Is mathematics discovered, developed, or both?

[Macoleco]

Basically the title. What do you think? Time to debate.

[GrandizerII]

Basically the title. What do you think? Time to debate.

Mostly (if not completely) developed.

[brewer]

Developed.

[Tiberius]

All of math was developed, even the bits we "discovered". There was a thread a few years ago about whether Pi was discovered or invented.

Its a bit of both. Pi as a number was discovered, however the way it's defined (as the ratio of a circle's diameter to its circumference) is entirely invented. Circles are a human invention; they don't appear in nature.

[Macoleco]

But what if we compare it to physics, or chemistry? It is discovered, and developed at the same time.

If mathematics is the purest science there is, isnt mathematics the study of the universe as a whole? I believe there is a bit of both.

[RozKek]

I'd just say they're all man-made models and concepts used to describe things, they don't exist in themselves as something.

[Alex K]

In my opinion, ideally, it is a discovery: one sets certain axioms as a starting point and discovers the tree of theorems which results from them. In reality, a lot of developing is going on, namely shaping and formulating the concepts with which one works. For instance, the concept of connectedness or simple connectedness was developed by mathematicians to express certain properties of sets. The rules which sets have this property and what other properties such sets have, is something that is discovered by formulating conjectures and proving them.

[vorlon13]

Go do some arithmetic in base -pi.

Then get back to me . . . .

[Whateverist]

In my opinion, ideally, it is a discovery: one sets certain axioms as a starting point and discovers the tree of theorems which results from them. In reality, a lot of developing is going on, namely shaping and formulating the concepts with which one works. For instance, the concept of connectedness or simple connectedness was developed by mathematicians to express certain properties of sets. The rules which sets have this property and what other properties such sets have, is something that is discovered by formulating conjectures and proving them.

Pretty much the way it seems to me too.  The discoveries all take place within a domain we've completely contrived.  Learning mathematics feels like discovering immutable facts but those facts depend entirely on the definitions we've chosen.

[axiom of congruence]

the concept of mathematics is a human invention, but the relations we can see in nature are things we can discover. nature has not just one mathematical property, it has only mathematical properties. so even though we created the concept of mathematics and use a human language to describe the patterns we see, we can still find more and more patterns in nature that follow the same principles.