"The infinite in mathematics is alway unruly unless it is properly treated"

James Newman addresses the unruliness of infinity in mathematics, highlighting both its profound allure and its inherent dangers. Infinity is, at once, a concept that elevates mathematical thought to astounding heights and a source of paradoxes and confusion when dealt with carelessly. The infinite naturally appears in many mathematical contexts: in the endless process of counting, the boundless extension of geometric lines, or the limitless precision between any two numbers on the real number line. These manifestations tempt the mathematician to reason as if infinity were just another number or tangible entity. However, infinity, by its nature, eludes such direct treatment.

Historically, the naive handling of infinity led to logical paradoxes and contradictions, especially in the development of calculus and set theory. Calculus, pioneered by Newton and Leibniz, initially manipulated infinitesimal quantities and infinite sums as if they were concrete numbers, yielding remarkably accurate results but also spawning foundational confusion. Only with the later formalization of limits by Cauchy and Weierstrass, which established rigorous criteria for convergence and continuity, did mathematics acquire the tools to handle the infinite consistently.

In set theory, Georg Cantor revolutionized mathematics by developing a precise language to describe and compare different sizes of infinity. His work exposed counterintuitive properties, for example, the set of all rational numbers is infinite, as is the set of real numbers, but the latter is a larger type of infinity. By introducing strict definitions and axioms, Cantor and later mathematicians brought order to processes involving infinity, preventing logical chaos and ensuring that mathematical results remain reliable and meaningful.

The infinite thus serves as both a peril and a possibility in mathematics: it inspires exploration and theoretical advances, but only under strict formal rules does it yield coherent, dependable results. Careful treatment transforms infinity from an unruly concept into a powerful element of mathematical reasoning.