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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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    $\begingroup$ What is more important in a water molecule, the oxygen or the hydrogen? $\endgroup$
    – Asaf Karagila ♦
    Jun 22, 2014 at 8:53
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    $\begingroup$ @AsafKaragila Well said! :) $\endgroup$
    – Shaun
    Jun 22, 2014 at 9:57
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    $\begingroup$ what is more important in a glass of water, the behaviour of water molecules, or the chemical composition of a water molecule? $\endgroup$
    – Ittay Weiss
    Jun 22, 2014 at 11:28
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    $\begingroup$ 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. $\endgroup$
    – user142299
    Jun 23, 2014 at 1:32
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    $\begingroup$ @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). $\endgroup$
    – William Hilbert
    Jun 23, 2014 at 4:35

3 Answers 3

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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.

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    $\begingroup$ 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. $\endgroup$
    – William Hilbert
    Jun 22, 2014 at 7:49
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    $\begingroup$ 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). $\endgroup$
    – user14972
    Jun 22, 2014 at 8:05
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    $\begingroup$ 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. $\endgroup$
    – Mariano Suárez-Álvarez
    Jun 22, 2014 at 10:28
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    $\begingroup$ @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 ..." $\endgroup$
    – celtschk
    Jun 22, 2014 at 19:38
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    $\begingroup$ @celtschk it is at least highly improbable that somebody would randomly choose some axioms and that something interesting will come out of it. $\endgroup$
    – Ittay Weiss
    Jun 22, 2014 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.

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    $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. $\endgroup$
    – Mark Fantini
    Jun 22, 2014 at 9:31
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    $\begingroup$ @Fantini, as far as the question stated in the title of the post (which is the only question in the post), this is an answer. $\endgroup$
    – Mariano Suárez-Álvarez
    Jun 22, 2014 at 9:38

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