Seth Ananda

Pythagoras, Neoplatonism, Maths, Geometry.

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Hence Arithmetic is the source of that preestablished harmony between reality

and language that we can not not believe after almost four centuries of astonishing

achievements, but we must even say that, neither tendentially, syntactic

representation can thoroughly mirror reality, become someway iconic. And this

because it is marked in its basic principles with a preestablished disharmony, that

is even its hidden evolutive principle.

It plays the role of source of never ending paradoxes well recognizable ever since

the beginning of formal thinking. Negation, truth and being ground an

antinomical argument, from the “negative judgement paradox” (impossibility of

asserting falsity), through the “liar paradox” (contradictory nature of self-asserting

falsity), to set-theoretical paradoxes and to Gödel's and Tarski's limitative




MATHEMATICS. (Dipartimento di Matematica, Università di Bari).


Oh yeah - so my research that I first sent to Borzacchini - I then sent it to math professor Joe Mazur and he said it was very impressive and I should submit it to the most read math journal - the MAA.


I did - but again I am critiquing the very foundation of math from the perspective of music theory! So the article was rejected without comment and then Mazur recommend I try a Indian History of Science journal.

Edited by pythagoreanfulllotus

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Not to continue the derail of Seth's thread here, but I feel it is worth speaking to the perspective that has been brought up by pythagoreanfulllotus.


Ignoring the body of experience to be found with Zhongyongdaoist with both Plato's works and his familiarity with Kingsley, which I feel is a foolish choice to make, I have looked for outside critiques of Kingsley's work. In the course of my attempt to vet the things that Kingsley has claimed, and on which pythagoreanfulllotus has based his argument, I came across a critique of Kingsley's work by Dr. John Bussanich, a Professor of Humanities at the University of New Mexico. He points out a number of inconsistencies of handling of history that Kingsley commits, as well as assumptions that he relies on the reader accepting to make his case. A number of these assumptions go against commonly accepted interpretations of historical texts. The paragraph that most brought to light the problems with accepting Kingsley as the sole reason to discredit the work of Plato and the Neoplatonists is here:


The full article can be found here:

The writer's credentials can be found here:


Hopefully this will put to bed the continued postings quoting Kingsley's work and derailing Seth's original request for information and allow for some more constructive replies.



This person is assuming to know Kingsley's intentions - and is practicing ad hominems - and then he projects the wrong definition of Metis onto Kingsley!


I have already quoted this discussion of Metis above and yet this writer is projecting the Platonic definition of Metis onto Kingsley!


I quoted above already that Plato defined metis as "cunning reason" but that for the orthodox Pythagoreans Metis did not mean to use words in a cunning way that is misleading.


So this author then takes the Platonic definition of Metis and turns it around to say this is what Kingsley is doing. It is an ad hominem attack that assumes to know Kingsley's intentions or motives and obviously Kingsley would disagree with being called this.



Kingsley...he’s cunning--like his Orphic masters, or so he claims. ...Invoking mētis, the archaic,magical, cunning intelligence he admires, Kingsley aims to shock and outrage scholars, to drive most of them away and, at the same time, to seduce the divinely foolish among us."


So the writer is claiming Metis means writing things down in a cunning way that tricks the reader -- a reader who also is already foolish.


This type of reasoning is terrible!! It makes assumption based on assumption based on a psychological projection by the writer onto Kingsley as an ad hominem.



which confirms that his interest in divination by dreams is not purely academic.


He says this is "exasperating." Why? He writes as if the above is a bad thing.....Lots of creative people get their insights from dreams - there's a whole book about leading scientists who have solved problems by dreams - the Creative Process.



I will return to Kingsley’s claim that mystical experience requires myth and esoteric initiation both for its attainment and for communicating it.


Again this is faulty logic - Kingsely does not make this claim!! The writer is projecting onto Kingsley.




Unfortunately, Kingsley doesn’t tell us where he stands amid the disputes of anthropologists and historians of religions about which are the essential and which the peripheralfeatures of shamanism. He is content to cite various phenomena and practices: descending to and

returning from the underworld, reviving the dead, removing souls from bodies, controlling theweather, knowing how to use the power of plants, prophecy, law-giving, magical healing,bilocation, and possession by gods or other powers which induces ecstatic states. But he does notspecify whether the shaman’s soul leaves his sleeping, inert body or whether a god or spiritenters and displaces the shaman’s consciousness—or both. The lack of a clear taxonomy of possession that applies to all these phenomena is a serious handicap even for sympathetic readers.


This whole paragraph completely negates what the author just quoted about Kingsley previously! Kingsley is stating that the Pythagoreans have a philosophy of nondualism!! The author then states that Kingsley should point out which practice was their main focus and why - this is a total misunderstanding of nondualism.

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O.K. so he later quotes Kingsley on Metis and then the author states:




This is not the standard picture of

spiritual masters that emerges in religious literature!


