[Note: This essay presumes basic familiarity with the qabalah. To see an introductory essay on the qabalah and the Tree of Life, please click here and enter the frame for Qabalah.]
It has been my observation that while many people talk about various systems of numerology, few have actually thought through the logic of them. My main purpose in this essay is to try to bring into focus the relevant aspects of arithmetic and how they may relate to numerology in practice, rather than to examine any system in great detail. Hopefully, this will allow those interested in pursuing any given system in depth to do so more carefully. I have some familiarity with the Hebrew and Pythagorean systems, so I take them as the relevant examples.
There are three basic viewpoints on numerology, namely:
1) The main or sole purpose of the study is to get the student in the habit of thinking associatively and to break down the habit of linear thought in the same fashion as koans are meant to. The systems themselves are essentially arbitrary.
2) The numerological traditions contain various secrets or doctines within them, but these are placed in an ad hoc fashion without an overarching pattern.
3) The numerological traditions use the structure of arithmetic in an efficient and complete way, so that the meanings of and relations between numbers follow a set of fixed rules.
These viewpoints are not mutually exclusive, and the second can be fitted with either of the other two without much difficulty. The first and third, however, tend to pull in opposite directions in terms of how one approaches things. One can pursue both seperately within a single system, but the results are unlikely to be complementary.
Also, the third viewpoint raises the obvious question of whether any traditions actually use numbers in this way. In the case of the Pythagorean tradition, certain such rules are central to the tradition. In the case of the Hebrew tradition, one has to judge whether any proposed set of rules actually fits with the material. It is my opinion that this is the case, as will be discussed later. Other systems would naturally have to be judged on similar grounds. First, we need to consider the logic of various possible rules.
The first thing to be considered is the base in which one is working. The I Ching and the Odus of Ifa essentially use a binary system, though without any arithmetic manipulations and with only a finite set of possibilities. The ancient Mesopotamians used a base 60 number system and may or may not have had a numerological system related to this. The Mayans used a base 20 system. Most systems use base 10, as this is the usual number system in most cultures. To simplify things we will restrict ourselves here to base 10.
In any such system, 1 is bound to represent unity or the origin of things in some sense, and 10 manifestation or completion. The numbers 2 and 3 can be expected to have some sort of polarty, though the details will vary. This is true both because it is natural to have a polarity emerging from unity and because 2 and 3 are prime whereas 4=2*2, thus 4 introduces a new consideration. In the case of the sephiroth of the Tree of Life, 2 and 3 are the root male and female principles. The the Pythagorean system, 2 is considered female and represents division, 3 is considered male and represents the trinity, which is active and spiritual. The 2nd and 3rd Hebrew letters are somewhat similar to the Pythagorean take on the numbers, though this aspect of things is more complex.
The numbers 4 and 8 are equal to 2*2 and 2*2*2, thus their meanings can be presumed to derive from that of 2. In the Tree of Life, the sephiroth numbered 2,4,8 are paired with the first three odd primes 3,5,7 to give the side pillars. In the Pythagorean system, the powers of 2 can be taken as relating to the number of vertices a line segment, square, and cube, thus reflecting space in various dimensions.
The number 9 is equal to 3*3 and thus can be expected to be an elaboration of the meaning of 3. Yesod is in some sense a lower version of Binah, representing the first step beyond the mundane as Binah is the major step and being the vehicle through which the higher sephiroth manifest. The empyrium, represented by 9 in the Pythagorean system, is the most completely divine of the numbers, elaborating on 3 as active divinity.
The number 6=1*2*3=1+2+3 can be expected to represent a synthesis of the root aspects of the system, representing the perfected soul in the Pythagorean system and a similar idea as Tiphareth.
The number 5=10/2 can be expected to represent that below which combines with 2 to give manifestation, something along the lines of free will. In the Pythagorean system, 5 represents the pentagram, microcosmic spirit. One of the main early meanings assigned to Geburah was that of restriction, in contrast to the free flowing of Chesed, which also fits the bill.
The number 7 is a wild-card of sorts.
