But while "people can't decide if 2+2 is 4 or 4" seems like something totally ridiculous, I'm sure you're aware that if we were to treat it as a cyclic group of 4, with the order of the set of positive integers, and we set 1,2,3, as equivalent to our base-10 1,2,3, but "5" to represent in our new group (which would not normally be a member of Mod(4), btw) what we represent as "0," then 2+2 can easily equal 5. That is something that completely conforms to group theory, and I'm sure one can just as easily prove it in set theory or ring theory. And yet it would've been entirely impossible to write intelligibly without the context it is sitting in. We must let go of this drive for some higher laws or some political necessity of logical structures: as I said, this amounts to Nihlism! Let us create ugly proofs, messy theorems, things that push the boundaries of intelligibility! Your aesthetic beauty and political necessity will turn into my excremental shit!
Complete nonsense. You do not need AC for CFG. The fact thar CFG can be programmed proves that, as literally nobody disputes finite AC.
>and many, many people in math and theoretical physics simply like the ZFC because of its so-called "aesthetic beauty," and not because we might produce better or more interesting results by holding onto notions of context-free grammars.
CFG exist with and without ZFC. AC is equivalent to Zorn which implies every VS has a basis. AC is enormously important to many Analysis and functional analysis results.
You got it the wrong way around people rail against AC because it is equivalent to well ordering and well ordering is obviously false (my oppinion).
>Every time you start doing a ZF-C, you get some very cool formulas and proofs even if they are "messy."
What? Every proof in ZF-C is trivially a result in ZFC. What are you on about? Do you not understand that adding an independent axiom (true here) just adds more provable statements?
>The entire point of what you are arguing against
I am specifically arguing against people like you who are hallucinating mathematical results and talking about them as if they had a shred of knowledge about them. This is embarrasing.
>then 2+2 can easily equal 5
This is literally just redefining what "=" means. It is a linguistic amiguity in mathematics that operators are "overloaded". Garbage tier argument.
>That is something that completely conforms to group theory, and I'm sure one can just as easily prove it in set theory or ring theory.
You are delusional. Do you have any idea what group theory, set theory or "ring theory" are? SPOILER: all statements in group theory are statements in set theory. Do you even know what a ring is?
>We must let go of this drive for some higher laws or some political necessity of logical structures: as I said, this amounts to Nihlism!
First and foremost you need to stop talking about mathematics. This is why postmodernists are dangerous, you make grand proclamations about things you have no idea about.
If I really had "no idea" about this subject we would not even be having this discussion and you wouldn't have put so much effort into replying. I admit I am not an expert in pure math, but clearly I have learned enough to talk about it with you and to force you to engage with me.
In any case, you seem more interested here in "taking down" what you perceive as postmodernism than critically engaging with the idea that CFG is something that relies on axioms which create rigid structures of logic that are unable to be questioned as they make it difficult to explore areas of math that do not conform to them. You say that ZF-C is trivial but there are significant portions of mathematics research dedicated to finding proofs without the AC and seeing if the AC is even necessary in constructing any axiomatic system, which would have wide reaching implications. I am not confident that the CFG is something that can exist without the AC (but I cannot here prove it), I don't think the fact that someone built a computer which follows very specific rules of interpretation can interpret context-free grammar proves much of anything besides that someone in a certain social context constructed something that relies on that social context.
>>then 2+2 can easily equal 5
>This is literally just redefining what "=" means. It is a linguistic amiguity in mathematics that operators are "overloaded". Garbage tier argument.
It is not a "linguistic ambiguity," you never specified the context for your claim. The arrogance on display when we are discussing the fundamentals of mathematical reasoning that something as simple as 2+2=4 cannot be questioned or made non-trivial shows that you are committed to a worldview where its more important that things "make sense," than that we critically engaging with those things.