This again is a projection onto Kingsley - he is twisting the meaning of what Kingsley wrote - the author is trying to claim Kingsley is promoting something that is with bad intentions when the opposite is true!!


The real meaning of Metis is based on nondualism which the author does not understand.



magical manipulation for selfish ends


This is what the author considers to be Metis - but that is the wrong Platonic view of Metis as I have already documented about.



Mētis was reintroduced, after long obscurity, into academic discourse by two experts on Greek mythology, Marcel Détienne and Jean-Pierre Vernant. The English translation of mētis in their work is “cunning intelligence”. Toulmin does not shy from the negative connotations of the word cunning, insisting on a neutrality reminiscent of the kind of moral neutrality leadership theorists often refer to in the introductions to their work.


As I mentioned Stephen Toulmin also corrects this error about Metis.



Similarly Toulmin, who had been one on the pioneers in introducing the concept of phronesis into the discourse of sociology, in his book Return to Reason takes up mētis in terms of Michael Polanyi’s differentiation of tacit and explicit knowledge as a way of broadening our understanding of knowing. Toulmin is aware of the difficulty in taking up mythology: “If we have inarticulate, pretheoretical, or untheoretical experience, so be it”. He is fully aware that mētis is “at the pragmatic, non-verbal extreme”.

Edited by pythagoreanfulllotus

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This dichotomous opposition between thinking and magic is a gross oversimplifcation, it seems to me.


So here the author reveals that he really does not understand meditation and visionary states!!


This is my whole point - he wants to claim that the NeoPlatonists are mystics - and that's fine - but on the other hand he thinks you can have thinking and magic at the same time. What kind of "magic" is he talking about?


He just quoted Kingsley in a very misleading manner:



theurgic ritual and magic are “the culmination of philosophy, as away of attaining what thinking alone was incapable of ever achieving: the raising of men to thegods and the divinization of the soul” (APMM 302).


Really? Why did the author not just fully quote Kingsley and instead the author adds magic to the Kingsley quote. That is not a sign of good scholarship. We have to just take the author's word for it when it would have been using the same number of words to quote Kingsley. The author has not provided us with any evidence to support his claim.



Kingsley’s valorization of magic...


Again this is a total projection on the part of the author based on his misunderstanding of nondualism - Kingsley does not place magic in some special position and yet this is what the author is misleadingly claiming.

Edited by pythagoreanfulllotus

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Numenius’ critique of the Academy and its scholastic disputes are enlisted in the attempt to identify the pure gold of Pythagorean teaching

(304), while neglecting Numenius’ point that the differences between Pythagoras and Plato were matters of style not doctrine and that pure Platonism is Pythagoreanism (O’Meara 13). Assessing the varying attitudes towards Plato among the Neoplatonists is beyond the scope of this essay. Suffice it to say that Kingsley bends and twists the evidence to support his contention that Plato is a mere cog in the Pythagorean transmission machine, and, in fact, a cog with broken teeth, so to speak, which causes it to sputter and creak for centuries.



This paragraph is twisted -- a complete warping of Kingsley!! Why would Kingsley use scholastic disputes to identify the pure gold in Pythagorean teaching? Kingsley teaches nondualism which can not be taught using words, much less, disputes. That is Kingsley's whole point about Platonic philosophy as I already quoted - because it is focused on this wording as disputes it misses the focus on the practice of meditation without words.


So then the author says Kingsley neglects Numenius stating that pure Platonic philosophy is the same as Pythagoreanism - this is a strawman argument! Kingsley never states that scholastic disputes provide the "gold" of Pythagorean philosophy and so how could Numenius be correct about Plato being Pythagoreanism?


So then after the author has twisted Kingsley around and warped him completely the author then claims this is what Kingsley does. haha. It's hilarious. What is his evidence? it's "beyond the scope of this essay." Nice deferral to no evidence!!

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Early and Late Pythagoreans surpass Plato, Aristotle, and their successors in their “emphasis not only on practical application but also on detailed knowledge and observation as means to achieving understanding and mastery of cosmic principles, involvement in magic and ritual, and the attachment of primary importance to healing in all its facets.”


So this is really the main crux of the whole argument against Kingsley - and the author here is correct to perceive the double standard that Kingsley is applying - seeing rationalism of Platonic philosophy as bad and then saying that the PreSocratics used their magic for "practical application" --



they will strike some as tendentious,


"Expressing or intending to promote a particular cause or point of view, esp. a controversial one:"


So what is this particular cause that Kingsley is promoting?


The practical application in the PreSocratics is fundamentally different then the Late Pythagoreans.


I have already quoted Kingsley's Ph.D. stating that Archtyas - a "Late" Pythagorean - "radically reinterpreted" Pythagorean philosophy so as to give it a materialist meaning!!


