/sci/ - Science & Math

Doesn't causality suffer from the same issue? Either there's an infinite causal chain (the regressive argument, a cosmological contradiction) or an effect with no cause (the axiomatic argument)?

Why not say it like a human being
>even if you know something, you can't know if you know it
i'm fairly certain the epistemologist's answer is that not only are you not obligated to show that you know that you know, but the meaning of knowing whether you know something is non-trivial.

>>11498445
Because you're a retarded faggot who is the perfect example of a pseud. I don't know why you're even here. Perhaps /diy/ is more your speed, especially the thread about making a noose.

bump

>>11498416
Sure, it's great to keep in the back of your mind but at some point you have to ignore it since we've found no way of reconciling with it and may never.

>>11500163
If you don't think about it you never will anyway. Most just look at it and decide that they can't.

>>11500183
By this trilemma, any logical argument can be debunked by presuming we exist in a sub-universe and from the perspective of the super-universe all of our logic is faulty. But this is paradoxical because we can't use faulty logic to come to the conclusion that our logic is faulty.
So really, this trilemma is worth ignoring until Jesus comes down to say hi or we become OP's pic related.

>>11498416
That excludes input from the outside world. (Observation/experiment)

>>11498416
I don't get how "axioms are not defended". It seems clear that we can defend the axiom "every element of a group has an inverse" just by saying that this is part of what we mean by the notion of "a group". In general we can argue that axioms are there to explain exactly what we mean by a concept.

>>11498416
>The Münchhausen trilemma is that there are only three options when providing further proof in response to further questioning

Prove it.

>>11498416
In the third case
>The axiomatic argument
if the proposition B is proven from a set of axioms A, then we have proven the meta-proposition A -> B. There are some problems with the axiomatic method in mathematics. By Gödels second incompleteness theorem it is impossible to prove the consistency of a consistent formalized system which contains elementary arithmetic. Many things can be said about this, but it is not a good reason to give up on mathematics.

>>11498416
Mathematics is the list of strings you can produce with the following rules (and nothing else)

we can view it as the list of strings of the form # = § where #,§ are strings

-We start with constant symbols S K T ( ) =
lower case letters x y z will be referred as "variables" below; a term will be any string not featuring the character = nor T and formed by putting characters among the seven characters S K ( ) x y z next to each other.

-axioms: Tu; Tv; Tw; Tx; Ty; Tz; TS; TK
-implication rules:
Tx -> Ty -> Tx(y)
Tx -> Ty -> K(x)(y)=x
Tx -> Ty -> Tz -> S(x)(y)(z) = x(z)(y(z))
Tx -> Ty -> Tz -> x=y -> x(z) = y(z)
Tx -> Ty -> Tz -> x=y -> z(x) = z(y)
Tx -> x=x
Tx -> Ty -> x=y -> y=x
Tx -> Ty -> Tz -> x=y -> y=z -> x=z

A theorem is either an axiom, or a string % such that there is an implication rule as above ending by % and whose all leftmost strings theorems, or a string % such that there is another string § and a variable whose every occurence has been substituted by a term in order to build %.

Example: the following sequence establishes S(K)(K)(x)=x as a theorem
TS; TK; Tx; Ty; Tz; S(x)(y)(z)=x(z)(y(z)); S(K)(y)(z) = K(z)(y(z)); S(K)(K)(z) = K(z)(K(z)); S(K)(K)(x) = K(x)(K(x)); K(x)(y) = x; K(x)(K(x)) = x; TK(x); S(K)(K)(x) = x

>>11500274
intuitively, Tµ means "µ is a well written, term of the language"
SK-combinator calculus (the above) without any variables is known to be a Turing complete language, and so in spite of its drastic apparent limitations, the construction we made can encode all mathematics (every "normal" formula of sentence in maths is the abreviation of a complicated string like the above).

>>11500277
>>11500274
also, remove Tu;Tv;Tw from the axiom list ...

>>11500270
You are misstating Gödel. It is impossible to prove the consistency of a consistent formalized system which contains elementary arithmetic, with a proof within the same system. Also there is a requirement on the axioms being recursively enumerable for this to be true. In the theory which has the standard model of arithmetic as its only model, it can prove itself consistent.

>>11500318
>You forgot
>"within the same system"
Yes, thank you. That is crucial.

