A summary:
The criteria for usefulness implies that there are mathematical sentences that you want to be able to be true or false. The idea that these sentences should be true or false presupposes the existence of the object the sentence represents, because the current axioms can't give it a truth-value, so there must be an abstract object that has the same properties as well as a truth-value.
The statement isn't weaker, but it still implies the existence of real mathematical objects that are independent of humans.
It makes them true or false in the new set-theoretic universe that you haven't constructed, but have presupposed exists.
suppose someone will murder you unless you attest that god is real. suppose they are willing to drug you and use psychological tactics to the point of actually convincing you god is real. this makes it useful to claim god is real, or even to believe god is real.
does this make god real in some metaphysical sense?
or rather, "the reality of god" as a question with an answer, i suppose. but you can apply this to any communicable concept. someone might drug you and beat you until you have a desired brainstate.
are you sure metaphysical reality isnt another word for conceivable in brains?
oh by the way i would like to hear your thoughts on this:
P=/=NP is a computer science conjecture that computer scientists think is probably true, but they haven't proved it yet. this conjecture seems like it should have an "actual" truth value, since it makes a strong statement about what computer algorithms can exist in the world, and it can be disproved by counterexample. but there is a remote possibility (i dont think many people believe this) that it is independent of our axiomatic system; then either
- there are no counterexamples, but we cannot prove there are none.
- there are counterexamples, but we cannot prove any of them are counterexamples.
it seems to me it must be one or the other, that is, "undecideable" is not satisfying. what is your take? is this some example of metaphysical truth?
False analogy. You stating God is real doesn't presuppose that there is an absolute truth-value of the statement in a formal language. It is useful to you because it prevents harm.
The addition of an axiom does presuppose there is an absolute truth-value of mathematical sentences that can't be proven in your current set-theoretic model. The usefulness is to gain access to those mathematical objects that cannot be constructed in that model without the added axiom. Thus, they preexist construction by humans and are real objects.
This is why constructivists do not accept the axiom of choice. It was added to ZF to allow for the existence of the reals, among other things, but the reals couldn't be constructed in ZF and they still can't be properly constructed in ZFC. Constructivists maintain humans create mathematics. By accepting choice, you accept that we don't construct mathematics in any way, but discover it.
Read the link I sent. 2.4.4 is enough for this and is only a few short paragraphs, but the entirety of 2.4 may help you understand why this explicitly refutes constructivism
What? No. Undecidable is a perfectly fine conclusion and gives us plenty of mathematical knowledge to springboard off of
If you'd read the link, you'd know exactly why this isn't true.
Umm, no, it doesn't at all. It makes a statement about the decidability of whether there exist mathematical methods to solve NP problems in polynomial time. And there are plenty of mathematicians who don't believe that or believe it's undecidable
Not necessarily
This is believed by many mathematicians
isnt this because you are shoving my claim into a constructivist paradigm? its not like i claimed to be a constructivist. its like i said i supported gay marriage and therefore assumed i must support welfare as well.
i think the constructivists are silly. im cool with the axiom of choice. if due to philosophical considerations the axiom of choice implies mathematics is discovered for some definition of "discovered", then thats fine too.
i think ur quibbling on the definition of usefulness. useful in the sense that you can use the new math to build something cool. useful in the sense that your new math lets you make better physical predictions. useful in the sense that your math gets you published and famous. useful in the sense that your new math makes you happy. useful in the human context.
beyond that if you want to say mathematics actually exists as a mathematical blob why cant i say the mathematical metaphysical blob just has two branches; one where you assume the axiom and one where you dont. im just telling the mathematicians to look at the more promising branch, for their human definitions of promising.
i think youve interpreted me as saying:
- statement A is "true but unprovable" !
- to prove statement A, we need axiom B.
- take axiom B, because it is useful to access the preexisting truth A.
usually i am skeptical that "true but unprovable" is meaningful. i gave P=NP as an example where true but unprovable makes sense to me, because the truth value has effects. can you help me understand?
like if P=NP were true but unprovable that means that for each NP-complete problem there exists an algorithm that solves it in polynomial time, but the algorithm is made of arcane magic and you can never prove it solves the problem correctly in polynomial time, though you could empirically test it in principle.
