Comment on Theories on Theories
sparkyshocks@lemmy.zip 5 weeks agoExactly.
HERE’S A THEOREM: IF IT’S PROVEN, IT’S TRUE EVERYWHERE, FOREVER
But at the same time, even if it’s true everywhere forever, it might still not be provable, because Gödel.
even if it’s true everywhere forever, it might still not be provable, because Gödel.
No. Gödel’s completeness theorem says that if something is true in every model of a (first-order) theory, it must be provable. Gödel’s incompleteness theorem says that there exists statements that are true sometimes, and these can’t be provable.
The key word is “everywhere”.
I think saying that a theorem is true presumes the axioms upon which it is based and so the entire system is “true everywhere forever”.
I often find it helpful to think of chess as my axiomatic system. When we say the king is in checkmate, it presumes that we accept all the underlying rules of chess. And these pieces that theoretically form a checkmate will always do so forever… Assuming the regular rules of chess, etc.
When you put things in terms of chess, these “deep” statements about “math” often become banal.
But that’s math.
I see philosophy as a place to make nonrigorous arguments.
Wait do you think Bertrand Russell and Alan Turing and Kurt Gödel weren’t making philosophical arguments?
They are clearly mathematical. Starting with definitions and axioms and deriving from there using mathematical statements.
They are clearly mathematical.
Sure. But they’re also philosophical. The categories aren’t mutually exclusive. Basic set theory (which is both mathematics and philosophy).
They all debated the question what being mathematical means there whole lives.
I see philosophy as a place to make nonrigorous arguments.
It’s the other way around: math is where your just ignore questions about what makes sense, what knowledge is, what truth is, what a proof is, how scientific consensus is reached, what the scientific method should be, and so on. Instead, you just handwave and assume it will all work out somehow.
Philosophy of mathematics is were three questions are treated rigorously.
Of course, serious mathematicians are often philosophers at the same time.
You’re just covering my third paragraph. Yes, everybody is a philosopher because we don’t have the tools to do away with philosophical arguments entirely yet.
I explicitly refer to your second paragraph.
I covered those other arguments in another top level comment in this same thread. Yes, you absolutely can argue computer verified proofs. They are very likely to be true (same as truth in biology or sociology: a social construct), but to be certain, you would need to solve the halting problem to proof the program and it’s compiler, which is impossible. Proofing incompleteness with computers isn’t relevant, because it wasn’t in question and it doesn’t do away with it’s epistemological implications.
yetAnotherUser@discuss.tchncs.de 5 weeks ago
Worse: If the chosen axioms are contradictory, then the theorem is effectively worthless.
And it is impossible to know whether axioms are consistent. You can only prove that they are not.
ytg@sopuli.xyz 5 weeks ago
You can go deeper. To prove anything, including the consistency or inconsistency of a theory, you need to work within a different system of axioms, and assume that it is consistent, etc.