Dear Bill,
What a delightful essay! You should consider moonlighting as a science writer (what is a physicist-in-residence, anyway?)
"..I reject physical infinity, for three reasons. First, mathematically, it
makes computations intractable. Second, operationally, I do not know how--even in principle--how
to observe, measure or manipulate physically infinite objects or systems. Third, conceptually, it
embodies a viciously unphysical ontology, namely, that physical constituent parts can equal each
other and the physical whole from which they derive."
These are all good reasons, but may I suggest that infinities in physical theories may have a useful role to play that is in my opinion still greatly under-appreciated: I think that at least in some (perhaps, with enough imagination, in all meaningful) cases in which they occur, they may be telling us that we are not looking at the physical situation at hand in "the right way".
The paradigm example to me is the Lorentz factor. For v=c it is infinite, and so presumably one of the unfortunate victims of your effort to eradicate its kin from physics. But what if we look at its inverse: The inverse of the Lorentz Factor tells us how much the proper time changes with respect to coordinate time. In fact, because of the mathematical form of gamma we can get it to tell us more: How much of the proper time is "projected" unto coordinate time (as I'm sure you know, one can easily see this by drawing the appropriate triangle that illustrates
[math] \tau\times(\gamma^{-2}=1- \beta^2)^{1/2}[/math]
In that case, if we take the triangle relationship seriously, gamma=1 tells us that all of the object's proper time is "projected" unto the observer's coordinate time and gamma=infinity tells us that none of it is "projected" unto the observer's coordinate time, or, in other words, that the object's proper time is orthogonal to the coordinate time if we were to assign unit vectors to the abstract plane spanned by the two time parameters . This is of course consistent with the fact that null vectors are orthogonal to time-like vectors.
Orthogonality is one of those situations which commonly involves zero and infinity, and seems to have been what lurked behind this infinity. Orthogonality is also a basic conceptual staple of physics, and so I suspect that there is something conceptually very clear and thoroughly physical behind many infinities in physics in a similar manner, but not very well recognized as such.
I'd be interested to know what you think of this argument, and whether it leads you to modify your categorical rejection of infinities in physics.
Best wishes,
Armin