# What are some results that shook the foundations of one or more fields of mathematics? [closed]

An example would be the proof that $\sqrt{2}$ is not rational, which was a violation of some fundamental assumptions that mathematicians at the time made about numbers. Another would be Russell's paradox, which proved that naive set theory contains contradictions. I'm looking for a list of examples of similar results which forced mathematicians to change widely held beliefs and rebuild the foundations of their field. How did mathematicians at the time rearrange their foundations to account for newly proven results?

There must be lots of examples.

EDIT

The question was put on hold as too broad. Hopefully this narrows it a bit. I am looking for examples where most mathematicians believed that a certain foundational set of propositions was true, and then a result was proven that showed that it was false. Most theorems do not fit the bill. Wiles' proof of Fermat's last theorem was a historic achievement, but as far as I am aware, it did not force Wiles' contemporaries to reevaluate their fundamental assumptions or beliefs about mathematics. Gödel's incompleteness theorems, on the other hand, showed that Hilbert's program was unattainable, causing a fundamental shift in our understanding of notions like provability.

EDIT

I don't understand why this keeps being closed. It seems to be generating quality answers and the answerers seem to be able to tell what is being asked. Maybe some commenter could let me know?

## closed as too broad by user642796Jun 8 '15 at 14:57

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• I wonder if the Taniyama Shimura conjecture would apply. – DanielV Jun 8 '15 at 0:10
• I wonder if the solution to Zeno's Paradoxes would count. – Asaf Karagila Jun 8 '15 at 0:11
• A lot of these answers are redundant...maybe we should make a community Wiki answer that incorporates them all. Oh, and with links to more detailed resources! E.g., Gödel's Incompleteness Theorems, non-Euclidean geometry, etc. – André 3000 Jun 8 '15 at 0:46
• @SpamIAm Big list questions, at least on MO, are usually one example per answer. That would also alleviate the redundancy problem. – Kimball Jun 8 '15 at 1:47
• I understand the question may have many possible answers, but I think it is sufficiently narrow that we can distinguish an answer from a non-answer in general, no? – crf Jun 8 '15 at 8:48

• Goedel's incompleteness theorems - forcing one to understand the limitations of formal languages.
• Skolem's theorems - forcing one to reconsider everything they new about first order logic.
• Robinson's infinitesimals - showing they really do exist.
• Cantor's set theory - creating a heaven for us all.
• The discovery of non-Euclidean spaces - settling Euclid's fifth, and opening the door to much of modern geometry.
• The existence of transcendental numbers - showing how shaky our initial understanding of the reals was.
• The insolvability of the general $n$-degree polynomial, $n\ge 5$, by radicals - for the result itself and the far reaching tools and techniques created.
• It is interesting that a lot of these are negative results - the fifth postulate is independent, Skolem"s theorem shows the limits of first order logic to specify any theory, the inadequacy of taking roots for solving polynomials, and even transcendental numbers are a sort of negative result. – Thomas Andrews Jun 8 '15 at 12:11
• @ThomasAndrews That's a rather pessimistic view (pun intended)! Aren't most results, interpreted a particular way, negative results? – Chappers Jun 8 '15 at 13:41
• But many of these arose because we were trying to solve a problem (in the case of angle trisection, or proving the fifth postulate from the others, many many centuries of effort) and saying, "Nope, you can't do that!" Sure, most yes answers can be made no answers in the technical sense, but most of these arose because somebody asked how to do something, and really really wanted to find a way to do so. @Chappers (Not seeing any accidental pun, so if no pun was intended, then there is no pun at all.) – Thomas Andrews Jun 8 '15 at 13:50
• @ThomasAndrews I think that effect is due to the question. A positive answer is very rarely going to shake the field - people ask the question so that a positive answer matches their expectations. – Asvin Jun 8 '15 at 14:22
• I don't know, I don't think a positive or negative answer to Fermat's Last Theorem is earth-shattering either way (though the mathematical results used in the proof my be.) Same with Goldbach or the twin prime conjecture. If we proved $P=NP$, that would be an earthshattering positive result. :) @Asvin – Thomas Andrews Jun 8 '15 at 14:25

Galois (and Abel) showed that one can only get so far in solving polynomials using radicals.

