Michael S. Howerd Les mathématiques Divinatoires Jacques Halbronn (en anglais)

Commentaires « binaires » sur les Mathématiques Divinatoires (1983) de Jacques Halbronn (en anglais) Posté par nofim le 6 mars 2024 Binary patterns in games and divination Posted 20th October 2012 by Michael S. Howard This thread is a continuation Huck’s post at viewtopic.php?f=11&t=873#p12778, into more general reflections that don’t quite fit the narrow focus of that thread-title. It is also an attempt to resume, on a more general level, a very interesting discussion from the « 5 x 14″ thread initiated by Marcos (mmfilesi) about tarot, chess, and geomancy, which then branched off into a « Collection Geomancy » thread. I will give links at the proper points.My immediate impetus, besides Huck’s post mentioned, is some thing else. I have been looking at Jacques Halbronn’s 1983 book Les Mathematiques divinatoires. Its first three chapters are about tarot, geomancy, and the I Ching, looking at the structure of each. He sees each of them as binary, and the first two even as « manichean », in that there are fundamental divisions between « good » elements and « bad » ones.TAROTHalbronn has five good vs. five bad in tarot (p. 46). In 4 cases the sum of the numbers of the Marseille cards is 22. In the fifth case, involving the unnumbered Fool, Halbronn assigns the number 11, as the average between two natural and frequent placements, 0 and 22; That number is also the average of all the numbers, with 10 below and 10 above. Positive/Negative 6Lover/16Maison-Dieu 7Chariot/15Devil 9Hermit/13Death 10Fortune/12Hanged Man 11Strength/11Fool The images are also complementary (pp. 48-49). The Lover card has a flash from above resulting in conjunction; the Maison-dieu has a flash from above resulting in rupture. The Hermit wears a heavy cloak, while Death is as nude as he can be. The Chariot shows victory; the Devil is a similar configuration showing enslavement of the losers. Strength shows a person confronting and mastering an animal; the Fool shows the person running away from an animal. The Wheel of Fortune signifies changes; the Hanged Man signifies immobility, a dead end. Halbronn also identifies 7 other pairs that are complementary in the sense that each completes the other in adding up to 22; he doesn’t attach « positive » or « negative » labels to the members of the pair. 21World/1Bateleur 20Judgment/2Popess 19Sun/3Empress 18Moon/4Emperor 17Star/5Pope 14/Temperance/8Justice But I find it tempting to go further than Halbronn. It seems to me that maybe even these seven consist of good/bad pairs, with a few assumptions. First, you have to make all of Pope, Popess, Emperor, and Empress « bad » cards. I think that is a reasonable assumption, if you assume that the Popess = the Church. Humanists mostly endured these figures as evils that they had to endure, and small business people such card makers probably had no use for them at all. Sometimes they were already undifferentiated, although probably for a different reason: the « four papi ». Humanists regularly castigated all four (unless one of them happened to be the superior of the humanist’s patron). By Etteilla’s time, it was acceptable to simply replace all of them (except the Bateleur, who as Magus would have been too dear to his heart) with others that were quite different, which he did even before the Revolution. And second, I think a case can be made for putting Justice, even though a virtue, in the « bad » camp. Justice is bad if taken to an extreme, either without enough mercy or with too much mercy. Temperance, i.e. Moderation, is better, as not only a virtue in itself but as that which guides all the other moral virtues (from Aristotle). Alternatively, if we see the mixing of water with wine as symbolic of the Eucharist, it becomes Mercy, which traditionally complements the severity of Justice. So we get: positive/negative 21World/1Bateleur 20Judgment/2Popess 19Sun/3Empress 18Moon/4Emperor 17Star/5Pope 14/Temperance/8Justice To these one could add the four suits, of which two, coins and cups, are « good », and two, batons and swords, are « bad ». This sometimes shows up in the sequences, which are sometimes reversed, for trick-taking purposes, and it is certainly true for the interpretations, starting with Etteilla, the larger numbers being worse than the smaller ones for batons and swords. I think the same is true of the pictures on the Sola-Busca number cards. Such a structure of the tarot would be in line with the prisca theologia (ancient theology) doctrine of the times (16th-18th centuries), of a pre-Greek, pre-Hebrew theology common to all civilized nations, which the « dualist » Zoroaster was one of the first to put in writing. GEOMANCY Halbronn’s second chapter is about geomancy. That system of divination seems to be an Arab invention. It borrows much of its terminology from