The question What seemingly innocuous results in mathematics require advanced proofs? prompts me to ask about conjectures or, less formally, beliefs or intuitions, that turned out wrong in interesting or useful ways.

I have several in mind, but will provide just one here now, as an example.

For centuries mathematicians tried to show that Euclid's parallel postulate followed from his others. When Lobachevsky, Bolyai and Gauss discovered that you could do interesting geometry just as consistent as Euclid when the parallel postulate failed a whole new world was open for exploration.

One example per answer, please. If you want to post several, answer repeatedly.


Conjectures that have been disproved with extremely large counterexamples?

Ethan Bolker
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    Re. the example, a relevant book is also [János Bolyai, Non-Euclidean Geometry, and the Nature of Space](https://www.maa.org/press/maa-reviews/j-nos-bolyai-non-euclidean-geometry-and-the-nature-of-space). From wikipedia's [page](https://en.wikipedia.org/wiki/J%C3%A1nos_Bolyai): "*He became so obsessed with Euclid's parallel postulate that his father wrote to him: "For God's sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life". János, however, persisted in his quest...*". – dxiv Aug 19 '17 at 00:40
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    @dxiv Thanks. Changed Bernoulli to Bolyai, which is historically better I think. – Ethan Bolker Aug 19 '17 at 00:43
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    @dxiv : Yes, and ironically it did end up getting to his head as his father warned him (as far as I know). Not that he failed in his goal of building non-euclidean geometries, but that others got most of the recognition for the work during his life time. But at least we know about him at this day! – mathreadler Aug 19 '17 at 11:20
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    Does survivorship bias count? – Rohinb97 Aug 20 '17 at 05:51

18 Answers18


From coding theory --

If you want to transmit a message, say 0 or a 1, through a noisy channel, then the natural strategy is to send messages with redundancy. (A noisy channel takes your message and brings it to its destination, though with a chance of scrambling some of its letters.)

For example, suppose I wanted to transmit to you $0$ or $1$ -- $0$ for upvote and $1$ for downvote, say, though this narrative choice is mathematically irrelevant.

Suppose our channel is such that the chance of a single transmitted zero changing to a one is $.1$. If I send you the message $00000$, as long as 3 of the zeros remain zeros, you will be able decode my message by majority voting with a greater probability than if I just sent you a single $0$.

(Simlrly to hw you cn rd ths?)

The collection $\{00000,11111\}$ is what a code is, and $00000$ is called a codeword.

By continuing this way (for example by sending a million zeros to transmit the message zero), you would be able to drive the probability of a miscommunication as small as you want, however, at the cost of requiring a longer and longer transmission time to send the same message (zero or one).

If you call the rate this ratio ( the number of messages you send ) / (the number of symbols in your message), this naive coding method described lets you make the error small but at the cost of sending the rate to zero.

According to sources I've read (but how historically accurate they are I cannot say), practitioners in the field of communications (e.g. telegraph engineers) thought that this trade off was absolute - if you want to reduce errors, you must drive the rate to zero.

This is a plausible thing to conjecture after seeing the above example - or anyway you could probably get people (students) to nod along if you told showed them this example and told them this false conclusion confidently. (Conducting such an experiment would be unethical, however.)

However, in 1948 Shannon proved a remarkable theorem saying that this trade-off was not so absolute. (The noisy channel coding theorem.) Indeed, for any channel there is a number called the "capacity" (which in the discrete case can be computed from the probabilistic properties of the channel as an optimization problem over probability distributions on a finite set), and as long as you do not try to send at a rate better than this capacity, you can find ways to encode your messages so that the error becomes arbitrarily small. He also proved that this capacity bound was the best possible.

One catch is that you need to have a lot of messages to send; if you want to send one of two messages over your channel (and then close the channel for business), then you really cannot do better than the naive method above. Shannon's proof is asymptotic, asserting that as the size of the set of possible messages goes to infinity, you can (in principle) find encoding methods that make the error low - morally, you can keep the code words far from each other without using too many extra bits to pad the transmissions against errors, because there is a lot of extra room in higher dimensions.