So the author here has made the error again of grouping together "Early" and "Late" Pythagoreans as if they were equivalent.


Notice again he does not actually quote Kingsley on this - he adds Early and Late to the Kingsley quote.


This is a fundamental point that misleads the readers.


So the whole point is that Archytas was not a real Pythagorean and yet the author here claims he is a "Late" Pythagorean.

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Remember I have already discussed this!!


Both Plato and Archytas were relying on Philolaus and therefore were not “orthodox”


Orthodox Pythagoreans did not use 9/8, not in the sense of Archytas. Why? Because the ratio 9/4,
reduced to 9/8, is not of the Pythagorean Tetrad based on the “orthodox” perfect fifth “Great
Dragon Tuning.”



Orthodox Pythagorean theory recognizes five consonances: fourth, fifth, octave,
twelfth, and double octave; and these are represented by the multiple and
superparticular ratios [n + 1 : n] from the tetrad. The number 8 obviously does not
belong to the tetrad.


André Barbera, "The Consonant Eleventh and the Expansion of the Musical Tetractys: A Study of
Ancient Pythagoreanism," Journal of Music Theory, 1984.


And so Archytas and Plato were not real Pythagoreans.

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I really just want to find a simple book on maths or geometry that show the Mystical relationship from a Neoplatonist or Pythagorean perspective and that starts simply.


Seth I didn't understand what you were looking for from your first post. My interest since high school has been in the application of modern mathematics to ancient philosophy and I recommended Ouspensky's treatment of four dimensional space for that reason. Aside from that I don't care that much for Ouspensky either and have sometimes both regretted that he met Gurdjieff and wondered what he might otherwise have achieved.


In the early eighties two works stimulated an interest in ancient arithmetic, one pythagoreanfulllotus has mentioned and that is McClain's The Pythagorean Plato and and the other Oscar Marcel Hinze's Tantra Vidya.


McClain has a website where PDFs of his two fundamental works, the aforementioned Pythagorean Plato and The Myth of Invariance can be downloaded. It is here:


To my surprise Tantra Vidya is still available:


I also found Greek Mathematical Philosophy by Edward A. Maziarz and Thomas Greenwood a useful supplement. It is more academic in its orientation, but fills in some interesting details. These are just some of the books that I read at the time. Unfortunately I am not familiar with a single book. The above books would be a good, stimulating beginning.


This is in a sense an indirect reply to pythagoreanfulllotus' gleeful taunt:


So you have not mentioned Archytas? haha. Apparently you are unfamiliar with the mathematics of Plato.


Maybe you want to read Ernest McClain's book the Pythagorean Plato.


I already have, thirty or so years ago.


Perhaps if pythagoreanfulllotus could give us more information on what exactly lead hm to this conclusion:



When I was in 10th grade in my "enriched geometry" class I never accepted the Pythagorean theorem as true because I knew the true music Pythagorean origins for it.



It might be easier to understand him.


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A good source material on the Pythagorean teachings (I have in my own bookshelf) are "The Pythagorean Source book and Library" by Kenneth Sylvan Guthrie. Is more like an anthology that complies different source-texts from these historical times thus giving a fairly objective view on some aspects of the teachings. It is also includes the book "Life of Pythagoras" that is a kind of Biography over his life.


I have read a lot about this stuff, but long time ago, so I have not everything fresh in my mind so forgive me if I am a bit off ;)


But my distinct feeling about it is that the Pythagoreans were really a great authentic spiritual movement, maybe the only one in Greece around this times. They were practical people, much like the Essenes, lived in harmony with everybody and nature, they were vegetarians, and they were focused on deep states of meditation and healing therapies that involved chantings and music. They worked as efficient energy-cultivators, much like the Daoists and contemporary Buddhists (in India about same time). They encountered much friction and difficulties in dealing with some authorities (that couldn't accept some of their radical and free views) that helped them in their process of selfobservation and selfdiscovery. Pythagoras is now associated much with mathematics and metaphysics, but this is only one side of his movement. However the teachings of the significance behind numbers, geometry, energy-ratios, sacred ratios (like the Golden), angulation, was an important part of the teachings.


To my opinion this movement was powerful like the Buddha-movement in India about the same time. However less is known about it.


Later on the teachings were compiled by different "rational philosophers" like Plato, Aristotele, etc., that had a more intellectual left-brain-sided approach to it. From there on the Greek philosophy started with its Aristotelian "classificative, deductive logic" / system that really ruined the mystic and magic side of anything.

Edited by Jonas
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Let me add a little more to my previous post... Like you I myself have always been interested in finding genuine people that follow the original teachings of Pythagoras but not found any. However to understand Pythagoras I think there is a need to differ it from neo-platonism, since it was added a lot of speculations and parallels later on from other sources, and especially in todays newage-versions of it.

