Dear Conrad,
I liked much your essay, I gave it a high rate and I included it in the (second) list of best essays of my review. You seem to gather unanimity here, so I think the interesting question is: who would think otherwise (believe in physical infinity) ? You wrote in your comment that you "started out believing in physical infinity" yourself. Was it just a default position by lack of precise ideas ? Do you know any physicists having a firm belief in physical infinity ?
If I had to express a bet with respect to cosmology, I would opt for the idea of a spherical universe, that I see as the simplest and most natural way a universe can be created. Indeed, I hardly see the sense and possibility of an infinite universe (how can it start in the first place ?), and I don't believe I have clones anywhere. So, since we already verified the surprising fact that the cosmological constant is extremely small (compared to its microphysical causes) but nonzero, I see it natural to expect a similar property for the curvature of the universal "geography".
But the other question is that of the infinitely small. You seem to assume that nobody takes seriously the idea of a physical infinity in the infinitely small, as all we can measure is approximations. But I do think that there are many people whose views logically imply the existence of a physical infinity in the infinitely small, even if they are not ready to admit it. What I mean here is that they have mutually contradictory beliefs and they fail to notice the contradiction.
Precisely, I see only 3 possibly coherent views with respect to the infinitely small:
1) A digital universe, made of pixels (or the like), where continuous geometrical symmetries are only an emergent property.
2) A quantum universe, where the (usually called "paradoxical") properties of quantum physics are accepted as actually describing how things are, and finally understood as not really paradoxical since they are the solution of this other paradox : the reconciliation of continuous geometrical symmetries with the absence of actual infinity in the infinitely small. This is achieved by the fact that the continuous symmetries (such as rotations of a local object) are not acting over an actually infinite list of really distinct states, but over the continuous values of probabilities for the system to appear in one or another state if it is measured. This means to reject physical realism, as the continuity of the transition between the possibilities for 2 states to be identical or distinct, means that there is no physical reality of which state a system exactly is in. I commented this further in pages 5 and 6 of my essay.
3) A classical continuous universe, which logically means to admit an actual infinity of physically distinct possible intermediate states between 2 states. A typical example is Bohmian mechanics. Supporters of such views may hope to keep this compatible with practical finiteness, i.e. that this actual infinity only concerns the ontology that, at the same time, they wish to deny on an effective level, where it would behave as a potential infinity only. Namely, they expect the effects of the whole infinity of decimals of their "hidden variables" are not popping up in finite times. However I do not see it clear if they can really find a coherent theory satisfying that property and that would be compatible with known physics (quantum field theory). For details, see in my criticism of Bohmian mechanics, the section "Problem 2 : the nonsense of deterministic randomness".
