Why demonstrations are important in mathematics? [closed]

Question

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

Answer 1

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

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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.

Answer 2

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