What is maths? "Maths is the study of ______"?

Question

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?

It is difficult to answer, see http://en.wikipedia.org/wiki/Definitions_of_mathematics — Amzoti, Jan 05 '14 at 14:41
Mathematics is the deductive science which has historically grown out of the study of numbers and geometric figures. — user119191, Jan 05 '14 at 14:42
Maths is just deducing one damned thing after another, and often not caring if it has any real life applications. — Lost1, Jan 05 '14 at 14:44
@fkraiem I'm not implying maths is the study of some particular thing. I just want an all-encompassing definition for what qualifies as maths and what doesn't — hb20007, Jan 05 '14 at 14:45
Jay Orear said “Physics is what physicists do late in the night”. The same applies to math. Why would you need a definition? There isn't one. — egreg, Jan 05 '14 at 14:51
@egreg Funny quote, but it simply postpones the problem to defining "physicists" (or, "mathematicians"). — Did, Jan 05 '14 at 14:52
@Did Of course you have to start from some arbitrary assumption, or you'll fall in a regressus ad infinitum — egreg, Jan 05 '14 at 14:53
@egreg In the case at hand, there are other definitions. Albeit imperfect, these might be more informative. — Did, Jan 05 '14 at 14:55
@Did I don't think so. Read the answers given so far: none is minimally satisfactory, apart from Sylvester's quotation. — egreg, Jan 05 '14 at 14:57
@egreg I have, and I disagree with your characterization. Sorry. — Did, Jan 05 '14 at 14:59
it is the study of axiomatic-deductive systems, where knowledge is organized from less complexity to major one and in its state-of-art is always conjecturing new facts from old, to be formalized in the scheme: "statement-proof". — janmarqz, Jan 05 '14 at 15:00
See http://math.stackexchange.com/a/425085/48510 for my answer to an earlier incarnation of the same question, — Andreas Blass, Jan 06 '14 at 05:22

score 8 · Accepted Answer · answered Jan 05 '14 at 14:41

8

Math is the study of deductions from assumptions.

answered Jan 05 '14 at 14:41

Asaf Karagila

393,674

2

That would have been my initial answer, but then it would make Aristotle's Metaphysics "math". – Martin Argerami Jan 05 '14 at 14:55
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Martin, you can't spell metaphysics without math (jumbled across the word, that is). $$\underline{\sf M}\rm e\underline{\sf ta}\rm p\underline{\sf h}\rm yiscs$$ – Asaf Karagila Jan 05 '14 at 15:09
:D ${\ \ \ \ \ }$ – Martin Argerami Jan 05 '14 at 17:10

score 4 · Answer 2 · answered Jan 05 '14 at 15:02

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:

Prove a theorem
Find a counterexample to a conjecture
Come up with a productive conjecture
Create a new subgame, perhaps inspired by something encountered in the world (a somewhat unusual and somewhat risky move, but often very rewarding)
Create a new variant of the game (a rare and risky move)
Teach others to play the game, who themselves go on to play it well

score 2 · Answer 3 · edited Jan 05 '14 at 14:53

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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

edited Jan 05 '14 at 14:53

Did

279,727

answered Jan 05 '14 at 14:51

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Mathematics is not a science; science demands faith in inferential statistics; mathematics suggests faith in the consistency of Peano arithmetic, ZF theory, etc. – dfeuer Jan 05 '14 at 14:55
@dfeuer faith is faith. – Pineapple Fish Jun 07 '20 at 19:41

score 1 · Answer 4 · answered Jan 05 '14 at 14:48

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.

I don't know about patterns. There is a discipline called Chaos Theory :P — Lost1, Jan 05 '14 at 14:53

hb20007 · Answer 5 · 2020-11-25T20:18:21.463

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.

What is maths? "Maths is the study of ______"?

5 Answers5