Namadev
jmd said:
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Neo-pythagorism and Tarot : Alain Bougearel's theory
Namadev
Hi all,
Well, the pythagorean strcucture has araised a debate marvelously resumed by JMD in another thread.
Maybe will you like to find the different elements I offered on :
http://www.lejournaldedemain.com/article.php3?id_article=736
Well, the pythagorean strcucture has araised a debate marvelously resumed by JMD in another thread.
Maybe will you like to find the different elements I offered on :
http://www.lejournaldedemain.com/article.php3?id_article=736
jmd
I personally have no doubts that various considerations were at play in the milieu in which Tarot arose.
Specifically, I too consider that it arises in a very syncretic neo-platonic and neo-pythagorean context.
With regards to the linked article (in French), Ficino, Pico della Mirandola and others, and their respective reverence towards the very deemed 'sainthood' of such earlier 'semi-gods' as Hermes Trismegistus, Zoroaster, Moses and Orpheus - each of whom were considered to somehow be implicated in the later incarnation of Christ in the body of Jesus - is also of note.
That 78 is a base-12 triangular number, that 22 may be placed in an 'incomplete' top-view four-sided pyramid, that so many of the numbers have a relation to base-4 geometrical figures, and that each of the 'key' numbers evident in Tarot may be displayed in a triangular number exhibited as a semi-square (an important consideration which is also partially lost when the triangular 78 is displayed in the also expected equilateral triangular form) - to each of these there is no doubt.
It should be noted that the number 22 is not itself pyramidal (the closest pyramidal number is 20) - though would have also been seen to have the same visual pyramidal impact as depicted on Namadev's image.
We therefore seem to be both agreeing and disagreeing on various points.
The first is whether the structure of Tarot is reflected in Namadev's discovery. In that I personally see little doubt: Tarot's structure may indeed be presented in this highly instructive geometrical way.
In the thread Interesting comment by Ronald Decker [which I'll have to take some time in the near future and possibly split into two threads - when I have a little more time], I also attach Namdev's sketch.
As a modern neo-pythagorean reflection, it is wonderful and meriting much further reflection.
That is not, from my perspective, at question.
What I nonetheless remain totally unconvinced by, however, is that this very model was also somehow direcly involved in Tarot creation.
That pythagorean, neoplatonic, hermetic, kabalistic and myriad other considerations were living at the times under consideration is clear. That this specific model also was, however, is far from evident - and I guess to me still rings as unlikely.
It does provide, however, avenues for further research - and that is important.
Each of the numbers involved (4, 10, 12, 26, 22, 28, 36, 55, 56, 72, and 78) were (and are) of course, important - and we may find either text or image which reflects such... but not in any way similar to that given by Namadev.
There is, then, a difference between the important contribution and discovery given by Namadev which reflects similar considerations which were also deemed important in earlier times (and which some of us similarly value in the present), and whether this specific way of depicting the 78 into a diagonally-severred-square (rather than equilateral) triangle, and its components breaking into those depicted 4 x 10, 16, and 22 parts was considered by anyone involved in Tarot's 'birthing'.
Specifically, I too consider that it arises in a very syncretic neo-platonic and neo-pythagorean context.
With regards to the linked article (in French), Ficino, Pico della Mirandola and others, and their respective reverence towards the very deemed 'sainthood' of such earlier 'semi-gods' as Hermes Trismegistus, Zoroaster, Moses and Orpheus - each of whom were considered to somehow be implicated in the later incarnation of Christ in the body of Jesus - is also of note.
That 78 is a base-12 triangular number, that 22 may be placed in an 'incomplete' top-view four-sided pyramid, that so many of the numbers have a relation to base-4 geometrical figures, and that each of the 'key' numbers evident in Tarot may be displayed in a triangular number exhibited as a semi-square (an important consideration which is also partially lost when the triangular 78 is displayed in the also expected equilateral triangular form) - to each of these there is no doubt.
It should be noted that the number 22 is not itself pyramidal (the closest pyramidal number is 20) - though would have also been seen to have the same visual pyramidal impact as depicted on Namadev's image.
We therefore seem to be both agreeing and disagreeing on various points.
The first is whether the structure of Tarot is reflected in Namadev's discovery. In that I personally see little doubt: Tarot's structure may indeed be presented in this highly instructive geometrical way.
In the thread Interesting comment by Ronald Decker [which I'll have to take some time in the near future and possibly split into two threads - when I have a little more time], I also attach Namdev's sketch.
As a modern neo-pythagorean reflection, it is wonderful and meriting much further reflection.