In the Pythagorean system, as well as most popular approaches to numerology, numbers can be reduced by adding their digits. Repeated reductions leaves one with a number from 1 to 9. It is sometimes claimed that this also applies in the Hebrew system, though my observation is that this completely fails to work in practice. In any event, addition and multiplication are conserved under reduction, so all statements of addition or multiplication can be considered to be elaborations on those involving numbers from 1 to 9. For example, any number reducing to 4 multiplied by any number reducing to 5 gives a number reducing to 2. If one is using such a system, the first thing to do is to work out the meanings of the basic statements of addition and multiplication. Of note are the facts that 9+X=X, 9*X=9, 3*3=3*6=6*6=9, 3*1=3*4=3*7=3, 6*1=6*4=6*7=6, 3*2=3*5=3*8=6, 6*2=6*5=6*8=3. 9 can thus be expected to represent the divine which both absorbs everything into itself and leaves everything unchanged, and 3 and 6 are thus an intermediate stage on the way to that divinity, in some sense or other.
If addition is taken to generally be significant, then one must consider that all integers ultimately reduce to 1+1+...+1, so that there is only one generating number in this regard. If we consider, say, 65=1+64=2+63=...=32+33, it seems clear that this will quickly reduce to a muddle if there isn't some ordering pattern behind the 32 additions. The notion of reduction by adding digits provides such an order and conversely tends to imply that addition is generally significant. There may be other ways to make addition work out, but none that are apparent to me.
In contrast, the only ways to write 65 as a product of two integers are 65=1*65 and 65=5*13. Numbers which can only be written as a product of two integers as X=1*X are called primes, and there are an infinite number of them. Any integer can be written as a product of primes in a single way, ignoring the order in which they are multiplied. This gives and infinite number of generators which can be combined in any quantities. Aleister Crowley has claimed that prime factorization is the main principle underlying the Hebrew system, a claim which I believe stands up to an analysis of the material.
Also important is the notion of analyzing the digits of a number. If reduction is in force, then this is simply a special case of addition. If not, then we are left with interpretting the digits as a string of symbols. In the case of the binary systems mentioned above, and also for the 3-5 digit numbers in the Enochian calls according to what I've received, this is the entirety of how the numbers are to be interpretted, and it is not really useful to regard them as numbers at all.
In the Hebrew system, the digits may derive meaning either from the corresponding sephiroth or the corresponding letters or a combination. Of special interest are numbers which are glyphs of the Tree or parts of the Tree. In this regard, we can take 12331, 120331, 121331, and their reverses as the largest examples, with it being permissible to add digits and for Daath to be written as 0 or 1 or 4 when taken together with the supernals or 8 when taken together with the lower 7. Thus 61 = ANI (ego) = AIN (nothing) represents the Zar Apin (sephiroth 4-9) above Malkuth, 47 represents the supernals including Daath above the lower 7, etc. Malkuth may also be represented by 4=Daleth/elements or 5=Heh(final)/pentagram. Note that this gives the three major divine names, AHIH=21, IHVH=26, and ADNI=65 as covering the entire Tree in overlapping layers. It does not seem that all numbers which might be taken as glyphs should be, but many should. Also worth mentioning, in conjunction with interpretting digits, is Crowley's doctrine of reversals, which says that a number and its reverse are connected, and, in particular, the digits should be interpretted in the same way.
The above are the main ways that arithmetic structure is used, but there are a few others that enter in. The sum of the integer from 1 to X is considered significant in both the Pythagorean and Hebrew systems. This comes out to X*(X+1)/2 and thus has X or X/2 as a factor. The product of the numbers from 1 to X, written X!, may also be considered significant. Kameas, squares that use all the digits from 1 to X*X, X being the length of a side, and whose rows and columns all add to the same things, are significant in the Hebrew system. If the length of the sides X is odd, then the central square has value (X*X+1)/2, the rows and columns add to X times this, and the whole square adds to X*X times this. If X is even, then the average value of two squares is X*X+1, rows and columns add to X/2 times this, and the whole to X*X/2 time this. These numbers -- (X*X+1)/2 for X odd, and X*X+1 for X even -- seem to represent something necessary for that which X represents to manifest. Kameas are traditionally only used for the planets, numbered 3-9, but in my experience the same principles work for higher numbers. Any number of other things that are interesting from a number theoretic point of view could be introduced, but one has to be careful that one is elaborating on the underlying symbolic structure and not simply playing with numbers.
Given the material above, one should be able to apply it to the basic Pythagorean system, which can most simply be summarized as 1=root, 2=division, 3=trinity, active divinity, 4=elements, 5=microcosmic spirit, 6=realized soul, 7=planets, 8=space or the crystal heaven containing the planets, 9=the empyrium. In his book Magical Alphabets, Nigel Pennick uses the theory that the Greek letters correspond to their order in the alphabet and their gematric values. I can't vouch for or against the historical authenticity of this, but it does produce a workable system which the reader might like to explore.
