First, I think it's great that you're taking the time to answer comments. It's rare that authors take full advantage of the conversational potential the internet allows, as opposed to more traditional forms of publishing.
I consider myself a philosopher with only an elementary understanding of mathematical foundations, but I've always found Godel's use of the liar's paradox to undermine Bertrand and Whitehead's Principia especially brilliant. I tend to be drawn to subversive intellectual accomplishments more than straightforward constructive projects. Though I also greatly respect the work of those like Gell-Mann and Feynman who have worked on the nuts and bolts of the Standard Model, which is probably humanity's greatest mathematical project.
I'm not a huge fan of mereology (http://plato.stanford.edu/entries/mereology/), but for a while I've felt that it's central occupation, the relation of parts to wholes, is the same problem underlying metamathematical concerns. Things like, what constitutes a unit? Does the property of divisibility only inhere within some objects, or all of them? In reference to math, we could ask these questions more like, how do we bound objects in order to pick them out for counting? How do we maintain the identity of an object after it is unbound into parts or fractions? And how do those fractions become independent units themselves?
Seems like the question of how math is applicable to the world has to grapple with these types of questions at some point. If only because I place our denoting of physical objects as units prior to our employment of those units for counting. I'm only basing that on evolutionary assumptions, but I don't think it's a stretch as far as assumptions go.
My usual initial response to these questions is that we've adopted a method of differentiating units based on what we discern to be separate objects using visual perception. As it seems the dominant human sense, and especially tuned to delineating the edges of macroscopic objects. I'm certainly not the first to suggest that if we used a different faculty than vision for getting around in the world, then our math, as well as a lot of other things, might be quite different. Thus, we're left with a kind of chicken-and-egg situation. Does our visual field separate objects because objects really are separable (and not necessarily just at our normal scale of perception), or do we think of objects as separable because we've had such success with a visual system that stresses the distinction between objects?
Just one more thing. I take the problems of units to be the inverse of the problems of infinity. Instead of just asking how can there be an endless succession of things, we might also ask how a thing can be separate from everything else?
More information about formatting options