That is not, from my perspective, at question.
What I nonetheless remain totally unconvinced by, however, is that this very model was also somehow direcly involved in Tarot creation.
That pythagorean, neoplatonic, hermetic, kabalistic and myriad other considerations were living at the times under consideration is clear. That this specific model also was, however, is far from evident - and I guess to me still rings as unlikely.
It does provide, however, avenues for further research - and that is important.
Each of the numbers involved (4, 10, 12, 26, 22, 28, 36, 55, 56, 72, and 78) were (and are) of course, important - and we may find either text or image which reflects such... but not in any way similar to that given by Namadev.
There is, then, a difference between the important contribution and discovery given by Namadev which reflects similar considerations which were also deemed important in earlier times (and which some of us similarly value in the present), and whether this specific way of depicting the 78 into a diagonally-severred-square (rather than equilateral) triangle, and its components breaking into those depicted 4 x 10, 16, and 22 parts was considered by anyone involved in Tarot's 'birthing'.
Huck
Tarot existing in a 4x14 + 22 - structure prior to 1423 cannot be proven and is far away to be likely.
In 1423 the deck with 4x14-cards existed - so we know from San Bernardino. Perhaps we may assume, that this game-structure existed with some dominance in all Italy.
When somebody had pythagorean ideas, when he increased the number of trumps or created a trumps-suit .... then it happened after 1423 and he didn't create the total structure in one attempt, but added just to an already existent structure (4x14).
The reason for the choice of the specific number of trumps does not know only a "pythagorean interpretation", but also others. Boiardo was near to oriental languages (with 22 letters), Galeazzo Maria's life had a personal 22-years-phenomen. Both persons are near to Trionfi-production and we've evidence of their actions.
Also near is the phenomen of the throw of 2 dices, which contains "21 results". Dices belong to a game, Tarot is also a game. We know already of very early Domino-cards in China. As Domino uses the same mathematical structure as 2 dices ... no surprize.
Against this 3 near ideas a consideration of general pythagorean phenomena (1+2+3+4+5+6) or-and (1+2+3+4+5+6+7+8+9+10+11+12), especially when they are accompanied by complicated ideas, seems to be an "overkill" of the problem. 4x14 already existed ... and the simple aim was just to add an unspecified number, not more.
In the standard Tarot we got a 21+1, in the Minchiate a 40+1, in other playing card forms we see a reducement of the number cards and in them the 4x14-structure (and with them the Pythagorean idea) is dead anyway.
The 5x14-variation, which is the likely predevelopment for the standar Tarot version, didn't know the idea also.
In 1423 the deck with 4x14-cards existed - so we know from San Bernardino. Perhaps we may assume, that this game-structure existed with some dominance in all Italy.
When somebody had pythagorean ideas, when he increased the number of trumps or created a trumps-suit .... then it happened after 1423 and he didn't create the total structure in one attempt, but added just to an already existent structure (4x14).
The reason for the choice of the specific number of trumps does not know only a "pythagorean interpretation", but also others. Boiardo was near to oriental languages (with 22 letters), Galeazzo Maria's life had a personal 22-years-phenomen. Both persons are near to Trionfi-production and we've evidence of their actions.
Also near is the phenomen of the throw of 2 dices, which contains "21 results". Dices belong to a game, Tarot is also a game. We know already of very early Domino-cards in China. As Domino uses the same mathematical structure as 2 dices ... no surprize.
Against this 3 near ideas a consideration of general pythagorean phenomena (1+2+3+4+5+6) or-and (1+2+3+4+5+6+7+8+9+10+11+12), especially when they are accompanied by complicated ideas, seems to be an "overkill" of the problem. 4x14 already existed ... and the simple aim was just to add an unspecified number, not more.
In the standard Tarot we got a 21+1, in the Minchiate a 40+1, in other playing card forms we see a reducement of the number cards and in them the 4x14-structure (and with them the Pythagorean idea) is dead anyway.
The 5x14-variation, which is the likely predevelopment for the standar Tarot version, didn't know the idea also.
Namadev
jmd said:I personally have no doubts that various considerations were at play in the milieu in which Tarot arose.
Specifically, I too consider that it arises in a very syncretic neo-platonic and neo-pythagorean context.
It should be noted that the number 22 is not itself pyramidal (the closest pyramidal number is 20) - though would have also been seen to have the same visual pyramidal impact as depicted on Namadev's image.
Hi,
Stricto sensu, 22 is a hexagonal pyramidal number.
