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(This question might be too vague, feel free to edit or vote for closing.)

In math there are usually lots of non-existence theorems. When someone presents such a theorem, one natural response is "why shall I even care", or "why should such a thing be impressive".

The problem is, in the case of a non-existence theorem, usually all examples are trivial. If you tell some undergrad non-constant bounded entire functions don't exist, he/she will probably reply with a shrug. Similar thing happen to me and my friends when we talk about some fancier theorems (or when I see a paper stating such a non-existence theorem). I feel like it's really hard to convince people (or convince myself) "a priori this thing could exist, however by this awesome theorem it doesn't."

I think people would be impressed if all hypothesis look innocent, like the one in Liouville's theorem on entire functions, or maybe the fact that people have seen the existence of differentiable bounded non-constant functions helps. In general it is not necessary that all hypothesis look friendly. How would one figure out whether a non-existence theorem is a good one or is true just because one of the hypothesis is insanely strong?

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Question is rather vague. There are no solutions to the equation $x^n+y^n=z^n$ in positive integers $x,y,z,n$ if $n>2$. Why do we care?The proof is kind of cool. –  Felipe Voloch Oct 27 '11 at 20:18
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I'm not sure the concept is well-defined. In your specific example, you could just re-write this as "Bounded entire functions are constant," so in this sense, every theorem of the form "Blah always happens" is a non-existence result. Even among the most famous non-existence results, say the non-existence of a radical formula for solving the quintic, is just an existence result in the sense that it proves the existences of a non-solvable qunitic. Or is this a non-existence result, since it shows the non-existence of a sub-normal series of S5...you see my point. –  Cam McLeman Oct 27 '11 at 20:20
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You mention Liouville's Theorem. In my impression it not typically, or at least not always, presented as a nonexistence theorem. One can say: If the function is nonconstant and entire, then it is unbounded. Or, if the function is bounded and entire, then it is constant. Perhaps, this is a half-way answer to your question. If possible, rephrase the nonexistence theorem and see whether the implication seems interesting. –  quid Oct 27 '11 at 20:21
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Any theorem of the form "all A such that B satisfy C" can be equivalently considered as non-existence theorems: "there is no A such that B and not-C". Whether the result is interesting depends, I guess, on whether people looking at things that satisfy some of the hypotheses might wonder whether they could satisfy the others. –  Robert Israel Oct 27 '11 at 20:29
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I think that some of this is missing the point. Of course, we all know that there are logically equivalent ways of stating propositions using different quantifiers, etc. The question is about presentation. –  MTS Oct 27 '11 at 20:49
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6 Answers 6

What I noticed is that the infamous question "so what?", though seldomly asked directly, can be read in the eyes of the audience every time when the following three (rather common) conditions are satisfied:

a) there is nobody in the audience who ever thought of the question himself

b) there is no catchy picture or phrase in the presentation

c) some non-zero special knowledge is needed to understand even the statement of the theorem.

It doesn't really matter much what kind of result is being presented: a non-existence one, or something else though condition (b) (the only one you really have some control over during the talk) is much harder to violate when you are presenting a non-existence theorem. Indeed, you, apparently, cannot provide a fascinating example of an entity that doesn't exist in mathematics. Neither cannot you conclude with "we still do not understand a lot about the behavior of this fascinating object" when the object does not exist. And the whole thing often looks about as exciting as a report of a treasure hunting expedition that ends up with "so, after spending many days questioning the locals, climbing, digging, etc., we can safely conclude that there has never been anything of value in that area".

Still, one can be impressed with a pure impossibility theorem.

My favorite non-existence result is the impossibility to find an elementary antiderivative of $\frac{e^x}{x}$. It is useless to try to impress a layman with it. The appreciation I have comes from the fact that in my case (a) and (c) were violated.

First, when I was a student our calculus recitation sections consisted for one semester almost exclusively of finding some tricky antiderivatives. Many of those were of the kind where changing a single sign or a coefficient would result in an unsolvable problem and there were some misprints in the assignments. That made me wonder whether it might be possible to solve the problem as given. Of course, our teachers claimed that $\int \sin(x^2)dx$ and such cannot be presented by a neat formula but they never gave us even the slightest hint why. We saw, of course, that no standard integration trick worked but that might merely mean that there are some new tricks to discover. I was smart enough not to try what was claimed to be proven impossible but not smart enough to prove that impossibility or even to have a decent idea about how such things could be proved in general. I thought of asking my teachers about it but I suspected (rightly, as it turned out) that they knew it no better than I and no analysis textbook I was reading gave me the slightest clue.

