[FRIAM] square land math question

[Jon Zingale]

SDG is a rather cool example of where the point notion can be radically
different than classically handled by Euclid. From the man himself, Anders
Kock[1]:

"Euclid maintained further that R was not just a commutative ring,
but actually a field. This follows because of his assumption: for any two
points in the plane, either they are equal, or they determine a unique
line.

We cannot agree with Euclid on this point. For that would imply that
the set D defined by

D := [[x ∈ R | x^2 = 0]] ⊆ R

consists of 0 alone, and that would immediately contradict our

Axiom 1. For any g : D → R, there exists a unique b ∈ R such that
∀d ∈ D : g(d) = g(0) + d · b"

Gotta love Kock.

[1] Synthetic Differential Geometry: https://users-math.au.dk/kock/sdg99.pdf
Also, the paper of his I am currently entrenched in to investigate further
some
ideas in the Instrumental Goal versus Evolutionary Function discussion:
https://arxiv.org/pdf/1105.3405.pdf



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