[chriswarbo]

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.

[QuaternionsBhop]

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.

[srean]

He maybe considered contrarian but his math is sound.

[zzless]

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.

[lich_king]

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.

[srean]

"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.

[fn-mote]

To clarify:

“Sound” != proof of soundness in the same way that the Riemann Hypothesis being true is not the same as RH being proven.

[srean]

Not a bad analogy. Damn good.

[kstrauser]

> "Sound" means free of contradiction with respect to the axioms assumed.

Gödel wept.

[adrian_b]

If with an axiomatic system there are undecidable propositions, that is not the same with the axiomatic system being contradictory, i.e. where you can prove that a proposition is both true and false.

An undecidable proposition is neither true nor false, it is not both true and false.

A system with undecidable propositions may be perfectly fine, while a contradictory system is useless.

Thus what the previous poster has said has nothing to do with what Gödel had proved.

Ensuring that the system of axioms that you use is non-contradictory has remained as useful today as by the time of Euclid and basing your reasoning on clearly stated non-contradictory axioms has also remained equally important, even if we are now aware that there may be undecidable things (which are normally irrelevant in practice anyway).

The results of Gödel may be interpreted as a demonstration that the use of ternary logic is unavoidable in mathematics, like it already was in real life, where it cannot always be determined whether a claim is true or false.

[srean]

Indeed. Soundness and completeness are different things.

There are two well accepted definitions of soundness. One of them is the inability to prove true == false, that is, one cannot prove a contradiction from within that axiomatic system.

[sublinear]

> 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).

[Razengan]

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.

[ajkjk]

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.)

[lupire]

That math exists, but it is annoying to work with.

[ajkjk]

any particular reference to what you're thinking of? I am aware of some writings on finitist or constructivist mathematics but they have not quite seemed to get at what I want (in particular doing away with explicit infinities does not require doing away with excluded middle at all, which is what most of that literature seems to be concerned with).

[donkeybeer]

How would you do limits or analysis?

[ajkjk]

I think it's just a perspective shift. The main idea is that you can't ever measure a real number, only an approximation to one, so if two values differ by less than the resolution of your measurement they are effectively the same. For example consider the derivative f(x+dx) = f(x) + f'(x) dx + O(dx^2). The analysis version of the derivative says that in the limit dx -> 0 the O(dx^2) part vanishes and so the limit [f(x+dx)-f(x)]/dx = f'(x). The 'finitist' version would be something like: for a sufficiently small dx, the third term is of order dx^2, so pick a value of dx small enough that dx^2 is below your 'resolution', and then the derivative f'(x) is indistinguishable from [f(x+dx)-f(x)]/dx, without a reference to the concept of a limit.

[donkeybeer]

Yes but like I was thinking more how you'd do any kind of "and it vanishes" or even "becomes sufficiently small" with a gappy number system as it would have to pass through gaps where "undefined" non-rationals exist.

[ajkjk]

I guess my stance (which is not very well-developed or anything) is that you try to learn to live with the gaps: define everything in terms of only what you can measure and it no longer matters whether a number is rational or irrational, or infinitesimal vs small-but-finite, because you can't tell. Instead of saying "it vanishes" as an absolute statement you say "it appears to vanish from my perspective".

[don-bright]

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.

[keeganpoppen]

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.