Yes. Computational complexity is a problem in physical computations. NP versus P is irrelevant in the Platonic realm.Skepdick wrote: ↑Tue Jun 10, 2025 8:24 amOK, 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.
Resolution of the question as to whether math is discovered or invented
Re: Resolution of the question as to whether math is discovered or invented
Re: Resolution of the question as to whether math is discovered or invented
I am not asking you about computational complexity. You keep running for the clouds every time.
Look at your feet for once.
I am asking you if the data-type and truth-value of X =X exists in the Platonic realm.
Re: Resolution of the question as to whether math is discovered or invented
At first glance, yes. Why not? There could be problems with the proposition, but that would require more context and a deeper investigation. If you are telling me that there are possibly weird corner cases, I believe you.
Re: Resolution of the question as to whether math is discovered or invented
Why do you need context, dude? Just consult the Platonic realm!
What corner cases?
Does x=x have a definite data type and truth-value in the Platonic realm; or not?
Re: Resolution of the question as to whether math is discovered or invented
I have never investigated the matter, but at first glance "x=x" is a legitimate logic sentence and provably true in first-order logic or first-arithmetic:
PA ⊢x=x
You do need context, because the following is not necessarily true in any arbitrary T:
T ⊢x=x
Is equality even supported in such arbitrary T?
Re: Resolution of the question as to whether math is discovered or invented
That's not true.
PA ⊢x=x IFF x is a natural number.
Equational reasoning is the collection of ALL such languages! Where x is an unbound variable; and x=x is reflexively true!
It's a fixed point computation.
Re: Resolution of the question as to whether math is discovered or invented
Which is why I didn't ask you about PA.
I asked you about ANY T ⊢ x=x. For any x.
Re: Resolution of the question as to whether math is discovered or invented
As I have already pointed out, context does matter. In the context of an arbitrary T, you cannot assert anything really, because it ultimately depends on how T is defined. Is it at least first-order logic? Or is it something else? If it is something else, how is the associated language defined and what does it allow?
Re: Resolution of the question as to whether math is discovered or invented
It's so peculiar for a Platonist to demand context; or to insist that truth depends on definitions.godelian wrote: ↑Tue Jun 10, 2025 9:19 am As I have already pointed out, context does matter. In the context of an arbitrary T, you cannot assert anything really, because it ultimately depends on how T is defined. Is it at least first-order logic? Or is it something else? If it is something else, how is the associated language defined and what does it allow?
Can you even distinguish between languages where x=x is defined and x=x is undefined?
How is "defined" defined?
Re: Resolution of the question as to whether math is discovered or invented
What exactly do you want to "prove" about Platonism? I am not personally the physical incarnation of Platonism. I just subscribe to it.Skepdick wrote: ↑Tue Jun 10, 2025 9:24 amIt's so peculiar for a Platonist to demand context; or to insist that truth depends on definitions.godelian wrote: ↑Tue Jun 10, 2025 9:19 am As I have already pointed out, context does matter. In the context of an arbitrary T, you cannot assert anything really, because it ultimately depends on how T is defined. Is it at least first-order logic? Or is it something else? If it is something else, how is the associated language defined and what does it allow?
Can you even distinguish between languages where x=x is defined and x=x is undefined?
How is "defined" defined?
Re: Resolution of the question as to whether math is discovered or invented
I am not trying to prove absolutely anything. I am trying to understand what exactly you have subscribed to.
The label seems completely vacuous. How do you subscribe to nothing?
It's not even that Platonism is false - it's that there's nothing there to be true or false about!
Re: Resolution of the question as to whether math is discovered or invented
Platonism is a particular take on the ontology of mathematical objects. You can find an entire page on it here:
https://en.wikipedia.org/wiki/Mathematical_Platonism
or here: https://plato.stanford.edu/entries/plat ... thematics/
These pages are very clear. I don't think that anybody complains about these explanations. They are not vague or ambiguous at all.Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.
Re: Resolution of the question as to whether math is discovered or invented
If it depends on HOW it's defined then it is NOT "independent of us and our language, thought, and practices."godelian wrote: ↑Tue Jun 10, 2025 9:19 am As I have already pointed out, context does matter. In the context of an arbitrary T, you cannot assert anything really, because it ultimately depends on how T is defined. Is it at least first-order logic? Or is it something else? If it is something else, how is the associated language defined and what does it allow?
You are claiming to subscribe to Platonism, but what you are advocating for is NOT Platonism.
When you preach contextuality; and "it depends".... you sounds a lot like a formalist; or a constructivist.
You sound like somebody who believes one thing; but says another.
You sound like somebody who believes the stuff is invented; not discovered.
Re: Resolution of the question as to whether math is discovered or invented
Choosing another T merely means choosing another map. Of course, you will see different things.Skepdick wrote: ↑Tue Jun 10, 2025 9:54 amIf it depends on HOW it's defined then it is NOT "independent of us and our language, thought, and practices."godelian wrote: ↑Tue Jun 10, 2025 9:19 am As I have already pointed out, context does matter. In the context of an arbitrary T, you cannot assert anything really, because it ultimately depends on how T is defined. Is it at least first-order logic? Or is it something else? If it is something else, how is the associated language defined and what does it allow?
You are claiming to subscribe to Platonism, but what you are advocating for is NOT Platonism.
When you preach contextuality; and "it depends".... you sounds a lot like a formalist; or a constructivist.
You sound like somebody who believes one thing; but says another.
You sound like somebody who believes the stuff is invented; not discovered.