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Felix Klein once said,

Mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof.

Till now I thought the opposite. I thought that it is the rigorous methods of proof that requires more ingenuity because in my opinion is is 'quite easy' to make conjectures but in many cases unimaginably difficult to prove it.

Consider the following viewpoints,

View 1: Every conjecture is intuitive at the beginning. But it is useless unless we prove it because simply deducing consequences if it had been true isn't worth a mathematician's time, perhaps. What if any of the deduced consequences doesn't contradict any of the established theorems?. So what in the end in essential, the rigorous proof of course.

View 2: It's not true that simply deducing consequences if it had been true isn't worth a mathematician's time. What if in course of this study (though logical but perhaps not always practical) the mathematician has derived a contradiction with the established theorems? And also, let's grant for some time that rigorous methods of proof is mostly important but you must have something to prove and that something must come from intuition.

Probably this question is not best suited for this site, but I am eager to know the thoughts of my fellow MSE users on this subject.

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William Hilbert
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    What is more important in a water molecule, the oxygen or the hydrogen? – Asaf Karagila Jun 22 '14 at 08:53
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    @AsafKaragila Well said! :) – Shaun Jun 22 '14 at 09:57
  • Both are important. I like the way Asaf says it! – Sawarnik Jun 22 '14 at 11:12
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    what is more important in a glass of water, the behaviour of water molecules, or the chemical composition of a water molecule? – Ittay Weiss Jun 22 '14 at 11:28
  • @Ittay: What is more important in a glass of water, the behavior of the water molecules or the chemical composition of the glass? – Asaf Karagila Jun 22 '14 at 14:30
  • Non-Euclidean geometry came into being exactly by deducing consequences if the parallels axiom were not true (actually in the hope of finding a contradiction). Only after that had been done, people realized that this indeed led to a valid theory. – celtschk Jun 22 '14 at 15:08
  • I wonder if the statement was really intended to be an accurate general overview of the whole of mathematical progress since Pythagoras, or if "those who..." basically means Gauss. – Steve Jessop Jun 22 '14 at 17:51
  • Or another way to look at it: those who most radically advance mathematics, necessarily must be distinguished by extraordinary intuition. Rigour need not be strongly correlated with this, so they will tend not to also be distinguished by extraordinary rigour. If you found an important area of mathematics and provide the central theorems, but offer proofs that need a bit of work on the details later, then whoever works on the details usually doesn't get the glory. However, I note that your question isn't "is this quotation true?", and this doesn't answer what you did ask ;-) – Steve Jessop Jun 22 '14 at 17:59
  • @celtschk I never said "nobody wakes starts deducing theorems from a very famous set of axioms, about which there is nothing random, with the aim of finding a contradiction so as to proof a particular (non-random) axiom is a consequence of other (non-random!) axioms". – Ittay Weiss Jun 22 '14 at 18:55
  • @IttayWeiss: And I never claimed you said that. My comment above was a reply to the question (specifically referring to "View 1"). If it had been a comment to your answer, I would have attached it to your answer. – celtschk Jun 22 '14 at 19:34
  • @celtschk I'm sorry, I didn't notice the location of the comment. My apologies. – Ittay Weiss Jun 22 '14 at 21:32
  • @IttayWeiss: Never mind. – celtschk Jun 22 '14 at 21:37
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    This question seems a bit confused, and opinion-based. There are the questions, (a) do proofs or (discovering?) theorems require more ingenuity?, (b) do proofs or intuition on theorems advance mathematics more?, and (c) which of proofs and theorems is more fundamental to mathematics as a field? The title of the question, taken alone, seems to me to be entirely opinion-based as it does not admit an objective answer. –  Jun 23 '14 at 01:32
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    @NotNotLogical: I have thought of the question in the following way. Loosely speaking, rigor has a limit, the limit of axioms. So, presumably, anyone who has radically advanced mathematics must be rigorous. But what advances mathematics is not how rigorously they proved this or that theorem, it is what they have proved, what was the idea behind the proof; and of course, that Idea must come from intuition (or experience, whatever it may be called). – William Hilbert Jun 23 '14 at 04:35
  • This question reminds me of Terry Tao's post on the pre-rigorous, rigorous, and post-rigorous stages of a mathematician. – Samuel Jun 26 '14 at 20:26
  • @Samuel: Thanks for the link. – William Hilbert Jun 27 '14 at 08:47
  • @AsafKaragila hydrogen is the more studied element in chem and phys and shows up much more than oxygen. Just because they both appear in water doesn't mean they are 'equal.' Likewise, sometimes the proof is more useful because it provides a proof pattern while the theorem, whose proof is often handwaved, has important results. – user29418 Feb 16 '22 at 13:41

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Both and neither. The reason we like theorems and proofs are not because of the facts or the justification that they really are facts, but rather the understanding they provide.

The following is an excerpt from Thurston's incredible and highly readable essay On Proof and Progress in Mathematics, which I strongly suggest taking a look at.

For instance, when Appel and Haken compuleted a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.

On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn't something they really wanted after all. They discover by this kind of experience that what they really wanted is usually not some collection of "answers" - what they want is understanding.

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Proofs in mathematics are like brush strokes in the art of painting. They are certainly important, but you rarely look at an art work to appreciate the brush strokes.