That was in Russia back in 80's, so you could not just google "Liouville theorem" up or download any fancy book you wanted from the web. So, I remained completely ignorant until much later, when I was a postdoc at MSU, I visited the University of Toledo and met Rao Nagissetti who gave me some papers of his. To my great surprise, those were about impossibility to solve some quadratic differential equations in elementary functions. The old memories returned at once and I read those papers overnight. The ideas were completely novel to me though I have read Lang's algebra by then and wasn't afraid of Galois theory and such; I just viewed all that as something infinitely remote from any analysis question. That was one of the few times in my life when I was genuinely impressed: the proof lay not beyond my technical abilities (it isn't technically difficult at all) but beyond the range my imagination.

I leave it to you to derive a moral from this story (if it has any). I want only to say that I have managed to forget many things I learned but I doubt I'll ever forget the Liouville theorem. I played with it a bit too over the years (See http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2471315#p2471315 for my latest attempt. It also shows that today some kids are as curious about these questions as I was 25 years ago).

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Well, this is sort of an answer, I just have no memory of the participants or the area of mathematics, just that it may have been in Yemon's world. But 25 years ago. Postdoc $\varepsilon$ met established Professor $Y,$ and, pleased to meet the great man, proceeded to give a sketch of the proof of his main result. (I did not use $X$ because...) $ \; \; Y \; $ listened for a while, then said, "That's a very nice result, let me see if I can disprove it." $Y$ then proceeded to rattle off a sequence of examples and figured out in what way each was not a counterexample to the new result. $\varepsilon$ said later that he learned more about his result in that 20 minutes than in all of graduate school. He went on to work with $Y$ for some time.

This has seemed the right response to various MO things, I have just been embarrassed about remembering no names, places, or much else. It still works as urban legend, though.

EDIT: it is not clear to me how I came to hear of this, but Henry Cohn has pointed out that this must be Keith Kendig and Hassler Whitney, see the second page in this pdf: HASSLER_WHITNEY a short essay by by S. Swaminathan, also in French on page 3 of the pdf.

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The preface of Keith Kendig's book Conics tells this story about his interaction with Hassler Whitney (including various details like the 20 minutes, so I bet it is the story you remember). Amusingly, the theorem wasn't one of Kendig's results, but rather Bezout's theorem about the number of intersections of plane curves. –  Henry Cohn Oct 27 '11 at 22:48
    
Cool! Thank you, Henry. I can't say the name Kendig links to this memory for me. But then, the name Manin was of no help when Keith Conrad filled in the details on my part rendition of the Coleman-Manin story. –  Will Jagy Oct 27 '11 at 22:54
    
This story implicitly answers the problem of the student who shrugs upon hearing the theorem on bounded entire functions. By the time it occurs in a class it's very natural, but presented with it cold, the reaction of any student of normal inquisitiveness would be astonishment and an attempt at a counterexample. A student who isn't excited about celebrated classical results is perhaps best advised to find a field of study more to his liking. Incidentally, a result that has allured many a student into mathematics is, so to speak, a non-existence non-theorem: the continuum hypothesis. –  Greg Marks Jul 6 '12 at 18:19
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A recurring theme in the answers and comments that you have received so far is that the question is ill-posed since many (or perhaps all) non-existence claims can be recast as universal claims. While this is logically correct, I sympathize with your question and have thought about it a bit myself. While Liouville's theorem can correctly be phrased as the statement that bounded entire functions are constant, it really feels like a non-existence theorem: our experience with one variable calculus over the reals suggests that there ought to be a wealth of bounded complex differentiable functions, but in fact there aren't that many.

Here is why I think non-existence theorems are useful, and why I think it is useful to think about them as non-existence theorems. Mathematics is fundamentally all about starting with a huge untamed wilderness of possibilities and hacking our way through it until we have completely understood all possible behavior. This is why every area of mathematics has its own classification program which often motivates much of the research in that area; we are not content to just have a list of examples of a particular mathematical object unless we know that the list is in some sense comprehensive. Even when we can't classify some class of objects - such as finitely presented groups - we aren't satisfied until we can prove that we can't classify them.

