This “Thelemic” mathematics6So called because it was first described in The Book of the Law. is founded on the simple conception of the point and the circle; where both are viewed as simple mathematical objects, each self-defined.  That is, the concept of the point being the first place on a linear scale (1, 2, 3) is discarded in favour of a system in which there is no initial concept of “straightness”.  The circle and point are seen as having no meaning unless defined in relation to each other.  So the mathematical point, which has nothing but “position”, is nothing without relation to the circle, which is pure “positionability”.  This circle evidently has no implicit magnitude, or size, but is unlimited in respect of every category of thought: it had better be referred to as a sphere, since that more amply conveys the idea of “pure circularity”.

   Now the “point” is no longer the basis of our idea of “magnitude”; but magnitude is defined as “that which distinguishes the point and circle”.  Therefore magnitude is not related to extension in one direction, but extension in every possible category.  And it is now seen that the idea of a “dimension” as a linear thing, to which other “dimensions” share “same-power” relationships,7e.g., extension at right-angles, in Euclidean geometry.  The model here presented is more in line with Gauss’ notion of “action”, i.e. linear action, circular action, extended circular action, etc.  This thesis, however, brings the idea of “action” to include even the point; and so of course the concept of extension is quite altered.  It is hoped that the nebulousness of the following discourse will not prevent sincere discussion of these points. is quite unnecessary.  Therefore I will the term “category” to mean “mode of manifestation”.  It is the concept of pure relation.

   Now we have three cognate conceptions, which are by nature self-defining.  The classification under the headings of 0, 00 and 000 is such because they are the basis of number, and not numbers in themselves:

   0) The point; the basis of “position”.  This is the only possible conception of zero, since it has no qualities.
   00) The sphere; the basis of the “unbounded”.  This is the only possible conception of infinity, since it is the all-encompassing.8These “0” and “00” are the philosophical “A”, and the implicit “Not-A”; where A = 0.
   000) The point and sphere as a complementary unity, the basis of “relation”.  The “category”.9These are then defined as it were by each other: the point is “Not”, unless there be “No-Thing”; and when one posits that they “are”, it is merely to say “Nothing Is Not”.

   This ultimate trinity is really a unity, which unity is the fundamental conception of the number one:

   1) The number one is neither a point nor a sphere, nor their relation, but all three combined in one.  It is rather difficult to pin down in thought, since it is everything, but is at once single.  It is existence, but it does not in itself manifest; for to manifest would imply division.  Therefore it can only be described as “potential”.  Now this is not to assert a quality to it, from the realms of the imagination; and if I were to attempt to do that, it would mererly be a logical basis for criticism.  Therefore it is only described as “potential” in a clearly defined way; and that, as the basis of all the numbers.  And one is only able to say that it is the basis of the numbers from the obvious inference that, as the unity must have everything “in it”, as it were, it can potentially generate the rest of the numbers.

   2) One cannot formulate the number two from the number one, since the number one is the unity.  To say “the unity can be doubled” is a contradiction in terms.  However, we already have the dyad, the number two, since we have two distinct concepts: the “not” trinity and the unity.  The simple confession that, in order to conceptualise the unity, we must do so with relation to nothingness, gives us our simple concept of the number two.  This “two” is then Not-One!  Joking aside, that is all that can be said of it, really.  Yet we get some sense of it as an intelligible concept by noting that, since it is two things in one, it does not imply a line connecting two points (as in traditional maths), but implies the concept of phase [0<–>1], and therefore or potential vibration; and that, not along an axis, but throughout all possible categories.

   3) We find that “potential vibration” implies some space wherein the system of “potential vibration” can subsist.  Thus the number three might be termed “potential vibration space”; which is evidently cognate with our conceptual sphere (00), but of a different order of being; in the same way that “potential” is cognate with the point (0), and “potential vibration” is cognate with the concept of “relation” (000).  These three (1, 2, 3) evidently form as it were a reflection of our Not-Trinity.

   The numbers 1, 2 and 3, being the Not-Trinity as known through the Not-Trinity, are shown forth in the numbers 4, 5 and 6, which form another triad.

   4) Being the unified conception of 1, 2 and 3, the number four is a complete definition of matter, since the binding, as it were, of the concept of “potential vibration space” gives in each of its three terms respectively our conceptions of mass, energy and size.  The concept of “potential”, when “realised”, is the very root of our concept of mass; and “real” vibration is energy, just as “real” space is our concept of “size”.  Matter cannot be defined except by these concepts, and has no need of any other concepts.  The number four specifies matter.

   5) As with the formulation of the numbers two and one, the combined conception of the number 4 (matter: mass, energy and size) is referred back to the previous trinity (“potential vibration space”).  The number five, then, allows the possibility of defining matter in three ways, which together compose the concept of “motion”:

   6) As with the number 3, we are now forced to conceive of some way in which these ideas (of matter and motion) can become realised.  This is of course the conception of Time; which is then seen as the “space” in which matter can be manifest (or not), and the quality which makes motion possible.

   Here again, the numbers 4, 5 and 6 form a complete trinity.  As before, the next trinity is conceived through the totality of this one.

