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 -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/
