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/sci/ - Science & Math

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10653708 No.10653708 [Reply] [Original]

Does Godel's incompleteness theorem have, in practice, any serious implications?

It seems more likely (according to most mathematicians) that mathematics is incomplete, rather than inconsistent. Being unable prove this doesn't really bother anyone.

But more importantly, this doesn't really affect the rest of mathematics. For instance, a triangle's inner angles won't stop adding up to 180 degrees just because of Godel's findings.

Claiming that the incompleteness theorem is a BIG DEAL is like claiming that because we can utter paradoxes, then language is fucked.

>> No.10653712

The consequence is that statements that we strongly believe is true, such as the Riemann Hypothesis, might be impossible to prove.

That's a big deal.

>> No.10653853

>>10653712
Nah that's the least of the worries. There are very simple statements in formal logic which we know are true by intuition but cannot be proven true. It reminds me of the parallel postulate in geometry. We know it's true intuitively (in Euclidean geometry) but the "proof" is not as rigorous as the proof that a line can be made by joining two points (this is an axiom). It doesn't stop anyone from going about assuming that it is a fundamental fact of geometry.

Inconsistency is the real problem, but there's not even a hint that mathematics is inconsistent.

>> No.10654575

>>10653708
>It seems more likely (according to most mathematicians) that mathematics is incomplete, rather than inconsistent.
Gödel's incompleteness theorems don't imply mathematics itself is incomplete or inconsistent. They apply to sufficiently expressive axiomatic systems, not mathematics.
Gödel was an extreme Platonist and he would've been super-depressed if he found out how often people today misinterpret his findings like that. He was shutting down the formalist attempts to put mathematics into a box, not identifying a limitation of mathematics. His findings are meant to show mathematics is *more* than just an artificial formalist game you can write down the rules for and be done with.

>> No.10654588
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10654588

>>10654575

>> No.10654602

>>10654588
Platonists don't believe Forms exist in the same way material objects do. Anyone expecting to see the idea of a number (either infinite OR finite) as an object in the material world would not be a Platonist. They wouldn't belong to any sort of coherent philosophical camp at all I'm aware of because that's an inane expectation. Neither Platonists nor nominalists argue for that.

>> No.10654614

>>10654588
Get out, retarded anime poster.

>> No.10654887

>>10653708
>>10653853
>>10654575
>>10654602
pseuds
back to >>>/lit/

>> No.10655051

>>10654588

d-dont make the loli cry anon ;(

>>10654614

fuck you

>> No.10656484

>>10654575
>Gödel's incompleteness theorems don't imply mathematics itself is incomplete or inconsistent. They apply to sufficiently expressive axiomatic systems, not mathematics.
But.. .but... mathematics IS a sufficiently expressive axiomatic system.

>> No.10656486

>>10656484
>But.. .but... mathematics IS a sufficiently expressive axiomatic system.
No, it's more than that. Which is the entire point. Formalists wanted to write down everything that accounted for mathematics so they could finally put the issue to rest. What Gödel did was prove their efforts to do that would not work.

>> No.10656499

>>10656486
>What Gödel did was prove their efforts to do that would not work.
That does not mean the alternative will work any better, though.

>> No.10656502

>>10656499
>That does not mean the alternative will work any better, though.
I have no idea what you're talking about. Gödel didn't have an "alternative." He just wanted to shut down the formalists and shit like Principia Mathematica. Which he did.

>> No.10656506

What if we just come up with new axioms and prove the unprovable and then sort of fit it together with the old axioms? Then all the bases are covered, everything is provable one way or another, just not all at once.

>> No.10656510

>>10656502
>I have no idea what you're talking about. Gödel didn't have an "alternative."
And yet you say that mathematics is more than just a sufficiently expressive axiomatic system. So what way of doing mathematics do you have in mind, then, that does not have the limitations a sufficiently expressive axiomatic system would? In >>10654575 you (I assume it's you?) say that Gödel also thought that his limitations on formal systems are not a limitations on mathematical as a whole -- does that not at least suggest (if not quite imply) him having such an alternative vision in mind as well?

>> No.10656512

>>10656510
>Gödel also thought that his limitations on formal systems are not a limitations on mathematical as a whole -- does that not at least suggest (if not quite imply) him having such an alternative vision in mind as well?
No, it doesn't suggest or imply that at all. This is all that happened:
>Formalists: What if we could pin down the foundations of mathematics once and for all!
Gödel: You can't, here's proof.
None of that requires Gödel create his own failed attempt to put mathematics in a box. I don't know where you're getting this from.

>> No.10656527

>>10656512
>I don't know where you're getting this from.
I'm getting it from:
>His findings are meant to show mathematics is *more* than just an artificial formalist game you can write down the rules for and be done with.
Maybe I'm overinterpreting here, but I read "more" as implying that Mathematics As A Whole does not have the limitations that formal systems do. Which is something that the incompleteness theorems certainly do not rule out, but they do not rule it in either.

>Formalists: What if we could pin down the foundations of mathematics once and for all!
>Gödel: You can't, here's proof.
One man's modens ponens is another's modens tollens. You could read the proof as saying that formal systems are an imperfect approximation of Platonic Mathematics, but an equally valid conclusion is that Platonic Mathematics is itself limited, and formal systems CAN capture it accurately.

>> No.10656530

>>10656527
>One man's modens ponens is another's modens tollens.
modus {ponens,tollens}, derp. I should not be posting mathematics before my first coffee.

>> No.10656536

>>10656527
>"more"
If you make system A, there's a Gödel sentence that basically amounts to "this sentence can't be proven in system A." Mathematics isn't one system. "More" doesn't need to mean anything mystical or impressive. It's "more" in the sense that there exists mathematics beyond what was encapsulated by Principia Mathematica, if for no other reason than there's a Gödel sentence that's both true and can't be proven by Principia Mathematica.