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

[godelian]

What does the empty set correspond to in Platonic reality?

As soon as you express it in language, you commit to a map already. By representing the territory, it is no longer the territory but a map.

But I am genuinely committed to the ontological non-existence of the empty set!

You can't represent nothing.

In the Von Neumann representation of the natural numbers {} corresponds to 0. So, you actually can represent "nothing". Both {} and 0 represent it.

The empty set exists is one model/map.
The empty set does not exist is another model/map.
Clearly those two models/maps correspond to DIFFERENT territories.

They could be representations of different parts of the territory.

So when I asked you about the cardinality of the non-standard territories; I didn't ask you about the cardinality of the maps!

Each nonstandard map represents a nonstandard territory. They are 1to1.

Of course, none of the above can be constructed. I am perfectly okay with that. I suspect that you are not.

Are you OK with it? Any map can be constructed. That's what ALL maps are. Constructions.

Yes, the map is a construction trying to represent the unknown territory. With the Platonic territory, we cannot see the territory itself. We always have to use a map and look at that instead. Platonism is the belief that the territory really exists, even though we cannot see it directly.

[Skepdick]

In the Von Neumann representation of the natural numbers {} corresponds to 0. So, you actually can represent "nothing". Both {} and 0 represent it.

Woah! Woah! Woah! STOP!

What is the content of the axiom?

0 IS a natural number; or 0 REPRESENTS a natural number?

They could be representations of different parts of the territory.

So.... my previous question becomes even more relevant!

What is 0? A natural number; or a representation of a natural number?

Each nonstandard map represents a nonstandard territory. They are 1to1.

OK but the axiom of PA doesn't say "0 REPRESENTS a natural number" !!!!

It says "0 IS a natural number"!!!!!

That is an ontological statement!

Yes, the map is a construction trying to represent the unknown territory.

What? How? How do you represent an unknown?

What are you representing?

PA makes existence claims, not representation claims!
Representation requires existence!

With the Platonic territory, we cannot see the territory itself.

So what the fuck are you representing?

We always have to use a map and look at that instead. Platonism is the belief that the territory really exists, even though we cannot see it directly.

How the fuck do you USE a map before CONSTRUCTING it?

And how do you CONSTRUCT a map of a territory you can't even see?!?

[godelian]

0 IS a natural number; or 0 REPRESENTS a natural number?

0 is a symbol. It is not the natural number itself.

It says "0 IS a natural number"!!!!!
That is an ontological statement!

0 is a symbol. Not a natural number.

With the Platonic territory, we cannot see the territory itself.

So what the fuck are you representing?

Something unknown. We do have alternative maps, though.

And how do you CONSTRUCT a map of a territory you can't even see?!?

By systematizing the basic beliefs that you have about the unknown territory.

[Skepdick]

0 IS a natural number; or 0 REPRESENTS a natural number?

0 is a symbol. It is not the natural number itself.

Sorry, that's incoherent. Symbols are variables whose value is equal to their literal representation.

So the VALUE of the symbol 0 IS 0. Of course you know this, because when you type the symbol 0 in any computational REPL you get... 0 in return!
The symbol 0 EVALUATES to 0 e.g itself!

So what the fuck is a "natural number"?

0 is a symbol. Not a natural number.

You just negated Peano's first axiom.

You know what happens in classical logic where you negate an axiom; right?

BOOOOM!!!!!

Something unknown.

OK... so how do you know this "unknown" thing is a natural number?

By systematizing the basic beliefs that you have about the unknown territory.

So you are INVENTING/CONSTRUCTING a map based on nothing but personal whims; and without even knowing whether the things you are representing exist?

You are NOT discovering anything about any Platonic reality.

Q.E fucking D.

Platonism: I construct systematic beliefs about unknowns and pretend I'm discovering eternal truths.
Constructivism: I construct systematic mathematical frameworks and admit that's what I'm doing.

We are both constructivists. Only one of us is intellectually honest about it.

Are you willing to be honest about what mathematics actually involves, or do you need to wrap it in mystical stories to make it feel profound to skeptical newcomers?

Which is the better narrative for sociological marketing and PR?
Which is more likely to appeal to modernity?

Mystical story: "We channel eternal truths from invisible realms"
Honest story: "Humans have constructed incredibly sophisticated, powerful, and beautiful formal systems that can model reality, prove theorems, and solve problems"

[godelian]

We are both constructivists. Only one of us is intellectually honest about it.

As I have mentioned already, I am not categorically opposed to constructivism. I just don't require examples of things like undefinable or uncomputable numbers. As far as I am concerned, they exist, even though it is not possible to give an example. Furthermore, I happily accept the LEM unless it creates insurmountable trouble, which it sometimes does, but most of the time, it actually doesn't.

[Skepdick]

We are both constructivists. Only one of us is intellectually honest about it.

As I have mentioned already, I am not categorically opposed to constructivism.

This is not about the content of your beliefs. Whether you accept or reject constructivism is IMMATERIAL to the fact that constructivism is what you DO in way of Mathematical practice.

I just don't require examples of things like undefinable or uncomputable numbers.

You don't require examples of anything! Not even computable natural numbers!

Not even 0!

Never mind that. What sort of example could a Platonist even offer (even to themselves!!!) given that Platonic reality is inaccessible?

As far as I am concerned, they exist, even though it is not possible to give an example.

As far as you are concerned it is not even possible to know that they exist.

Furthermore, I happily accept the LEM unless it creates insurmountable trouble, which it sometimes does, but most of the time, it actually doesn't.