One of the things the Pythagoreans were aware of are the energy-forming aspects of Sound.

In today times Robert Monroe investigated the power of sound on the psyche with his Hemi-Sync-system, also Hans Jenny and his Cymantics are worth to mention and Google. You can see some real videos and experiments on how sound affects (creates in) matter. (Just Google it on you tube).

In ancient cultures the Word and Sound were associated with a mouth / seed, as the initial point of any creation. You state your intent with Words and then something is created. That is the foundation (FIRST-ONE) point in all magic.

Also relations of tones, harmonies, creates different states in ourselves. We listen to a sad melody and become sad, and opposite. Words and music carries information and geometry, but also geometry carries information (words) in the reverse aspect. Contemplation on the Mandelbrot Fractal has made me discover some esoteric principles and parallels in there. This fractal is never "outdated" or a "fashion-thing" since it comes from true mathematics, a simple formula like z = z^2+p and is the same in the whole Universe (that's why mathematical language is so pure and universal). Pythagoreans divided number in masculine and feminine (odd and even). If they had known about imaginary numbers and fractals I am sure they would have been very fascinated by all of that.

You ask for some contemporary views on it, but be careful, just Pythagoreanfullots states... watch out for home brewed newage-materials.

In my understanding the Pythagoreans took on themselves a lot of hardships, like working hard in the soil, moved around a lot (didn't make themselves comfortable in great houses), observed Mouna (silence), Meditated regularly in early mornings and evenings, and listened to lectures by seniors. They observed very high morality and ethics. They studied mathematics, relations, music, scales, the laws of frequencies.

I highly recommend the quality-book "Sacred Geometry" by Robert Lawlor. I had read it about three times, and still I reflect upon the lessons therein. Its a great book and can be sound on Scribd. Its not overly esoteric, quite scientific and objective, and that's why I like it!

A good site about some Pythagorean / Greek info is

There is also reference to the Pythagorean Pentacle (Pentagram) and a lot of useful information.

Edited by Jonas

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Well, the math today is way more advanced and I am curious what would  Pythagoras or the other ancient teachers say about the mystical implications of our mathematical knowledge.

Some books that you may find interesting:


Tuning, timbre, spectrum, scale - W. Sethares (music related, there are discussions about Pythagoras and his scale, also other traditional JI scales in it)


Kappraff, J. "Connections: The Geometric Bridge between Art and Science

Kappraff, J. “Beyond Measure: Essays in Nature Myth, and Number  (This guy is math professor and  has many other similar books, but I've only these two and they are very good.)


The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics - C. A. Pickover

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Ken Wheeler seems to me, a strong Neo-Platonic Metaphysician.  Not to mention a profound translator of greek, latin and ancient Pali. 


I find his musings on the metaphysics of the nature of light and his Pali translations (particularly those of the origins of citta and self) to be stunning and wonderful to engage with, they also resonate with my own experience regarding same...  his channel theoria apophasis is worth a perusal.

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Well, your topic caught my interest, and my reply here is probably not what you are looking for, but it might be related food for thought that might help:

A few years ago I discovered, or re-discovered a possibly lost formula for the area of a Right Triangle. At the time I thought that it must be that somebody else knew this previously, because it is so simple. This lead me to search online, which to my dismay, I have not yet found evidence, testimony or online anything stating that even anyone in ancient history knew this formula, and I am not going to pay anybody to do a literature search. It probably exists as an exercise in an old math book somewhere, but so far, as I said, I haven't found it.

This led me to investigate Euclid's Elements and the interesting history of the attempts to solve the Delian problem - doubling the volume of a cube using "pure" geometry. The problem was in fact solved more than 2000 years ago by Greek mathematicians, but Plato criticized those solutions as being the result of cheating - they were not found using "pure" geometric techniques because they didn't use only a compass and an unmarked straight edge.

From what I read, the Delian problem is considered one of the three most important problems in math in the last 2000 years because attempts to solve it apparently led to other developments. Many major mathematicians of the last 2000 years apparently tried their hand at it, until somebody proved finally that the problem cannot be solved using only a compass and an unmarked straight edge.

I arrived at "NordaVinci's Formula for the Area of a Right Triangle" originally using algebra. Then after researching how the ancient Greeks thought about geometry, I came up with the visual proof by re-arrangement, and still am of the opinion that at  least the Greeks of 2400 years ago must have known this, but have not found any evidence of that. So until I find out who might have actually previously pointed this out, I am calling it "NordaVinci's Formula for the Area of a Right Triangle."

Here are two brief blog posts I wrote about this with pictures:

Edited by NordaVinci

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Posted (edited)

Perhaps the most important learning from Pythagoras is that the numbers are alive.   


I used to build economic mathematical models professionally and I soon learned to feel when numbers were unhappy - and put in the effort required to fix their problem



Edited by Lairg

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