One well-known approach to making sense of infinite and infinitesimal quantities is nonstandard analysis (NSA), where the real number system R is extended to *R, which includes “numbers” that are larger than every real, and also ones that are smaller than every positive real and yet are themselves larger than 0. The latter (small ones) and their negatives become the infinitesimals in common use in physical reasoning. This was the aim of Robinson [20]: to develop *R and to show that in fact the familiar intuitive arguments using infinitesimals then become quite rigorous.

But infinitesimals are not the same thing as zero; they are simply very very close to zero; one might say that they form a kind of fuzzy zero—and more generally, that each real r has about it a band of new numbers (r plus any infinitesimal) that “coat” r so closely that for ordinary purposes r and its coat are indistinguishable.15

A key point is that, while being in zero’s coat, an infinitesimal e nonetheless has a well-defined reciprocal 1 /e, which is infinite (larger than every real). We still do not have a reciprocal for zero itself, but perhaps we can dispense with that, and when a variable “approaches” zero we may try to regard it as being in zero’s coat rather than being zero itself. More generally, the coat of a real r then provides stand-ins for r, which are r-ish in more or less degree (but all of them are r-ish and not s-ish for any other real s).

As Robinson has shown, *R can be given a very rigorous definition, so that it remains an algebraic field and respects the “usual” mathematical properties of R. These properties are given sharp characterization, roughly as follows: for any sentence S that can be expressed in a particular formal language L (including much of standard math notations, for instance +, x, constants, =, <, V, set-membership, etc.—but NOT using a symbol for R itself), S is true when interpreted as being about elements in R iff it is true about *R.16 Now this “transfer principle” between R and *R is the basis for a great many applications of NSA.17 But results of such applications—at least when those results are interpreted as being about R (or more precisely about the “set-theoretic superstructure for R”)—generally are theorems that can also be proven (though maybe less easily or intuitively) without NSA. One of the suggestions we are raising here is this: perhaps *R (or its superstructure) can be taken seriously as a model of physical reality, to see whether this sheds light on infinities that arise in physics.18

One very nice (traditional) application of NSA is the delta function, which now can be defined an as actual (non-fictional) function from *R to *R. For instance, given an infinitesimal e, let S(x) = 0 for all numbers (in *R) that lie outside [—e/2, e/2], and let S(x) = 1/e for numbers in that interval. The graph of such a function then is an infinitesimally thin, infinitely high rectangle, and the area under it is exactly e x 1/e = 1. And then the integral of S(x) times any function *f from *R to *R (that is an appropriate extension of an integrable function f on the reals), gives f(0)—or more precisely, gives the average value of *f in that interval, which is itself in the coat of—and so normally indistinguishable from—f(0). [1] [2] [3] [4]

But now the product of any two such delta functions from *R to *R is unproblematically another function from *R to *R. There is a tradeoff, however. For we must choose a particular delta function to use in a given application, rather than opt for the distributional approach that lumps many such together.[5]

  • [1] I apologize for introducing the term coat for this; already in use are: monad, haze, cloud, halo.My excuse is that a coat of paint is thin, hugs close to its target, and is not to be touched by otherentities (at least while wet).
  • [2] Details can get a bit complicated; see [7].
  • [3] There are by now dozens of books and hundreds or articles on the subject of NSA in general andapplications of the transfer principle in particular. See for instance [2, 5].
  • [4] See [12] for a rare exceptional—but alas all too preliminary—treatment of NSA’s nonstandarduniverse itself as having physical significance, in this case to QFT.
  • [5] Further investigation (I am unaware of any work on this topic) may reveal advantages to particular“natural” choices for a delta function in particular applications. For now I simply point out one fromRobinson’s book (p. 138): —= exp(-X-). For real values of e this is just an ordinary Gaussian,which arises quite naturally in many situations, and has very nice mathematical properties. Possiblyin the nonstandard realm it will also play a helpful role. Note that this is not claimed to resolveissues about non-linear applications where integration properties of products arise.
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