foundations of mathematics The reexamination of infinity

Although mathematics flourished after the end of the Classical Greek period for 800 years in Alexandria and, after an interlude in India and the Islāmic world, again in Renaissance Europe, philosophical questions concerning the foundations of mathematics were not raised until the invention of calculus and then not by mathematicians but by the philosopher George Berkeley (1685–1753).

Sir Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany had independently developed the calculus on a basis of heuristic rules and methods markedly deficient in logical justification. As is the case in many new developments, utility outweighed rigour, and, though Newton’s fluxions (or derivatives) and Leibniz’s infinitesimals (or differentials) lacked a coherent rational explanation, their power in answering heretofore unanswerable questions was undeniable. Unlike Newton, who made little effort to explain and justify fluxions, Leibniz, as an eminent and highly regarded philosopher, was influential in propagating the idea of infinitesimals, which he described as infinitely small actual numbers—that is, less than 1/n in absolute value for each positive integer n and yet not equal to zero. Berkeley, concerned over the deterministic and atheistic implications of philosophical mechanism, set out to reveal contradictions in the calculus in his influential book The Analyst; or, A Discourse Addressed to an Infidel Mathematician. There he scathingly wrote about these fluxions and infinitesimals, “They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?” and further asked, “Whether mathematicians, who are so delicate in religious points, are strictly scrupulous in their own science? Whether they do not submit to authority, take things upon trust, and believe points inconceivable?”

Berkeley’s criticism was not fully met until the 19th century, when it was realized that, in the expression dy/dx, dx and dy need not lead an independent existence. Rather, this expression could be defined as the limit of ordinary ratios Δy/Δx, as Δx approaches zero without ever being zero. Moreover, the notion of limit was then explained quite rigorously, in answer to such thinkers as Zeno and Berkeley.

It was not until the middle of the 20th century that the logician Abraham Robinson (1918–74) showed that the notion of infinitesimal was in fact logically consistent and that, therefore, infinitesimals could be introduced as new kinds of numbers. This led to a novel way of presenting the calculus, called nonstandard analysis, which has, however, not become as widespread and influential as it might have.

Robinson’s argument was this: if the assumptions behind the existence of an infinitesimal ξ led to a contradiction, then this contradiction must already be obtainable from a finite set of these assumptions, say from:

But this finite set is consistent, as is seen by taking ξ = 1/(n + 1).