Mathematics: created or discovered?

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Are Mathematicians discovering mathematical structures, laws,etc... that, in some sense, already exist or are they creating same?

Fractal geometry, in nature, has existed for non- trivial time periods and Benoit Mandelbrot delineated the mathematics for us. Did he discover or did he create?
Pal Erdos believed that all of the correct mathematics already exist in a metaphorical "book of proofs" and that the role of the Mathematician is to uncover them for humanity.

What about pure abstract math(s) for which there is no application and no apparent relationship with reality?

What is your opinion on this philosophical question?

The principles of natural mathematics, just like the principles of natural physics, are discovered.

Creativity is for art, fantasy, and religion.

What he said.

Someone did not sit down and arbitrarily decide to invent the concept of zero, he discovered that adding a zero in to some equations can dramatically change the results (the obvious examples are division and multiplication, dividing by zero nets you an impossible number and multiplying by zero will ALWAYS give you a zero).

I created math

Creativity moves science along when reason fails...or else we would only understand Newtonian physics.

care to elaborate?

6.853

Your response is not an elaboration IMO

It seems like a perfectly valid elaboration to me.

It depends what you mean. While playing a game such as ZFC, a mathematician really is discovering which theorems can be proven in the game.

However, the game was created by mathematicians in the first place.

Here is an analogy: a chemist might create a new chemical, and then discover things about it through experimentation. The "chemicals" that a mathematician creates and studies are formal systems of first-order predicate logic. However, the people who make the chemicals and the people who study the chemicals aren't usually the same people. The former are set theorists and logicians, and the latter are specialists in various mathematical fields.

But I should add that the chemicals are not just randomly generated. They are created for a certain purpose, and if they are not fit for purpose, nobody studies them. This explains why mathematics is useful for science. It's because that's what we care about, so that's what we make the chemicals for. There are all sorts of other weird chemicals that we will never study, and they are simply seen as pathological.

Mathematics is discovered in the same way that physical laws of nature are discovered.

However, mathematical notations and systems for describing mathematical concepts are created. We think of the number 0 as denoting "nothingness" - ie no pencils, no slices of pizza, no marbles in a jar. In a different society, 0 might mean infinity and infinity might mean 0.

I'm stating this obvious but you get my point. Mathematics is discovered.

Agreed.

The only times creativity is ever applied to maths are when someone uses "creative accounting" in an attempt to conceal embezzlement or tax fraud.

the field of mathematics was created to discover the relationships of nature.

This doesn't explain the many times that pure math which nobody thought could be applied in any practical way actually turned out to be very useful for a practical purpose which did not exist when that mathematical idea was discovered.


It takes a lot of creativity to discover mathematical truths.

No, just diligent research supported by an already strong foundational understanding of valid mathematical principles.

Mathematics is not about what you feel, it's about what you know, and what you do with that knowledge.

I can't completely agree. Just because you're working within a logically structured framework doesn't mean you can't be creative. Coming up with proofs in many cases requires creative thinking. Also, some arguments are considered more beautiful and/or elegant than others.

There are also plenty of areas of mathematics like set theory / category theory where the choice of axioms and definitions is a bit up in the air and mathematicians of the 20th century were at least partially motivated by striving for an aesthetically elegant foundation. Of course this subjectivity disturbed some mathematicians like Von Neumann who eventually stopped working on set theory on that account.