/sci/ - Science and Mathematics

>>3625

>Doesn't its state something like: within any formal system of logic there will nessarrily be inherent and unavoidable contradictions.

Yes, in a way, it says that formal systems of logic cannot be both consistent and complete. There will either be unavoidable contradictions, or unprovable statements. It doesn't break maths ber se, but it means you can't expect to just derive everything given enough time.

>>3628

Reminds me of a similar thing from computer science (which might actually be related but I wouldn't know if it is) that says that almost all decision problems are uncomputable. Because the set of all possible programs maps to Natural numbers whereas the set of all possible inputs maps to the real numbers. And the Reals is a bigger infinity than the natural numbers so there will never be enough programs to go around.

>>3637

I don't know if this is what you're referring to, but Turing's work was actually very closely related to Godel. The idea of computability is something like the incompleteness theorem applied to sets of instructions. That there are some problems that might have some answer, but fundamentally cannot be computed.

Then again, I didn't even understand the Wikipedia version of the explanation, so I could have misunderstood something.

>>3621

To be completely fair, we don't yet know everything that is unsolvable. Just because one thing is completely unsolvable doesn't mean that we won't be able to prove a lot of other theories.

>>3668

That's true.

But this really made me wonder if, for example, turbulent flow has a general solution at all.

>>3671

Every measurable phenomenon can be modeled to an arbitrarily high degree of accuracy by some computable mathematical model, even systems which have sensitive dependence. This is trivial because you can simply "hard-code" highly detailed observations into the model in the form of constants calculated from experiment. For chaotic systems this still works, you just have to do far more experiments to get better constants.

This is all cheating of course. What you really want to us find a mathematical structure which us simple to describe and compute with and describes the system in question to high accuracy without any of these "weasel constants". This is the entire challenge in doing science.

What this article is probably saying, although I didn't read it, is that some beautiful mathematical model some physicists were considering has something of physical import being undecidable or uncomputable (as in logically possible to have a finite-time algorithm for computing) in their favored axiomatic system. This has no real significance for the physical world, it's just an artifact of trying to efficiently model it.

>>3677

Fucking autocorrect.

You're not retarded, you should be able to get the intended meaning.

Just trying to remind you guys that fucking Wikipedia is not accurate. Many ideas are misrepresented there. Matter from different sources is pasted there about a single topic, but there is no organisation. Fucking read the published papers or read recommended books.

>>3683

Wikipedia is often quite accurate when it comes to math and science. If you are having trouble making use of wikipedia then you need to work on your research skills.

The 'philosophical implications' of the incompleteness theorem (or other scientific principles like the uncertainty principle) really aren't that profound, or even surprising to people who actually study math or science.

I agree with you though, science has developed an almost cult-like following of useless faggots who do little but spew half-baked drivel about things they don't understand. It's almost sad; becoming a scientist or even a scientifically literate individual in the current year necessarily entails a whole lot of cringing at the rest of your species.

I didn't read that fucking clickbait but I still have a strong scientific optimism. Natures absurdity hasn't stopped us yet.

>>3677

>For chaotic systems this still works

No it doesn't, because every single thermal instability and quantum uncertainty will lead to divergent error.

>>3697

Yes it does. All that that uncertainty means is that if you try to "brute force" a model you will have to make a log of observations to make the model accurate for even a short time. This is in contrast to something like Newtonian mechanics, where you might model the motion of a car in an idealized setting and your model might stay fairly accurate to a real car's motion for hundreds of hours before becoming inaccurate.

For an example of how chaotic systems are modeled in this fashion, see large scale weather system modeling using supercomputers.

>>3698

Should read "a lot of observations".

>>3698

And I hope you're aware that even observations are subject to those uncertainties. Usually in theory a measurement is assumed to be immune to quantum effects, but in reality it is not so.