Also, you are welcome to send me private messages, but I don't see any reason why my messages should be censored.

but thats what i come here for

I'm still trying to figure out wtf just happened here. lol

I think it's because I belittled his profession. If so, sorry about that (honestly).

I was doing a bit of reading to find out how accurate it was to say that the Greeks really were all about the study of abstract worlds blossoming from arbitrary axioms for their own sake, which cannot possibly be "wrong", as Reid more or less presumed, or if they actually were engaging (at least initially) in something closer to modelling physical, geometric shapes, as I had argued was likely the case.

The interesting thing is, while it's clear that the Greeks loved abstract perfection and deduction, it's not exactly clear why they became interested in geometry as a topic in the first place. Was it surveying? I would have assumed so. Certainly earlier civilizations cared a lot about trial and error experiment, such as the Egyptians, who (if memory serves) likely would have used geometric formula to help them build structures such as the pyramids.

But unfortunately, we may never know, simply because there aren't enough surviving texts to tell us.

stop fighting or I swear to god, I will crash this car right into the side of a school bus

Here's what I mean by this: sure, lot's of interesting mathematics can exist in each of various choices of incompatible axiom systems. Of course I wouldn't presume to pass judgement on one system being right or wrong, But at least initially, well, axioms may be introduced for ostensive purposes, but which ultimately fail to fulfil their promise.

The debate surrounding axiomatic set theory is actually pretty germane here: I seriously doubt that Cantor (well, actually, Hilbert) would have thought that his Continuum Hypothesis would be proven to be independent of axiomatic set theory, as it was by Paul Cohen. So in the end, an enormous amount of consternation surrounding questions of axiomatic set theory turned out to be completely meaningless. In the end, though, the technique invented to show this ("forcing") turned out to be far more interesting and powerful, leading to a whole host of new and unexpected research projects.

So sure, there might be lots of interesting mathematics that comes out of badly formulated, or, shall we say, "not even wrong"

[quote=Doron Zeilberger]
Likewise, Cohen's celebrated meta-theorem that the continuum hypothesis is "independent" of ZFC is a great proof that none of Cantor's א-s make any (ontological) sense.
[/quote]

questions, which mathematicians and logicians attempt to formulate axiomatically, and then get their asses kicked. But again, I am not passing judgement on this or that choice of axiom from a modern, retrospective view of mathematics. It would be stupid for me or anybody else to argue otherwise.

But in the historical sense, without the benefit of hindsight about which axioms really lead to interesting mathematics? Axioms can totally be "wrong" (read Jon`C's original post if you are confused about the sense in which he and I understand the word "wrong" here). There have been a bunch of projects in topology and geometry from the early 20th century, during a period of time where mathematicians were utterly obsessed with axioms, but in the end were shown to be a complete waste of time, because they unexpectedly and unfortunately turned out to be trivially equivalent to existing axioms.

Finally, let me leave you with the provocative opinion of the great electrician, Oliver Heaviside, whose operator calculus was only later rescued, by combinatorialists, long after it had been cast aside by snobbier mathematicians as "not rigorous" enough.

[quote=Oliver Heaviside]
In the preceding, I have purposely avoided giving any definition of 'equivalence.' Believing in example rather than precept, I have preferred to let the formulae, and the method of obtaining them, speak for themselves. Besides that, I could not give a satisfactory definition which I could feel sure would not require subsequent revision. Mathematics is an experimental science, and definitions do not come first, but later on. They make themselves, when the nature of the subject has developed itself. It would be absurd to lay down the law beforehand.
[/quote]

And yes, I understand that I have a serious deficit in so-called "soft", interpersonal skills.

http://www.personal.psu.edu/~j5j/IPIP/

Reid, if there is one thing I especially respect about you, it is that you always take the time to read through people's posts and respond methodically.

But since you were clearly at wits' end with me by the time I made the last lengthy post on this page, I just wanted to say that I would be completely OK if you didn't respond to it. I won't interpret it as conceding anything in the argument, and we can agree to move on. I mean this seriously, and I am not trying to imply anything snide.

