/sci/ - Science & Math

>>11383688
they can just work on what witten and other string theorists say

Who?

>>11383688

Absolutely no one should have ever been surprised that mathematical truth cannot be equated with theoremhood in some finite axiomatic system.
An infinitude of mathematical truths are uninteresting trivia, with no obvious route to being proved. I think even Gauss knew this.
E.g check if the digits of square root of 47 is contained in the decimal expansion of pi^101. Well, we can just calculate out the sequences of digits and check. If it's true, you can find the digits that match, but no amount of just grinding out the digits and checking will ever prove it: there are always more digits to check. If it's not true, same problem, you cant disprove it. That is an unprovable mathematical fact. It is also a very, very, very uninteresting one.

All Gödel did was find a clever way to construct a provably unprovable mathematical fact, given any consistent and finite set of axioms to work with. The work is clever piece of technicality but in no way profound. It should have come as no surprise at all. And surely it never really impacts most of mathematics, except for some narrow branch.

At Hilbert's time it would have made more sense to have this sentiment that mathematics took some dark turn due to the incompleteness theorem, because it killed Hilbert's meme program. But it has been a century since then, it should've been clear that Hilbert's vision was too idealistic, and of course he was overconfident. Godel's theorem marks the boundary of Enlightenment, not its refutation (boundary that first rate mathematicians should've seen coming long before it was reached). Again I think philosophically, this stuff was known since Gauss who spent a lot of time staring at natural numbers, if we drop Godel's paper on Gauss's office, he would skim over it and wouldn't have cared much for it and continue sipping his tea and read his geodesic survey and cartography, which would actually impacts math and physics century later (differential geometry, General Relativity, etc)

I fucking hate when people first hear about Godel's theorems and completely misinterpret them.

Also, if you haven't already, it's definitely worth taking the time to learn the formal theory surrounding the incompleteness theorems. Even if you aren't super into/"good at" math, you should still be able to learn the subject, so I would highly recommend doing so to anyone who hasn't yet.

>>11383924
I'm relatively innumerate, but based on the articles and videos I've seen, the premise is that no amount of axioms could create a perfectly-closed system.
I'm curious how it gets misinterpreted, I've read that Godel was a strongly-convinced theist, so I imagine this had implications for that

>>11383776
>All Gödel did was find a clever way to construct a provably unprovable mathematical fact, given any consistent and finite set of axioms to work with.
How do you do that?

>>11383776
cringe

>>11383688

There's lots of other fun stuff to occupy your time with, and lots of other things which can be resolved. Read the final chapter of Gödel's Proof for perspective on this, sometime. Oh who am I kidding. You're 20 and you don't know anything about anything.

>>11383776

You're namefagging as "Aeris Turtle" about Gödel. Please go away.

>>11383924

>I hate it when people opine stupidly about a subtle theory so I want more people who are incable of properly understanding or representing the theory to sorta-read it so that the conversation will be improved---

>>11384132
>>11384125
this is one of the faggiest, most unironic high-on-your-own-farts pretentiousness I've seen
who the fuck says "opine" that isn't trying hard to come off as intelligent

>>11384040
Yeah it's pretty cringey when people who struggle with exercises from Velleman's How To Prove It rave about Godel's theorems.
Those who believe mathematics can be reduced to formal logic in this day and age probably have autism, and I would say the incompleteness results are useful for keeping these special folks from wasting their time, much like the energy conservation law keeps special folks from spending their entire life trying to construct perpetual motion machines just because there are some combinations of cogs and wheels they haven't tried.
But then again, most mathematicians don't have this sort of autism, and clearly the average student of physics can cope with conservation of energy.

>>11384009
You can assign numerical codes to formal arithmetic expressions then locate the 'this sentence is false' expression using the diagonal argument.