His proof opened up a huge amount of research into explicit constructions of codes that approach his bound, which has connections to many other topics in mathematics and computer science.


If you want to learn the precise statement, the book Cover T. M., Thomas J. A., Elements of Information Theory, is excellent.

Juhani Heino
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Elle Najt
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  • Actually I think Shannon's theorem is not so surprising. In a phone conversation, for example, the more background noise you have, the louder and slower you have to speak for the other side to understand. So from this naive standpoint, it's not surprising that there is no absoulute tradeoff. – GregT Aug 23 '17 at 10:55
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    @GregT you just re-stated the incorrect intuition that Shannon *dis*proved. – N. Virgo Aug 23 '17 at 13:33
  • @Nathaniel Maybe I misunderstood something, but what I wanted to say was in line with the answer: "for any channel there is a number called the "capacity" (which in the discrete case can be computed from the probabilistic properties of the channel as an optimization problem over probability distributions on a finite set), and as long as you do not try to send at a rate better than this capacity, you can find ways to encode your messages so that the error becomes arbitrarily small" I also say that you need not drive the rate to zero. – GregT Aug 23 '17 at 14:23
  • @GregT Depending on the channel, there is a quantity called the power which affects the capacity; maybe this is what you are getting at. Mathematically the idea is like this: The Gaussian channel takes in a real number, and the person on the other end receives that number plus an independent mean zero Gaussian of some fixed variance. If you send very large numbers (i.e. of large power / $L^2$ norm), you can send a lot of information with very low error. In order to have a realistic model, you need to limit the average power you use. You can read about this in MacKay "Information Theory..." – Elle Najt Aug 23 '17 at 14:56
  • @AreaMan Yes. Thanks! – GregT Aug 23 '17 at 14:58

The first thing that came to mind was the proof that a general expression for the roots of a quintic (or higher) doesn't exist (Abel-Ruffini and Galois). People in previous centuries had thought it might be possible, and searched for the solution. The area of maths that Galois sparked was revolutionary.

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Kurt Gödels Incompleteness theorems in mathematical logic came as a shock to many mathematicians in a time when formalistic optimism ruled:

The theorems state that a sufficiently powerful theory always must have statements which are possible to express but impossible to prove / disprove, making mathematics a forever unfinishable "game" or puzzle as any of those unprovable statements can be chosen to be added as an axiom.