I'm not sure I can say I enjoyed this conversation, but I mostly have myself to blame for that. I don't think I was as angry as you presumed me to be, though. At any rate, I certainly don't resent you, and I shouldn't take for granted the amount of politeness you normally display here. (But I suppose everyone has limits to the amount of rudeness they can reasonably tolerate.)

Yeah, so obviously axioms can be wrong: https://en.m.wikipedia.org/wiki/False_premise

This applies to Euclidean geometry no less than syllogisms. If the axioms of Euclidean geometry are false, one can still make deductions that logically follow from the premises. But even though the reasoning that leads to the conclusion may be sound, the conclusion will still be false because it is based on a false premise. So even though one can make demonstrations based on the axioms of Euclidean geometry, the axioms can still be false. It doesn't mean the reasoning that leads to the conclusion is incoherent or illogical. It just means the conclusion is also false.

Also, I don't know why it took me 10 minutes to realize Jon`C just called us children.

Ahem.

Not to start another flamewar, but I think there is a subtle distinction here that gets lost if we simply restrict ourselves to talking about "wrongness" as logical inconsistency. Although this is also a reason to say that an axiom system is 'wrong'.

But in a big way, axioms can be wrong if they don't capture what they are intended to capture, despite being consistent. Mathematicians in the 21st century such as Reid don't have much reason to care about this, though. In fact, playing games with axioms is not a good way to do mathematics, and most mathematicians learn to work within them without fuss.

[quote=John von Neumann]
Young man, in mathematics you don't understand things. You just get used to them.
[/quote]

Most mathematicians just assume ZFC and forget about it. For all intents and purposes, I don't think most mathematicians do any significant work by worrying about whether or not the logical axioms that underlie their work are 'good' choices. In fact a great deal of mathematics in axiomatics is simply to show that axioms are equivalent. Which is a total waste of time, actually.

Actually, most mathematicians don't even care about (axiomatic) set theory at this point, and use category theory instead.

Perhaps a better counterexample is Praxeology, a system of psychological axioms that are each of them trivially wrong.

Er. I think the distinction you're saying got lost is the exact distinction I made?

Ooh. Maybe. Let me read this a little better, lol.

Okay, yeah, that's right.

And of course there's the rub, from Reid's point of view: mathematics isn't supposed to need to be about anything that could possibly be a false premise, because mathematics is simply a game of formal symbols, abstract concepts, etc.

OTOH, physics certainly does care about making false premises! For example, that we live in Euclidean space.

I wonder why the express rejection of empirical data in favor of "a priori" knowledge wasn't a red flag for Austrian economists. Not sure if there's anything that cries out louder that your theory is just plain wrong.

In a way, the root of my disagreement with Reid probably stemmed from the question of which of mathematicians or physicists get to "own" Euclidean geometry.

My personal opinion is that the best mathematicians and the best physicists are those who take deep inspiration from both fields. In fact a big sticking point for me in my interactions with Reid has been a belief (which I am somewhat unable to justify) that mathematics really isn't much without physics there to inspire it, especially when considered historically.

I wish I understood it better, but a lot of the deepest mathematics AND physics really just comes down to optimization. The calculus of variations is a topic that is deeply intertwined with physics, both historically and today, but also comes up in pretty much anything involving differential geometry (don't quote me on this). But on the other hand (and I wish I understood this better), even category theory is about optimization in some ways.

I think about this stuff, and I’m like, what if our universe is really seminormed, but the average over macro scale converges to the Euclidean metric by hilarious coincidence? Just for example. I’m not a physicist but I bet whatever underlying structure of spacetime is super ****ed up.

because they dumb a.f.

The first principle, the axiom of action: humans exist and act.

Nobody tell the metaphysicians.

Well since we seem to be done attacking each other for now, I'm still waiting for Wookie06 to come back and explain why the teachers should all have guns.

to protect children obvi

* for certain axiomatic definitions of "protect"

Elementary school: to serve and protect

Oh, that makes a lot of sense then. I assumed you were from Ontario for some reason.