References in English :
Number Theory Special Numbers Figurate Numbers Polyhedral Numbers
Hexagonal Pyramidal Number
A pyramidal number of the form , The first few are 1, 7, 22, 50, 95, ... (Sloane's A002412). The generating function of the hexagonal pyramidal numbers is :
The sequence is visible at :
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002412
Now, it's true that people normally call "pyramidal" the square pyramidal numbers .
Often, the tetraédical pyramidal numbers are forgotten.
So, maybe could we agree on :
22 is a pentagonal pyramidal number?
alain
Namadev
[Hi,
For those who are completely lost in these puthagorean numbers, a simple introduction is :
http://membres.lycos.fr/villemingerard/Geometri/NbGeomet.htm
For those who are completely lost in these puthagorean numbers, a simple introduction is :
http://membres.lycos.fr/villemingerard/Geometri/NbGeomet.htm
jmd
I stand corrected - 22 is a hexagonal pyramidal number.
I suppose I did restrict myself to tetrahedral (which the closest is 20) and square-based pyramids - simply because these are the two types previously mentioned.
22 is not, however, a pentagonal pyramidal number, but 'simply' a pentagonal one.
All these things mean, for those interested, is the following:
This, however, is quite different to the shape implied and drawn by Namadev in the diagramme previously provided...
I suppose I did restrict myself to tetrahedral (which the closest is 20) and square-based pyramids - simply because these are the two types previously mentioned.
22 is not, however, a pentagonal pyramidal number, but 'simply' a pentagonal one.
All these things mean, for those interested, is the following:
- a triangular, or square, or pentagonal number is one in which marbles (for example) can be arranged in that shape (triangle, square, pentagon);
a pyramidal number is one which builds from a shape upwards to a point. The two most common are the triangular-based pyramid (which in our numbers becomes what is called a tetrahedron), and a square-based pyramid.
This, however, is quite different to the shape implied and drawn by Namadev in the diagramme previously provided...
Namadev
Hi JMD,
We'll both stand corrected then.
So, let's use of the actual consensus for :
22 a pentagonal number drawing a figurative pyramid...
OK?
We'll both stand corrected then.
So, let's use of the actual consensus for :
22 a pentagonal number drawing a figurative pyramid...
OK?
Namadev
Huck said:Tarot existing in a 4x14 + 22 - structure prior to 1423 cannot be proven and is far away to be likely.
***Did I say something else?
See my comments about the TarotL History fact Sheet!
Huck :
In 1423 the deck with 4x14-cards existed - so we know from San Bernardino. Perhaps we may assume, that this game-structure existed with some dominance in all Italy.
When somebody had pythagorean ideas, when he increased the number of trumps or created a trumps-suit .... then it happened after 1423 and he didn't create the total structure in one attempt, but added just to an already existent structure (4x14).
The reason for the choice of the specific number of trumps does not know only a "pythagorean interpretation", but also others. Boiardo was near to oriental languages (with 22 letters), Galeazzo Maria's life had a personal 22-years-phenomen. Both persons are near to Trionfi-production and we've evidence of their actions.
Also near is the phenomen of the throw of 2 dices, which contains "21 results". Dices belong to a game, Tarot is also a game. We know already of very early Domino-cards in China. As Domino uses the same mathematical structure as 2 dices ... no surprize.
***Who said that the dominos theory wasn't interesting? I? On the contrary : see my posts on tarotL and LTarot.
***When did I reject the dices theory as exposed by the pythagorean John Opsopaus?
Huck :
Against this 3 near ideas a consideration of general pythagorean phenomena (1+2+3+4+5+6) or-and (1+2+3+4+5+6+7+8+9+10+11+12), especially when they are accompanied by complicated ideas, seems to be an "overkill" of the problem. 4x14 already existed ... and the simple aim was just to add an unspecified number, not more.
In the standard Tarot we got a 21+1, in the Minchiate a 40+1, in other playing card forms we see a reducement of the number cards and in them the 4x14-structure (and with them the Pythagorean idea) is dead anyway.
The 5x14-variation, which is the likely predevelopment for the standar Tarot version, didn't know the idea also.
***Yes, this is a probable and simple explication that I consider also.
Why not?
Now, I prefer to go on with my own theory even if I respect those of others..
BTW, I've gathered some data.
I had written on Tarot-Fr :
"En fait, la connaissance de cette divine proportionne ne saurait donc être seulement datée du franciscain Lucio Pacioli et de son traité "Divina proportione": il dit lui-même avoir pleine conscience d'une science très secrète "secretissima scientia".
http://tarots.free.fr/structure-fr/epilogue.htm
What is interesting is that modern scholars suspect Luca Pacioli of haven borrowed his pythagorean knowing to Piero della Francesca...
