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Good evening, I'm studying math and would like to know how important are mathematical proofs in the world and particularly in a school of mathematics

Thanks for your help

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See here and here. –  Zev Chonoles May 20 '13 at 2:15
This may also be interesting. –  Alex Becker May 20 '13 at 2:20
"The achievement of the mathematicians who found the Prime Number Theorem was quite a small thing compared with that of those who found the proof. It is not merely that in this theory you can never be quite sure of the facts without the proof, though this is important enough. The whole history of the Prime Number Theorem, and the other big theorems of the subject, shows that you cannot reach any real understanding of the structure and meaning of the theory, or have any sound instincts to guide you in further research, until you have mastered the proofs." —G.H. Hardy –  MJD May 20 '13 at 2:39
I don't really see the need of downvoting this question. I mean, many students that are just beginning with math just know the basics taught at high school (where proofs are often ignored) so that when they come to do real math, and proofs and all of that they get confused. So I think this question is indeed valid. Showing why proving the propositions as we do is important may help a beginner understand the way to think when doing math. –  user1620696 May 20 '13 at 2:45
@user1620696 It's certainly an important question, but maybe it could be focused a bit more? I think that many people generally dislike questions of the form "how important are...." It might be easier to answer if we were presented with some reason that proofs might not be important. –  Trevor Wilson May 20 '13 at 2:58
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closed as not a real question by J. M., Amzoti, vadim123, Chris Godsil, Shuhao Cao Jun 1 '13 at 5:32

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2 Answers

up vote 3 down vote accepted

Most of the deep proven results in abstract mathematics were proven with mathematical proofs that could not be derived by intuition alone.

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I agree: Kline’s mention of intuition is very misleading, it seems to me. For my own part at least, my intuition has led me astray more times than I would care to enumerate. –  Lubin Jun 1 '13 at 4:51
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If you are not

  • creating your own proofs
  • working out examples and doing computations
    (with the goal of gaining intuition and understanding, to better create your own proofs)
  • reading other people's proofs
    (with the goal of gaining intuition and understanding, to better create your own proofs)

I would say you are not doing mathematics. Now, there are other things that people often confuse with mathematics, such as

  • blindly following someone else's directions for doing a computation
  • blindly giving directions to a calculator or computer to make it do a computation

both of which have their purpose in the world, but they are not mathematics. Since proofs are an intrinsic part of mathematics, I would say they are of paramount importance (or, more accurately, the inextricable combination of intuition, understanding, and proofs is paramount).

To put it simply: math is really interesting, beautiful, and fun to think about! There's just a fundamental intellectual curiosity, a desire to understand. When we can condense that understanding into a statement which we are confident is true, and explain our justification for it, we have stated a theorem and given a proof. To me at least, and I expect to most other mathematicians, that is why proofs are important!

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition." - Morris Kline

What is the importance of mathematical proofs in the world? Well, lots of the mathematics that humans do is, or originally was, motivated by our desire to understand the world. So far, mathematics has been very useful for this. To quote Paul Dirac:

The physicist, in his study of natural phenomena, has two methods of making progress:

  1. the method of experiment and observation, and
  2. the method of mathematical reasoning.

The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success.

So, we have observed that, when we use mathematical objects as a model for the world around us, theorems about the mathematical objects seem to correspond to true statements about the corresponding real-world objects. Very useful indeed!

It is also well-known that mathematics has a very close relationship with computers (e.g., via cryptography, information theory, computational complexity theory). Proving things in mathematics can give us information and ideas for how to get our computers to do things most efficiently, or send signals with minimal chance of garbling the message, etc. Also very useful!

It's true that not all of mathematics has direct applications like these right now. But many surprising connections have been made where computer scientists or physical scientists realized that some bit of mathematics, which had previously been considered just an academic curiosity, was just what they needed to model their science. So there's always a chance that any mathematical theorem will end up being useful in the world, and a theorem is nothing without a proof telling you it's true.

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Thank for your wonderful answer. You mentioned that if somebody is not creating his own proofs, he is not doing mathematics. By this do you mean by proofs given as exercises in a text books or do you mean proofs of deep theorems we learn in mathematics? Creating a proof of a deep theorem in mathematics is a very difficult task. –  Mohan Jun 1 '13 at 3:29
@Mohan it doesn't have to be original to the world, but it should be original to you. If you know enough of an area to reconstruct an alternate proof of a theorem and decide what you do or don't like about the one you were originally given, then you really know the subject area. It doesn't have to be big, it just has to be yours. –  Robert Mastragostino Jun 1 '13 at 5:20
@Mohan: Robert put it very nicely, I absolutely include things that are given as exercises. As long as it's exercising your brain, it's good :) –  Zev Chonoles Jun 1 '13 at 5:36
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