A good illustration of what I mean is the classification of finite simple groups. At its heart, the result is a non-existence theorem: an important part of the proof was constructing all of the exceptional examples, but I think most of the 15000+ pages of the proof are dedicated to verifying that no other exceptional examples exist. By now I believe there actually are a few general theorems about finite groups that were proved using the classification theorem, but certainly not enough to justify the decades of labor that it took to complete the program. The reason that effort was worth it is because if it wasn't done then there would probably be a lot of group theorists out there right now up nights wondering if there is some crazy example of a finite simple group that nobody has thought of yet.

In the end mathematics is about building models for some sort of real-world phenomenon (even though it is difficult to trace some pieces of mathematics back to reality), and part of our job is to place constraints on what kinds of models are possible. It often pays off: the non-existence of bounded entire functions implies the fundamental theorem of algebra, a nontrivial existence result, and the non-existence of retractions from the disk to the circle implies the existence of a fixed point for any continuous self-map of the disk. I look at these results as examples of situations where we place enough constraints on a model to completely determine some of its behavior. This happens often enough to justify the overall philosophy in my mind.

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Speaking of the classification theorem, there is a huge difference between saying 'there are a finite number of sporadic groups', 'there are less than N sporadic groups, for some enormous N', and 'there are exactly 27 sporadic groups', and 'there are exactly 27 sporadic groups and we know what they are'. There are other examples in number theory where we can't even make the first statement, even though we only know a small number of examples (Fermat primes are my favourite). Making a contrast like this is handy. –  David Roberts Oct 28 '11 at 0:00
    
I think this answer comes closest to my own not-really-crystallised feelings. For instance, some problems in Banach and operator algebras ask if certain pathological things exist, not necessarily because such a putative example could be used for anything else, but because its existence would give us some kind of perspective on what we already know in the area, and its non-existence would give us hope that somehow all Boojums are of a form we feel we understand. –  Yemon Choi Oct 28 '11 at 0:57
    
But we don't need Liouville's Theorem to prove the fundamental theorem of Algebra, and we don't need the non-existence of retractions from the disk to the circle to establish Brouwer's fixed point theorem. –  Denis Serre Oct 28 '11 at 7:27
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But a nice application of Liouville's Theorem is the description of the graded algebra of meromorphic functions in ${\mathbb C}$ with two given periods $1$ and $\omega$. –  Denis Serre Oct 28 '11 at 7:34
    
I'm not too confident about my knowledge of history, but I think Liouville's theorem led to one of the first correct proofs of the FTA and the non-existence of retractions was the basis of the original proof of the BFT. I do not claim that non-existence theorems are unavoidable, only that the philosophy of finding truths by whittling away falsehoods actually does work. –  Paul Siegel Oct 28 '11 at 12:25
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One important class of "non-existence theorems", which includes Liouville's theorem as a model example, are the various rigidity theorems throughout mathematics that tend to have the general flavour of "any object in class X which is weakly regular, is automatically strongly regular and/or unexpectedly algebraic". Or, in non-existence form, "There does not exist any object of class X which is weakly regular, but not strongly regular". These results tend to be very powerful in applications (because one can then verify a strong regularity property simply by establishing a much weaker, and thus presumably easier to check, regularity property) and are also often important psychological bridges between otherwise disjoint categories (e.g. between the topological and smooth categories). It also often allows one to think of strongly regular objects in X as being "isolated" in some sense from the irregular objects, since there are no transitional objects of intermediate regularity.

Rigidity results can often be motivated as an unexpected converse: one can state the stronger and weaker notions of regularity, and show that the stronger implies the weaker with so much room to spare that it would seem ridiculous that the converse claim can be established - and yet it is true.

An elementary example of a rigidity theorem is the high school geometry theorem that any two triangles with the same side lengths, must in fact have the same angles as well and are thus congruent; or in non-existence form, one cannot find a pair of triangles with the same side-lengths but differing angles. Thus, triangles are rigid in a way in which quadrilaterals, for instance, are not.

