Of course they can.

DId you know that Greek geometry started as an emperical science? It's literally a theory of measurement, and, well, it actually presumes Newtonian physics, because the curvature of space-time is too small to notice.

They can be improperly formulated, i.e. conflict with previous axioms, but no, a properly stated set of axioms like Euclid's axioms can't be wrong.

That doesn't mean you have to accept Euclid's fifth axiom in all settings, it means when you accept it, in that context you can prove all of the statements that follow.



So what you mean to say is the Greeks misunderstood the relationship between geometry and reality, as in, they took geometry to be a physical science? I'm not sure about that, but if so, then yes, Euclidean geometry doesn't describe reality. For all we know, Minkowski spaces don't either. Mathematics is a structure we impose when approaching the world, and the validity of the model depends on how tightly it agrees with reality. But the mathematics can't be wrong, properly understood, that's absurd.

What's absurd is that you so badly misinterpreted the meaning of Jon`C's post.

All models are wrong, some are more useful than others. Are you that far gone a Platonist to think that math is somehow special and different than other models? Just because two sentient beings can arrive at the same mathematics doesn't mean it exists outside their heads except as the part of some philosophical theory. Which, you guessed it, is also in your head.

Your brain is just a prediction machine, get used to it. Mathematical frameworks can absolutely be wrong if they don't do what they say they do. And Euclidean geometry, as understood the way the Greeks did, is broken for precise enough measurements.

Look, nothing is "wrong" if you're allowed to patch up its interpretation up with further explanation every time a contradiction arises.

"Oh, the bible didn't literally mean that Jesus walked on water. No, that was just an allegory. Don't mind the fact that people probably used to take it as literal."

Mathematics isn't a model. Mathematics is the study of how to reach new, interesting conclusion after accepting some basic axioms, and the study of abstract things.

In my view, the origin of mathematical objects is in the a priori forms our consciousness has before understanding the world, and studying mathematics is uncovering the mechanistic parts of consciousness. In other words, it deals primarily with the form of how we are able to even model reality, not with content of the model itself. Physicists care whether Minkowski space or Euclidean space are good models, the mathematician gives no ****s.

You're misunderstanding my post. The mathematical validity of Euclidean geometry, and the validity of the axioms that give rise to it, are wholly independent of any attempt to apply that reasoning to real-world objects. I agree it's a worse model for space, but that doesn't make the axioms "wrong", it makes them "not as good for X purpose". If the purpose is accuracy, they're worse, but accuracy is also far less relevant and misleading when it comes to truth.



I never said math exists outside of our heads. Quite the opposite is my belief, in fact.



Maybe your brain is just a reductionist stemlord half-baked philosophy.



Yeah, except, we aren't the Greeks, and what the Greeks thought about geometry has zero impact on how mathematicians understand geometry today. I'm not sure why you're bringing that up, as it's just way off mark.

Who cares? I understood the part where you misunderstood Jon`C's post. Everything else is more fodder for you to make this more complicated in order to justify your original mistake.

I guess that depends on what "exists" means?

Euclidean geometry is 100% as valid today as it was when the ancient Greeks laid it down. They may have been wrong about how to interpret their geometric truths, but that doesn't mean they had to modify the geometry to make it compatible.

I didn't misunderstand Jon's post, I pointed out that he used axiom incorrectly. He probably knows that and was just speaking carelessly, which is whatever. I have no idea what you're doing now besides making a total ass of yourself.

Read this again and tell me you aren't confused about the intended meaning of the word "wrong" in context.

You do this a lot. You misinterpret people by inserting your own interpretation of the words they use, and accuse them of being "wrong", when all you've done is shifted the epistemic foundation in a way that is inconsistent with their intended meaning.

You can call it sloppiness on their part, but I call it pedantry on your part.

Also, there is a fixed point in your confusion here: the word "axiom" is itself axiomatic. You don't win the argument by changing the meaning of the word axiom.

Let me repeat Euclid's axioms, translated from the Ancient Greek:

• "To draw a straight line from any point to any point."
• "To produce [extend] a finite straight line continuously in a straight line."
• "To describe a circle with any centre and distance [radius]."
• "That all right angles are equal to one another." The parallel postulate:
• "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

None of these axioms at all pertain in any way to reality in anyway. They all explicitly deal with only mathematical objects. Calling them "wrong" is therefore a mistake. They aren't describing reality, they're laying out rules for a geometric space.

If one of the axioms said anything about the real world, then yes, they would be wrong, but Euclid never said or did that.

Maybe it was a bit pedantic, and unnecessary. Sure, but being ****ing stupid, pedantic and incorrect about basic mathematical ideas is far worse.

You're speaking nonsense right now.

OK, have fun talking to yourself, I'm out.

Try PMing me if you want to have a discussion, instead of acting passive-aggressive and shooting random aggro insults every other page in the thread. It makes you look like a huge *****.

Anybody who wants to understand my point can read my posts.

I still think you took a wrong turn in your interpretation of the word "wrong", so there's not much more to be said (already far too much has been said).

In any case, yeah, pretty much all linear algebra is done over Euclidean space, so named because the geometry of Euclidean space is consistent with the axioms laid out by Euclid.

Obviously, linear algebra has millions of applications to all sorts of things wherein the relevance of Euclidean space to modeling physical space is irrelevant.

Other than that, the specific set of axioms Euclid laid out aren't used very much.

My original post literally said "not sure what you mean by wrong", then went to explain my take on what it meant to help clarify what he was trying to say.

What he wrote was perfectly clear before your attempts to "clarify" it.

