categories: Re: Michael Healy's question on math and AI

[John Duskin <duskin@math.buffalo.edu>]

At the risk of putting my head on a platter (as Mike Barr 
said).....If you take the point of view of ``representable functors", 
an object P, together with an orderded pair of arrows (pr_A:P --->A, 
pr_B:P---->B) is a product of A with B iff for all objects T, the 
mapping Hom(T,P)--->Hom(T,A)xHom(T,B) defined by 
f|---->(f.pr_A,f.pr_B) is a bijection. If you compose this map with 
the bijection
(a,b)|--->(b,a): Hom(T,A)xHom(T,B)--->Hom(T,B)xHom(T,A), you get that 
P, together with the ordered pair of arrows ( pr_B:P---->B, pr_A:P 
--->A) represents a "product of B with A". and this is different even 
if we are talking about a product of A with itself. In other words, 
the "product we usually are thinking about"  is universal for ordered 
pairs of arrows (a:T--->A,b:T--->B). If this means that "ordered 
pair" needs to be made a primitive notion within the underlying logic 
(as the Bourbakists did), so be it, because the simple translation of 
the above statement out of "set theoretic" terms" into "category 
theoretic" terms (no Hom-sets allowed) needs the notion of "ordered 
pair" in order to state it independently of any overarching "theory 
of sets".