Comment by chriswarbo
5 days ago
Norman Wildberger takes this to the extreme with Rational Trigonometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
It eschews angles entirely, sticking to ratios. It avoids square roots by sticking to "quadrances" (squared distance; i.e. pythagoras/euclidean-distance without taking square roots).
I highly recommend Wildberger's extensive Youtube channels too https://www.youtube.com/@njwildberger and https://www.youtube.com/@WildEggmathematicscourses
He's quite contrarian, so I'd take his informal statements with a pinch of salt (e.g. that there's no such thing as Real numbers; the underlying argument is reasonable, but the grand statements lose all that nuance); but he ends up approaching many subjects from an interesting perspective, and presents lots of nice connections e.g. between projective geometry, linear algebra, etc.
I also invented this! There is cool stuff like angle adding and angle doubling formulas, but the main downside is that you can only directly encode 180 degrees of rotation. I use it for FOV in my games internally! (With degrees as user input of course.) In order to actually use it to replace angles, I assume you'd want to use some sort of half angle system like quaternions. Even then you still have singularities, so it does have its warts.
He maybe considered contrarian but his math is sound.
With all due respect, no, it isn't. His drivel against set theory shows that he didn't even read the basic axiomatic set theory texts. In one of his papers, he is ranting against the axiom of infinity saying that 'there exists an infinite set' is not a precise mathematical statement. However, the axiom of infinity does not say any such thing! It precisely states the existence of some object than can be thought of as infinite but does not assign any semantics to it. Ironically, if he looked deeper, he would realize that the most interesting set theoretic proofs (independence results) are really the results in basic arithmetic (although covered in a lot of abstractions) and thus no less 'constructive' than his rational trigonometry.
Almost every critique of the axiom of infinity is philosophical. I don't think you can just say "the axiom is sound, so what's your point". And you don't even get to claim that because of Godel's incompleteness theorem.
The axioms were not handed to us from above. They were a product of a thought process anchored to intuition about the real world. The outcomes of that process can be argued about. This includes the belief that the outcomes are wrong even if we can't point to any obvious paradox.
"Sound" means free of contradiction with respect to the axioms assumed.
If you can derive a contradiction using his methods of computation I would study that with interest.
By "sound" I do not mean provably sound. I mean I have not seen a proof of unsoundness yet.
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> It precisely states the existence of some object than can be thought of as infinite but does not assign any semantics to it
Can you elaborate on this? I think many understand that the "existence of some object" implies there is some semantic difference even if there isn't a practical one.
I really enjoyed Wildberger's take back in high school and college. It can be far more intuitive to avoid unnecessary invocation of calculation and abstraction when possible.
I think the main thrust of his argument was that if we're going to give in to notions of infinity, irrationals, etc. it should be when they're truly needed. Most students are being given the opposite (as early as possible and with bad examples) to suit the limited time given in school. He then asks if/where we really need them at all, and has yet to be answered convincingly enough (probably only because nobody cares).
Stuff like this is what really interests me in trying to imagine how differently aliens might use things that we consider to be immutable fundamentals.
personal theory: I think there's going to turn out to be a parallel development of math that is basically strictly finitist and never contends with the concept of an infinite set, much less the axiom of choice or any of its ilk. Which would require the foundation being something other than set theory. You basically do away with referring to the real numbers or the set of all natural numbers or anything like that, and skip all the parts of math that require them. I suspect that for any real-world purpose you basically don't lose anything. (This is a stance that I keep finding reinforced as a learn more math, but I don't really feel like I can defend it... it's a hunch I guess.)
That math exists, but it is annoying to work with.
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How would you do limits or analysis?
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I had this feeling of alien math when I went thru his videos on ancient Babylonian math. They were very serious about the everything divided by sixty stuff. Good times.
he sounds awesome. even though i’m sure i would view him as a total kook, he’s the kind of kook that life is brighter for everyone with his existence.