Euclid’s axioms are wrong because they purported to be irreducible facts about reality. They aren’t. They describe a particular kind of geometry which exists only within our minds, based on axioms that were chosen to be convenient for us, rather than a faithful representation of something we’ve experienced or something that can be said to have been created by God rather than by man.

This reveals Jordan Peterson’s absurdity. Euclid’s axioms were wholly created by a man, and describe a phenomenon that never existed, that only exists within the imagination of man. Yet those axioms can be used to formulate proofs, something even children are capable of doing. Where is God in this process, in the invention of Euclidean geometry? Nowhere.

And of course axioms can be wrong, just look at the axiom of choice ****show. Mathematicians debate whether axioms should be used or excluded all the time. A lot of the axioms that have been chosen turn out to be equivalent too, so they were never axioms to begin with (like copying line segments in Euclidean geometry).

I suppose I'm just getting at a semantic distinction, because I don't see any evidence of that in the axioms themselves. I don't doubt that Euclid believed, as in, spoke meta-mathematical commentary on what he believed he was doing, that he was arriving at reality itself, and that the mathematics he laid out were perfect. But there's a gap between the axioms themselves and what one believes the axioms to represent.

Where'd you get the idea that axiom of choice is a "****show"? Accepting the axiom of choice is completely optional, and both branches of accepting or rejecting it lead to wholly valid mathematical systems. There's literally nothing wrong with having two separate versions of mathematics which don't meet in all of the same places.

Like, you can reject the group axioms. You can do tons of mathematics without believing in the existence of groups. There's literally nothing a priori wrong with doing so.

The only reason people might not want to accept the axiom of choice is that accepting it conflicts with meta-mathematical conceptions about what's "right" in mathematics. It creates non-measurable sets and weird "paradoxes", yeah, but nothing that's actually inconsistent or invalid.

Showing that a statement can be proved as a theorem of other axioms doesn't mean the statement can't be an axiom, it just makes it a pointless axiom.

It's a stylistic concern to try and "cut down" the amount of axioms to the minimum needed.

Speaking of "wrong", holy crap how did I miss this



Ever heard of Maxwell's equations? Modern theoretical physics and pretty much all of signal processing can be traced back to the work of Maxwell more than anything else. It's a fantastic theory has stood up well even to quantum mechanics for the most part.

Also, on page 5: read and weep, mathematicians:

P.S.,



at your own risk

[quote=Doron Zeilberger]
In my ultrafinitist weltanschauung, the great significance of both Gödel's famous undecidability meta-theorem, and Paul Cohen's independence proof is historical (or as Cohen would put it, "sociological"). Both are reductio proofs that anything to do with infinity is a priori utter nonsense, debunking the age-old erroneous belief of human-kind in the actual (and even potential) infinity. Granted, many statements: like "m+n=n+m for all (i.e. "infinitely" many) integers m and n" could be made a posteriori sensible, by replacing the phrase "for all" (when it ranges over "infinite" sets) by the phrase for "symbolic (commuting) variables (or rather letters) m and n". We have to kick the misleading word "undecidable" from the mathematical lingo, since it tacitly assumes that infinity is real. We should rather replace it by the phrase "not even wrong" (in other words utter nonsense), that cannot even be resurrected by talking about symbolic variables.
[/quote]

In particular:




There absolutely is something wrong with a theory that becomes complicated if the ostensible reason for the theory is invalidated by said complexity. I can't remember the name the philosopher who said it, but basically: nothing ever useful in philosophy has ever been said that couldn't be said concisely. If your theory itself is experiencing exponential growth in complexity, it's time to trash the theory itself rather than sit there and rationalize it with further amendments.



It seems you are completely ignoring the role of modelling mathematical theories as an applied activity, within the field of mathematics.

Maybe you don't understand: mathematicians aren't trying to be useful to you, and owe nothing to you. We don't need the stuff we do to be taken the next day and applied in your projects. And in this vein, a huuuuge amount of academic work is not useful to you, most studies won't be useful to you, if it's useful to anyone at all. If mathematicians want to work with physicists or not (they do want to and do work with them), that's not up to you, nor do mathematicians owe it to you.

In case you haven't got it, the point is, your views are irrelevant to what mathematicians do, and no amount of quotebooking is pumping the brakes on that. Mathematicians do what they do because they like doing it, the theories are fascinating and cool, and we derive enjoyment out of them. We don't need you to like it (or be able to understand it) for it to be worth doing.

lol what do I have to do with this?

Many people have an undergraduate phase where they like to dictate to everyone how mathematics should be done. Most get over it by the time they graduate.

You're the one who brought up mathematics. Your point about axioms is still wrong.

I should say, you're the one who introduced mathematics considered as a cloistered academic club. Jon`C was making a very simple and easy to understand point.

PM me if you have anything else you want to discuss.

Why? What are you afraid of?

Incidentally, this is a completely off-topic ad hominem that didn't address a single thing I wrote. But I can't say I'm surprised that you don't understand your mistake if you haven't by now.

For the record, I ****ed up the quote I was trying to paraphrase. I should have said: "nothing ever useful in philosophy has ever been said that couldn't be said concisely."

I'm not afraid of the discussion, but since you're intent on forcing a flame war it shouldn't **** up the forum.

If you want to discuss axioms, I'm willing to do it, but since your level of aggro on me on this forum has been substantial I don't expect you to stay rational for long enough to parse an argument. So go to PM, and maybe instead of desperately trying to win the approval of the other members we can speak like adults.



You've given plenty of criticisms of how mathematics should be done. You're a pipsqueak, biting at the ankles of giants.

I am perfectly willing to carry on this discussion with you if you try not to take it so personally. Look, you're wrong. Axioms can be wrong. We can agree to disagree on this, because I can see you aren't going to admit your mistake.