Jonathan
You state that, "You and Kronecker say integers are basic; Penrose and Rucker think that Imaginary numbers are bits of Math which pre-dated their discovery."
Step into my parlor-- If, as Kronecker says, integers are basic, we should recall that he insisted that God made the integers. Instead, I call upon logical circuits to produce integers, and from there we're off and running. All the rest (of mathematics) was 'made by man', through the effort to connect rational ('ratio-based') numbers, upon which all men could agree and reach a consensus, to irrational (no ratio) numbers, which spanned the logic bridge all the way to the "understanding" bridge, which only is achieved in the consciousness field of the universe, anchored to local "logical machinery" sustained by "living machinery" that is physically real.
All of the logical steps of math can be instantiated either via symbols or constructs, and I view these as originating in the brain. One probably must depart (or at least withdraw) from such to realize Korzybski's attempt to "transcend words entirely, and to work from a consciousness where we run the word machine." The major point of semantics is that apples and oranges must be handled carefully. If physicists must be careful of the Map and Territory issue, how much moreso mathematicians, who deal only with Maps of Maps?
Finally, you state: "So if the fundamentality of geometric constructions is allowed, I'll have an easier time believing in what you are trying to prove, because this construction gives it a conceptual basis.
I suppose the question is whether the universal consciousness field could physically self-induce a vortex motion, or whether it required the permission of a Platonic "ideal universe" in which case, had it failed, we'd be up the proverbial creek, wudn't we.
I don't see why the universe needs "pre-existing" permission to behave as it does.
"Penrose and Rucker think that Imaginary numbers are bits of Math which pre-dated their discovery." Jonathan, I don't even know what this means.
Penrose states 13 times (I counted) that complex numbers "are magical" and I certainly share this feeling. But I do not find any need or desire to assume that this magic "pre-existed" physical reality in any sense. I just don't see it.
Of course we are blessed that the range of behavior includes the production of physical particles, and these can be used to build the world as we know it. But if a photon can propagate through space and time as we currently conceive it to do, then we can, by writing symbol sequences down, derive all of the logic and math necessary to describe the behavior using complex numbers, where the imaginary number typically relates to a ninety degree rotation, that is, an orthogonal direction, which requires something that maintains orthogonality. This 'i' keeps 'x' and 'y' from mixing and merging, and allows the wave to propagate along 'z' while preserving the orthogonality of the 'x' and 'y' components. Of course you know that we can formulate the same thing using two-dimensional vectors and never introduce 'i' at all.
Why, if one can invoke physical reality, including a consciousness field capable of being aware of and freely affecting physical reality, and can therefore produce integers, and from there all mathematico-logical constructions, would one even *want* to have, let alone require, some other ideal universe that will forever remain outside of space and time, and hence be non-physical. What is the difference in this and an insistence on God, biblical or otherwise? God at least gives us a ground for morals (missing otherwise). What does a Platonic ideal universe give us?
Some other author talks about "where music goes when we're not listening to it" I don't view these as well posed problems. They have the general form: "If a man says something in the forrest, and there is no woman around, is he still wrong?"
I'm not a professional philosopher, but these seem like 'category' errors to me.
I have been struck by how many Platonists there are coming out of the woodwork, and ask myself: why? I think that it is related to the theme that I wrote on Terry Padden's page, where physicists have become almost drunk on ornate maps. And further, if one buys the "collapse of the wave function" physics, then one has become accustomed to believe in some "possibility space" which is essentially mathematical, not physical. In short, it seems as if there is a tendency not to believe in physical reality. That is one reason I ground my physics on gravity and consciousness, rather than someone else's abstraction du jour.
Jonathan, you say "But I think it would be hard to prove that something like the Mandelbrot Set doesn't exist outside the observable universe." Does this mean you think it would be easier to prove that it does?
Somewhere else you discuss the Tao. In those terms, I view the material-based entities as "the ten thousand things". Given them, the problem for me has always been to account for awareness and free will. Specifics about the ten thousand things themselves and their interactions, interest me only peripherally. It is awareness and free will that I strive to grasp, and how these interplay with the things. The current consensus is that *somehow* consciousness "emerges" or "arises" from these things. When someone can explain to me just how that happens, I'll have to reconsider my theory of the consciousness field.
BTW, congratulations on sticking to the wall and not sliding back down. You deserve it.
Edwin Eugene Klingman