1)Luca Pacioli
http://www.ac-poitiers.fr/arts_p/b@lise14/pageshtm/page_4.htm
Ce tableau, de Jicopo de Barbari, où Fra Luca Pacioli explique un théorème, fait apparaitre le partage " en extrème et moyenne raison " (la " divine proportion ").
On y retrouve en effet, le nombre d'or : Si E est la projection orthogonale sur (D C) de l'extrémité de l'index de la main gauche du moine on a : DC / DE = .
Par ailleurs, le pouce et l'index gauches de Fra Luca Pacioli partage la hauteur du livre selon la section dorée.
Fra Luca Pacioli : moine franciscain et mathématicien (1445 - 1517 Rome) . Il a écrit en 1498 le livre De Divina Proportione, consacré au nombre d'or, ses propriétés mathématiques, ses attributs esthétiques et même certains aspects mystiques .
2)http://www.loc.gov/exhibits/vatican/math.html
Piero della Francesca,1416-1492 De quinque corporibus regularibus
In Latin
Parchment
1480s
The early Renaissance artist Piero della Francesca developed a mathematically rigorous system of perspective on which he wrote the treatise De prospectiva pingendi. His interest in mathematics increased as he grew older and late in his life he wrote two other treatises, a Trattato d'abaco, on algebra and the measurement of polygons and polyhedra (solids), and "De quinque corporibus regularibus," on the five regular polyhedra, which survives only in this unique manuscript from the library of the Duke of Urbino. The figures are said to be by Piero himself. Shown here are the inscriptions of an icosahedron (a solid composed of twenty equilateral triangular faces) in a cube, and of a cube in an octahedron (a solid of eight equilateral triangular faces).
3)**A first survey of Plato's solids
http://www.dartmouth.edu/~matc/math5.geometry/unit6/unit6.html
jmd
Again, there is no doubt that the regular or platonic (and the lesser known 13 archimedean) solids, the golden mean (Phi), and various inter-relation they have formed (and forms) part of important geometrical considerations.
I fail to see, however, how these specifically relate to the thesis.
For those interested, the bottom part of the logo for the Association for Tarot Studies (click on my 'website' button below) is taken from Dürer's partially truncated hexahedron (ie, cube) as presented in his Melencolia I. That very image also has a Jupiter 'magic' square there-on.
Yet each of these - magic squares, platonic and archimedean solids, and triangular, square, pentagonal or other numbers considered or discovered by geometrical means - though highly important in our own further deepening of understanding of both the world and of the mediaeval milieu in which Tarot arose again bears importance in what may have been at play in the construct of either individual design or the whole.
But the specific motif Namadev presents seems a later mathematical discovery of the inter-relation existing within the whole (the 'holon' of Tarot, if you like) than one which was one of its driving force.
What is highly fascinating is that the basic relative dimension of the decks available generally FAIL to exhibit what may have been expected: golden rectangular proportions.
Rather, they seem to suggest either one of two possibilities: the more highly probable double square; and the less likely but also probable relation of (approx.) 6.7:12 - ie, the overall relation as depicted by the height-to-width ratio of Namadev's pattern.
Now that aspect may, perhaps (and to my reflections), be more important than the internal divisions Namadev provides.
I fail to see, however, how these specifically relate to the thesis.
For those interested, the bottom part of the logo for the Association for Tarot Studies (click on my 'website' button below) is taken from Dürer's partially truncated hexahedron (ie, cube) as presented in his Melencolia I. That very image also has a Jupiter 'magic' square there-on.
Yet each of these - magic squares, platonic and archimedean solids, and triangular, square, pentagonal or other numbers considered or discovered by geometrical means - though highly important in our own further deepening of understanding of both the world and of the mediaeval milieu in which Tarot arose again bears importance in what may have been at play in the construct of either individual design or the whole.
But the specific motif Namadev presents seems a later mathematical discovery of the inter-relation existing within the whole (the 'holon' of Tarot, if you like) than one which was one of its driving force.
What is highly fascinating is that the basic relative dimension of the decks available generally FAIL to exhibit what may have been expected: golden rectangular proportions.
Rather, they seem to suggest either one of two possibilities: the more highly probable double square; and the less likely but also probable relation of (approx.) 6.7:12 - ie, the overall relation as depicted by the height-to-width ratio of Namadev's pattern.
Now that aspect may, perhaps (and to my reflections), be more important than the internal divisions Namadev provides.
