## The Metaphysical Principles of the Infinitesimal Calculus

René Guénon

Guénon’s early and abiding interest in mathematics, like that of Plato, Pascal, Leibnitz, and many other metaphysicians of note, runs like a scarlet thread throughout his doctrinal studies. In this late text published just five years before his death, Guenon devotes an entire volume to questions regarding the nature of limits and the infinite with respect to the calculus both as a mathematical discipline and as symbolism for the initiatic path. This book therefore extends and complements the geometrical symbolism he employs in other works, especially *The Symbolism of the Cross, The Multiple States of the Being*, and *Symbols of Sacred Science*. According to Guénon, the concept ‘infinite number’ is a contradiction in terms. Infinity is a metaphysical concept at a higher level of reality than that of quantity, where all that can be expressed is the indefinite, not the infinite. But although quantity is the only level recognized by modern science, the numbers that express it also possess qualities, their quantitative aspect being merely their outer husk. Our reliance today on a mathematics of approximation and probability only further conceals the ‘qualitative mathematics’ of the ancient world, which comes to us most directly through the Pythagorean-Platonic tradition.

One of the ways contemporary ‘scientism’ encroaches upon metaphysics is through an irresponsible use of such terms as ‘zero’ and ‘infinity’. Mathematics as ‘pure abstraction’ seems to the uninformed like a kind of ‘mystical language’ empowering scientists to unravel the secrets of the universe and to express them in precise terms that allow technicians—whom we like to call ‘wizards’—to transform them into devices of awesome complexity and power.

Guénon agrees with Plato that mathematics can furnish symbols for metaphysics. But it cannot fulfill this function if metaphysical principles in themselves are not understood—which they clearly are not by the vast majority of modern mathematicians and scientists. Guénon sees Leibnitz, discoverer of the infinitesimal calculus and major critic of Descartes, as a significant modern philosopher of mathematics, whose insights are worth placing in a metaphysical context—something Leibnitz himself tried to do. Guénon helps make these efforts more metaphysically intelligible, and transposes them to a higher level. In *Metaphysical Principles of the Infinitesimal Calculus*, René Guénon points the way toward a rediscovery of mathematics as the handmaiden of metaphysics.

### Table of Contents

Infinite and Indefinite—The Contradiction of 'Infinite Number'—The Innumerable Multitude—The Measurement of the Continuous—Questions Raised by the Infinitesimal Method—'Well-Founded Fictions'—'Degrees of Infinity'—'Infinite Division' or Indefinite Divisibility—Indefinitely Increasing and Indefinitely Decreasing—Infinite and Continuous—The 'Law of Continuity'—The Notion of the Limit—Continuity and Passage to the Limit—'Vanishing Quantities'—Zero is not a Number—The Notation of Negative Numbers—Representation of the Equilibrium of Forces—Variable and Fixed Quantities—Successive Differentiations—Various Orders of Indefinitude—The Indefinite is Analytically Inexhaustible—The Synthetic Character of Integration—The Arguments of Zeno of Elea—The True Conception of 'Passage to the Limit'

### About the Author

**René Guénon** (1886–1951) was one of the great luminaries of the twentieth century, whose critique of the modern world has stood fast against the shifting sands of intellectual fashion. His extensive writings, now finally available in English, are a providential treasure-trove for the modern seeker: while pointing ceaselessly to the perennial wisdom found in past cultures ranging from the Shamanistic to the Indian and Chinese, the Hellenic and Judaic, the Christian and Islamic, and including also Alchemy, Hermeticism, and other esoteric currents, they direct the reader also to the deepest level of religious praxis, emphasizing the need for affiliation with a revealed tradition even while acknowledging the final identity of all spiritual paths as they approach the summit of spiritual realization. His greatest contributions are a blindingly lucid exposition of the principles of orthodoxy and traditional metaphysics, an uncompromising critique of the deviation of modernism, and a breath-taking view of the polyvalence of traditional symbols. Implicit in these three *genres*, as in all Guénon's writing, is the need for personal affiliation with an orthodox tradition as a precondition for a bona fide spiritual practice that might lead, at least in principle, to the intellectual intuition of which he speaks. Little known in the English-speaking world till the recent appearance of his *Collected Works* in translation, Guénon has nevertheless long been recognized as a veritable criterion of truth by a vanguard of remarkable writers who evince that rare combination: intellectuality and spirituality. Regarded by leading scholars as the first truly authentic interpreter of many Eastern doctrines in the West, Guénon never tired, in face of the seemingly inexorable process of dissolution in the twentieth century, of pointing to the transcendent unity of all religious faiths and the abiding Truth that contains them all.