"Hi Nathan,
Welcome to the forum. I don't think I have talked with you before. (Although I suppose Alzheimer's is sort of like insanity in that you can't tell if you have it yourself.)"
Thank you, I'm glad to be here. These are my first posts, although I used to subscribe to Mel Bartel's atm list ages ago.
"In other words, numbers would not exist in the system unless and until they had been specificially defined (i.e. conceived). Hence I probably qualify as a constructionist or an intuitionist."
That would make sense if one (the One) were to proceed from axiom to plenum according to an ordinal program (implying a temporal process or at least one analogous to it). One of the precepts [I think] you hold is that given an algorithm, it must be processed before it's result exists. If the algorithm produces an infinite series, then there is no result. (To be rigorous, the series needs the property that for all n, there exists m > n such that Sm Sn. We could also restrict the discussion to Cauchy convergent series since that is the hard analytic requirement for the series to have a definite solution.)
This is where I see the anthropomorphic bias. I don't grant that God must process an algorithm to it's conclusion in any iterative fashion. Later on, you claim that the math doesn't have such a bias, but you do invoke a choice of A. Choice [pi] on the basis of a Zenophobic [pi] temporal difficulty. [I crack myself up!]
"(except for one exception: I believe it is a fact that thought happens.)"
That's fine for finding oneself, but then we discover an apriori Universe against which our problems of perception, cognition, and induction spring from psychology and the human condition. To cogitate from some ergo is putting Decartes before the horse [OK, that was lame].
"'that begs me to ask about unseen falling trees.'
I'm not sure exactly what question you are begged to ask, but let me guess that it is the old trick question of whether a falling tree makes any sound if there is no one around to hear it."
I was alluding to a more general question about whether the Universe exists whether or not we are paying attention to it. This goes to your (paraphrased) statement that a number does not exist until conciously realized. There is a much deeper consideration about whether any such cognitive realization is ontological at a primal rather than merely a cognitosymbolic existence. However, if you would have it, I'd grant that where human sentience is insufficient for arithmogeny, God's mind is apt. This of course leads back to whether God can conceive the sum of an infinite series and whether such conceptions as the complex field over the reals have already been worked out.
"I went to the shelf to grab my copy of Whitehead and Russell's 'Principia Mathematica [...]'"
I just finished Russell's (I'm going to try underlines...) The History of Western Philosophy. and was so pleased with his style that I'm going to read the Principia next.
"As it turns out, Kronecker and I lost [the battle over the preposterousness of actualizing infinite regress]. Mathematicians have been following Cantor now for over a hundred years. But I, for one, still reject the notion."
I, for one, have a sense (I hate using the word 'believe') that a truly comprehensive, coherent, and closed theory of physics and reality in general will describe a suprafinite lattice of events. I use the term suprafinite to mean that from any given perspective the number of events in it's past light cone is finite, but that a transfinite cardinality is achieved when viewed from the extemporal viewpoint of total reality.
"Warm regards,
Paul"
Taken in the most sincere sense,
 sfn
