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.

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srean 8 hours ago | parent | next [-]

He maybe considered contrarian but his math is sound.

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zzless 4 hours ago | parent [-]

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.

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lich_king 2 hours ago | parent | next [-]

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.

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srean 4 hours ago | parent | prev | next [-]

"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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fn-mote an hour ago | parent [-]

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 33 minutes ago \| parent [-]
		Not a bad analogy. Damn good.

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sublinear 3 hours ago | parent | prev [-]

> 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).

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Razengan 2 hours ago | parent | prev [-]

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.