I liked the history lesson, but I think the analogy to modern anxieties about AI was a bit stretched. It's the first I've heard that "analysis" was meant to be akin to an intelligent mind's methods. This read more like a modern re-imagining of a time when mathematics was in upheaval with new methods.

I do not think it is much of a reach, Leibniz by that point had already conxeptualized the computer and believed that all questions could be answered by it.

funny how now the opposite of "synthetic" is no longer "analytic" but "organic"

or actually "natural" (vs "synthetic")

Very interesting article. It somewhat reminds of current differences in academic philosophy where some distinguish between "analytical" and "continental" schools.

I think this is relevant. I just read a section of Alain Sokal's Fashionable Nonsense, and it was extraordinarily reactionary in it's contents. It was clear to me that Sokal had made no attempt to understand the underlying systems of thought that the so-called "postmodernists" employed in their writings, or immerse himself in the intellectual history that would give him the context to understand it. He just read the mathematical formulae or scientific claims, decided if they were "true" or not, and did not engage in serious attempts at interpretation or bring up secondary literature or any critiques of the work (because I guarantee that every single "postmodernist" listed in that book has received dozens and dozens, if not hundreds of critical articles on their work, by people who very well understand their systems of thought.) In Sokal's world, things are true or they're not true, they're rigorous or they're unrigorous. Writings that might question normative political institutions, ways of thinking about the world and ways of organizing social reality (all of "postmodern" writing), are simply cast as unintelligible and any need for serious intellectual engagement is wiped away.

This x1000. I am truly amazed, also, that so many people that are quite intelligent could fall in for such a paper-thin, sloppy "they don't believe what we believe! let's completely talk past each other" bad faith argument and think they somehow "owned" the targets.

Hyperspecialization has done massive damage to the overall intellectual health of society.

What Sokal did is roughly equivalent to a humanities scholar creating a spoof hit piece to show how some mathematician did not employ a literary metaphor correctly in an article about geometry.

The only reason it was even remotely successful is because the western (particularly American) perspective is so heavily biased toward STEM at the expense of basically producing individuals who can adequately solve practical problems but who (on average, there are exceptions) have the humanities knowledge of a stick and revere their favorite Netflix show or animated superhero blockbuster as though it were high art on par with anything hanging in the Guggenheim.

Well, I like the Guggenheim but I still feel there is a place for critiques of popular media and literary merit/appreciation of certain Netflix shows (though I am pretty tired of capeshit, aside from The Boys and shows of that ilk). But I by and large agree, even if I'm biased because I chose to study literature and learn all about "postmodernism."

But look there is something about logical positivist styles of thinking that was very attractive to many of the people Sokal attacks (which is why he is even able to attack them in the first place!), and also to me. I can't tell if that's because I have some deeply repressed control and power fantasies (probably), or because there is something to the analysis of the structure and order of logic and language in a rigorous fashion that reveals something about the order of social and psychological life, even if unintentionally. If these more radical authors used the mathematic and logical formulae in this manner, I don't see how a criticism of "it's poorly defined" works, because anything would be poorly defined in an algebra of society, psychology, history--if you only defined it in terms of contemporary mathematics and logic which seeks to divorce itself from those things.

And modern literary education isn't much better. My inner-authoritarian wants every humanities major to get a rigorous classical education and required courses in logic. Trying to break history is very hard when you have no consciousness of it whatsoever, and the humanities disciplines today are almost designed to keep students away from contributing their work to a "rigorous" STEM field, as they are deemed too reactionary to be worth critiquing or engaging with. It is for the very reason I listed above that its imperative humanities students receive the same basis of education as the STEM majors, and vica versa!

>things are true or they're not true, they're rigorous or they're unrigorous

Mathematics (more specifically, meta mathematics) has the ability to talk about these issues. "What is truth" quickly becomes an asinine question, if you don't at least accept some meta framework in which the question becomes actually meaningful and so far the only field which I which actually has the ability to coherently question itself is mathematics.

The problem with post modernism I have aren't the ideas specifically, but the fact that if you see the world as filled with mutually incohrent narratives talking about anything becomes very hard. But post modernist throry gives insane people the thought that they can write anything and it becomes justified, because narrative coherence can not exist anyways. Nothing could be further from the "truth".

>But post modernist throry gives insane people the thought that they can write anything and it becomes justified, because narrative coherence can not exist anyways.

That's not really the argument (and there is no coherent "unity" of postmodernism anyway, Foucault, Derrida, and Lacan, even if they all occupied the same intellectual spheres, had different projects). If there is an argument, its that by dint of its "place in discourse," any statement adds and also modifies every other statement in the great network of social life, so the moment you begin to talk about anything or interpret anything you begin to move away from the thing. But the idea of an original, pure object of critique is seen as suspect to begin with as well, since such a notion is always composed with further discourse, which, in a sense, replaces the object of discourse and negates it. The fact that there are these "definite negations" in language, every time we speak, is not taken as some great horror but instead is meant to be affirmed as an act of human creativity: there is no bottom to our logic, but we nevertheless always discover more and add to it! To try and enforce normative constraints on what is deemed logical also in the end constrains the limits of discourse and intelligibility: it limits the very possibilities of how we could order our society, and claims that the one we live in, which is currently grinding young men into meat in eastern Europe for the profits of large arms manufacturers, which arbitrarily targets and imprisons the most impoverished and downtrodden of its citizens and forces them into slave labor, which participates in the regular everyday exploitation of labor and suppresses and controls any resistance to its hegemony with constant propaganda and moralism for the weak, and direct violence towards the strong and well-organized--it claims THAT society is always somehow "logical," "rationally organized," and "the best system that exists." If we accept that this totalitarian nightmare society as the best that we can have, then we are effectively nihilists and we have abandoned any idea that we have the power to make a better, stronger, more creative, more free, more fulfilling life for ourselves.

>That's not really the argument

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".

I know Mathematics is respected for introducing notions of context-free grammar for recursive definitions, so everything could always be defined rigorously, but even that requires the Axiom of Choice for it to make sense in set-theoretical terms, 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. Every time you start doing a ZF-C, you get some very cool formulas and proofs even if they are "messy." Quantum Field Theory is messy, still gives us workable results (I'm not an expert on physics or anything btw, so don't dig into that statement to much). The entire point of what you are arguing against (which I will not even claim is some unified postmodernism), is that even inventing a core of logical principles stipulates a certain logic of power that both enforces and obscures itself at the same time as natural and/or necessary. It can't be natural...most of our highly-educated know that already, so instead its a political necessity: like you said, if we abandon these standards "Pure anarchy reigns."

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!

>I know Mathematics is respected for introducing notions of context-free grammar for recursive definitions, so everything could always be defined rigorously, but even that requires the Axiom of Choice for it to make sense in set-theoretical terms

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.

>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.

>f 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.

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'm very shocked your post hasn't been reported, you've directly gone against several of the guidelines listed here: https://news.ycombinator.com/newsguidelines.html

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.

You are a pseudointellectual.

> 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.

> Mathematics (more specifically, meta mathematics) has the ability to talk about these issues.

Yup. See e.g. https://en.wikipedia.org/wiki/Synthetic_differential_geometr... for another way in which the opposition between "analytic" and "synthetic" methods OP mentions can play out.