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Could we develop a geometry which does not depend on the metaphysics of real numbers, continuity, infinity?  Or still have infinity, but make it more like Poincare's, a direction (like a time axis).

There are vast realms of such geometries, going back to projective geometries over finite fields and the like, and to general topology. Lie groups (including the one-point compactification of the complex plane) and Lie algebras. Banach spaces. Measure theory. Spaces of fractal dimension. Minkowski space. Differential geometry of Einsteinian spacetime. Non-commutative geometries in quantum mechanics (von Neumann algebras over Hilbert space). Many more.

A feature we're looking for is "accessible to grade schoolers" i.e. we don't want you already out the other end of some lengthy pipeline wherein brainwashing has already occurred.

Spherical geometry. The trig gets a bit complicated, but even in kindergarten children can see that an octant of a sphere is a triangle with three right angles. As long as you don't bring in the big words.

I understand that some elite schools get into jiggering with the fifth postulate (Euclid's) even pre-college,

That's doing it the hard way.

<debate>

Resolved:  "Irrational numbers are of course morally superior to the rationals as all the best constants (e, phi, pi) are irrational, even transcendental if we're lucky."

I'll see your e, phi, and pi, and raise you 0, 1, -1, and i. e^i*pi + 1 = 0 Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. The integers were made by God. All else is the work of man.--Kronecker

Pro, con or stand aside?  Come prepared next Tuesday.

</debate>

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Kirby

Or you could say Phi (golden mean) is the Phirst Phractal (certainly the recursivity is there in the algebra).

Kirby Urner ндсжег воss

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-- Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin Silent Thunder is my name, and Children are my nation. The Cosmos is my dwelling place, the Truth my destination. http://earthtreasury.org/

-- Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin Silent Thunder is my name, and Children are my nation. The Cosmos is my dwelling place, the Truth my destination. http://earthtreasury.org/