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    Does it say anything about the detection of these problems? For all theorems that are impossible to prove, is it possible to prove that they are impossible to prove? – Enrico Borba Aug 19 '17 at 04:30
  • @EnricoBorba : Good question. I am no expert in neither logics nor mathematical history, just a happy layman who read about things like this as exciting stories when I was a kid. – mathreadler Aug 19 '17 at 09:21
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    Theorems are provable *by definition of theorem*. There are true statements one can make along the lines of the post (e.g. talking about truth in a model or backing off and talking about undecideable propositions), but the one in the post is not one of them. –  Aug 19 '17 at 11:25
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    @EnricoBorba: There is no algorithm that (correctly) decides which propositions are decideable and which are not -- for any algorithm attempting to answer this question, there are some propositions for which the algorithm either gives the incorrect answer or fails to give any answer at all. –  Aug 19 '17 at 11:25
  • Okay @Hurkyl you may have a point. I updated the formulation to *statements* from *theorems*. – mathreadler Aug 19 '17 at 11:38
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    Turing’s proof that the halting problem is undecidable (closely related to this answer and the follow-up question by @EnricoBorba) is itself one of the most important fundamental results in computer science. A lot of other proofs show that, if something could be done, it would give us a way to solve the halting problem. – Davislor Aug 19 '17 at 15:31
  • @Davislor That is really cool, I have never thought about the halting problem as something of the same kind as Gödels theorems. – mathreadler Aug 19 '17 at 15:52
  • @mathreadler Turing titled the paper where he proved it, "On Computable Numbers, With an Application to the Entscheidungsproblem." Here’s a good lecture about how the Halting Problem and Incompleteness relate: http://www.scottaaronson.com/democritus/lec3.html – Davislor Aug 19 '17 at 16:04
  • @Davislor I expected a computer science paper about the halting problem to not pop up until the 50s or 60s at the earliest. I mean when we had any computers at all. – mathreadler Aug 19 '17 at 16:18
  • @mathreadler Then Alan Turing was twenty or thirty years ahead of his time! – Davislor Aug 19 '17 at 16:19
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    @mathreadler Well, we _did_ have computers, but they had wetware rather than software, and notepads and pencils rather than hard disks. Uf they were lucky, someone would give them a hand-cranked adding machine. – Spencer Aug 19 '17 at 19:11
  • @EnricoBorba: We can say something stronger than Hurkyl's response to your question. For any useful formal system $S$ that interprets arithmetic (so that the incompleteness theorem applies to it) and is consistent, $S$ cannot prove any sentence of the form "$S$ cannot prove ...", otherwise it would easily prove "$S$ cannot prove a contradiction", which is impossible by the (Rosser) incompleteness theorem. – user21820 Aug 20 '17 at 07:39
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    @mathreadler: For a direct connection between the halting problem and Godel's incompleteness theorem, as well as between the weaker zero guessing problem and Rosser's strengthening, see [this post and the linked blog post](http://math.stackexchange.com/a/1895288/21820). – user21820 Aug 20 '17 at 07:42
  • @Davislor: The blog post I'm referring to in the above comment is by the same Scott Aaronson, 5 years after the lecture you linked to. =) – user21820 Aug 20 '17 at 07:43
  • @Hurkyl "There is no algorithm that (correctly) decides which propositions are decideable and which are not." -- This applies to the set of all propositions. If I take a subset of that set, the halting problem may or may not remain relevant. For instance, for the subset {"The successor of 0 is 1"} of all propositions, every proposition is provable. EnricoBorba is specifically asking about a subset of all propositions that ask about the provability of other propositions. I suspect that the halting problem applies, but a proof is in order. – Him Aug 21 '17 at 12:52
  • @Scott: Even after rereading it and trying to construe his words that way, I cannot see how to interpret EnricoBorba's comment to this answer in that fashion. –  Aug 21 '17 at 12:56
  • @Hurkyl, EnricoBorba's questions specifically hinges on the set of propositions: "For all theorems that are impossible to prove, is it possible to prove that they are impossible to prove?" (where by 'theorems', Enrico means 'sentences'). An immediate example is Gödel's unprovable sentence in a system. This sentence is impossible to prove, but it is provably impossible to prove, so the sentence "Gödel's sentence is unprovable" is provable. Do there exist sentences which are unprovably unprovable? Probably Gödel's argument can be easily extended to show such a thing.... – Him Aug 21 '17 at 13:14
  • @Scott: Okay, I see how to construe things. But that's precisely what my comment addresses -- if $\mathrm{Decideable}$ is the collection of decideable statements, then there does not exist an algorithm for deciding membership in $\mathrm{Decideable}$. If you have an oracle for this problem, it's pretty easy to construct an algorithm for producing a *complete* theory extending arithmetic: start with Peano axioms and iterate through every proposition one by one. Each proposition you find that cannot be decided by your current axioms you add to the list. –  Aug 21 '17 at 13:34
  • @Hurkyl, such a system would not be consistent, because if X is undecidable, then ~X is also undecidable. – Him Aug 21 '17 at 13:35
  • @Scott: If you encounter $X$ first, you add it to your axioms; when you reach $\neg X$ in a later iteration, it will be decideable and you don't add it to your axioms. Similarly if you encounter $\neg X$ first. –  Aug 21 '17 at 13:38
  • @Hurkyl, this is no guarantee that the resulting system will be consistent, because what if you get sentences X, A and B; ~X isn't added as an axiom, but it happens to follow from A and B. Still the system isn't consistent. Indeed, I suspect this specific line of reasoning cannot be amended, because if there was an "add em all as axioms" sort of solution, then a similar thing would apply to the set of all statements. Viz. go through all the statements, add X, don't add ~X. – Him Aug 21 '17 at 13:42
  • @Scott: If $~X$ follows from $A$ and $B$, then $~B$ follows from $X$ and $A$, so you wouldn't have added $B$ to the list. (Assuming Peano arithmetic is consistent) the proof you get a consistent set is straightforward: you start with a consistent set, and if you started any iteration with a consistent set, you either leave it alone or you add something undecideable from it, so you finish the iteration with a consistent set. –  Aug 21 '17 at 13:44
  • @Scott: And there *is* an "add em all as axioms" solution -- or more simply just take your axioms to be all true statements about the natural numbers. Godel's theorem doesn't forbid complete sets of axioms -- what Godel's theorem says is that for any such thing, you cannot decide membership in the list of axioms by algorithm. –  Aug 21 '17 at 13:47

