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The Metaphysical Principles of the Infinitesimal Calculus
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.
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.
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.
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.
152 pages, Paperback
First published December 27, 2003
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About the author
René Guénon
290 books760 followersRené Guénon (1886-1951) was a French author and intellectual who remains an influential figure in the domain of sacred science,traditional studies, symbolism and initiation.
French biography : http://arlesquint.free.fr/rene%20guen...
http://www.index-rene-guenon.org/
French biography : http://arlesquint.free.fr/rene%20guen...
http://www.index-rene-guenon.org/
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Displaying 1 - 9 of 9 reviews
September 26, 2018
(Video : Wolfgang Smith on René Guénon's Principles of Infinitesimal Calculus)
A rather overlooked book by Guénon unfortunately because of it's special subject - it is in fact an excellent example of linking back an specific field (here, infinitesimal calculus) with it's principles. Thus, errors automatically disappear, and a much more coherent picture emerges.
It can be compared, in a way, to what Ibn Ajiba did in his commentary on the Ajurrumia by demonstrating the link between Arabic grammar subtleties and higher spiritual meanings.
Precise mathematical points are discussed and put in their rightful perspective here. The "infinite"/"indefinite" confusion, the continuous/discontinuous and others. Many of Leibnitz's assertions (also Jean Bernoulli and Pierre Varignon) are here supported or refuted, many ideas here are given a clean shave and are pushed to their limit (!) in order to show their (in)consistency.
"[...] These observations will allow us to understand more precisely in what sense one can say, as we did at the beginning, that the limits of the indefinite can never be reached through any analytical procedure, or, in other words, that the indefinite, while not absolutely and in every way inexhaustible, is at least analytically inexhaustible. In this regard, we must naturally consider those procedures analytical which ,in order to reconstitute a whole, consist in taking its elements distinctly and successively; such is the procedure for the formation of an arithmetical sum, and it is precisely in this regard that it differs essentially from integration. This is particularly interesting from our point of view, for one can see in it, as a very clear example, the true relationship between analysis and synthesis: contrary to current opinion, according to which analysis is as it were a preparation for synthesis, or again something leading to it, so much so that one must always begin with analysis, even when one does not intend to stop there, the truth is that one can never actually arrive at synthesis through analysis. All synthesis, in the true sense of the word, is something immediate, so to speak, something that is not preceded by any analysis and is entirely independent of it, just as integration is an operation carried out in a single stroke, by no means presupposing the consideration of elements comparable to those of an arithmetical sum; and as this arithmetical sum can yield no means of attaining and exhausting the indefinite, this latter must, in every domain, be one of those things that by their very nature resist analysis and can be known only through synthesis.[3]
[3]Here, and in what follows, it should be understood that we take the terms 'analysis' and 'synthesis' in their true and original sense, and one must indeed take care to distinguish this sense from the completely different and quite improper sense in which one currently speaks of 'mathematical analysis', according to which integration itself, despite its essentially synthetic character, is regarded as playing a part in what one calls 'infinitesimal analysis'; it is for this reason, moreorever, that we prefer to avoid using this last expression, availing ourselves only of those of 'the infinitesimal calculus' and 'the infinitesimal method', which lead to no such equivocation."
No need to be good at math here, as this study refers specifically to the principles.
To be noted that Guénon has written a thesis in 1916 called "Examen des idées de Leibnitz sur la signification du Calcul infinitésimal" and that Principles of Infinitesimal Calculus was released in 1946.
Highly recommended. And a definite re-read sometime in the future.
A rather overlooked book by Guénon unfortunately because of it's special subject - it is in fact an excellent example of linking back an specific field (here, infinitesimal calculus) with it's principles. Thus, errors automatically disappear, and a much more coherent picture emerges.
It can be compared, in a way, to what Ibn Ajiba did in his commentary on the Ajurrumia by demonstrating the link between Arabic grammar subtleties and higher spiritual meanings.
