Taking Physical Infinity Seriously
Abstract The concept of infinity took centuries to achieve recognized status in the field of mathematics, despite the fact that it was implicitly present in nearly all mathematical endeavors. Here I explore the idea that a similar development might be warranted in physics. Several threads will be speculatively examined, including some involving nonstandard analysis. While there are intriguing possibilities, there also are noteworthy difficulties.
Keywords Non-standard analysis • Infinity • Physics
Infinity plays a central role in mathematics, and arguably always has—despite occasional negative characterizations (even by some of the most esteemed practitioners). Today surely there is little question about its importance in the minds of the vast majority of mathematicians. There is also very wide appreciation of the idea that whither goes mathematics, there also goes physics (and often the other way around). And yet in physics the notion of infinity plays a rather curious “fix-it-up” role, rather like duct tape, that is brought out for use whenever needed but then put firmly back
'In  Martin Davis includes a discussion of infinity in mathematics in terms of imaginative powers of our minds (my words, not his), and (partly) justifies this by analogy with physics— somewhat the reverse of my point here, but one I am equally sympathetic to.
My thanks for helpful comments and clarifications from: Paulo Bedaque, Juston Brodie, Jeff Bub, Jean Dickason, Sam Gralla, Dan Lathrop, Carlo Rovelli, Ray Sarraga, and two anonymous reviewers—none of whom however is to be blamed for any errors or outrageousnesses that remain.
in the tool box again. Thus it is not kept front and center in actual physical models, quite unlike its now central and fundamental role in mathematics.
This is part of a much larger issue: how mathematics relates to physical reality. This involves many aspects that we will not touch on here, other than some brief comments. For instance, Wigner  regards it as “unreasonable” that there is such a strong connection between math and physics. And Kreisel  has considered whether quantities that are physically observable (according to a given physical theory) can be generated by a Turing machine; such a theory he calls “mechanical”. See also [1, 16, 22], all of whom discuss cosmological issues such as whether space is infinite in extent; Rovelli  in particular distinguishes—similarly to a distinction we shall draw—between infinite divisibility and infinite extent.
A related question is: what sort of universe is needed in order for there to be a possibility of mathematics at all? That is, not actual mathematical practice, but simply the possibility of “stuff” sufficient to allow, for instance, such things as sequences, records, relations. There would seem to be a requisite minimum level of temporality and spatiality even for natural numbers to have any meaningfulness. And, perhaps deeper: what counts as stuff, and what is it for stuff to “be”? But we will leave these questions aside, and return to our main theme.
Here I will describe a number of examples in which infinity is used explicitly in physics, and possible developments that these might suggest, including a few detours along the way. Yet I must add that, as a non-physicist, I also approach the broader topic with some trepidation; and while I have consulted a number of physicists in the writing of this paper, still any misconceptions are completely my own. I trust the reader will pardon any sense that I am throwing in the kitchen sink; this essay represents some possibly far-flung imaginings that perhaps do not fall altogether within customary styles in scientific writing.
The rest of this paper is organized as follows: We describe the examples just referred to above, to distinguish several modes of use of infinities in physics; next I review some ideas due to Jose Benardete on a Zeno-like puzzle about infinity, and some related issues concerning particles, densities, and spin; we then turn to nonstandard analysis as one methodology that appears to shed some light (in connection with Dirac delta functions), but has difficulties of its own.
-  One prominent example that will not be discussed at any length here are the divergent Feynmanintegrals (among others) of quantum field theory (QFT). See for instance the excellent Wikipediaentry for Renormalization .
-  I can’t resist noting that in roughly 1968-9 Martin Davis mentioned to me that in his estimationa huge unclarity underlay foundational issues in mathematics and in particular set theory: whatcounts as a thing?
-  That the topic is appropriate to a volume dedicated to Martin Davis, I justify with the observationsthat (i) Martin helped instill in me a general love for ideas on topics far and wide; and (ii) at leasttwo of Martin’s writings bear on related themes: nonstandard analysis  and quantum physics .I note that Rovelli  entertains an idea already present in , namely that of observer-dependentreference frames in quantum mechanics; and (personal note from Rovelli) this also apparently hascome up in writings of Kochen and Isham as well, all after Martin’s contribution appeared. See also for more on this theme.