>>11498416
I was shocked but happy when I read up on the philosophy of science. It was a shock to the system as previously I had worshipped science as the path to ultimate truth, rather than a set of thinking tools. It was good to knock it off it's pedestal and helped me really open up to much more creative ways of thinking about things.

>>11500330
You are welcome! If I may inquire, I would like to understand the background for your other comment to the effect that "there are problems with the axiomatic method". Maybe that is related to my comments before, that if you have any specific model in mind, like the arithmetic of natural numbers, then you cannot capture it?

>>11500370
>Maybe that is related to my comments before, that if you have any specific model in mind, like the arithmetic of natural numbers
Yes, precisely. The "problem" I was thinking of was Gödel's second incompleteness theorem. The first one is also a good contender. As you say, these results does not extend to all axiomatic systems. E.g. propositional logic is complete.

>>11500416
How is that a problem though? The fact that Group Theory is incomplete seems more like a strength than a "problem". In the sense that the same theory works for a lot of different interesting cases/models called "groups".
Your notion of a "problem" seems iffy, unless you can make it into a question, or even a conjecture. To say that the second incompleteness theorem is "a problem" isn't technically correct because you do not explain what you would consider "a solution".

>>11498416
but this is wrong, at least in mathematics. A demonstration can not rely on circular reasoning or a regressive argument, or it’s not valid.
A demonstration is always axiomatic and lays on principles that aren’t supposed to be the absolute truth but rather define rules for a logical langage.

>>11498416
https://www.lesswrong.com/posts/C8nEXTcjZb9oauTCW/where-recursive-justification-hits-bottom
https://www.lesswrong.com/posts/zmSuDDFE4dicqd4Hg/you-only-need-faith-in-two-things

>You only need faith in two things: That "induction works" has a non-super-exponentially-tiny prior probability, and that some single large ordinal is well-ordered. Anything else worth believing in is a deductive consequence of one or both.

Is Le Metabolic Privilege Man right?

>>11500349
>much more creative ways of thinking about things
Do elaborate

>>11500459
>To say that the second incompleteness theorem is "a problem" isn't technically correct because you do not explain what you would consider "a solution".
The "solution" to the "problem" would be to be able to use a formalized system which contains elementary arithmetic to prove its own consistency. Maybe it is better to call this a "restriction" than a "problem". A restriction is only a problem if you want to go where you cannot. Gödel's incompleteness theorems are restrictions neither of the quality nor the quantity of mathematical truth, but rather the quantity that we can access through rigorous methods.

>The fact that Group Theory is incomplete seems more like a strength than a "problem". In the sense that the same theory works for a lot of different interesting cases/models called "groups".
Interesting even though I do not understand what you mean to be honest.

>>11500526
Recursion is just the belief in the right to make copies of strings and produce new strings using clearly defined rules. See Post's system or abstract rewriting.

>>11500655
He's using the term "recursion" differently in this case though. Recursive justification is something like this:
>I believe A is true.
>Why do you believe in A?
>I believe in A because B.
>Why do you believe in B?
>I believe in B because C.
And on and on and on.

>>11498416
So basically, what every three year old does.

>>11498416
>have a group of multiple people you trust and there are no cia niggers
>have them seperated and watching with telescopes or binoculars
>get another person to hold up x fingers
>ask everybody what they saw, without asking them what they were meant to see first
>find out they all give the same answer

While I understand how saying "1+1=2" isn't "true". How does saying "Assuming Peano postulates to be true, then 1+1=2 is true" isn't an absolute truth? Basically, if I have a proposition P which is a theorem in some theory T, how isn't the statement "If T is true, then P true" an absolute truth? I get it that it's just axiomatization with more steps, but while P is only true on T, the whole statement is just true. Someone please explain why I'm wrong. Also, please recommend books on logic and epistemology to a math fag.

>>11498416
Brainlet. No epistemologist takes the munchhausen trilemma seriously. Also, the axiomatic and circular paths arent even problematic neccessarily. Read kant.

>>11498416
Psyop detected

>>11500518
So then mathematics uses the third option, axiomatic.

Suppose we have demonstrated that it is impossible to prove any truth.

Then we have proven that it is true that 'it is impossible to prove any truth.'

This is a contradiction and thus the statement that it is impossible to prove any truth must be false.