Maybe not the answer you expected on a mathematical site, but it feels fitting.

In the 19th century, scientists were convinced that all matter obeys Newtonian laws. The only small problem was a pesky anomaly: all equations predicted that black bodies should glow bright blue, but in reality they don't.

Max Planck was one of the people who tried to solve it. He tried using the equation


with $E$ for the energy, $n$ for any integer, $f$ for the particle's frequency, and $h$ for an arbitrary constant.

Planck's assumption was not justified by any physical reasoning, but was merely a trick to make the math easier to handle. Later in his calculations Planck planned to remove this restriction by letting the constant $h$ go to zero. [...] Planck discovered that he got the same blue glow as everybody else when $h$ went to zero. However, much to his surprise, if he set $h$ to one particular value, his calcuation matched the experiment exactly (and vindicated the experience of ironworkers everywhere). --- Herbert Nick, "Quantum Reality"

That's how Planck discovered Planck's constant, and as a side effect, quantum physics.

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    +1 Quantum mechanics was one of the examples I had in mind, as much for the loss of Newtonian determinism as for the quantization. Although it's more physics than mathematics it led to much new mathematics. – Ethan Bolker Aug 20 '17 at 00:07
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    @EthanBolker yes, it is widely known that the discovery of quantum physics was surprising, so no wonder you (and probably others) thought of it. What I didn't know before I read the cited book (which I highly recommend, btw) is that it didn't start with a deep theoretical insight, but with some whimsical playing with the maths. – rumtscho Aug 23 '17 at 10:52

The idea that the motion of every celestial body must be a perfect circle, and that, when it wasn’t, the way to “save the appearances” was to add epicycles around the cycles, turned out not to be a useful theory.

However! It turns out that calculating epicycles (in a slightly different way than the Aristoteleans historically did) is equivalent to finding the Fourier transform of a planet's motion, and converting it from the time domain to the frequency domain.

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  • I would contest that adding epicycles turned out not to be a useful theory. Ptolemy's model of the solar system was empirically sound and the standard theory for a lot longer than other successful theories like Newton's theory of gravitation. It would take significant advances in optics and some 1300 years to make observations which would challenge his view of the heavens. That's basically as successful as a theory can get. – CyclotomicField Dec 25 '17 at 13:24

As you wanted separate answers for each suggestion, another interesting one is the proof that angle trisection is impossible. People had been searching for a method to trisect an angle for about two thousand years, and the answer came from the most unlikely of places at the time - the relatively arcane area (at the time) of abstract algebra.

