The Metaphysical Principles of the Infinitesimal Calculus (Collected Works of Rene Guenon)

Gunon'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, Gunon 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 Gunon, 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. Ren Gunon (18861951) 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. Title ......Page 5 Copyright ......Page 6 Contents ......Page 9 Editorial Note ......Page 11 The Works of Rene Guenon ......Page 13 Preface ......Page 15 1 Infinite and Indefinite ......Page 21 2 The Contradiction of Infinite Number' ......Page 29 3 The Innumerable Multitude ......Page 33 4 The Measurement of the Continuous ......Page 39 5 Questions Raised by the Infinitesimal Method ......Page 45 6 'Well-Founded Fictions' ......Page 49 7 'Degrees of Infinity' ......Page 55 8 'Infinite Division' or Indefinite Divisibility ......Page 61 9 Indefinitely Increasing; Indefinitely Decreasing ......Page 68 10 Infinite and Continuous ......Page 74 11 The 'Law of Continuity' ......Page 78 12 The Notion of the Limit ......Page 83 13 Continuity and Passage to the Limit ......Page 88 14 'Vanishing Quantities' ......Page 92 15 Zero is not a Number ......Page 97 16 The Notation of Negative Numbers ......Page 103 17 Representation of the Equilibrium of Forces ......Page 109 18 Variable and Fixed Quantities ......Page 114 19 Successive Differentiations ......Page 117 20 Various Orders of Indefinitude ......Page 120 21 The Indefinite is Analytically Inexhaustible ......Page 125 22 The Synthetic Character of Integration ......Page 129 23 The Arguments of Zeno of Elea ......Page 134 24 The True Conception of'Passage to the Limit' ......Page 138 25 Conclusion ......Page 142 Index ......Page 145

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, Guénon 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.

Infinite & indefinite The contradiction of 'infinite number' The innumerable multitude The measurement of the continuous Questions raised by the intinitesimal method The 'well-founded fictions' 'Degrees of infinity' 'Infinite division' or indefinite divisibility Indefinitely increasing & indefinitely decreasing Infinite & continuous The 'law of continuity' The notion of the limit Continuity & passage to the limit The 'vanishing quantities' Zero is not a number The notation of negative numbers Representation of the equilibrium of forces Variable & 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'. “After a careful reading...I believe that your book [Religion in the Modern World] is exceptionally good.... It is most important first of all to understand deeply and live one’s own tradition, not confusing it with what is foreign to it, if one is to seriously appreciate other traditions and distinguish in them what is close to one’s own and what is, perhaps, irreconcilable with one’s own.... I am very grateful for your important and thoughtful book, and I am sure you can see I am in the deepest possible sympathy with your views.” --Thomas Merton, from his correspondence with Lord Northbourn PROCEEDING IN A MANNER INVERSE to that of profane science, and in accordance with the unchanging perspective of all traditional science, we must before all else set forth the principle that will allow us almost immediately to resolve the difficulties to which the infinitesimal method has given rise, without letting ourselves be led astray by potentially interminable discussions, as indeed happens in the case of those modern philosophers and mathematicians who, by the very fact that they lack this principle, have never provided a satisfactory and definitive solution to these difficulties.