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It is a somewhat philosophical question. I personally believe that the importance of math is due to its usefulness and lots of applications. Mathematics is used in everywhere nowadays; as Ian Stewart said in his book Letters to a young mathematicians, if we put 'MATH IN IT' stickers on everything that used math, then it will be hard to find a thing without the sticker. In this point of view, math is important because of its useful results.

However, some of the most sophisticated results of math, such as Fermat's last theorem, is not that useful or applicable in other fields. Despite their seemingly unusefulness, I think these results are also important because they have extended the boundaries of human knowledge. Knowing is important because it gives us some answers about who we are and how the universe is like. In the ancient times, people thought that the earth is flat and is the center of the whole universe. But now we know that the earth is round, and we are probably not the major characters in the history of universe. In this sense, mathematics is important because it gives us truths.

Now let me give you my personal anecdote. I was occasionally talking about the proof assistant Coq to my analysis professor. Of course Coq cannot prove everything, but she asked me whether I want all the problems in math to be solved by computers, supposing the proofs of computers are just like those of human mathematicians that no one can distinguish the proofs by computers from those of humans. I said yes, since I thought whether it is done by computers or humans, knowing the truth is important. But my professor said she don't like it. She thinks proofs made by computers are not as beautiful as those of humans. She thinks the importance of mathematics is in the joy of solving problems, and the beauty of mathematics comes from the efforts put in when solving problems. In this point of view, unknown mathematical truths are something like undisturbed nature that we have to prevent computers from pervading mathematical world.

Now I am confused. Where does the importance of math come from? Is using computers to prove theorems damage the very nature of mathematics?

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closed as primarily opinion-based by user127096, rschwieb, achille hui, Asaf Karagila, Claude Leibovici Apr 12 '14 at 4:52

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

Perhaps you should look into foundations and logic, especially the rigorous definition of "proof". I believe there are fairly old theorems (~100 years) that show limitations on what computers can even theoretically do. – A l'Maeaux Apr 12 '14 at 3:31
@Daniel, 100 years isn't "old"... – vonbrand Apr 12 '14 at 8:54

A huge proponent of using computers for math proofs is Doron Zielberger. You can read his opinions here.

Computer generated proofs have been in much controversy though- and it really depends on which person you are talking to; everyone have their own opinions.

For me- I do math, not just to find "truth" or beauty, but because I can't imagine my life without math. However, that's only a personal feeling.

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All of that is secondary. The main reason is: because it's fun.

As for the computer question, I don't really like the idea, honestly. Math is a fun game, and if you start using cheat codes, you are depriving me of my fun game.

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Math is the most contradiction-free, relatively true, succinct source of information in existence. Also beautiful if you like weird symbols, commuting diagrams, and symmetries. Also, we could be in a mathematically defined abstract universe right now. As in we don't exist except in an abstractly defined structure. Once our math is up to par we can recursively create sub-universes that seem to exist to us and unquestioningly exist to the beings inside. Computers are a tool, and also can be tools at times. :)

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Personally I view much in the same way you mentioned. I view it as a tool for finding truths about the universe. Math and physics can be used to figure out where we came from. How we got here. I doesn't answer the question why we are here but math can point us in the right direction towards this meaning by showing us how we got here. Mathematics is truth. Provable, solid truth. I must also agree with your professor however. Computer generated proofs may not be as beautiful as proofs generated by humans but what should it matter? Beauty is subjective and is ultimately not the purpose of mathematics, simply a consequence. The importance of math is, in my opinion at least, to probe the unknown, figure why things the way they are, to find hidden truths in the universe. But ultimately math is what you yourself make of it. There is no one answer to this question. Everyone is going to have their own philosophy when it comes to mathematics.

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