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  • So trisection is splitting equally in three or what does it mean? – mathreadler Aug 19 '17 at 11:47
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    @mathreadler https://en.wikipedia.org/wiki/Angle_trisection – Hong Ooi Aug 19 '17 at 12:50
  • @mathreadler You can construct a 60 degree angle with a straightedge and compass (line-drawing tool and circle-drawing tool). You can also bisect any given angle with straightedge and compass, so you can construct 30 degree angles, 15 degree angles, etc. People then wanted to know if it's possible to *tri*sect a given angle with straightedge and compass. However, it turns out that this is impossible, since it has been shown that 20 degree angles cannot be constructed with straightedge and compass. – Akiva Weinberger Aug 20 '17 at 04:24
  • Yes I realize now there actually is no better word in english than compass for that circle drawing tool. In my mind a compass is the magnetic field sensitive tool for navigation and orienteering, not circle drawing. – mathreadler Aug 20 '17 at 04:50
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    @mathreadler If you want to be pedantic, it is really called "pair of compasses". English is a bit clumsy here. – IS4 Aug 22 '17 at 09:56
  • @IllidanS4 : Yes, we should invent a new word. How about a *circle pencil*. No it's got to be fancier than that, I presume. – mathreadler Aug 22 '17 at 10:02
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    @mathreadler In Latin, it is called "circinus", might be a good start. – IS4 Aug 22 '17 at 10:08
  • “If they be two, they are two so / As stiff twin compasses are two.” – Davislor Dec 24 '17 at 21:50

Here is one, as pertains to:

beliefs or intuitions, that turned out wrong in interesting or useful ways.

Paul Cohen was toying with ideas around a "decision procedure" that could take as its input a mathematical assertion, and create as its output a mathematical proof or refutation. Here are two relevant quotations directly from Cohen:

At the time these ideas were not clearly formulated in my mind, but they grew and grew and I thought, well, let’s see—if you actually wrote down the rules of deduction—why couldn’t you in principle get a decision procedure? I had in mind a kind of procedure which would gradually reduce statements to simpler and simpler statements. I met a few logicians at Chicago and told them about my ideas. One of them, a graduate student too, said, “You certainly can’t get a decision procedure for even such a limited class of problems, because that would contradict Gödel’s theorem.” He wasn’t too sure of the details, so he wasn’t able to convince me by his arguments, but he said, “Why don’t you read Kleene’s book, Metamathematics?” (Albers & Alexanderson, 1990)

As to the relevance of this quotation to the question at hand, this mistaken "belief or intuition" reared its head later on in Cohen's work:

Because of my interest in number theory, however, I did become spontaneously interested in the idea of finding a decision procedure for certain identities . . . I saw that the first problem would be to develop some kind of formal system and then make an inductive analysis of the complexity of statements. In a remarkable twist this crude idea was to resurface in the method of ‘forcing’ that I invented in my proof of the independence of the continuum hypothesis. (Albers & Alexanderson, 1990)

Specifically, one could speculate that it was Cohen's lack of familiarity with mathematical logic (i.e., the incompleteness theorems) that allowed him to form naive beliefs/intuitions which, by his own account, were useful insofar as they reappeared when he developed his approach to forcing.

The interview from which the above two excerpts are derived is worth a read; as a piece of shameless self-advertising, I included both quotation pulls, along with a fair bit more, in the following paper:

Dickman, B. (2013). Mathematical Creativity, Cohen Forcing, and Evolving Systems: Elements for a Case Study on Paul Cohen. Journal of Mathematics Education at Teachers College, 4(2). Link (no pay wall).

Benjamin Dickman
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    You mention “proof of the independence of the continuum hypothesis,” which could be another example itself. The disproof of the naïve intuition that, specifically, there are more rational numbers than integers led Cantor to write the first proof by diagonalization. – Davislor Aug 19 '17 at 05:46
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    (To be clear, those^ are two separate examples ) – Akiva Weinberger Aug 20 '17 at 04:28
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    @AkivaWeinberger Yes. I could probably have been clearer about it, but that was what I was trying to say. – Davislor Aug 20 '17 at 08:04

Euler, Gauss, Riemann, Lagrange, Poincare and various other eminent mathematicians were horrified by the infinite and deemed it to be something that only some mad philosopher ought to deal with. This view was indeed even heralded among the Greeks, for they thought that the art of mathematics deals only with the certain and the precise, and that surely dealing with something as outrageously paradoxical as the infinite would only result in the destruction of such a steadfast discipline.

