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I'm still a student and something funny that I've seen is that many mathematicians evade the question, "What is mathematics?" And when they don't evade it, every person gives a different definition. So I was wondering, and I think this is the best place to ask: What is mathematics specifically?

I think it is a very curious question and I just want to see what you think.

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    $\begingroup$ What are mathematics? - What is art? $\endgroup$ – Lucian Sep 24 '14 at 14:36
  • $\begingroup$ If it is true that everyone gives a different definition, how can you expect to get a specific definition? $\endgroup$ – graydad Sep 24 '14 at 14:41
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    $\begingroup$ The best answer to that question I know is: mathematics is what mathematicians do. $\endgroup$ – Timbuc Sep 24 '14 at 14:43
  • $\begingroup$ Why uncommon? Especially for everyone that is related to mathematics... $\endgroup$ – Did Sep 24 '14 at 14:56
  • $\begingroup$ This question has nothing to do with mathematics. $\endgroup$ – Marc Bogaerts Sep 24 '14 at 15:33
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My very short answer: Mathematics is a very broad subject that seems to have grown out of questions of number and shape. By now it has expanded to include questions of pattern, structure, and relationship.

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I think math could be defined as the study of axioms, and theorems you can derive from those axioms.

For example, arithmetic. The axioms are stuff like "zero exists" and "if $a=b$ and $b=c$ then $a=c$" and "every natural number has a number after it" and "$a+1$ means the number after $a$" and "$a+(b+c)=(a+b)+c$", etc.

And using the axioms, you can prove that $1+1=2$:
$a+1$ means the number after $a$.
Thus, $1+1$ means the number after $1$, which is $2$.

And, building off of that, you can prove that $2+2=4$:
$2+2=2+(1+1)$, by the previous theorem.
That equals $(2+1)+1$, since we can move parentheses around.
Since "$a+1$" means the number after $a$, we have:
$(2+1)+1=3+1=4$.

And geometry is what we get with Euclid's axioms. (Technically, there is non-Euclidean geometry, too...) etc.

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  • $\begingroup$ I forgot to prove that $1$ exists. Proof: "$1$" is defined to be the number after $0$. Since $0$ exists, and every number has a number after it, $1$ exists. (Similarly for $2$, $3$, and $4$.) $\endgroup$ – Akiva Weinberger Sep 24 '14 at 15:47

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