   7) Having formulated an universe wherein matter, motion and Time are conceived as possible, one can only put forward the notion of light as their sum total.  The “dark” universe of 4, 5 and 6 (wherein is only the possibility of matter, motion and Time) is only appreciable by the force of its own interaction, which must be light, for the simple reason that when one tries to formulate the question: “How can matter move in Time?”, one can’t do so without as it were “picturing the scene” (implying light).  I beg the reader’s indulgence, and request him or her to try to formulate this question whilst excluding light.  There is some barrier which cannot be pierced, unless one has this illumination!  I suggest that “light” is then something that is the reflection of everything in everything else.  Of course, this does not necessarily mean that there is no better concept than “light”; it’s just that one can’t conceive of any.  This “light” is implicit and defines the very concept of spatial separation; and so the sense of this number is “that which separates and defines motion, matter and Time”.

   8) Now by referring light[-separation] back to matter-motion-Time,10Using the same process as for 2 and 5. we arrive at the concept of the rate of separation of matter-motion-Time.  These compose the concept of the speed of light, which may be said to regulate the rate of Time, of motion and of matter.  The “rate of matter” would seem rather awkward to describe – it does not mean the rate of vibration of molecules in matter (which is the rate of motion), but would seem to allow a differentiation between types of matter (e.g. quarks, hadrons, electrons, etc.).11The reader will note the connection with Relativity Theory.

   9) This number is then the binding together of the concepts of “speed” and “that which separates” (light).  Since we are to conceive of a universe in which “separation” and “speed” are contained, what we have is the concept of a holding together of the force of separation of things; which is gravity.12As an acceleration which contains the expansive force of the progression of number.  This expansion is cognate with the Einsteinian notion of the “cosmological constant”.  The Newtonian equation for the cosmological constant is the formula of a simple harmonic oscillator; which is a linear model of the concept of “relation” as “vibration”.

   In summary, it must be noted that we have used the same simple rules throughout.  From our initial conception (the point), we have inferred another (the sphere); and from these two, we have inferred their relation in a third.  This gives us another “point”, from which we may infer a second (being the relation of the first unit with the preceding triad), which gives us a third wherein they may relate; and so on.  By the rules of this system, we must now posit the number 10 as the synthesis of all the above: the Universe itself.  This seems to be the end of the series, since we have found all of the basic conceptions necessary to make an universe (though we don’t know how they may be related, except as units on a scale).  It seems complete, since we now have a trinity of triads formed our initial Not-Trinity; thus expressing as it were the permutations of that original trinity.  There is no need to go beyond the number ten to create the universe.

   However, there obviously are numbers beyond ten; and if we wish to proceed, we had better be aware that we are beings who live in number ten, as it were.  Therefore any attempt to conjecture about the following numbers must be based solely on our [individual] experience of the first ten numbers.  They should tally with our actual observations of the universe.  It is important to note that from here on in, we must admit the possibility that individual interpretation may vary (and by this I mean, depending upon the mass, speed, etc., of the observer); yet we may make certain inferences from the rules of the system now established.  We know that the rule is union of opposites in a third, which together make a triad.  Three triads form a trinity of triads, which crystallizes out.  It is therefore plausible that, since we are examining the universe as a whole, the numbers 1–10 inc. form the first point of a “decadic” triad (1–30) which then makes up a trinity-of-“decadic”-triads (1–90), crystallizing in the number 91;13These trinities of triads (1–10 & 1–91) may form two parts of a trinity-of-trinities-of-triads (if the rule may be expanded indefinitely, as it seems it should); but it seems that (given our knowledge of the Enochian system) this enumerates to 441. which may represent the [impersonal forces of the] Universe’s experience of us.

   We must now turn from straight mathematics to the consideration of all this in relation to the Enochian system of Dr. John Dee and Sir Edward Kelley.

6 So called because it was first described in The Book of the Law.

7 e.g., extension at right-angles, in Euclidean geometry.  The model here presented is more in line with Gauss’ notion of “action”, i.e. linear action, circular action, extended circular action, etc.  This thesis, however, brings the idea of “action” to include even the point; and so of course the concept of extension is quite altered.  It is hoped that the nebulousness of the following discourse will not prevent sincere discussion of these points.

8 These “0” and “00” are the philosophical “A”, and the implicit “Not-A”; where A = 0.

9 These are then defined as it were by each other: the point is “Not”, unless there be “No-Thing”; and when one posits that they “are”, it is merely to say “Nothing Is Not”.

10 Using the same process as for 2 and 5.

11 The reader will note the connection with Relativity Theory.

12 As an acceleration which contains the expansive force of the progression of number.  This expansion is cognate with the Einsteinian notion of the “cosmological constant”.  The Newtonian equation for the cosmological constant is the formula of a simple harmonic oscillator; which is a linear model of the concept of “relation” as “vibration”.

13 These trinities of triads (1–10 & 1–91) may form two parts of a trinity-of-trinities-of-triads (if the rule may be expanded indefinitely, as it seems it should); but it seems that (given our knowledge of the Enochian system) this enumerates to 441.