Out of such glorious names, one may pick out a saying of Gauss in particular;

I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction. [In a letter to Schumacher, 12 July 1831]

The authority of Gauss was rather strong, and for a long time, only finite magnitudes and lines were meddled with. But the great Cantor sought to demolish such traditions. He proved that it was possible to introduce into mathematics definite and distinct infinitely large numbers and to define meaningful operations between them. And with that, the whole of mathematics went through a truly remarkable change.

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  • Saying that those guys were *horrified* seems a bit much... – Mariano Suárez-Álvarez Aug 19 '17 at 22:09
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    Euler, really? He blatantly used the infinite in many nonrigorous arguments. – Akiva Weinberger Aug 20 '17 at 04:30
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    It would be more accurate to say that some modern mathematicians might be horrified by Euler's use of the infinite. For instance, to show that $1/3 + 1/7 + 1/8 + 1/15 + 1/24 + \dots + 1/(m^n-1) + \dots = 1$, he starts with "Let $x = 1 + 1/2 + 1/3 + 1/4 + 1/5 + \dots$". [HEDI-2005-02](http://eulerarchive.maa.org/hedi/HEDI-2005-02.pdf) –  Aug 20 '17 at 14:12
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    Of course, some (most notably [Wildberger](http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf)) still question the "definiteness" and "distinctness" of the infinitely large numbers, as well as the meaningfulness of the operations between them. "If you have an elaborate theory of ‘hierarchies upon hierarchies of infinite sets’, in which you cannot *even in principle* decide whether there is anything between the first and second ‘infinity’ on your list, *then it’s time to admit that you are no longer doing mathematics.*" – Wildcard Aug 23 '17 at 05:18

The idea of infinitesimal numbers goes back to the ancient Greeks, before being reformulated by Newton and Leibniz. The lack of rigor in their formulation led to standard analysis, and then the investigation of whether infinitesimals could be part of a rigorous theory led to nonstandard analysis. A false dawn, then three major theories over the course of millennia!

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  • Sounds interesting. Do you have any sources where I can read about the greeks infinitesimals? I know about the turtle and Achilles(?) and they showed the limit of a geometric series $\sum_{k=0}^\infty \frac{1}{2^k}$ was 2 but I don't know how they tried to build infinitesimals – mathreadler Aug 19 '17 at 11:47
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    Archimedes’ work was lost until the 20th century, when this work was rediscovered: http://www.archimedespalimpsest.org/ – Davislor Aug 19 '17 at 15:23

The Hirsch conjecture stood for $50$ years: Any two vertices of the polytope of $n$ facets in dimension $d$ can be connected by a path of at most $n−d$ edges. (This is equivalent to the $d$-step conjecture: the diameter of a $2d$-facet polytope is no more than $d$.) Santos constructed a counterexample to the Hirsch conjecture in $d=43$:

Santos, Francisco. "A counterexample to the Hirsch conjecture." Annals of Mathematics, Volume 176, pp. 383-412 (2012). arXiv:1006.2814 (2010). (arXiv abs.)


Added. The Polynomial Hirsch conjecture is the subject of a PolyMath project, and remains open.

Joseph O'Rourke
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    How is the counterexample useful or interesting in any way other than disproving the conjecture? – Ethan Bolker Aug 19 '17 at 13:33
  • @EthanBolker: The diameter of polytopes is important for the simplex method (and polytope theory more broadly), and Santos' counterexample is a step toward understanding how the diameter behaves. The "polynomial Hirsch conjecture," which has more direct relevance to the simplex method, remains open. – Joseph O'Rourke Aug 19 '17 at 13:48
  • Why would anyone care about something in 43-dimensional space?! :) – DonielF Aug 21 '17 at 03:27
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    @DonielF: $43$ dimensions just means $43$ variables, and linear programs routinely solve problems in thousands of variables. – Joseph O'Rourke Aug 21 '17 at 10:51

Rejecting the idea that there was no square root of -1 led to the development of the complex numbers, and moving past the idea that there could be only one imaginary unit led to the quaternions. (You could consider the ancient intuition that the square root of 2 should be a rational number a similar case.)

