The Resurrection of Infinitesimals: Abraham Robinson and Nonstandard Analysis

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The Resurrection of Infinitesimals: Abraham Robinson and Nonstandard Analysis

Overview

For centuries prior to 1800, infinitesimals—infinitely small numbers—were an indispensable tool in the calculus practiced by the great mathematicians of the age. Between the mid-1800s and the mid-1900s, however, infinitesimals were excluded from calculus because they could not be rigorously established. This changed in 1960, when Abraham Robinson resurrected their use with his creation of nonstandard analysis. Since that time, nonstandard analysis has had an important effect on several areas of mathematics as well as on mathematical physics and economics.

Background

An infinitesimal is an infinitely small number. More precisely, it is a nonzero number smaller in absolute value than any positive real number. But merely defining a mathematical entity does not guarantee its existence. For example, we can define an obtuse-angled triangle as a triangle all of whose angles are greater than 90 degrees. But such triangles do not exist (in Euclidean geometry)! In the case of infinitesimals, there are no real infinitesimals, since given any positive real number a, a/2 is a smaller positive real. To accommodate infinitesimals we must extend the real numbers.

This idea of extending a mathematical system in order to obtain a desired property not already present is common and important in mathematics. For example, while the positive integers are prehistoric, the other number systems, such as the integers, rational numbers, real numbers, and complex numbers, arose over the centuries as human constructs. Thus, the integers were introduced so as to make sense of numbers such as -1, the real numbers to give meaning to numbers like √2, and the complex numbers to accommodate such numbers as √-1. In each case, these numbers were introduced because they turned out to be useful.

This was also the case for infinitesimals—or differentials, as Gottfried Wilhelm Leibniz (1646-1716) called them. They were indispensable in the calculus of the seventeenth, eighteenth, and early nineteenth centuries. For example, to find the slope of the tangent line (later called the derivative) to the parabola, y = x² at the point (x, x²), seventeenth-century mathematicians would argue as follows:

Let e be an infinitesimal. Then is a point on the parabola infinitesimally close to (x, x²), hence the tangent line to the parabola can be identified with the line joining these two points. Its slope is by canceling e. Finally, 2x + e can be identified with 2x since e is infinitesimally small compared to 2x and can therefore be deleted.

Of course the answer is correct since the derivative of x² is indeed 2x. But what about the method used to obtain it? It was severely criticized even in the seventeenth century. Canceling e, the critics argued, meant regarding it as not zero; but deleting e implied treating it as zero. This is inadmissible, they rightly claimed. In a trenchant critique of infinitesimal methods, the philosopher George Berkeley (1685-1753) called such e's "the ghosts of departed quantities," arguing that "by virtue of a twofold mistake one arrived, though not at a science, yet at the truth."

Most mathematicians, however, were unperturbed by such objections. Although they recognized that their methods were logically questionable, these methods yielded correct results. Leibniz, for example, said of his differentials (infinitesimals) that "it will be sufficient to simply make use of them as a tool that has advantages for the purpose of calculation, just as the algebraists retain imaginary roots with great profit." In Leibniz's time complex numbers (imaginary roots) had no greater logical legitimacy than infinitesimals. These are two important examples of a common phenomenon in mathematics, namely the use of objects before their existence is rigorously established.

The nineteenth century ushered in a critical spirit in calculus, in which fundamental concepts were reexamined and put on a logical basis. This resulted in the replacement of the logically problematic infinitesimals with limits, defined rigorously by Karl Weierstrass (1815-1897) in the mid-nineteenth century in terms of the Greek letters epsilon, ε, and delta, δ.

About a century after Weierstrass had banished infinitesimals "for good"—so we all thought until 1960—they were brought back to life as rigorously defined mathematical objects in the nonstandard analysis conceived by the mathematical logician Abraham Robinson (1918-1974). His idea was to provide a rigorous development of calculus based on infinitesimals rather than on limits.

While standard analysis—the calculus we inherited from Weierstrass (and others)—is based on the real numbers R, nonstandard analysis is grounded in an extension of the real numbers called "hyperreal" numbers, R^*. The hyperreal numbers contain infinitesimals, where (by definition) e ε R^* is infinitesimal if e ≠ 0 and -a < e < a for all positive a ε R. They also contain infinite numbers, since if e is an infinitesimal, 1/e is an infinite (hyperreal) number.

Robinson was inspired to create nonstandard analysis in part by his work in the newly emerging subfield of mathematical logic called model theory. Here is how he put it:

In the fall of 1960 it occurred to me that the concepts and methods of contemporary Mathematical Logic are capable of providing a suitable framework for the development of the Differential and Integral Calculus by means of infinitely small and infinitely large numbers.

It is ironic that infinitesimals were excluded from calculus in the nineteenth century because they proved to be logically unsatisfactory, and they were rendered mathematically respectable in the twentieth century thanks to logic. Robinson was very gratified that it was mathematical logic that made nonstandard analysis possible. The great mathematical logician Kurt Gödel (1906-1978) valued Robinson's work because it made logic and mathematics come together in such a fundamental way. The contemporary mathematician Simon Kochen echoed this: "Robinson, via model theory, wedded logic to the mainstream of mathematics."

Robinson was also guided in his work in nonstandard analysis by a sense of history. He saw it in the tradition of the great analysts Leibniz, Leonhard Euler (1707-1783), and Augustin-Louis Cauchy (1789-1857). In fact, he argued that "Leibniz's ideas can be fully vindicated" by his own rigorous theory of infinitesimals.

Leibniz, as we mentioned, tried to justify his work with infinitesimals on pragmatic grounds—that it yielded correct results. He also attempted to rationalize his handling of infinitesimals with a rather vague principle of continuity—that (in our language) properties of the reals also hold for the hyperreals. But as mathematicians of the seventeenth century realized, not all properties of the former carry over to the latter. For example, the Archimedean property, which says that given real numbers a and b, withb positive, there exists an integer n such that nb > a, does not hold in R^*. For if a = 1 and b = e, a positive infinitesimal, then e < 1/n for every positive integer n, by definition of an infinitesimal, so that ne < 1 for all n. Robinson observed that

What was lacking at the time [of Leibniz] was a formal language which would make it possible to give a precise expression of, and delimitation to, the laws which were supposed to apply equally to the finite numbers and to the extended system including infinitely small and infinitely large numbers.

It is the working out of this program for which Robinson is responsible. More specifically, he 1) provided a rigorous construction of the system of hyperreal numbers R^* in which he was able to prove the existence of infinitesimals; and 2) formulated a transfer principle that gave formal expression to Leibniz's principle of continuity and thus rendered precise those properties that are transferable from the reals to the hyperreals. He accomplished both tasks with the aid of mathematical logic.

Impact

What has nonstandard analysis accomplished? First, it has supplied a rigorous presentation of calculus based on infinitesimals that, some have argued, is much preferable to the standard treatment via limits. Second, the methods of non-standard analysis have been introduced into important branches of mathematics aside from calculus, such as topology, differential geometry, measure theory, complex analysis, and Lie group theory. They have also been applied in functional analysis, differential equations, probability, areas of mathematical physics, and economics. These inroads of the subject, in such a short time-span, are indeed most impressive.

ISRAEL KLEINER