"Discover" or "invent" in mathematics context [closed]

A very theoretical question I came across with friends some time ago. I am not a native English speaker, so I may be misunderstanding some details, but as I understand:

1) Discovery relates to a concept which existed prior to ... well "discovery" but was unknown before the act.

2) Invention relates to a creation of a concept which never existed before the act

Following from this, are mathematical concepts such as sinus "discovered" or "invented"? In one sense, being a fundamental property hence could only be discovered. On the other hand, the logical metaphor was created by humans, hence invented.

Which one is right and which one is wrong and why?

This is a very old question, even among mathematicians who are accomplished native English speakers. There are definitely aspects of discovery as well as invention in math, and some things lend themselves more to one than the other. For example, it is more appropriate to say that notation is invented, rather than discovered. It's the other way for proofs.

In general, if you're not sure which one, it's a little safer to say discover, as it is a more "humble" term; but in casual use, either one is usually acceptable.

Discover is typically used among mathematicians. Most mathematicians are typically Platonists internally, even though the Platonist view has no real credibility among philosophers or those that study the philosophy of mathematics.

Since most mathematicians are atheists (based on my experience in grad school and in working as a math professor), it would be more consistent for them to speak of inventing theorems, but they just don't. I suspect that the reason for this is that in the process of doing mathematics it is expedient to think as if this stuff is simply out there.

I think neither verb is entirely appropriate for mathematical concepts in ordinary English. You can't "discover" a square root, nor can you "invent" a radius: they just are, I'd say. I suppose you could use these verbs, depending on context, but I suspect that it would nearly always be better to avoid the situation.

If I said "Pythagoras discovered the Pythagorean theorem", that would be possible; but it'd sound a tiny bit childish, as though I weren't aware of the very problem you present here. The same applies to "invent".

(If you like Kant, you will probably agree that mathematical concepts are best considered models created by the mind in which we shape our sensory impressions. We have always used these models subconsciously in various ways: discovering or inventing them means just becoming aware of patterns in the ideas we have about reality.)

You might use to conceive of:

Surprisingly it was Pythagoras who first conceived of the Pythagorean theorem.

It depends on what you believe.

A. Mathematical concepts exist. (we find them out)
B. Mathematical concepts don't exist. (we build them)

If you believe A then you can use "discover" else you can use "invent". Even if you believe A you can still use "invent" but that won't be consistent with your belief.

If you believe that A is true then B is false for you and if you believe B is true then A is false for you.

There is no way to decided which one is actually true. That is why this debate can go on forever.

In real life, people don't really care which one (A,B) is true. So they are comfortable with either word.

Such questions are bread and butter of philosophers.

All things are discovered. All possibilities exist. We only discover these possibilities. Does a composer create a symphony? All of the notes are already there. The composer merely disovers a pleasing sequence. Did Hans Lippershey invent the telescope, or did he observe properties of light and materials and discover that when certain materials were arranged that a pleasing image is produced. Did Jack Kilby invent the integrated circuit? The physical properties of conductors already existed. All the properties of physics and chemistry, the laws of motion, and of mathematics, and of economics and the universe already exist, just waiting to be discovered.