I've been slowly going through Divine Proportions: Rational Trigonometry to Universal Geometry, by Norman J. Wildberger. Wildberger is a mathematician, and a f

I've been slowly going through Divine Proportions: Rational Trigonometry to Universal Geometry, by Norman J. Wildberger. Wildberger is a mathematician, and a finitist. This means that he doesn't believe in infinity in the modern mathematical sense (I'm sure he won't dispute that the integers are unbound, for example). Which means he does not believe that Real numbers are properly defined either, or that limits are really a thing. So he put his money where is mouth is and invented a branch of trigonometry that only uses rational numbers, by replacing length and angle with square distance (which he calls "quadrance") and the square of the sine (which he calls "spread"). All of the above is just what motivated him, what's interesting is that the resulting maths itself is all correct and quite nice to go through. It basically boils down to saying "hey, it's called trigonometry for a reason, so maybe it makes more sense to make actual triangles the fundamental unit, not circles," and working your way from there. Personally I'm kind of curious if his approach might be more practical for computer implementations too, since all number representations on computers are either rational numbers or approximations of other numbers via rational numbers.