You do when you claim mathematics exists in the human brain and not as some independent metaphysical object
Alright, then mathematics doesn't exist in the human brain and has a different metaphysical status
It's well-defined by Gödel. Read the link. Usefulness is defined by giving access to provable mathematical objects that are currently unprovable. The degree of usefulness is how much you can then prove from the new axiom.
Holy shit read the link. It talks all about it: you can show that ZFC with an axiom that shows the CH is true and ZFC with an axiom that shows the CH is false are isomorphic. This has been considered.
And if you mean to assume ZFC and also ZF2 (as Zermelo called it), then you assume that two universes of sets exist, and necessarily exist independent of the human mind. See Zermelo, A new proof of the possibility of a well-ordering, 1908.
Not all do. Constructivists do not. Only mathematical realists do. In fact, in Russell, The regressive method of discovering the premises of mathematics, 1907 Russell says that the best framework for mathematics is borrowed from empirical science: mathematical axioms may not be intuitive as previous formulations of set theory supposed, just as the premises of physics aren't intuitively obvious.
Thus, we should choose axioms that are justified by ability to entail, explain, and systematize the intuitive mathematical propositions. This is very similar to the premises of constructing a scientific theory to explain an observation and then test the robustness. But instead of empirical methods on material objects, you use first and second order logic on mathematical structures that are necessarily independent of the human mind, because they are presupposed without construction.
Just FYI, constructivism has all but died out. Almost all work done in constructivist mathematics is done by realists to either try to construct objects from realist mathematics or prove they can't be constructed. Your opinion of "it'S ALl tHe brAiN" is dead in the water.
Wrong. I never asserted statement 1. I said that assuming a mathematical sentence should have a truth-value assumes an abstract, independent mathematical structure in which it has a truth-value.
I never made any argument about true but unprovable. If it's unprovable, it has no truth-value. But you saying it should be added because it's useful to have a truth-value is a decidedly realist stance and rejects constructivism.
READ THE FUCKING GÖDEL LINK IT'S ONLY A FEW FUCKING PARAGRAPHS
This is impossible. Unprovable implies it has no truth-value in your mathematical structure
No, you literally can't. If P=NP but you can't prove that it's true, no algorithm constructed in our set-theoretic universe can br proven to give the correct solution to an NP problem if it operates in polynomial time. It may work some of the time, but it may not work other times. It literally can never work for every formulation of the NP problem
i already said this. i claimed it was all in the brain so you labelled me a constructivist, then youre telling me consructivism is dead. it doesnt logically follow.
- You believe X.
- Constructivists believe X.
- Constructivism is dead in the water.
- Your opinion is dead in the water.
but also, i said that its probably fine to suppose mathematics "exists outside your brain" and you are discovering it or whatever. as long as it doesnt confuse you.
if you test it and it doesnt work sometimes, then it is not a valid counterexample. it would have to work every time, you just cannot prove it does just by running it over and over (though you maybe convinced with high probability.)
is the universe set theoretic?
i think there is a computer program running right now that halts if and only if some undecideable problem is true. intuitively it does seem whether it halts or doesnt halt has a truth value outside of the human mind. is this good enough? have i reached the insight you were trying to get me to make?
Constructivism being all but dead is independent of my claim that you're a constructivist. The former is an assertion about how many constructivists there are and how robust their arguments have been to critique. The latter is an assertion that your fundamental principle that "everything comes from the brain" is a constructivist principle.
No, that's not the argument. I'll help:
- You believe x.
- Believing x definitionally makes a constructivist
- You believe y
- Believing y necessarily makes you a mathematical realist
- Your philosophical system is contradictory and incoherent. Either mathematical objects are independent of humans or you don't get the axiom of choice or any other extrinsic extension of ZF.
Then there are metaphysical objects that are not contained in the brain and are not the product of the brain. So your assertions about everything else that supposedly is contained in the brain are not well-founded whatsoever.
You're the confused one. Read the fucking link.
That's the point I just made. You can't make an empirical test of the truth-value of P=NP in ZFC if it is independent of ZFC.
Nonsensical question.
Then it will never halt in finite time, by definition.
This doesn't make sense because it cannot halt, and you can't prove that given infinite time that it would halt. That's how unprovability works.
I don't know, do you now understand why your assertion of all human thoughts being contained in the human brain is not well-founded and contradicts your other assertions regarding math, science, and philosophy?