/sci/ - Science & Math

Knowledge is generally synthetic rather than analytic so what's the difference?

>>7412869
there are only intuitions, even behind analytic ''knowledge''

The analytical judgement is what is obvious, obvious by its meaning, what is intuitive, what requires not a proof. The other kind of judgement is the synthetic one, where to accept a statement as truth, we must see at least one (valid) evidence, one valid proof, once a criterion of validity is chosen. Evidently, we fall back on the intuition, for each step of this proof must be intuitive or possesses a proof.

>>7412860
No.

Godel's incompleteness says we can cook up a function "Th" such that Th(n, A)="the nth true theorem of the mathematical system A" once A becomes sufficiently 'complex'. Basically it says there is no general cheating method in finding proofs, that's all.

>>7412913
>tfw everything really does fall back to intuitions
horrifying

>>7412960
>horrifying
yet

>>7412933
>Basically it says there is no general cheating method in finding proofs, that's all.
Can we cook up a function that at least sometimes gives us the correct proof?

>>7412933
the solution is to stop doing arithmetic in FOL,

boom

l,

>>7412933

You're understating the implications a bit. The trajectory of mathematics before Incompleteness was toward a envisioned Hilbertian sort of final axiomatization that would eventual allow all of mathematics to be "solved" on the same basis, and someone setting out to develop a new field would never have to stop to ask "what set of axioms shall we use?" because there would just be one, the only true basis for all mathematics.

The fact that incompleteness shat all over that dream means that everybody is now sort of working to their own rulebook. Without incompleteness there would be no Based Wildberger off in the woods with a knife, blazing his own little trail of mathematics without reals. There wouldn't be some theories that take the axiom of choice and some that don't. etc. Its really a radically different landscape than we would have if a complete, consistent axiomatization were to exist.

>>7413410
>and someone setting out to develop a new field would never have to stop to ask "what set of axioms shall we use?" because there would just be one, the only true basis for all mathematics.
this pleases our god


how shit you actually talk about him in your last paragraph .

>>7413410
>the only true basis for all mathematics.
Exists. see: wildhamburger

>>7412860
Do you know you can't know nuffin?

there is no Ignoramibus

>>7413440
THE ONE TRUE GOD
PATRON SAINT OF /SCI/

>>7414243
>imblying

In the First Order Logical Language of Arithmetic, there are four desirable properties we would like for our Theory of Arithmetic:

1) We would like it to be consistent.
2) We would like it be complete
3) We would like it to contain Peano Arithmetic
4) We would like it to recursively enumerable

What Godel's Incompleteness Theorem ultimately says is simply that there is no theory in this language that has all four of these properties. However, it is possible to show there are theories with any three.

Go to /lit/ this is a science board. If you wanna post about 'lmao nothings real' or 'why should I livr' then get the fuck out of this board with your illogical bullshit

>>7414251
>axiomatic axioms are axiomatic axioms

>>7414495
Go away, retard.

>>7414283
Why do we want 4? Just so that it's computable?