Resolution of the question as to whether math is discovered or invented

[Flannel Jesus]

This is the greatest revelation in human history and no one here at this "philosophy" forum, can see it. It's surreal how no one is capable of basic independent reasoning

What's the greatest revelation? That mathematics is invented?

[accelafine]

Whether it's discovered or invented, it still has an 's' on the end.
Thorry about your lithp.

Fun fact, the term "math" predates "maths".

https://www.thesaurus.com/e/grammar/math-vs-maths/

Amazing how google can always find what you want it to find, no matter what question you ask it 'Dikshinry.com'

[accelafine]

Whether it's discovered or invented, it still has an 's' on the end.
Thorry about your lithp.

You're the only person that has ever said that I have a lisp.

AND you completely missed the point

This is the greatest revelation in human history and no one here at this "philosophy" forum, can see it. It's surreal how no one is capable of basic independent reasoning

I can't read it because the word 'math' makes my brain hurt. Can't you just say 'mathematics'? It's such a nice word. You took the time to write a whole essay so it seems silly to skimp on one word

So which is it? Discovered or invented? You need to let the scientific community know immediately!

[Flannel Jesus]

Whether it's discovered or invented, it still has an 's' on the end.
Thorry about your lithp.

Fun fact, the term "math" predates "maths".

https://www.thesaurus.com/e/grammar/math-vs-maths/

Amazing how google can always find what you want it to find, no matter what question you ask it 'Dikshinry.com'

Nice cop out.

[godelian]

...

That's a really long-winded way to go about it.

All axioms are invented.
All consequences thereof (theorems) are discovered.

Agreed.

However, even though axioms are indeed essentially invented, you still need to "discover" which sets of axioms are going to turn out to be incredibly influential. Of all the inventions, PA and ZFC ended up massively dominating the field. It was therefore a question of "discovering" that they would.

[Skepdick]

...

That's a really long-winded way to go about it.

All axioms are invented.
All consequences thereof (theorems) are discovered.

Agreed.

However, even though axioms are indeed essentially invented, you still need to "discover" which sets of axioms are going to turn out to be incredibly influential. Of all the inventions, PA and ZFC ended up massively dominating the field. It was therefore a question of "discovering" that they would.

And then Gödel ruined the dream.

Turns out sculpting apart the necessary from the sufficient conditions was even more useful; and everything's just a game of more and more powerful coding schemas. Now we understand the logical structure of mathematical theories at a deeper level than the previous "build everything from one axiomatic base" approach.

Meta-mathematics. Computer science. More and more powerful coding schemas/programming languages...

Potato/potatoh.

[vamvam]

That's a really long-winded way to go about it.

All axioms are invented.
All consequences thereof (theorems) are discovered.

Invention is when from infinite enumerated sequence of axioms you pick some. Discovery is when you browse through infinite enumerated sequence of consequences.

[Skepdick]

Invention is when from infinite enumerated sequence of axioms you pick some. Discovery is when you browse through infinite enumerated sequence of consequences.

You can go in either direction... you can discover some consequences then invent axioms for them; or you can invent some axioms and discover their consequences.

Welcome to the game where all of your conclusions are never stronger than your premises.

[vamvam]

What I want to say is that difference between invention and discovery reflects the way common people see the world. From mathematical perspective the difference is much smaller if any.

[godelian]

Invention is when from infinite enumerated sequence of axioms you pick some. Discovery is when you browse through infinite enumerated sequence of consequences.

For reasons of biological survival, animals have a limited and heuristic version of informal mathematics built into their biological firmware.

https://en.m.wikipedia.org/wiki/Informal_mathematics

Informal mathematics means any informal mathematical practices, as used in everyday life, or by aboriginal or ancient peoples, without historical or geographical limitation. Modern mathematics, exceptionally from that point of view, emphasizes formal and strict proofs of all statements from given axioms. This can usefully be called therefore formal mathematics. Informal practices are usually understood intuitively and justified with examples—there are no axioms.

Aboriginal mathematics and logic is good enough for plucking fruit and chasing wild animals.

In spite of going to school where they may somewhat attempt to learn axiomatic mathematics and logic, in daily life, the vast majority of people tend to fall back on aboriginal methods.

The choice of axioms in PA and ZFC was certainly also inspired by the widespread practice of aboriginal finger counting.

[roydop]

This is the greatest revelation in human history and no one here at this "philosophy" forum, can see it. It's surreal how no one is capable of basic independent reasoning

What's the greatest revelation? That mathematics is invented?

That it's wrong. The invention aspect is the ongoing misinterpretation of the message expressed by the number system

[roydop]

Whether it's discovered or invented, it still has an 's' on the end.
Thorry about your lithp.

You're the only person that has ever said that I have a lisp.

AND you completely missed the point

This is the greatest revelation in human history and no one here at this "philosophy" forum, can see it. It's surreal how no one is capable of basic independent reasoning

I can't read it because the word 'math' makes my brain hurt. Can't you just say 'mathematics'? It's such a nice word. You took the time to write a whole essay so it seems silly to skimp on one word

So which is it? Discovered or invented? You need to let the scientific community know immediately!

You don't deserve this knowledge. You're not ready for it

[luberti]

...

That's a really long-winded way to go about it.

All axioms are invented.
All consequences thereof (theorems) are discovered.

This is going in a better direction.

But note that "All axioms are invented" is still false. After you described a class of axiomatic systems, you may discover them without any inventing.

