[chriswarbo]

Norman Wildberger's YouTube channels are the most thorough I've seen ( https://www.youtube.com/@njwildberger and https://www.youtube.com/@WildEggmathematicscourses ).

There are hundreds of videos, organised in playlists, from undergraduate lectures ( https://www.youtube.com/playlist?list=PL55C7C83781CF4316 ) and research seminars ( https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ) all the way to basic fundamentals like how to think about counting (e.g. https://www.youtube.com/watch?v=Puk-ipOTiD4&list=PL5A714C94D... )

The reason I find them fascinating is that Wildberger doesn't agree with some of the conventional approaches, in particular with the use of infinity and taking limits. This leads him down interesting paths (e.g. Rational Trigonometry and Algebraic Calculus), which (a) show the process of mathematics (exploring, making definitions, building up in different directions, etc.), whilst (b) remaining mostly grounded and approachable (e.g. no appeals to inscrutable lemmas from abstract research areas).

For example, he's recently been making videos about "multisets" (computer scientists would call them Bags), their arithmetic (where "adding" is union, and "multiplying" is pairwise/cartesian product of the elements), and how this generalises: from an algebra containing only empty bags (trivial, but self-consistent; behaves like zero), to bags of zeros (behaves like natural number arithmetic), to bags of natural numbers (behaves like polynomial arithmetic), to bags of polynomials (behaves like polynomials in arbitrarily-many variables) https://www.youtube.com/watch?v=4xoF2SRp194

[jdkee]

"The reason I find them fascinating is that Wildberger doesn't agree with some of the conventional approaches, in particular with the use of infinity and taking limits."

So no transfinite ordinal analysis or large cardinals? Hard to take him seriously.

[chriswarbo]

More than that: no Real numbers, no pi, no square root of 2, no sine/cosine, etc.

It's similar to 'reverse mathematics' (trying to find the minimum set of assumptions required to prove a known result)

[fn-mote]

The square root of 2 does not require any limiting process. It's the hypotenuse of a right triangle with legs of length one.

[chriswarbo]

> It's the hypotenuse of a right triangle with legs of length one.

Yes, we can construct such a line segment; but line segments are not numbers.

We don't actually need "legs of length one" (which pre-supposes some system of units); all we need is the ratio of the lengths of the sides. However, finding lengths requires the ability to take square roots, which would either make this a circular definition (e.g. that √2 = √2 / 1), or requires the limit of an infinite process (like Newton's method, or equivalent).

Instead, it's much easier to count the areas of the squares on each leg (1 and 1), and add them together to get the area of the square on the hypotenuse (1 + 1 = 2). No need for lengths, so no need for square roots, so no need for √2.

Wildberger abbreviates 'area of the square on a segment/vector' as the 'quadrance' of that segment/vector (defined as the dot-product with itself). Likewise we can avoid angles by taking ratios of quadrances (e.g. 'spread' is defined via a right-triangle as the quadrance of the opposite side / quadrance of the hypotenuse); together this gives rise to a whole theory of Rational Trigonometry, which gives efficiently computable, exact answers; works in arbitrary fields (except for characteristic two), and with arbitrary dot-products/bilinear-forms (e.g. euclidean, relativistic, spherical, etc.). Here's Wildberger's textbook on the subject http://www.ms.lt/derlius/WildbergerDivineProportions.pdf

[fatneckbeard]

no computer can calculate that exact distance, which is kind of Wildberger's point.

Infinities are very interesting but the non-infinite maths have kind of got neglected over the past 100 years. I had to memorize Laplace transforms in college but never heard of Fairey sequences until I watched his videos.

People get upset at him but he's basically just having fun seeing how far you can go in Math without infinity. It's quite interesting to a certain audience (like myself).

[fn-mote]

> So no transfinite ordinal analysis or large cardinals

You could insist on sticking with the Axiom of Countable Choice if you wanted to avoid some of that.

I'd say it's pretty hard to avoid thinking about 'infinity' though.