The Metaphysical Principles of the Infinitesimal CalculusGuénon'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. |
Contents
Preface | 1 |
Infinite and Indefinite | 7 |
The Contradiction of Infinite Number | 15 |
The Innumerable Multitude | 19 |
The Measurement of the Continuous | 25 |
Questions Raised by the Infinitesimal Method | 31 |
WellFounded Fictions | 35 |
Degrees of Infinity | 41 |
Continuity and Passage to the Limit | 74 |
Vanishing Quantities | 78 |
Zero is not a Number | 83 |
The Notation of Negative Numbers | 89 |
Representation of the Equilibrium of Forces | 95 |
Variable and Fixed Quantities | 100 |
Successive Differentiations 103 | 111 |
Various Orders of Indefinitude | 106 |
Infinite Division or Indefinite Divisibility | 47 |
Indefinitely Increasing Indefinitely Decreasing | 54 |
Infinite and Continuous | 60 |
The Law of Continuity | 64 |
The Notion of the Limit | 69 |
The Indefinite is Analytically Inexhaustible | 108 |
The Arguments of Zeno of Elea | 120 |
Conclusion | 128 |
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Common terms and phrases
absence of quantity according Acta Eruditorum apply Bernoulli chap conceived conception consequently consideration considered contradiction contrary correspond decrease indefinitely definition Descartes determined quantities discontinuous distinction division domain elements envisage essentially example existence expression fact fictions final term finite fixed and determined fixed quantities fractional numbers geometric Guénon idea implies incomparably increasing indefi indefinite sequence indefinitely decreasing quantities indefinitude indeterminate form indivisible infinite division infinite number infinitesimal calculus infinitesimal method infinitesimal quantities infinitists infinity insofar inverse Jean Bernoulli latter law of continuity least Leibnitz limit logical magnitude mathematical mathematicians means metaphysical moreover multiplicity multitude nature negative numbers never nite nonetheless notion null obviously ordinary quantities passage point of view possible precisely principle pure question ratios reality regard regular polygon Reign of Quantity RENÉ GUÉNON respect result rigorous sense sequence of numbers sequence of whole speak symbol things tion true truly unit variable quantities whole numbers word ZENO OF ELEA zero