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I can fill in the blank by just listing the different fields of maths but my goal is to define all of mathematics.

An answer that I would've accepted a few years ago is "Maths is the study of numbers." But certain fields like topology and game theory do not have much to do with numbers.

A mathematician proposed "Mathematics is the study of the meaning behind numbers" but again that's assuming there can't be maths without numbers.

Another: "Maths is the study of abstraction" but that's just … abstract.

So what is maths?

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    $\begingroup$ It is difficult to answer, see en.wikipedia.org/wiki/Definitions_of_mathematics $\endgroup$ – Amzoti Jan 5 '14 at 14:41
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    $\begingroup$ Mathematics is the deductive science which has historically grown out of the study of numbers and geometric figures. $\endgroup$ – user119191 Jan 5 '14 at 14:42
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    $\begingroup$ Why should math be the study of some particular thing? $\endgroup$ – fkraiem Jan 5 '14 at 14:42
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    $\begingroup$ It's not mathology. $\endgroup$ – Karl Kroningfeld Jan 5 '14 at 14:44
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    $\begingroup$ Math is just high falutin' pattern making. $\endgroup$ – Alexander Gruber Jan 5 '14 at 14:46
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Math is the study of deductions from assumptions.

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    $\begingroup$ That would have been my initial answer, but then it would make Aristotle's Metaphysics "math". $\endgroup$ – Martin Argerami Jan 5 '14 at 14:55
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    $\begingroup$ Martin, you can't spell metaphysics without math (jumbled across the word, that is). $$\underline{\sf M}\rm e\underline{\sf ta}\rm p\underline{\sf h}\rm yiscs$$ $\endgroup$ – Asaf Karagila Jan 5 '14 at 15:09
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I've said it before, and I'll say it again (like a bleepin' broken record):

Mathematics is a collection of massively multi-player creative games that have collectively been played by tens of thousands of people on every inhabited continent since before the dawn of history. As befits such a game, mathematics has several different ways to score points:

  1. Prove a theorem
  2. Find a counterexample to a conjecture
  3. Come up with a productive conjecture
  4. Create a new subgame, perhaps inspired by something encountered in the world (a somewhat unusual and somewhat risky move, but often very rewarding)
  5. Create a new variant of the game (a rare and risky move)
  6. Teach others to play the game, who themselves go on to play it well
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Mathematics: Science that studies by means of deductive reasoning the properties of abstract entities (numbers, geometric figures, functions, spaces, etc.) and the relationships established between them.

Translation of the definition of mathematics from the French dictionary Larousse

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    $\begingroup$ Mathematics is not a science; science demands faith in inferential statistics; mathematics suggests faith in the consistency of Peano arithmetic, ZF theory, etc. $\endgroup$ – dfeuer Jan 5 '14 at 14:55
  • $\begingroup$ @dfeuer faith is faith. $\endgroup$ – Pineapple Fish Jun 7 '20 at 19:41
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Math is the study of natural patterns and offers a way to describe nature.

(This is just a way I like to look at it.)
Linear algebra deals with the translation of space and the place in space.
Group theory is about the study of (rotational, translational) symmetries.
Number theory comes from counting objects and dividing objects in groups.

Of course, a lot of these topics generalize 'nature', but topics like topology are useful to describe nature too.

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    $\begingroup$ I don't know about patterns. There is a discipline called Chaos Theory :P $\endgroup$ – Lost1 Jan 5 '14 at 14:53
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Here’s another rather poetic answer:

[Math] is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of… existence.

— James Joseph Sylvester

And another (since everything in mathematics can be pretty much defined in set-theoretic terms):

Math is the study of things that can be described as sets.

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