astrology. Another source suggests that it may be a « poor man’s astrology », a way of casting a pseudo-horoscope without having to know anything about the stars at the time of the querent’s birth (see the « poor man’s astrology » quote at viewtopic.php?f=11&t=505&hilit=geomancy#p6957), You just need a way of randomly generating a few numbers, and some rudimentary mathematical ability: specifically, being able to add a few single-digit numbers and recognize whether they are are even or odd. After the first ones, moreover, all you’re adding is 1′s and 2′s. I won’t go into the details, as you can read about them anywhere, but here is a digest. The sequence starts with four « mothers ».Then, going row by row, you get a « daughter » configuration by adding the dots in each row of two of the « mothers ». If the result is 1, you put one dot in the « daughter » row; if it’s 2, you put 2 dots. If it’s 3, you put 1 dot; if it’s 4, you put 2 dots. Then you get a « niece » (sometimes « nephew ») by adding dots again, and from them four final figures: two witnesses, a judge, a conciliator. There are exactly 16 possible « binary tetragrams » generated by such means: Image (Image from http://www.renaissanceastrology.com/ast … ncy.html#B) Out of these 16, 8 pairs can be formed, in which the pattern of each is the reversal of the pattern of the other. (In the above, the reversal of the one in the lower right is the one in the lower left; otherwise, they are next to each other. And if you can’t make out the words, don’t worry; I’ll give them as we go.) Not only do the paired designs complement each other geometrically, but in most cases the words associated with the pairs are related and form binary oppositions between « internal » and « external », of which the internal are « good » and the external « bad ». Halbronn says (p. 59) « L’interioritie serait [i.e. were] benefique et l’exteriorite malefique ». (He cites Hadji Kamballah, La Geomancie tradionnelle, p. 16.) The « internal » are called « dakhila » in Arabic, and the « external » are « kharidjah ». So we have Fortuna Major = nosrat ou-i-dakhilah = internal victory. while Fortuna Minor = Al nosrat ou-i-kharidjah = external victory. Well,right away there is a problem: it is not clear how « external victory » is bad. In fact, Halbronn or his source is oversimplifying. As presented by anthropologist Wim van Binsbergen (http://www.shikanda.net/ancient_models/ … 201996.pdf, found by Marcos), the Arabs actually had four conditions: good, good-neutral, bad-neutral, and bad. Fortuna Minor is actually « good-neutral », good but not as good as Fortuna Major. So we must reformulate Halbronn’s generalization: the two sides of the opposition are better and worse relative to each other. Let us go on. Caput Draconis (Head of the Dragon) = internal threshold Cauda Draconis (Tail of the Dragon) = external threshold One is internal expansion of consciousness; the other is outward expansion into illusion, as good vs. bad (I get this from http://www.shikanda.net/ancient_models/ … 201996.pdf, p. 15). Next: Aquisitio [Acquisition] = taken internally Amissio [Loss] = taken externally The other pairs (p. 57) are Puer [Boy]/Puella [Girl], Tristitia [Sadness]/Loetitia [Joy], Carcer [Prison]/Coniunctio [Union], Rubeus [Red]/Albus [White], and Via [Road]/Populus [People]. Rubeus and Albus are aggressive masculine and gentle feminine. Carcer means Prison, i.e. Separation, the opposite of Conjunction. Prison is negative, joining together usually positive. Tristitia is Sadness, Loetitia is Joy. For Via, i.e. Road, and Populus, i.e. People, one is the journey and the other a destination, i.e. a city (one meaning of the Arabic, as listed by van Binsbergen). For this last, it’s not clear from these words whether either one is positive or negative. GEOMANCY AND TAROT Halbronn does not propose that tarot originated from geomancy. One might wonder whether the divinatory tarot in Balogna of the mid-18th century, with its single-word card interpretations, might have been influenced by geomancy. But the words (DDD p. 49) are different, and there is not much bad/good dichotomizing. I also wonder if there are any parallels between the structure of geomancy and that of lot books. Geomancy had a lot of prestige in the late Middle Ages and Renaissance. For example, three of the dozen or so small book collections enumerated by Susan Connell in her article, « Books and their owners in Venice » (Journal of the Warburg and Courtauld Institutes 35 (1972), pp. 163-186) contained a book listed as « geomantia ». Other systems might not have wanted to blatantly contradict this system. What do you say, Huck? The influence of geomancy on the « books of destiny » of 18th-19th century France and England is more probable.Van Binsbergen cites a « Napoleon’s Book of Fate » that explicitly uses a five row 32 figure geomantic system. He