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    Note that there is more than one square root of -1 even in the complex numbers :) – psmears Aug 19 '17 at 09:50
  • Yeah, *i* and *-i*. I phrased that badly. – Davislor Aug 19 '17 at 15:10
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    I'm not convinced about your assertion of the origin of quaternions. – Kimball Aug 20 '17 at 08:17
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    I’m no expert, but what I’d read is that William Rowan Hamilton, who thought of complex numbers as “algebraic couples,” set out to create a theory of “triples.” He found out that this didn’t work, so he tried four dimensions, and came up with the fundamental equations of quaternions on October 16, 1843. No doubt it was more complicated than that, but thinking about what would happen if there were more than one imaginary unit was a necessary step. I might be mistaken, however. – Davislor Aug 20 '17 at 13:52
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    You're correct in the outline of Hamilton's work (in your comment above), but I will mention that the preface of his actual original work is *fascinating* to read. He considered Algebra to be the science of pure *time,* and defined algebraic numbers (i.e. the number line) to be *comparisons between steps through time.* And then set out to apply the same comparison principles to points in space and steps through space—with results that are recognizable in modern vector algebra, but much cleaner conceptually (though less symbolic) in his writings. Completely ingenious. – Wildcard Aug 23 '17 at 05:23
  • @Wildcard That *does* sound fascinating. I admit I have never actually read it. Thank you for the recommendation. – Davislor Aug 23 '17 at 08:46

Polya's Conjecture

What is it?

Polya's conjecture states that $50\%$ or more of natural numbers less than any given number $n$ have an odd number of prime factors. More details on the conjecture can be found here and here.

How is it Interesting or Useful?

It's pretty useful because the size of the smallest counter example is $906,150,257$. This means it serves as a valuable lesson to younglings about why mathematicians have to use proofs instead of empirical verification. This value is amplified by the fact it's a pretty easy idea to explain to even elementary school students.

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    This answer does not quite capture the spirit of the question. All conjecutres which were open for a lot of time could have the same "use" you mentioned. – Kartik Aug 19 '17 at 16:38
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    Yeah, this rather fits perfectly the linked question about extremely large counterexamples. – Leif Willerts Aug 19 '17 at 18:31

Perhaps the problem of squaring the circle fits the bill? It is the challenge of constructing a square with the same area as a given circle by using only compass and straightedge (and a finite amount of time), and dates back, in some form, at least to the Babylonians. The solution requires the construction of the number $\sqrt{\pi}$. In 1882, Ferdinand von Lindemann showed that the undertaking is impossible by proving that $\pi$ is transcendental, hence non-constructible. Interestingly, the only property of $\pi$ that comes into play in Lindemann's proof is the good old $e^{i \pi} + 1 = 0$.

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The intuition that deterministic equations lead to predictable behavior isn't tenable in view of chaos theory. The famous quote from Laplace:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

Poincaré discovered that with anything less than perfect knowledge (e.g. only knowing the data to finitely many decimal places) this isn't true. In something as simple as the three body problem there are solutions which exhibit sensitive dependence on initial conditions: arbitrarily small changes in initial conditions lead to radically different long-term behavior. If our knowledge of the initial conditions is only out to a certain number of decimal places, then the initial conditions we use will differ by a small amount from the true initial conditions, hence any long-term predictions we will make of the system will be worthless as predictions.

John Coleman
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I have recently read some material on the surface tiling problem, and some results looked really interesting. Perhaps one of the most interesting ones is Rice's Pentagonal tiling (although this might not be considered as a counterexample to a conjecture, but anyway...)

But the more relevant problem is Keller's conjecture, which states that

In any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. For instance, in any tiling of the plane by identical squares, some two squares must meet edge to edge.

While it seems intuitively obvious in two or three dimensions, and is shown to be true in dimensions at most 6 by Perron (1940). However, for higher dimensions it is false.

