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TimeSkip
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So, I don't think anyone has addressed the question posed in the title; but, is complexity in mathematics in your opinion determinate?
Thanks.
Thanks.
TimeSkip said:So, I don't think anyone has addressed the question posed in the title; but, is complexity in mathematics in your opinion determinate?
Thanks.
Namely a precise amount of steps to call a proof either least (qualitatively less 'complex') or where a necessarily least amount of steps has been made in the proof of the theorem.stevendaryl said:I think you haven't made it clear, to me, anyway, what you mean by "complexity" and "determinate".
TimeSkip said:Namely a precise amount of steps to call a proof either least (qualitatively less 'complex') or where a necessarily least amount of steps has been made in the proof of the theorem.
Can you conceptualize 'complexity' if this is hard to still understand or at least quantify it in regards to proofs?
Well, yes. All I'm saying is there even any syntactical sense in talking about a "growing alphabet" even though we don't really know how complexity should be defined in we cannot begin to ascertain in any quantitative aspect the notion of a proofs complexity, instead resorting to something of the sort in talking about a proofs 'length', whatever that may or may not mean?Stephen Tashi said:The development of mathematics often involves inventing definitions and proving theorems using the new terminology. Using new terminology allows statements to be written in a shorter form. Is this phenomena represented when we compute "Turing Complexity"?
Yeah, syntax would be the question in point here. I know that Gödel pretty much disproved that there's any universal syntax to be applied in terms of formalism. Yet, still it seems important to say that the size of the alphabet matters, in terms of the provability or even decidability of algorithmic equations like God's algorithm.Jarvis323 said:I think that the answer is yes, granted that you have ##T##, the set of provable theorems. Because you could compute this function by simply testing all possible proofs one by one from smallest to biggest, and checking if they prove the theorem or not, and then stop when they do. There is one catch, which is that we don't know what the domain is, because we don't know if some theorems are provable or not. But say you had a proof of a theorem and you wanted to know if it were the shortest possible one, just test every string in the language that is smaller and see if any of them are shorter proofs of the same theorem.
The other question you seem to be asking is whether the size of the alphabet matters. Say you had the provable theorems ordered according to the size of their minimal proofs in a given language. Would the order change if the language changes? It could change by having different syntax, and a larger or smaller alphabet. Yes, I think that a theorem may have a smaller or larger minimal proof relative to another theorem in the different language.
I think fresh_42, already answered it here:nuuskur said:My brain is in knots, I can't follow anything at all, it should go like this
[tex]
A_1\rightarrow A_2\rightarrow \ldots\rightarrow A_{n-1} \rightarrow A_n,
[/tex]
but instead it's going more like this..
[tex]
A_1 \rightarrow A_2 \overset{?!}\rightarrow A_3?, B_1? \rightarrow \ldots \rightarrow A_n (?) \rightarrow \ldots
[/tex]
The more philosophy flavored forums might entertain the notion of arguing based on foggy intuition spawned semi-definitions, but you are constantly assuming we all know what (the hell) you mean by every single piece of terminology you put out here. This is not mathematics, not even on a meta level.
Worse still, you have the auauaudacity to point out that nobody has actually answered your question. Have you considered that your question is incomprehensible?