Precise mathematical points are discussed and put in their rightful perspective here. The "infinite"/"indefinite" confusion, the continuous/discontinuous and others. Many of Leibnitz's assertions (also Jean Bernoulli and Pierre Varignon) are here supported or refuted, many ideas here are given a clean shave and are pushed to their limit (!) in order to show their (in)consistency.
"[...] These observations will allow us to understand more precisely in what sense one can say, as we did at the beginning, that the limits of the indefinite can never be reached through any analytical procedure, or, in other words, that the indefinite, while not absolutely and in every way inexhaustible, is at least analytically inexhaustible. In this regard, we must naturally consider those procedures analytical which ,in order to reconstitute a whole, consist in taking its elements distinctly and successively; such is the procedure for the formation of an arithmetical sum, and it is precisely in this regard that it differs essentially from integration. This is particularly interesting from our point of view, for one can see in it, as a very clear example, the true relationship between analysis and synthesis: contrary to current opinion, according to which analysis is as it were a preparation for synthesis, or again something leading to it, so much so that one must always begin with analysis, even when one does not intend to stop there, the truth is that one can never actually arrive at synthesis through analysis. All synthesis, in the true sense of the word, is something immediate, so to speak, something that is not preceded by any analysis and is entirely independent of it, just as integration is an operation carried out in a single stroke, by no means presupposing the consideration of elements comparable to those of an arithmetical sum; and as this arithmetical sum can yield no means of attaining and exhausting the indefinite, this latter must, in every domain, be one of those things that by their very nature resist analysis and can be known only through synthesis.[3]
[3]Here, and in what follows, it should be understood that we take the terms 'analysis' and 'synthesis' in their true and original sense, and one must indeed take care to distinguish this sense from the completely different and quite improper sense in which one currently speaks of 'mathematical analysis', according to which integration itself, despite its essentially synthetic character, is regarded as playing a part in what one calls 'infinitesimal analysis'; it is for this reason, moreorever, that we prefer to avoid using this last expression, availing ourselves only of those of 'the infinitesimal calculus' and 'the infinitesimal method', which lead to no such equivocation."
No need to be good at math here, as this study refers specifically to the principles.
To be noted that Guénon has written a thesis in 1916 called "Examen des idées de Leibnitz sur la signification du Calcul infinitésimal" and that Principles of Infinitesimal Calculus was released in 1946.
Highly recommended. And a definite re-read sometime in the future.
April 23, 2021
This book is rather short and it revolves around a single subject: how do we calculate a limit of 'infinite' series (reading the book will explain the quotes).
If you are like me, and modern mathematical notions of zero or infinity seem weird for you, then this is the book for you. Hint: these are not numbers in the general sense of the word, these are more like functions.
Furthermore, Rene Guenon shows how, by treating our subject seriously and watching closely for any unwanted implicit assumptions, it is possible to "deprofanize" any coherent body of ideas, be it a scientific theory, theological work, esoteric doctrine, etc.
I recommend starting with 'The Reign of Quantity' before reading this book.
If you are like me, and modern mathematical notions of zero or infinity seem weird for you, then this is the book for you. Hint: these are not numbers in the general sense of the word, these are more like functions.
Furthermore, Rene Guenon shows how, by treating our subject seriously and watching closely for any unwanted implicit assumptions, it is possible to "deprofanize" any coherent body of ideas, be it a scientific theory, theological work, esoteric doctrine, etc.
I recommend starting with 'The Reign of Quantity' before reading this book.