Lakatos, in Proofs and Refutations, compares mathematics to physics. In physics a theory is meant to model some part of reality and truth flows from reality to the axioms, meaning that if the theorems of the theory, i.e., the predictions, turn out to disagree with the model, i.e., with reality, then we change the axioms, not the reality. The formalist view of mathematics (a la Hilbert) says that mathematics is just a game with symbols of a piece of paper, where simply one sees what comes out of the theorem. The flow of information is then from the axioms to the model. However, there is a bit of a problem with that harsh approach, since after all there must be something that guides us in the selection of axioms. Nobody wakes up in the morning, randomly chooses some axioms and starts deducing theorems. And if somebody does that, no journal would (or should, at least) publish such work. The formalist point-of-view offers no answer to the meta-question "why these axioms?". According to Lakatos the flow of information in mathematics is the same as in physics. A mathematical theory is supposed to model something. Not a part of reality, but something none-the-less. The role of the mathematician is to decide when the theorems of the theory disagree with whatever it is that is modeled, and when that happens, one should change the axioms rather than reality. Personally, this is how I view your quote of Klein. Immense progress is made when the giants upon which future generations will stand realize that theorems are wrong, even though the proof is perfectly fine, and then they change the axioms by better ones.

Ittay Weiss
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    Mathematics doesn't need to agree with the reality, but Physics must. This is the basic difference between Mathematics and Physics. I think that in this respect Lakatosh is wrong. – William Hilbert Jun 22 '14 at 07:49
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    There is a very easy answer to "why these axioms?": I play a game because it is a game I want to play. And as a formalist, I'm willing to admit that I am playing a game, even if I think the game should be is a reflection of something else (e.g. reality). –  Jun 22 '14 at 08:05
  • @Hurkyl but why do you choose to play such games as group theory, linear algebra, Banach spaces, category theory, set theory, but not the fantastic game of "a set with three binary operations defined on it, all are associative, the first one is not commutative, but the the other two are, the first one distributes over the other two, and the other two satisfy $(ab)#(ba)=aaaaa*a#b#b#b#b#b#b#b#b$"? – Ittay Weiss Jun 22 '14 at 08:25
  • @WilliamHilbert have you even read Lakatos? Lakatos never claimed that mathematics must agree with the same reality that physics must agree with. Please read my answer more carefully. – Ittay Weiss Jun 22 '14 at 08:28
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    Connaiseurs do look at artworks to appreciate the strokes of the painter, and painters are well known for their characteeristc strokes. Only to the tourist who walks by at a somewhat hurried pace through the museum details such as strokes are not important. Likewise, anyone who intends to do mathematics seriously has to develop both a taste and an ability not only for proofs but for good proofs. The passer by is the only one who can afford not to dwelve in the stokes that make up math. – Mariano Suárez-Álvarez Jun 22 '14 at 10:28
  • I completely agree with that @MarianoSuárez-Alvarez Proofs, like brush strokes, are essential. Mastering proof techniques is essential. Developing new proof techniques is not essential but may open up closed doors. All of that is irrelevant though to a discussion of what matters more, the brush strokes or the work of art. One would not exist without the other, but just like an artist tries to convey something through a painting so does a mathematician convey something through a theory. The theorems compose the theory, and thus are the means of conveyance. – Ittay Weiss Jun 22 '14 at 11:26
  • @IttayWeiss: I referred to this quote, "According to Lakatos ...mathematical theory is supposed to model something. Not a part of reality, but something none-the-less. The role of the mathematician is to decide when the theorems of the theory disagree(?) with whatever it is that is modeled, and when that happens, one should change the axioms rather than reality." – William Hilbert Jun 22 '14 at 12:27
  • @WilliamHilbert but in your comment you said "Mathematics doesn't need to agree with the reality, but Physics must" yet you correctly quoted me saying that "not part of reality, but something non-the-less". So, what are you trying to say? – Ittay Weiss Jun 22 '14 at 12:32
  • @IttayWeiss: But you also said, "...In physics a theory is meant to model some part of reality and truth flows from reality to the axioms..." How does this become Not a part of reality in your answer afterwards? I referred to this reality. – William Hilbert Jun 22 '14 at 12:45
  • It's about the non-randomness of mathematics. There is a reason why we study the axiom systems we do, and not others. That reason is external to mathematics. How do mathematicians decide which systems of axioms (i.e., which theories) to develop? Why do we not even consider trillions of randomly chosen axioms? The answer must be that there is an external guidance. Just like the physical world guides physical theories, so, there seems at the very least, there is something that guides mathematicians in the choice of axioms. That something is the mathematical reality I was talking about. – Ittay Weiss Jun 22 '14 at 12:54
  • "Nobody wakes up in the morning, randomly chooses some axioms and starts deducing theorems." That's a pretty strong statement. Are you really sure not a single person has ever done exactly that? – celtschk Jun 22 '14 at 14:55
  • @celtschk if anybody ever did do that, did they get their work published? – Ittay Weiss Jun 22 '14 at 18:51
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    @IttayWeiss: (a) It doesn't matter. (b) If so, we will probably never know. They probably wouldn't have written in their article "One day we woke up and thought about these random axioms ..." – celtschk Jun 22 '14 at 19:38
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    @celtschk it is at least highly improbable that somebody would randomly choose some axioms and that something interesting will come out of it. – Ittay Weiss Jun 22 '14 at 21:33
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A theorem is a statement which has a proof.

You can probably answer your question yourself based on that fact.

Mariano Suárez-Álvarez
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