Is mathematics invented or discovered?

[nomadreid]

One of the possible false assumptions here is that mathematics is uniform enough to have the same answer for all of it. It is possible that parts of mathematics are invented, and parts are discovered. The people claiming one side often latch on to that part of mathematics that fits their choice to justify their claim. In this way the debate could go on uselessly forever.

 

[PeroK]

I'm not sure I agree that group theory was invented. The axioms of group theory were chosen the way they are because it leads to relationships between transformations that describe objects in our universe. In this sense I would argue that Galois saw that those axioms led to particularly useful and interesting structures. If it were purely invented, then you could choose any random set of axioms for your theory.

The monster group isn't an object in our universe.

 

[DaveC426913]

In this sense I would argue that Galois saw that those axioms led to particularly useful and interesting structures. If it were purely invented, then you could choose any random set of axioms for your theory.

You just self-corrected.

You could "choose any random axioms for your theory", but you don't - and the reason you don't is because certain "axioms lead to particuarly useful and interesting structures".


Look, all types of categorizing are human-invented. We might group green things together: leaves, caterpillars, bird plumage. But that is not a natural grouping. We invented that group by deciding that what was important was colour*. There is no such thing as a green group in nature. The same thing can be said about any form of grouping. The group's definition is defined by what some human decides is significant.

*actually, even that doesn't exist objectively. "Green" only exists in the mind.

 

[phyzguy]

The monster group isn't an object in our universe.

I agree it is not a physical object, but it is a well defined structure. Where does that structure reside if it is not in our universe?

 

[phyzguy]

Look, all types of categorizing are human-invented. We might group green things together: leaves, caterpillars, bird plumage. But that is not a natural grouping. We invented that group by deciding that what was important was colour*. There is no such thing as a green group in nature. The same thing can be said about any form of grouping. The group's definition is defined by what some human decides is significant.

But that's exactly my point. "Some human decides what is significant" by studying the objects around us. As you say, we are guided by "what is a natural gouping". We don't just make that up (i.e. invent it), we are guided by inventorying the things around us (i.e we are discovering them).

The structure of group theory was guided by how things transform in our universe. It wasn't just made up on a blank slate.