For example, it turned out via automated search that there are only seven 21-symbol (i.e., according to the publications, minimal) single axioms for Hilbert systems of propositional logic with only the classical operators {¬,→}. One was discovered by Carew Arthur Meredith in the 1950s, six more were discovered by Matthew Walsh at Northwestern University in 2021 (who also first claimed that 21-symbol axioms are minimal for this case, but didn't publish a proof or even the source code to support this claim).

Similarly, you could extend this further upwards:
1. What are we looking for?
2. Discover/Invent a definition we can use. (either, or)
3. Discover/Invent a system we can use. (either, or)
4. Explore a system. (e.g. discover its theorems)

Finally, in engineering, we have:

5. Build something based on what we have learned about certain systems.

It boils down on the specifics of where "discovery" ends and "invention" starts.

Quoting from Google:

a discovery is finding something that already exists but was previously unknown, while an invention is creating something entirely new

But that is insufficient since it now boils down to what is "entirely new" vs. what is not. Each of the steps may fall into different categories from case to case.

From ontological perspective, all abstract things exist in a way, but therefore nothing abstract could be entirely new, and only physical and mental objects could be "invented". But that's only one perspective. When taking a different approach, what would even qualify an abstract thing to "exist"? One could make arbitrary definitions, and I doubt it is possible to show any single one of them as particularly relevant, except the one according to which no abstract objects exist. But this would imply that nothing of mathematics is discovery (but all invention), including step 4. It would also imply that machines and algorithms are great at inventing things.

So this whole problem/thread is only about arguing semantic details of natural language and ontological terms. Moreover, it assumes lacks of variation and is therefore a false dichotomy. (However, what OP wrote is merely bias lacking evidence.)

[Skepdick]

But note that "All axioms are invented" is still false. After you described a class of axiomatic systems, you may discover them without any inventing.

How do you describe a class that doesn't already exist?

How did this class came into existence without the axioms which define it?

For example, it turned out via automated search that there are only seven 21-symbol (i.e., according to the publications, minimal) single axioms for Hilbert systems of propositional logic with only the classical operators {¬,→}. One was discovered by Carew Arthur Meredith in the 1950s, six more were discovered by Matthew Walsh at Northwestern University in 2021 (who also first claimed that 21-symbol axioms are minimal for this case, but didn't publish a proof or even the source code to support this claim).

And the search parameters/criteria to your search algorithm... are they discovered or invented?

Similarly, you could extend this further upwards:
1. What are we looking for?
2. Discover/Invent a definition we can use. (either, or)

Using discovered or invented acceptance criteria for it?

3. Discover/Invent a system we can use. (either, or)

Is the utility function for the system discovered or invented?

4. Explore a system. (e.g. discover its theorems)

Is the system being explored discovered or invented?

Finally, in engineering, we have:

5. Build something based on what we have learned about certain systems.

The methods of building/construction in engineering are always constrained by the specifcations of what it is we are building.

Are the specifications discovered or invented?

It boils down on the specifics of where "discovery" ends and "invention" starts.

Some would say that the distinction is artificial when operating under constraints.

But; of course the question remains: are the constraints discovered or invented?

[luberti]

But note that "All axioms are invented" is still false. After you described a class of axiomatic systems, you may discover them without any inventing.

How do you describe a class that doesn't already exist?

How did this class came into existence without the axioms which define it?

Under common ontological notions, you obviously cannot describe a class that doesn't already exist, since they all exist as abstract objects as soon as they are classes. Also a class didn't come into existence. It exists independently of time since it is neither a physical nor a mental object.

For example, it turned out via automated search that there are only seven 21-symbol (i.e., according to the publications, minimal) single axioms for Hilbert systems of propositional logic with only the classical operators {¬,→}. One was discovered by Carew Arthur Meredith in the 1950s, six more were discovered by Matthew Walsh at Northwestern University in 2021 (who also first claimed that 21-symbol axioms are minimal for this case, but didn't publish a proof or even the source code to support this claim).

And the search parameters/criteria to your search algorithm... are they discovered or invented?

Similarly, you could extend this further upwards:
1. What are we looking for?
2. Discover/Invent a definition we can use. (either, or)

Using discovered or invented acceptance criteria for it?

3. Discover/Invent a system we can use. (either, or)

Is the utility function for the system discovered or invented?

4. Explore a system. (e.g. discover its theorems)

Is the system being explored discovered or invented?

My point was that under common ontological notions, abstract objects cannot be invented but are always discovered. That's really trivial since they exist independently of time. Once could use different notions; what matters is what is true under which definitions.

Finally, in engineering, we have:

5. Build something based on what we have learned about certain systems.

The methods of building/construction in engineering are always constrained by the specifcations of what it is we are building.

Are the specifications discovered or invented?

It boils down on the specifics of where "discovery" ends and "invention" starts.

Some would say that the distinction is artificial when operating under constraints.

But; of course the question remains: are the constraints discovered or invented?

Again, under common ontological notions, physical objects come into existence, therefore they can be invented, whereas specifications as abstract objects cannot.

I think I made myself clear:

From ontological perspective, all abstract things exist in a way, but therefore nothing abstract could be entirely new, and only physical and mental objects could be "invented". But that's only one perspective. When taking a different approach, what would even qualify an abstract thing to "exist"? One could make arbitrary definitions, and I doubt it is possible to show any single one of them as particularly relevant, except the one according to which no abstract objects exist. But this would imply that nothing of mathematics is discovery (but all invention), including step 4. It would also imply that machines and algorithms are great at inventing things.

So this whole problem/thread is only about arguing semantic details of natural language and ontological terms. Moreover, it assumes lacks of variation and is therefore a false dichotomy. (However, what OP wrote is merely bias lacking evidence.)

So I don't see a point in continuing this since it is already solved and it concerns just basic semantics.