What the fuck?

By LEM: ∃0 ∨ ¬∃0

Does 0 exist; or not?

[Skepdick]

I happily accept the LEM unless...

This statement is the witness to the proof of the formal claim: ¬Classical(godelian)

There is no UNLESS. Your "principled" commitment to classical logic is actually opportunistic, not universal.

Conditional acceptance of LEM IS constructivism.

[godelian]

You don't require examples of anything! Not even computable natural numbers!

Not even 0!

You can prove the existence of 3 by counting till 3. So, 3 is on the map. You cannot do that by counting till zero. So, zero is not automatically on the map. You have to manually put zero on the map.

I am not against examples of mathematical objects. I just do not require individual examples when you can only put their entire collection on the map. There may be no procedure available to put the individual elements on the map.

As far as I am concerned, they exist, even though it is not possible to give an example.

As far as you are concerned it is not even possible to know that they exist.

Take for example the diagonal lemma. It proves the existence of collections of sentences but it may not be possible to pinpoint one single element in such collection. I am fine with that. Our inability to put a single element individually on the map does not mean that such element does not exist. There's just no procedure available to do that.

Does 0 exist; or not?

Zero is on the map for no other reason than we say that it is. Zero is a legitimate Platonic object in the territory because it is part of its structure.

[godelian]

Your "principled" commitment to classical logic is actually opportunistic, not universal.

It is not a principled commitment to classical logic but to Platonism.

[Skepdick]

You can prove the existence of 3 by counting till 3.

Counting till 3... starting from what?

So, 3 is on the map.

We've already established that anything you want is on the map. There's nothing special about 3.

You cannot do that by counting till zero. So, zero is not automatically on the map. You have to manually put zero on the map.

Eh? I can count till zero. -3, -2, -1, 0.

I can count til anything.
From anything.

I can count up to; or down to.
I can count as many objects as time permits me to count.

I am not against examples of mathematical objects. I just do not require individual examples.

What? So why do you need Peano's first axiom which puts zero on the map?!?

Take for example the diagonal lemma.

Child, you always keep running for the clouds where all the problems are right at your feet.

Zero is on the map for no other reason than we say that it is.

OK... so whether it is; or it isn't on the territory is immaterial!

Zero is a legitimate Platonic object in the territory because it is part of its structure.

What? Contradiction!

You just said that zero is on the map for no other reason than human construction/declaration!
Which necessarily entails that zero is NOT on the map because it's an object in Platonic reality!

[Skepdick]

Your "principled" commitment to classical logic is actually opportunistic, not universal.

It is not a principled commitment to classical logic but to Platonism.

But we have so far established that your Platonism is a completely vacuous label!

Can you go ahead and actually commit yourself to something; rather than twisting yourself in a pretzel every time I give you a counter-example?
Can you even commit yourself to the truth value of x=x existing in Platonic reality, independent of any mind?

Are you even sure you are committed to a Turing machine as your model of computation?!? To count ℵ₀ objects you need at least ℵ₀ time.
Is your computational model capable of supertasks? Turing machines are not.

https://en.wikipedia.org/wiki/Supertask

[godelian]

To count ℵ₀ objects you need at least ℵ₀ time.
Is your computational model capable of supertasks? Turing machines are not.

Time does not exist in the Platonic realm. Time is a feature of the physical universe.

ChatGPT: Doesn't it take an infinite amount of time to fully induct the set of natural numbers?

While it seems intuitive that fully inducting N would take infinite time, mathematical induction is not a time-based process — it's a logical principle. The fully inducted set of natural numbers exists because of the axioms and logic in foundational mathematics, not because we "finished" an infinite number of steps.

According to Platonism, mathematical objects such as numbers and sets are metaphysical objects. Therefore, mathematics is the deductive branch of metaphysics. While computation in the physical universe takes time and energy, this is not the case in the Platonic realm, because time and energy do not exist in it.

I do not appreciate physicalism or materialism in mathematics. I do not believe in it. I do not believe that it would be useful in any way. Therefore, I do not want to have anything to do with it.

[Skepdick]

Time does not exist in the Platonic realm. Time is a feature of the physical universe.

See! You are twisting yourself into a pretzel!
Which realm does the Mathematician exist in? The physical one; or the Platonic one?
which realm does the counting machine exist in? The physical one; or the Platonic one?

Can the counting machine perform supertasks, hypertasks and ultratasks?

Does it mean that count({1,2,3}) and count({1,2,3,...}) all produce instant answers?
Does it mean that any and all infinitary processes produce instant answers?

[godelian]

Does it mean that supertasks, hypertasks and ultratasks are all (in principle) computable?

Computations take time in the physical universe but not in the Platonic one.

Does it mean that count({1,2,3}) and count({1,2,3,...}) all produce instant answers?
Does it mean that any and all infinitary processes produce instant answers?

In the physical universe, no. In the Platonic realm, yes.

[Skepdick]

Does it mean that supertasks, hypertasks and ultratasks are all (in principle) computable?

Computations take time in the physical universe but not in the Platonic one.

Does it mean that count({1,2,3}) and count({1,2,3,...}) all produce instant answers?
Does it mean that any and all infinitary processes produce instant answers?

In the physical universe, no. In the Platonic realm, yes.

OK, so the truth-value of X=X, or X = Y is instantly decidable in the Platonic realm?
No matter how computationally complex X and Y; and no matter how long the reduction to their canonical form takes?!?

Is this what you are finally committing Platonism to? The truth-value of of an equational statement exists in Platonic reality - independent of any human mind.