That's been my experience. In my own experience undergraduate education at elite universities really is no better than many non-elite universities. I've taken undergraduate courses in Canadian universities, and at Ivy League universities, and I can honestly say that the education that I received in Canada was much, much better than the education that I would have received at an Ivy League school. But in neither case did it really matter when it came time to go to grad school: at grad school, I effectively had to start my education over and relearn from the ground up everything I had learned as an undergraduate.

I'm sure I would be better networked if I had a bachelors degree from an Ivy League school instead of where I went, but that's about the only advantage I can think of. Maybe also I would have a bigger ego. It seems like five to ten minutes of every lecture at Ivy League schools are dedicated to telling you how great you are by virtue of your association with as august an institution as such-and-such University.

Well, don't forget that it might have been easier.

Or maybe not. Maybe they're all super stressed out about failing and wasting all that money.

Going back to Jordan B Peterson's propensity to opine on subjects outside his domain of expertise: I believe I have located the actual coordinates of Mount Stupid.

Incidentally, I feel that the phrases that SMBC picked as examples are just too easy, making for a lost opportunity to slay some sacred cows. I'd have preferred some advanced level examples that not everybody might obviously recognize or immediately accept as terrible. OTOH, it's a comic strip for a mass audience, so.

many of those examples are little stray pieces of information that someone acquires by chance rather than information that someone acquires on the way to expertise

What I actually said was "properly formatted, non-contradictory axioms" can't be wrong, if you actually thought about what I wrote instead of making **** up.

No, because you've had a habit of dropping random insults on this forum, including "Reidtard" which is offensive on many levels, not just to me.

The point at hand was whether Euclid's axioms are wrong. Yes, you can find axioms that lead to contradictions. Euclid's axioms don't lead to contradictions, unless you have some shocking new proof they're completely consistent.

The point about reality doesn't bear on the axioms because the axioms don't pertain to Euclidean geometry as a model of reality. This is plain, and for some reason made you go insane because for some reason you can't accept that a space can exist apart from physics.

Saying Euclidean space depends on a false premise is stupid. Only a stupid person would say that and mean it sincerely.

Eh. Reid, see what I wrote above about false premises. You're assuming that axioms are incorrect when reasoning from them leads to contradictions. But that's not correct. It's also possible to reason correctly from incorrect premises. When reasoning from false premises, even if one's inferences are valid, the conclusion is still false, because the reasoning is based on a false premise.

For example:



The reasoning here is correct: the inference that Socrates is immortal is a valid inference based on the (major) premise that all men are immortal. But the conclusion is still false, even though the inference is valid, because the premise that all men are immortal is false.

In other words, the fact that someone can draw logically valid conclusions from premises (or axioms) does not make the premises correct. And, so too, the fact that a person can come up with proofs based on the axioms of Euclidean geometry doesn't validate or prove the axioms of Euclidean geometry.

Good grief. I'm going to bed.

Okay. Yes. You can make premises that are true, and premises that are false. This is basic p-q predicate logic stuff. Totally agree.

Where I don't agree is that this actually described mathematical axioms. I've already made it clear that Euclid's fifth axiom is optional. You can reject it if you want to work in more general geometries, or you can accept it if you want to work in Euclidean geometry. That decision, whether to take the fifth axiom or its negation, neither leads to a true->false statement. They're both entirely logically consistent.

I guess what I should say is, you're right that the form of argument can be correct from a false premise, but describing Euclid's fifth axiom as "false" or "true" is a nonsense description. "All men are immortal" is false because that's an empirically verifiable fact. There's ways of going out and checking: yup, someone died. That's not how it works in mathematics. There's no truth in accepting or rejecting it.

Maybe the problem is, because you personally think mathematics shouldn't be about formalism, then you're not liking the fact that all of this stuff can happen formally, without any regard for physical reality. Are you just uncomfortable with formal ideas?

Like, if you see a person use a power series to solve a recurrence relation, do you just boil in anger because the series doesn't converge, it's a "false premise"?

I should just establish right now: this is irrelevant to the main topic at hand. Axioms can clearly be wrong, and the fact that one can make reasonable and logically coherent deductions from axioms does not validate the axioms.