Deeper examples of rigidity theorems include

  • Liouville type theorems: solutions to a certain PDE which are "uniformly bounded" or "precompact" in some sense are necessarily constant (or trivial, or a soliton...). Such theorems have become a central part of the modern theory of critical dispersive PDE, see e.g. this survey of Killip and Visan. Similar themes also come up in Perelman's proof of the Poincare conjecture, when he finds that the asymptotics of Ricci singularities are in some sense governed by the very special solutions known as gradient shrinking solitons. Somewhat related here are the various elliptic regularity theorems, such as the theorem that every weakly harmonic function (or distribution) is strongly harmonic.

  • Hilbert's fifth problem: C^0 Lie groups are necessarily C^infty Lie groups (and in particular come with the rich algebraic structure of a Lie algebra, which is certainly not obvious at all if one only begins with C^0 regularity). This is actually part of a large constellation of related theorems, such as Cartan's theorem that any closed subgroup of a Lie group is again a Lie group, or Gromov's theorem that any group of polynomial growth is virtually nilpotent, or the Peter-Weyl theorem which asserts (among other things) that every compact group is the inverse limit of linear Lie groups.

  • Eliashberg-Gromov symplectic rigidity: roughly speaking, the C^0 limit of symplectomorphisms is again a symplectomorphism, which is surprising as one would naively imagine one would need something like C^2 control instead. I'm not an expert on this topic, but I understand that this is a fundamental theorem in symplectic topology. A related result which also has some rigidity flavour to it (and is certainly a non-existence theorem) is Gromov's symplectic non-squeezing theorem that one cannot sympletically map a large ball into a thin cylinder. (Hmm, Gromov's name is coming up a lot...)

  • Dynamical rigidity: For certain very special types of dynamics (such as homogeneous dynamics coming from unipotently generated groups), the only minimal closed invariant sets or ergodic measures are those that come from algebraic constructions, such as algebraic subgroups; Ratner's deep theorems on this subject are model examples of this phenomenon. These theorems can be incredibly powerful for establishing equidistribution or density of orbits in homogeneous spaces, for instance the Oppenheim conjecture can be solved as a quick corollary of these theorems (though this was not quite the historical chain of events, as Margulis' proof of this conjecture preceded Ratner's work by a few years). Another example of dynamical rigidity is superrigidity, that in some cases the action of a continuous group can be controlled by the subaction of a discrete lattice; this in turn is related to hyperbolic geometry rigidity theorems such as Mostow rigidity, though this is far from my own area of expertise. Kazhdan's property (T), which asserts that approximately invariant vectors in actions of certain groups must be close to genuinely invariant ones, is another related property of groups that certainly pulls in the direction of rigidity.

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Speaking of Gromov, his program of quasi-isometric rigidity is very much like what Paul Siegel describes in his answer below, wading through the incredible thicket of finitely generated groups and trying to impose order, classify, with a verrrry far off and dim hope of eventually proving that nothing more exists. –  Lee Mosher Jul 6 '12 at 19:52
    
Ah, yes, qi rigidity is another great example of the rigidity phenomenon in mathematics - quasi-isometries seem far too weak to preserve algebraic structure, and yet they often do to a surprising extent. For instance, an immediate corollary of Gromov's polynomial growth theorem is that any finitely generated group which is quasi-isometric to a virtually nilpotent group, is again virtually nilpotent. –  Terry Tao Jul 6 '12 at 20:20
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Maybe it's in the phrasing. For example, instead of saying "There does not exist a non-constant, bounded, entire function," you could say "A bounded entire function is constant," or "A non-constant entire function is unbounded."

So, instead of saying that an object cannot share two properties at the same time, the theorem can be presented as saying that having one property forces the negation of the other property. Of course, then there is the question of which way to go. Do you say (entire + bounded $\Rightarrow$ constant) or (entire + nonconstant $\Rightarrow$ unbounded)?

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"A general angle cannot be trisected using straightedge and compasses."

Wikipedia

Would it be better to phrase this positively somehow?

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Since there are many angles that can be trisected with straightedge and compass, I would work on the phrasing some more. Perhaps start with "There is no single method..." . Gerhard "Ask Me About System Design" Paseman, 2012.07.06 –  Gerhard Paseman Jul 6 '12 at 14:37
    
Or: “you can trisect a 90-degree angle, but not a 30-degree angle.” –  Lubin Jul 6 '12 at 16:45
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