The ubiquitous use of infinitely small quantities in mathematics dates back at least to the seventeenth century. Despite continuing qualms as to their legitimacy and their supposed elimination as a result of the thoroughgoing reform movement of the nineteenth century, "infinitesimals" have continued to be used, especially in applied mathematics. The logician Adolf Fraenkel gave what was no doubt the widely accepted view when he stated, "The infinitely small is only to be understood as a manner of speaking based on the limit concept, hence a potential infinite; it is a matter of variable … [positive] numbers or quantities that can ultimately decrease below any arbitrarily small positive value. A fixed [positive] number different from zero that can serve as a lower bound to all finite positive values is not possible" (1928, p. 114, my translation, emphasis in original). In 1960 Fraenkel's one-time student Abraham Robinson showed how to obtain just such a "fixed number" and thereby vindicated the discredited infinitesimal methods.
The benefits of the free use of infinitesimal methods were amply demonstrated by the success of Gottfried Wilhelm Leibniz's version of the differential and integral calculus and the continued use of these methods by the Bernoulis and especially by Leonhard Euler. Working mathematicians had no difficulty in knowing just which properties of ordinary numbers infinitesimals could be assumed to possess and just when it was legitimate to equate such quantities to zero. But the lack of any clear justification for these methods provided an opening for scathing attacks such as that of George Berkeley. The need for rigorous methods was felt by mathematicians themselves and eventually supplied (Edwards 1979, Robinson 1974).
Robinson's key insight was that the methods of model theory could be used to construct a powerful rigorous theory of infinitesimals. Thus, for example, we may consider a first-order language in which a constant symbol is provided as a "name" for each real number, a function symbol is provided as a "name" for each real-valued function defined on the real numbers, and the only relation symbols are = and <. Let T be the set of all true sentences of this language when each symbol is understood to have its intended interpretation. Let δ be a new constant symbol, and let W consist of the sentences of T together with the infinite set of sentences:
δ > 0
δ < 1, δ < ½, δ < ⅓, δ < ¼,…
Since any finite subset of W can be satisfied in the ordinary real numbers by interpreting δ as a sufficiently small positive number, the compactness theorem for first-order logic guarantees that W has a model. But in that model, the element serving to interpret δ must be positive and less than every positive real number (i.e., infinitesimal). The structure with which we began of real numbers and real-valued functions can readily be embedded in the new model. Thus if r is a real number and cr is the constant of the language that names r, we may regard the element of the new model that serves to interpret cr as simply r itself. Functions can be embedded in the same way. One speaks of the new model as an enlargement.
Moreover, because T ⊆ W, all true statements about the real numbers that can be expressed in our language are also true in this enlargement. A false statement about the real numbers is likewise false in the enlargement: If the statement S is false, then ¬ S is a true statement about the reals and hence is also true in the enlargement. It is this transfer principle, the fact that statements are true about the real numbers if and only if they are true in the new enlarged structure, that makes precise just when an assertion about ordinary numbers can be extended to apply to infinitesimals as well.
The enlargement will contain infinitely large as well as infinitesimal elements. This is readily seen by applying the transfer principle to the statement that every nonzero real number has a reciprocal. One may even speak of infinite integers; their existence follows on applying the transfer principle to the statement that for any given real number there is a positive integer that exceeds it.
The basic facts of real analysis can be established on this basis using modes of argument that would earlier have been quite correctly regarded as illegitimate. For example, the basic theorem that a continuous function on a closed interval assumes a maximum value can be proved by dividing the interval into infinitely many subintervals, each of infinitesimal length, and selecting an endpoint of such a subinterval at which the function's value is greatest (Davis 1977, Robinson 1974). By beginning with a more extensive language, it is possible to apply infinitesimal methods to branches of mathematics requiring a more substantial set-theoretic basis (e.g., topology, functional analysis, probability theory). It has even proved possible to use these "nonstandard" methods to settle certain open questions in mathematics.
For those with qualms concerning nonconstructive methods in mathematics, these infinitesimal methods are bound to seem unsatisfactory. Because the underlying language is built on an uncountable "alphabet," the use of the compactness theorem hides an application of some form of the axiom of choice. This in turn is reflected in a basic indeterminacy; we can establish the existence of enlargements but cannot specify any particular enlargement. Robinson himself has emphasized that although nonstandard analysis "appears to affirm the existence of all sorts of infinitary entities," one always has the option of taking the "formalist point of view" from which "we may consider that what we have done is to introduce new deductive procedures rather than new mathematical entities" (1974, p. 282, emphasis in original).
Davis, M. Applied Nonstandard Analysis. New York: Wiley, 1977.
Edwards, C. H., Jr. The Historical Development of the Calculus. New York: Springer-Verlag, 1979.
Fraenkel, A. Einleitung in die Mengenlehre, 3rd ed. Berlin: Springer, 1928; reprint, New York: Dover, 1946.
Robinson, A. Nonstandard Analysis, 2nd ed. Amsterdam, 1974.
Martin Davis (1996)