Before going on to discuss the Hebrew system in greater detail, a few words about Thelemic numerology seem in order. I haven't studied any of the proposed English gematrias, so I can't comment on them specifically. Crowley seems to have had the idea that Hebrew and Greek numerology are the same or could be synthesized, an idea I believe is false because they follow incompatible rules. A true Thelemic qabalah which relates to both systems must therefore have its own logic. Besides the proposed gematria(s) and the specific mentions of numbers in the text, I think that the verses might be expected to correspond to their numeration (an idea explored by Kenneth Grant in his usual haphazard fashion). The best evidence for this is verse I:25, "Divide, add, multiply, and understand." 25 divides as 5*5, 5+5=10, 5*2=10 (25=5^2 and has digits 2 and 5), thus we have a notion of independent movement indicated by 5 being elaborated or pursued until a realization of something higher represented by 2 is reached, and the two combine giving manifestation represented by 10. This goes well with the 25th path of the Tree, and especially with Crowley's Art card.
The pattern of dividing the sephiroth into pillars pairing powers of 2 and odd primes has already been mentioned. Daath may be connected with the number 11, as the 11th sephira. The number 11 also connected to Aleph as the 11th path and to the Qlipoth. It may be interpretted as a polarity of higher and lower, with Aleph emphasizing the higher, the Qlipoth the lower, and Daath both. Each letter relates to both its path number and its gematric value. The first ten letters' gematric values seem generally to refer to patterns on the Tree. Aleph refers to Kether, or to Daath as child of the supernals. Beth may refer to either Chockmah emerging from Kether or the pairing of Chockmah-Binah or of above and below the Abyss (note that ChKMH and GML both add to 73, BINH and ZIN to 67). Gimel refers to the supernals or, secondarily, to the pillars or mother letters. Daleth (DLTh=434 a glyph of the Tree) refers to either Daath as part of the supernals or Malkuth as elemental. Heh refers to Binah or Malkuth, as per IHVH, and thus does not fit the general pattern. Vav refers to the Zar Apin, sephiroth 4-9. Zayin refers to the lower 7. Cheth refers to the lower 8 including Daath. Teth refers to the upper 9. Iod refers to the entire tree.
One application of numerology is in the construction of magical formulae. I find that using MLLKA AKHKA IAKA, a formula I part received and part constructed, as a mantra, is useful in getting into the right frame of mind. MLLKA adds to 121=11*11 indicating Daath emerging from the supernals, the MLL indicating dividing the balances as per Crowley's interpretation of MIM and L taken as Libra, and KA adding to 21 the same as AHIH indicating something manifested out of unity. AKHKA adds to 47, Daath as child of the supernals within the Tree, the latters reflecting the KA being reflected into manifestation. IAKA adds to 32, the path number of Tau, indicating that something is manifested and mimicing Greek formulae starting IA such as IAO. 121+47=168=8*21 indicating the KA entering into the lower 8, 47+32=79 the value of the pillars Boaz and Joachim and indicating something built, and the all together they total 200, the value of Resh and indicating will in the world.
As an example of how a formula may be constructed, one starts by selecting a suitable number. 1653=1+2+...57=3*19*29=57*29 can be interpretted as 57=3*19: the material (3) constraints of creation (19 is a glyph of Malkuth receiving from the higher nine) allowing Hadit (9 of 29) to act in the divided world (2 of 29, see Crowley's Moon card). 16, path number corresponding to the Hierophant, causes the lower to reflect the higher (53). We need some letters with high gematric values to get up to 1653, and 4 Th's would seem appropriate. An M in the middle seems appropriate, leaving us needing letters adding 13 more. If we want to alternate Th's and non-Th's, then we need two letters adding to 13, and Ch and H seem best. We put the Ch before the M indicating reaching up and H after indicating fruitful manifestation, giving us the formula ThChThMThHTh. This may be used as a mantra or at appropriate points in ritual as desired. It is, of course, important to allow one's intuition as well as one's intellect to play a major role in this process.
Works Cited.
Crowley, Aleister. Liber 777 An edition of this is included in 777 and other Qabalistic Writings of Aleister Crowley, York Beach, ME: Samuel Weiser, 1973.
Pennick, Nigel. Magical Alphabets. York Beach, ME: Samuel Weiser, 1992.