August 10, 2022
This book explores some of the contradictions found in modern mathematics, which he attributes to the detachment of the subject from the metaphysical principles found in tradition. Apparent contradictions found in concepts such as negative and positive infinity, to state that between two numbers there's an infinite amount of number as such a finite infinite, zero which without principles would be taken as just something that is nothing, negative number aka a number less than nothing etc. He disentangles such a mess of confusion in several ways. For instance, distinguishing between geometry (and concepts such as magnitude) from pure arithmetic, as such stating that in pure arithmetic, there is no continuity but rather when used to represent magnitude and geometry of space and time concepts such as negative and decimal numbers arise, which is all fine and well however when useful conventions become attached to pure arithmetic that's when it becomes messy. Also, h proposes to switch the concept of infinitesimal in calculus by replacing it with indefinite.
Also, the secularisation of the concept of the infinite and detaching it from the absolute and its symbolism brings about the absurdity that one infinite is bigger than the other and that there are infinitely-many infinity bigger than infinity. For instance, infinity plus 1 is bigger than infinity cause if you map it 1 to 1 indefinitely, the prior will be bigger. Still, any number in infinity plus one is found in infinity. To have more than infinity is just absurd and a result of forgetting the principles.
Great book, but haven't read much of the literature about the subject, especially that about the continuum hypothesis, so can't really determine the value of the book.
Also, the secularisation of the concept of the infinite and detaching it from the absolute and its symbolism brings about the absurdity that one infinite is bigger than the other and that there are infinitely-many infinity bigger than infinity. For instance, infinity plus 1 is bigger than infinity cause if you map it 1 to 1 indefinitely, the prior will be bigger. Still, any number in infinity plus one is found in infinity. To have more than infinity is just absurd and a result of forgetting the principles.
Great book, but haven't read much of the literature about the subject, especially that about the continuum hypothesis, so can't really determine the value of the book.
June 5, 2023
This proved to be one of our favorite texts by Guenon! Definitely the most accessible (to us at least) of Guenon's strictly metaphysical works, this is primarily comprised of a critique of Leibnitz. The main point here is that the infinite should not be confused with the indefinite; most problems that Leibnitz runs into in advancing his infinitesimal calculus can be avoided when this confusion is cleared up, though there are also arguments against atomism, clarifying Zeno's paradoxes, and conceptualizing the 'passage to the limit' in a continuous series. Would definitely recommend reading this text BEFORE Guenon's other works in pure metaphysics, especially if you've studied calculus.
May 13, 2024
infinite stelle per questo libro
October 14, 2022
No comment ...
November 27, 2022
You can tell Rene Guenon was a mathematician before he converted to Sufiism. Probably one of the best of his books but also furthest from his usual subject matter.
April 23, 2024
The best book on mathematics, really had an impact on me.
October 4, 2023
It's a very interesting little book. It took me longer than I thought it would, it is a very technical and abstract exploration. One thing is, Guénon certainly seems a bit unfair to poor old Leibniz here; especially considering much of what he cites amount to private letters Leibniz wrote back and forth with his friends, where his embrace of a less-than-rigorous conceptualization of the calculation system in question would certainly have everything to do with conviviality, rather than a facile understanding of the metaphysical implications. I think it's definitely possible Leibniz's concerns actually reached a further horizon than Guénon's. For instance, he takes it as an impossibility that infinitesimals, "indefinite variables" as he puts it, have some real existence. Leibniz, on the other hand, clearly wrestled with exactly how "real" these variables actually are, granting that they are not fully real, but never admitting that they were complete fictions. This suggests that he had some higher understanding of them he did not wish to readily disclose, not that he understood them less than the Scholastics or the Hermeticists.
However, as usual, it is full of little asides and digressions which are chestnuts of interesting and intriguing esoterica. And the man clearly understood mathematics, better than the educators I encountered in public school as a child! I enjoyed it, even though his dogmatism keeps things on a certain set of rails the whole time.
However, as usual, it is full of little asides and digressions which are chestnuts of interesting and intriguing esoterica. And the man clearly understood mathematics, better than the educators I encountered in public school as a child! I enjoyed it, even though his dogmatism keeps things on a certain set of rails the whole time.
Displaying 1 - 9 of 9 reviews