A la, Penrose, if one believes in mathematical Platonism and such, then there seems to be a final system that could account for all proofs in it, no? — Posty McPostface
It seems intuitively obvious if you consider QM in infinite Hilbert Space or anything in infinite Hilbert Space. — Posty McPostface
Yes, I understand that much. But, having objects that can only exist in infinite Hilbert space isn't a mystery to you? — Posty McPostface
No, why? Perhaps you can explain your point of view to me. I didn't study much physics but I've studied functional analysis. Hilbert spaces aren't very mysterious at all. In fact when I learned that the mysterious bra-ket notation is nothing more than a linear functional acting on a vector, I felt enlightened, as if perhaps QM isn't that far beyond me. If you can explain to me what you're thinking I'm sure I'd learn something. Maybe there's a mystery I'm not appreciating. — fishfry
So it does not rule out the possibility that one might be able to prove 'everything' in a language with an uncountably infinite alphabet. — andrewk
A Platonist is someone who thinks that "out there" in the Platonic world there are sets, and that in that world of sets, Choice is either true or else it's false. There's a definite answer. — fishfry
But the world is every changing. How do they account for that? — guptanishank
Yeah, I meant how do the platonists who do think this way, think about it? — guptanishank
I did not know that about Godel. — guptanishank
The next question is, if formal systems can't get at the truth, what can? I don't know anything about what philosophers think about that. — fishfry
Wouldn't it have to be something like intuition? Or perhaps, on the other hand, new formalizations that are truer to intuition? — t0m
For me the finite and the computational are just nakedly "real" or "true." They are more persuasive than the philosophy that might try to ground them. You'll probably agree that it's the infinite that gives us trouble. — t0m
Tentatively this trouble seems to involve the gap between a fuzzy, linguistic concept and a mechanizable concept. There are limits to mechanization (halting problem, for instance), and yet mechanization is as Platonic as it gets? — t0m
But recursion need not be finite. Surely that is possible. — guptanishank
But a computational process, could go on forever? The symbols might be finite, but they are referring to something quite plausible. — guptanishank
The point is that modern set theory is the search for new axioms that are plausible and seem natural for the world of sets we have in our minds. In our intuition. Yes, it's ultimately driven by intuition. By our intuition about what the Platonic sets must be. Even if we're formalists in the end we must be part Platonist. — fishfry
I would say that the infinite is what makes math interesting! Otherwise it's just combinatorics. Balls in bins. Finite sets are boring. Also you need infinity to come up with a satisfactory theory of the real numbers. Which themselves are a philosophical mystery. — fishfry
Something that I find interesting is that even though we have all these crazy theories about humongous infinite sets; all of our reasoning is finitistic. Proofs are finite strings. The axioms and theorems are finite strings. The rules of inference are described with finite strings. You could program a computer to check if a proof is valid. This is a huge area of active research these days, they're doing amazing things.
So all we're really doing is playing around with finite strings of symbols. We tell ourselves it's "about infinity," but it really isn't. We are only pretending to be able to deal with infinity. That's one way to look at things. — fishfry
One thing:
How do we know that the statement we are proving outside the axioms is the same statement as inside the axioms? — guptanishank
A statement is comprised of all the axioms above it. — guptanishank
So in a way we are proving different statements to be true, even though they may look the same.
Any thoughts? — guptanishank
It seems very counter intuitive to me to use two different axiomatic systems for two different proofs of the same statement.
Could it not be that a statement is True under one axiomatic system and False under another then? — guptanishank
Well, we discussed that if a statement is unprovable, then we go outside the ambit of the axiomatic system to establish a larger meta system which can prove or disprove that statement, in the above discussion.One thing:
How do we know that the statement we are proving outside the axioms is the same statement as inside the axioms?
— guptanishank
I don't know what that means. Can you give an example? — fishfry
Of course 2+2 = isn't a statement, but 2+2 = 4, comprises of all the axioms needed to define 2, +, = and 4A statement is comprised of all the axioms above it.
— guptanishank
That doesn't correspond with my understanding of what a statement is. A statement in a formal system is simply a well-formed formula that may be true or false. For example "2 + 2 = 5" is a statement. "2 + 2 =" isn't. — fishfry
Let's say under ZFC we could not prove 2+2 = 4, as true. Then to prove Godel's theorems we would still need to establish that statement as true or false eventually. The thing is that truth or falsehood of the statement will vary according to the axiomatic framework we consider.So in a way we are proving different statements to be true, even though they may look the same.
Any thoughts?
— guptanishank
Don't know what you mean. Example? — fishfry
The thing is that truth or falsehood of the statement will vary according to the axiomatic framework we consider. — guptanishank
Provability and truth are two distinct notions? — guptanishank
How would you know if a statement is true without the proof? — guptanishank
Truth is a semantic notion. Given an axiomatic system, we choose some interpretation of the symbols, and then we see if the statement is true or false under that interpretation. — fishfry
How do you "see" this without proving? Mathematical statements can be notoriously hard to "see" if true or false. — guptanishank
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