Karen,
Thank you for your most gracious and informative response! I would have added this reply earlier, but I don't seem to get any kind of notification from FQXi when someone replies on anything except my own essay thread. I had to search manually for my own name, essay by essay to find responses. Argh! I must be missing something?
You are very generous about my critique points, and I deeply appreciate that you used them as an opportunity to make positive, constructive suggestions. That to me is the heart of good science! I note that some of my favorite comments from one fellow essayists were the ones in which he did his best to point out holes in my argument. Delightful! The points were valid and made me both think carefully and explain myself better.
I'm glad you liked my two title suggestions, and that they were constructive.
Regarding your number 4 non-perturbative criterion, I must confess that was the only one I wasn't quite sure of. Why? Well, there seems to be a deep "lumpiness" to both physics and our universe in general that lurks behind such powerful mathematical concepts as renormalization. Renormalization is not really as exotic or even as mathematical is it is in, say, Feynman's QED theory. What it really amounts to is an assertion that our universe is, at many levels, "lumpy enough" that many objects (and processes) within it can be approximated when viewed from a distance. That "distance" may be real space or some other more abstract space, but the bottom line is that this sort of approximation option is a deep component of whatever is going on. I say that in part because we are ourselves as discrete, independently mobile entities are very much part of this lumpiness, as are the large, complex molecules that make up our bodies... as are the atoms that enable molecules... as are the nucleons that enable atoms... and as are the fundamental fermions that make up nucleons.
This approximation-at-a-distance even shows up in everyday life and cognition. For example, let's say you need an AA battery. What do you think first? Probably you think "I need to go to the room where I keep my batteries." But your navigation to that room begins as a room to room navigation. You don't worry yet about exactly where in that room the batteries are, because that has no effect on how you navigate to the room. In short, you will approximate the location of the battery until you navigate closer to it.
The point is that the room is itself lumpy in a way that enables you to do this, but the process itself is clearly approximate. You could in principle super-optimize your walking path so that it minimizes your total effort to get to the battery, but such a super-optimization would be extremely costly in terms of the thinking and calculations needed, and yet would provide very little benefit. So, when the cost-benefit ratio grows too high, we approximate rather than super-optimize, because the lumpy structure of our universe makes such approximations much more cost-beneficial overall.
What happens after your reach the room? You change scale!
That is, you invoke a new model that tells you how to navigate the draws or containers in which you keep the AA batteries. This scale is physically smaller, and again is approximate, enabling tolerance for example of highly variable locations of the batteries within a drawer or container.
This works for the same reason that in Feynman's QED is incredibly accurate and efficient for modeling an electron probabilistically. The electron-at-a-distance can be safely and very efficiently modeled as a point particle with a well-defined charge, even though that is not really correct. That is the room-to-room level. As you get closer to the electron, that model must be replace by a far more complex one that involves rapid creation and annihilation of charged virtual particle pairs that "blur" the charge of the electrons in strange and peculiar ways. That is the closer, smaller, dig-around-in-the-drawers-for-a-battery level of approximation. In both cases, the overall "lumpiness" of our universe makes these special forms of approximation both very accurate and computationally efficient.
At some deeper level, one could further postulate that this may be more than just a way to model reality. It is at least possible (I personally think it probable) that this is also how the universe actually works, even if we don't quite understand how. I say that because it is always a bit dangerous to assume that just because we like to model space as a given and particles as points within it, those are in the end just models, ones that actually violate quantum mechanics in the sense of postulating points that cannot exist in real space due the quantum energy cost involved. A real point particle would require infinite energy to isolate, so a model that invokes such particles to estimate reality really should be viewed with a bit of caution as a "final" model.
So my bottom line: While formal formula (criterion 4) are great, our universe seems weirdly wired for at least some forms of approximation. I find that very counterintuitive, extremely fascinating, and likely important in some way that we flatly do not yet understand.
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Enough, I'm droning on again! Thanks again for your response, and I really like what you are doing. Your broader goals are great -- please keep at them!
Cheers,
Terry