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PaperclipPerfector  2d ago (text post)   9524 thread views

Culture War Roundup for the week of May 19, 2025

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As late as the 1990's, part of an old-school mathematical education was the idea that submitting a correct co-ordinate geometry proof when a classical one was available would get you full marks and the lasting scorn and derision of the examiner.

Funnily enough classical geometry can be made to admit a coordinate system over it so both classical and (basic) coordinate geometry are effectively isomorphic in the sense that C++ and Conway's game of life are isomorphic (both are Turing Complete). Both classical and coordinate geometry have a proof theoretic ordinal of omega and even more there's actually a canonical way to convert statements of coordinate geometry to statements of classical geometry and vice versa so it's not even like using coordinate geometry when there's a classical geometry proof is like using a sledgehammer to crack a nut. Both theories are equally powerful in what they can do, it's just that the coordinate geometry formalism is easier to build upon which in my view makes it superior.

(Can you tell I hated the geometry problems in olympiads?)

Funnily enough classical geometry can be made to admit a coordinate system over it so both classical and (basic) coordinate geometry are effectively isomorphic in the sense that C++ and Conway's game of life are isomorphic (both are Turing Complete).

The cry of the Intercal programmer. Whether classical geometry corresponds to Intercal and co-ordinate geometry to Python or the other way round is let as an exercise to the interested reader.

(Can you tell I hated the geometry problems in olympiads?)

So did I, but then I don't claim to be a mathematician.

I never did figure out why classical geometry was given such place in our high school curriculum. It seemed that any time I wanted to actually use geometry for anything, it was strictly better to use coordinate geometry and trigonometry. Then again, I still consider proofs to be a waste of time unless you want to study mathematics for its own sake.

Broke: Uses classical geometry to solve problem

Woke: Uses coordinate geometry and a bit of algebra

Bespoke: Converts problem into statement of Number Theory via its Godel numbering, then proceeds to directly prove it.

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