explains that 32 figures is « sufficient to accomodate all lunar mansions » (p. 53). This book, which he knows from a 1925 English edition, « does not seem to be older than the nineteenth century. » Such a system also exists in « the standard commentaries on Dante », he adds, unfortunately without citing examples. Perhaps they are older. In any case, Etteilla, although ignoring the « internal/external » distinction, manages to incorporate most of the words, or at least ideas, attached to the 16 traditional geomantic figures into his number-card keywords (for Etteilla’s list in various translations and versions, see my posts 15, 18, and 20 at http://www.tarotforum.net/showthread.ph … 963&page=2 and 21 at http://www.tarotforum.net/showthread.ph … 963&page=3). Another system is in the « Spiel Der Hoffnung » of 1799. Its words are different from those of geomancy and Etteilla, but still has the good/bad dichotomy (viewtopic.php?f=11&t=844&start=10#p12089). One way tarot might have arisen from geomancy is exemplified in the Michelino of 1420s Milan. This is where Marcos’s posts, starting at around viewtopic.php?f=11&t=365&hilit=binary&start=200#p6944, are very helpful. He says that Petrus Albanus wrote a major text of Geomancy at Pavia in the 14th century, probably the one listed in the Visconti Library inventory (http://trionfi.com/0/l/11/ ) as « Geomancia » (it is at http://books.google.es/books?id=fwY6AAA … &q&f=false. I get this from the « Geomancy » thread, short but very meaty.) Given Filippo Visconti’s superstitious nature, it is hard to imagine that he wouldn’t have known it. Petrus’s structure looked like this, in the excellent diagram that Marcos constructed: Image As you can see, it has a « relatively good/relatively bad » dichotomy for all the pairs except Via/Populus, for which both are « good-neutral ». (Marcos gives the specific passage in Albanus at viewtopic.php?f=11&t=365&p=6953&hilit=albanus.jpg#p6953.) For another schema, I have already given the link to the Arabic-based one on p. 15 of http://www.shikanda.net/ancient_models/ … 201996.pdf. As far as I can tell, the results for our purposes are the same as in Petrus. In another text, Bartholomew of Parma, 128-1300, Populus is positive and Via negative (Huck at viewtopic.php?f=11&t=505&hilit=geomancy#p6957). The journey is bad, the goal is good. Image It is easy to imagine the Michelino constructed with an eye to his sort of geomancy. For one thing, in each, 12 figures go together, the Olympian gods in one, the mothers-daughters-nieces in the other, with four of a nonconforming character. Another similarity (not dealt with by Marcos) is that in both geomancy and the Michelino there are 8 « better » figures and 8 « not as good » figures. In the Michelino, 8 are in the « good » suits of « virtues » and « virginities », and 8 in the « not as good » suits of « riches » and « pleasures ». I don’t see that the similarity extends any further than this. It might have, with individual cards corresponding to individual geomantic figures, but since we are not trying to do geomancy with the cards (we’re just playing a game), such a schema (although Marcos gives us one just as a possibility) is not necessary. Then we come to an ur-Cary-Yale, if there was one (before 1440), with 16 trumps. Does it have a similar binary structure, in the sense of « better » and « worse »? Its cards, as Marcos observes, are mostly in the « good » category. But even here I think we can make a distinction between more or less good. The seven virtues and the Petrarchan Chastity (i.e. the lady on the Chariot) are very good. The other cards–Emperor, Empress, Love, Death, Fame, Time, and Judgment, to which I add Fortune–are good sometimes, bad sometimes. CHESS We also have, in both the Michelino and the CY, something else, the parallel with chess, which also has 16 figures and many other binary features. Unlike geomancy, chess isn’t a system of divination, and so while it has features that might make for dichotomies between « good » and « not so good », these, unlike in a divinatory system, can be ignored. if desired. For chess there first of all is the black team and the white team; black is customarily associated with evil (even though in chess the white group is the aggressor). For each group, there is a good side–right–and bad side–left–for which there are two of the special pieces for each rank (royalty, bishops, knights, rooks). There is also the division into two rows: special pieces and undifferentiated pawns. Huck has compared the pairing of pieces between the two sides to a similar pairing of tarot cards in two early decks, the Cary-Yale and the Charles VI. The pairs are not the same as in geomancy, but there is no reason why they should be. It’s the structure that matters. I myself have analyzed the 22 card early tarot in terms of Giotto’s binary opposition of 7 virtues to 7 vices (viewtopic.php?f=12&t=848#p12119), deriving not from any divinatory system but from the struggle between virtues and vices in Prudentius’s Psychomachia. In a sense, the other part of my analysis, Petrarch’s 6 Triumphs, also has its binary oppositions: Love with its good and bad aspects, followed by Chastity good, followed by Death, bad, and Fame, good, Time, bad, and Eternity, bad or good. Halbronn does promote the idea that tarot originated from chess, based on the similarity of « Mat » to « Mate » (Arabic for « death ») and « Fil »–he says that’s a name for the Rook, or Elephant–to « Fol » (a rather tenuous connection, I think). Another consideration, which seems to me to apply to ordinary cards as well, is that three of the courts correspond to three of the special chess pieces. He says these ideas are not original with him, they are in a 1950 book by R. Ambelain, Les Tarots. And of course chess as played in India, with four sets of royalty, parallels the four suits in cards. THE I CHING Halbronn’s third chapter is on the I Ching. I don’t know much about its binary structure except that its configurations are made up of open and closed lines and go in pairs, too. I don’t read French well enough to follow Halbronn’s analysis without a lot of work. Wikipedia notices similarities between geomancy and the I Ching, and also differences: the I Ching uses binary trigrams, as opposed to geomancy’s binary tetragrams. The eight trigrams form four complementary pairs, in comparison to geomancy’s sixteen tetragrams forming eight pairs (image from http://en.wikipedia.org/wiki/Geomancy). Image Considering that the I Ching is Chinese and the tarot is European, the Arabs’ geomancy might be the link between the other two, historically. The 8 trigrams are expanded one way in the I Ching, another way in geomancy. You can read more at Huck’s post viewtopic.php?f=11&t=505&hilit=geomancy#p6940). KABBALAH Although neither a divinatory system nor a game, there are binary oppositions similar to those of geomancy in Kabbalah. The Seferotic Tree.has two sides, one of severity and the other of mercy, again associated with the colors red and white (for aggression and gentleness, just as in geomancy), starting with Chochmah and Binah. This characterization is especially clear in the Portae Lucis, (Gates of Light), as translated into Latin by Ricci and published in 1516 (http://www.literature.at/viewer.alo?vie … 3463&page=). So we have Chesed/Gedullah, Love, with Gevurah, Severity; and Netzach, an agent of love/mercy, with Hod, an agent of severity/justice. The other four sefirot are mixed (for documentation, see my essay at http://latinsefiroth.blogspot.com/). And the terminology of the Sefer Yetsira somewhat corresponds to that of geomancy. « Mother letters » which by this terminology suggest the generation of the rest, « single letters » and « double letters », like « daughters » and « nieces ». Also, the 32, the number of paths, is another power of 2. Both the Sefirotic Tree and the Sefer Yetsira are probably in some form earlier than any form of geomancy. However the details I have mentioned may have been post-geomancy. In general, however, are independent example of binary thinking (including a third, the synthesis), derived from the interaction of Hebrew and Greek thought. The Greeks had their own binary oppositions: hot/cold and dry/wet generated the four elements and humors; Aristotle even attributed to the Pythagoreans a doctrine of 10 basic pairs: « finite and infinite, even and odd, one and many, right and left, male and female, rest and motion, straight and crooked, light and darkness, good and bad, square and oblong » (http://history.hanover.edu/texts/presoc … ommentary2). CONCLUSION Admittedly, binary divisions are a natural, liable to spring up anywhere, without the necessity of prior history, especially good/evil, yes/no, and male and female: in Romance languages, everything is one or the other. This polarity makes a similar structure, in games and divination, a natural way of extending life into imagination. All the same, the specific ways in which these dichotomies are expressed in those areas in the time of the historical tarot makes the notion of mutual influence and history a suggestive one. Quote 16 Sep 2012, 23:14 User avatar Huck Re: Binary patterns in games and divination tools 2 The binary system is behind a lot of things … chess for instance. It’s in the lines of your ancestors, basic of all genealogy. It’s in he DNS-code. It’s in the most trivial ways, how everybody organizes his/her thinking. It’s in the computers. It’s in the dice results. It’s in Kabbala.Do you know this? http://trionfi.com/tarot/new-themes/sepher-yetzirah/ Image Huck http://trionfi.com Quote 17 Sep 2012, 02:22 mikeh mikeh Re: Binary patterns in games and divination tools 3 Well, I read it a couple of times, the article with the three operations. I