Cohen showing the continuum hypothesis is not provable, was a shocker and it changed set theory entirely. Arguably set theorists were close to adding $V=L$ to the "canonical" set theoretic axioms, but Cohen's work showed that there is a whole world outside of $L$!

And of course Cantor's work. Finding out that there are different infinities? Mind blowing.

Cauchy and the finitary definition of limits was a foundational cornerstone for modern analysis, which shook the foundations in a positive way.

• Seeing how this is the only answer that was downvoted twice (and one of the few which were downvoted at all), I'd be happy to discuss whatever bothers people here. Is it the overlap with other answers? The multiplicity? The writer of the answer? – Asaf Karagila Jun 8 '15 at 14:53
• I had some mysterious downvoted answers lately. Perhaps a troll? – Ittay Weiss Jun 8 '15 at 17:24

Gödel's incompleteness theorem -- any formal system with a reasonable level of descriptive power contains statements that are both true and unprovable. And these statements can be constructed.

The construction of hyperbolic geometry -- satisfies all of Euclidean axioms except the parallel postulate.

The Weierstrass function, continuous everywhere, differentiable nowhere. Very cool.

Russell's Paradox, showing that a naive viewpoint towards sets is not consistent (and destroying Frege's work at the time). While this didn't start the interest in developing the axiomatic method, it was certainly a major motivating factor once it was found and has shaped our foundations of set theory immensely.

The discovery of continuous functions that were nowhere differentiable caused a great tormoil among mathematicians in the $19$th century, since those functions (continuous) were considered to be well-behaved, except maybe at a few points. And we have Weierstrass function as an ilustration of that.

In order to give more historical background and mathematical insight here is an interesting paper.

• I think you mean 1800s. By 1900, analysis had mostly learnt how to deal with such entities. – Chappers Jun 8 '15 at 11:52
• Earliest nowhere differentiable continuous function was Bolzano 1820 (not published until 1920), but of course the Weierstrass example was 1872. – GEdgar Jun 8 '15 at 12:55
• @Chappers In history one determines a century from the first two digits $+ 1$. Then it is said to be $1900$'s as a reference to the century whence Bolzano around $1830$ ($18 + 1 = 19$) published a continuous function nowhere differentiable in a interval $[a,b]$, we say it was early $1900$'s. – Aaron Maroja Jun 8 '15 at 14:00
• I'm afraid that's simply wrong. "The nineteenth century" is the century between 1800 and 1900 (AD/CE, usually), with some discussion as to where the ends lie exactly. (since "the first century" is between 0 and 100, there being no zeroth century) But "the 1900s" means "those 4-digit years whose first two numerals are 19". Or it is also used in the way that "the 1910s" means "the years 1910–1919", to mean "1900–1909" Or as a more recent example, "the 1980s" are the years 1980–1989. (And I don't see why you want an apostrophe in there: none of possession, abbreviation or ambiguity are involved.) – Chappers Jun 8 '15 at 14:24
• @Chappers In my language is like this. I'll search it here and make the necessary modifications. – Aaron Maroja Jun 8 '15 at 14:27

The discovery of exotic spheres by John Milnor in 1956. These are topological spheres carrying a smooth structure different from that of a standard smooth sphere. Their discovery divided the study of manifolds into two disciplines: topological manifolds and smooth manifolds.

