Is mathematics invented or discovered?

[DaveC426913]

So from that standpoint, when did this invention of the human mind come into existence?
(1) When Galois first wrote about group theory?
(2) When people started studying simple groups?
(3) When someone first realized that this huge group existed?
(4) Since obviously nobody has conceptualized the monster group as a whole, does it exist even now?

Does it have to be exact moment?

Most inventions go through refinement over time. Is one point more valid than all the others? Or more of a Theseus's Ship writ temporal?

 

[PeterDonis]

The group - "the primes" - was invented.

In the sense that humans formed the concept and explicitly recognized its properties, sure.

But at least one of those properties--the fact that a prime number has no other factors besides itself and $1$--is the reason why evolution produced the time periods for cicadas that it did. That reason exists independently of humans and does not require humans to conceptualize it in order to drive an evolutionary process.

So while one could say we humans invented the part that we explicitly formulated, I don't think you can say humans invented the properties that drove evolutionary processes that happened independently of humans and without even requiring humans to exist. Those things, we discovered.

 

[PeterDonis]

when did this invention of the human mind come into existence?

Invention is a process, not an event. There is no instant of time when an invention "comes into existence". You are trying to draw sharp boundaries in a domain where they simply don't work.

 

[DaveC426913]

In the sense that humans formed the concept and explicitly recognized its properties, sure.

But at least one of those properties--the fact that a prime number has no other factors besides itself and $1$--is the reason why evolution produced the time periods for cicadas that it did. That reason exists independently of humans and does not require humans to conceptualize it in order to drive an evolutionary process.

So while one could say we humans invented the part that we explicitly formulated, I don't think you can say humans invented the properties that drove evolutionary processes that happened independently of humans and without even requiring humans to exist. Those things, we discovered.

Exactly correct. My point was that virtually any kind of grouping (though not all) is abstract and created by humans.

Like the monster group and Galois' theorems, where this started (for me) around post 33.

 

[jedishrfu]

There's a curious parallel here.

When great authors invent their characters and where they reside, they often find that the story begins to write itself ie they begin to discover things about their characters that they didn't know.

They keep writing because they're invested in their own story.

It seems invention and discovery go hand in hand.

With that thought in mind, our discussion here is like the blind men who describe their meeting with an elephant. Each comes away with a different description and yet others see the creature as an elephant.

Perhaps this a good time to close this thread.