didn’t understand the third operation, how the examples exemplify the operation you describe. Perhaps you could go through the opration step by step with an example. Quote 18 Sep 2012, 07:39 User avatar Huck Re: Binary patterns in games and divination tools 4 Alright.You have 64 hexagrams in I-Ching and 32 « ways of wisdom » in Sepher Yetzirah. I assume, you know the hexagrams.Operation 1:If you unite the complementary hexagrams … Example: 111111 unites with 000000 Example: 011111 unites with 100000 Example: 001111 unites with 110000 Example: 000111 unites with 111000 etc. … you have now 32 I-Ching elements (double hexagrams) … as you have 32 ways of wisdom in Sepher Yetzirah. Operation 2: The Sepher Yetzirah now gives the information, that there are 22 letters and 10 Sephiroth and these are the 32 ways of wisdom. Looking at the 32 I-Ching elements, one can observe, that the elements also are partable in 2 distinguished groups. 22 elements are « not balanced », 10 are « balanced ». « Balanced » is defined in the manner, that all hexagram pairs which have 3 yang and 3 yin in their 2 single hexagrams are now called « balanced ». These are 10 pairs, which naturally include 20 single hexagram. All other hexagram pairs are called « not-balanced ». Naturally these 22, cause 32 = 22 + 10. Example: 111111 / 000000 is not balanced Example: 111110 / 000001 is not balanced Example: 111100 / 000011 is not balanced Example: 111000 / 000111 <— is balanced Example: 110000 / 001111 is not balanced Example: 100000 / 011111 is not balanced So there a structural similarity between the system of I-Ching and the system of the « 32 ways of wisdom ». Both deal with 22+10 = 32. Operation 3: That, what creates difficulties. Sepher Yetzirah parts the 22 letters in 3 groups, and it parts the 10 Sephiroth in 3 groups. Letters 12 single letters 7 double letters (one of the 7 designed as center, the other as 6 directions) 3 mothers Sephiroth: 1 dominant 3 … more like elements 6 others (Later Kabbala sorted the Sephiroth according a 3-6-1 structure and used them in the life-tree: 3 are Kether (1), Chochmah (2) and Binah (3), and the single dominant element became Malkuth (10)) Commentaires This seems to have created the problems. The article speaks of a « rolling process » and a cycle. The I-Cing is usually presented as six lines, in which something is bottom and something else is top. The « rolling process » changes this form of vision into a cycle: 1 – 2- 3 – 4 – 5 – 6 – 7=1 – 8=2 – 9=3 – 10=4 – 11=5 – 12=6 – 13=1 …. etc. … or … 000001 000010 000100 001000 010000 100000 000001 000010 000100 001000 010000 100000 000001 etc. If I use this method on the 64 hexagrams of I-Ching, I get different groups (totally 14), but if I use it on the 32 pairs, I get only 9 groups, from which 4 are modified to 2 groups to form the 12-pairs-zodiac. Well, this is a complication … when the article was written, it was considered as too complicated. There are some other minor complications, which once took about 10 years to be solved … :-) … so these further complications were left aside. Maybe you detect them. Here’s a sort of improvement i-ching-02 dans numérologie Maybe, you need the help, that 100000 shall be read as hex. 24 and 011111 as hex. 44. And the other hexagrams are meant in the same reading direction. The first line stands (in the I-Ching appearing at the bottom) for the begin number (0 or 1) and the last number (0 or 1) for the 6th line (in the I-Ching at the top). Huck http://trionfi.com Quote 18 Sep 2012, 11:53 mikeh mikeh Re: Binary patterns in games and divination tools 5 Well, yes, that helped, once I read between the lines, so to speak.I get your rolling process. You put whatever was at the end of a six-digit number (or hexagram) at the beginning (bottom of the hexagon) and move everything over (or up) one space.One thing I didn’t get was how that principle applies to the group of 12 zodiac signs. But now I see that it applies in leapfrog fashion, the 1st to the 3rd to the 5th etc., and the 2nd to the 4th to the 6th, etc. Really it’s not one of 12, but two of 6. And that makes 12.Another thing I didn’t get was why you put « etc, cyclical » after each six digit number. But now I see. It’s at that point that the cycle repeats. And in the case of the « breath », there’s only room for one pair, because all the six-digit numbers obtainable by the rolling process are already used up in the first pair of the series. So this is about the mathematical properties of the 32 pairs of six digit binary numbers, that when you group the pairs so that the rolling process fits each group, they divide up into 4 groups of 6, 2 groups of 3, and 2 groups of 1, using up all the hexagrams. That’s not the way either the I Ching or the Sefer Yetsira makes its divisions, but the Sefer Yetsira comes close, in that it can be seen as (1 + 6) + 3 + (6 + 6) + 1 + 3 + 6.