Look I am just learning about some areas of pure math recently (like Group Theory referenced above), but I know my set theory fairly well and there was a reason I was using group theory and not set-theory, since it seemed easier to express the notion of a possible algebra where a set containing integers could add to something other than another positive integer that was ordered such that one element (4) would always come as the result of the addition of 2+2. I don't know why I'd express it in a set-theoretical definition when that would clearly be more obtuse than describing it as a cyclic group. The number system is completely arbitrary and you treat it as some sort of holy writ! Don't say "2+2 always equals 4 except in unique circumstances which we all know about but don't question common sense," the core of pure math is questioning commonsensical ideas of what should be provable or possible.
I am ridicoluing you. Everything you said was nonsensical or an inversion of the truth. No second year your would write blatant nonsense like that. Your post is again full of confidently asserted statements that are pure nonsense.
>with the idea that CFG is something that relies on axioms which create rigid structures of logic that are unable to be questioned as they make it difficult to explore areas of math that do not conform to them.
Idiotic thing to write. Who cares about CFG? It is a part of language theory. Its not even first order logic. The rest of the statement is just bizarre.
>You say that ZF-C is trivial
You are halucinating. I said every proof in ZF-C is a proof in ZFC. But since you know nothing about mathematics you do not know what that means.
>but there are significant portions of mathematics research dedicated to finding proofs without the AC and seeing if the AC is even necessary in constructing any axiomatic system
LOL, the question has always been answered. All countable mathematics is fine without AC. You don't even know what AC is about. AC is not neccesarry since ALL finite mathematics do not need it.
> I am not confident that the CFG is something that can exist without the AC (but I cannot here prove it),
Because you are an idiot. AC is only relevant in uncountable math, it never applies to CFG. Finite AC is like saying "the sky is blue".
> I don't think the fact that someone built a computer which follows very specific rules of interpretation can interpret context-free grammar proves much of anything besides that someone in a certain social context constructed something that relies on that social context.
You think that because you know very little about mathematics. And because you have no idea what AC or a CFG even is. To everyone with a basic math education this claim is obvious though.
> I was using group theory and not set-theory
In ZFC the only mathematical object is a set + a few primitives. All statements in group theory are set theoretical statements. 2 is a set + is a set. You don't know this because you are uneducated about mathematics, but every first semester learns this.
I could go on further. My critique of you is: You know nothing about the subject, yet you are talking extremely confidently. I really do not care what you have to say, as everything you just said shows a profound lack of knowledge and humility in regards to mathematics.
I was very surprised you didn't bring up Gödel, he is the usual goto for pseudointellectuals like you. Please stop talking about mathematics.
I really don't think that the kind of language your using is acceptable on HN and I'd prefer if you were trying to engage or teach me something instead of ridiculing me, as you said.
> I'd prefer if you were trying to engage or teach me something instead of ridiculing me, as you said.
You haven't shown the tiniest bit of humility in regards to mathematics. And I see absolutely zero reason to try to teach someone without a shred of humility.
The true danger of postmodernism is people like you. You believe because the "grand narratives" have broken down you can go arround and confidently talk about any topic you want, but you don't even know what the words mean.
How are you defending post modernism and at the same telling me what is and isn't the argument.
>To try and enforce normative constraints on what is deemed logical also in the end constrains the limits of discourse and intelligibility:
In that world all thought is incoherent. Pure anarchy reigns if people can not decide whether 2+2 is 4 or 5.
It isn't that I don't agree with the breakdown in centralized universal narratives, which lies at the core of postmodernism. But the only discipline which has succesfully grappled with it is mathematics. In all other disciplines it has largely reduced conversations to people talking about nothing, not even caring that they are only talking about mutually incoherent topics.
>it claims THAT society is always somehow "logical," "rationally organized," and "the best system that exists."
I agree, but if your conclusion from seeing the scientific socialism (inherited from the French revolution) is that coherence leads to human eradication then pure anarchism is the only consequence. The correct lesson would be to accept that narratives of "scientific government" inevitably exist to obscure power structures and create willing democratic populations who abdicate all individual thought to "experts".