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  • I'd question whether "tiling" can be intuitively understood *without* the direct implication that two tiles must meet face to face. – Wildcard Aug 23 '17 at 05:25

Gödel's completeness theorem: the fact that every tautology in a first-order language can be derived logically with rather simple and intuitive means. It shows that logic can do more than one would naively expect. For me, it was a shock when I learnt it.

If you say it's just my personal shock and as such not subject of the question — well, not really: there is a respectable tradition that implies logic is very limited, though this tradition is not within mathematics; it's within Russian artistic literature, and also within discussions of art in general (“truth and lie are not really dissimilar”, and so on). No contradiction, of course.

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A big surprise (at least for me) was that one can use linear transformations over simpler scalar fields to investigate more complicated types of numbers. This is what is done in representation theory. For example we can build the matrix

$$\left[\begin{array}{cccc}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{array}\right]$$ Can be used to represent the permutation $1\to4,2\to3,3\to2,4\to1$. In similar ways one can show any finite groups elements can be represented with some set of permutation matrices.

Another example from an infinite group is the Quaternions which can be represented as a matrix containing only the real numbers $a,b,c,d$:

$$a+b{\bf i}+c{\bf j}+d{\bf k} \rightarrow \left[\begin{array}{rrrr}a&-b&-c&-d\\b&a&-d&c\\c&d&a&-b\\d&-c&b&a\end{array}\right]$$

So we can do linear algebra over matrices containing already known numbers instead of having to program a new calculation machinery for each new type of numbers we invent.

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The example of Newtonian physics is given but here is another example, from physics, that made me shocked when I learned:

Schrödinger Wave Function is a mathematical description of the quantum state of a system and it gives most of the properties of a physical matters with respect to time and space; therefore it is one of the building blocks of quantum mechanics. It is represented by $\psi$ and time dependent Schrödinger Wave Equation is $$ iħ\frac{\partial}{\partial t}\psi(r,t) = \hat{H}\psi(r,t)$$ where $i$ is the imaginary unit, $ħ$ is called reduced Planck constant, that is $\frac{h}{2\pi}$, $r$ is the position vector, $t$ is the time, and $\hat H$ is the Hamiltonian operator. After Schrödinger's Wave Equation was published 1926, there were many comments and interpretations of the Wave Function $\psi$ from some of the most renowned scientists of that time:

Here comes the most interesting part according to me:

  • According to Einstein, Rosen and Podolsky, Schrödinger Wave Function does not provide a complete description of physical reality, therefore it is an incomplete theory (for more information, https://en.wikipedia.org/wiki/EPR_paradox). Even though this interpretation leads them to create EPR paradox (but later it is resolved) which takes part in development of quantum mechanics, their intuitions about quantum mechanics were proven wrong by the following event:

(Before that, as you know, without axioms mathematics would not have solid foundations)

  • Paul Dirac and John von Neumann gave a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space with Dirac-von Neumann axioms. After that, Schrödinger Wave Equations and Wave Function were mathematically proven and this showed that quantum mechanics is complete (at least mathematically) (for more information, https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics).

To sum up, although all of these renowned scientists have a role in development of quantum mechanics and they supported their intuitions with solid physical phenomena (thought experiments are included when their knowledge about physics is thought), they were not completely right when we see their arguments. To me, this is one of the best examples that show how physics (and mathematics, indeed) is developed by accumulation of incomplete (or even partially false) ideas.

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  • I don't recall quantum mechanics in the curriculum of my pure math degrees. – Asaf Karagila Dec 25 '17 at 08:58
  • I saw an answer on coding theory which is also not in the curriculum of pure math degree and newton's law either. Math is in everywhere in science and engineering, I know that quantum is not pure math but I don't think that restricting math with the curriculums is right thing to do. – ArsenBerk Dec 25 '17 at 09:09
  • I also saw the answer about discovery of Planck's constant, which is about quantum mechanics as well. – ArsenBerk Dec 25 '17 at 09:17