• The first example (which quite possibly was the cause for mathematics to be studied as a "safe" subject in the first place, was Parmenides, who (reportedly: the original source is unreadable) showed that by naïve philosophical reasoning there could only be one thing in the world. The conclusion, there being more than one thing in the world, was that all natural philosophy up till then had been built on nonsense, since you could deduce rubbish. (Parmenides was Zeno's teacher, to give you a reference point you have heard of.) Here is also born the unreasonable, devastating skepticism we still use in science, and particularly mathematics, to test theories and results to this day.
• The next quake in Greek mathematics that we know about is the discovery of non-comeasurable magnitudes. This is what you probably know as "irrational numbers". However, the Greek method of measuring line segments is to measure one with another, hence comeasure. You stack copies of one end-to-end, and copies of the other end-to-end, to find the point when they have a common length. (The actual algorithm is more complicated—look up anthyphairesis. Or, of course, Euclid's algorithm. Both give you the common unit measure-line, of which the line lengths are multiples) For the side and diagonal of a square, this process doesn't end, so the side and diagonal of the square are not comeasurable. This gives you serious problems when your number system is entirely based on comeasurability. (And nowhere has the word "irrational" been used. The Greeks could not have discovered "irrational numbers", because they had no way to express such an idea.) So does that mean there are different types of length? Well, eventually Eudoxus sorts that one out, by inventing the theory of proportion that we find in Euclid. Eudoxus is one of the smartest mathematicians ever, up there with Newton and Gauss, and this was a Big Deal.
• Hindu numerals. No, really. Totally changed commercial arithmetic. Allowed the purchase and sale of debt, so banking is invented and the world comes to an early end.
• Non-example: proving that the three classical problems (squaring the circle, doubling the cube, and trisecting the angle), cannot be done with ruler and compass. Even the Greeks suspected it was impossible (circle-squarer is used as a "Ha, what an idiot"-type insult in Aristophanes's play The Birds).
• Non-example: Infinitesimals (I mean the old sort, used from basically the 15 century on) are paradoxical. Although everyone knows this from about 1550 on, they carry on using them (carefully...) anyway. Because you still get the right answer for anything physically reasonable that you are considering.
• Curves are equations and equations are curves. This is the first revolution in geometry: Descartes's invention of algebraic geometry. Suddenly you can do geometry using algebra, and algebra using geometry. Before, they appeared to be two rather different things, algebra being far more suspicious and dodgy because it's not synthetic: you start with something unknown, then find it. After this, both can be used in tandem.
• The second revolution in geometry is Riemann. Very little interest is shown in hyperbolic geometry: Gauss doesn't bother to publish it, and Boylai and Lobachevsky's work is almost completely ignored when it isn't rubbished. But Riemann takes Gauss's theory of surfaces and a) completely changes complex analysis and the theory of abelian functions, freeing it from the plane, and b) explains how to a geometry with intrinsic curvature can work. This circulates through the Italian school, until eventually some chap called Einstein happens upon it.

Although possibly not the same level as the examples above, Hilbert's basis theorem received a lot of criticism (someone called it theology) since it was not a constructive proof.

• This actually led to a lot of 'ideological' reconstruction throughout the 1900's. During Hilbert's time there was a very strong belief that constructive proofs were the only valid ones. Very famous mathematicians took this point of view including Brouwer. This led to arguments between the two mathematicians which can be researched on the internet. – Eoin Jun 8 '15 at 0:18
• "This is not mathematics; this is theology", widely (mis?)attributed to Gordon, may be misunderstood or may be historically inaccurate, e.g. see here. – Bill Dubuque Jun 8 '15 at 0:27
• @BillDubuque Ahh, I would not doubt that it is misattributed. I can't quite remember where I first saw the quote. I think it was in Algebraic Geometry by Smith et al. (After looking it up, it was in this work. The reference they include is in Hilbert by Reid a biography printed in 1970, still decades after Hilbert). – Eoin Jun 8 '15 at 0:39
• For more on the massive mythology around this see Colin McLarty's Theology and its discontents: David Hilbert's foundation myth for modern mathematics – Bill Dubuque Jun 8 '15 at 1:01

The axiom of choice would be another such result--seemingly innocent in its statement but quite far-reaching in terms of its applications and implications.

• He asked for results. This is an axiom. (The proof that its proofless could count though.) – PyRulez Jun 8 '15 at 1:22
• Another way to address PyRulez's concern: while the axiom itself may not qualify as a result, some of the implications of the axiom might be said to have shaken the field, maybe the Banach-tarski Paradox and Zermelo's Theorem. – kuzzooroo Jun 8 '15 at 13:50

I am no expert but I believe Gödel's incompletness theorems should be included as they shook the widely held idea that every true result in mathematics is provable (within an axiomatic system + constraints), and this was something that surprised the likes of B. Russell and D. Hilbert amongst others.