left-continuous, can $f$ be nowhere continuous? and my answers to A search for theorems which appear to have very few, if any hypotheses and Real-valued function of one variable which is continuous on $[a,b]$ and semi-differentiable on $[a,b)$?

Dave L Renfro
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    Your intuition is correct. On page 307 in the 1982 version, the letterspaced paragraph is the sought-for "theorem". – Mirko Nov 10 '20 at 18:29
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    The corresponding place in the Italian version is on page 224: https://archive.org/details/fondamentiperla01dinigoog/page/n240/mode/2up – pavel Nov 11 '20 at 15:20
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Arrow's impossibility theorem comes to mind. To quote Wikipedia:

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

  • If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
  • If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
  • There is no "dictator": no single voter possesses the power to always determine the group's preference.

More in the spirit of the question: The set of fair rank-ordered electoral systems is empty.

Dirk
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The odd-order theorem states that every finite group of odd order is solvable, and the proof involves developing a very large theory explaining what the smallest counterexample looks like, and to ultimately deduce that it cannot exist.

The odd-order theorem has been formalised (pdf) in Coq, a computer theorem prover, and the formalisation is to date one of the largest bodies of formalised mathematics. This makes it appealing to AI researchers, who go and train their deep learning networks using the collection of theorems proved in the formalisation, hoping that one day computers will start to be able to compete with humans in the realm of theorem-proving.

I find it amusing that, as a consequence, these networks are being trained to recognise a whole bunch of facts about an object which doesn't exist.

Kevin Buzzard
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I do not know whether this applies to the spirit of the question. However for me one of the high points of an undergraduate algebra class was seeing Witt's elegant proof of Wedderburn's theorem: There are no finite non-commutative division rings.

I recall discussing this with a professor in graduate school who expressed slight regret about this theorem. He felt that algebra would be richer if there were finite non-commutative division rings.

M. Khan
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So, I remember my teacher telling the following story:

Erik Zeeman was trying, for 7 years, to prove that it was impossible to untie a knot in a 4-sphere. He kept trying and, one day he decided to prove the opposite: That was indeed possible to untie the knot - It took him only a few hours to do so

Eduardo Magalhães
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    Any idea how old he was? I don't know the history of the result that any 1-knot in 4-space is equivalent to an unknot, but for some reason thought this would have been known for a long time, even predating Zeeman's birth year. – Todd Trimble Nov 16 '20 at 20:24
  • I confess I don't know, My teacher just mentioned it and never laked about it again @ToddTrimble – Eduardo Magalhães Nov 17 '20 at 14:34
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    This example shows why mathematicians need to learn to argue both sides of the case at the same time - not a skill that people in other occupations usually have. (I can't avoid thinking of a certain about-to-be-ex president here,) – Paul Taylor Nov 19 '20 at 20:24
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While Siegel zeros are not currently an example, they will hopefully become one in the future.

Jakub Konieczny
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