It may be said that this appears to be a very obvious subject. But there are some deeper aspects I wish to share.
Firstly for the Ancient Greeks, Thomas Taylor tells us, 1 and 2 are not numbers. He reasons thus – to belong to a set, of which numbers is an example, it is necessary to possess all the properties of that set. The only properties which numbers share in common is that when a number is added to itself the sum is less than when that number is multiplied by itself. Since 3+3 is less than 3×3, similarly 4+4 is less than 4×4, likewise 5+5 is less than 5×5.
But this is patently not the case with 1 or with 2. For 1+1 is greater in quantity than 1×1, while in the case of 2 both the sum and the result are the same, yielding 4.
What we see is that an inversion occurs between the order found in 1 (1 is after all the origin of all numbers and therefore must be considered first) and all other numbers from 3.
We have in 1 a unique situation. A situation that is maintained by the intervention of 2 between 1 and the set of numbers that follow. 2 serves as a barrier, a defence against the invasion of numbers clamouring to reach 1.
So what is 2? For me, I have to say it this way, 2 is a mirror, it is the surface in which all of the unformed ideas in 1 are reflected into form in the form of numbers. It is Space, in which all forms, expressed in creation as numbers, take their Being. They are not Space, but they Occupy Space. Before those Beings can manifest they need somewhere to do it in. Space is that first level of expression and first expression of Consciousness. Space is the deepest level of Consciousness, some mistake it for a void and declare it is all there is and the end of their journey. But the 1 exists beyond that Void. For the Greeks that Void was called Night, and recognised as a Goddess. For the Hinds, this same being is called Kali, or Time (in a feminine aspect).
1 and 2 are isolates. They are unique in their own way, different from all others. This is not chaotic but the very beginning of order and structure. Following 2 the numbers fall into the categories of odd or even.
Since 1 is the origin of all things and contains within itself in potential all that later manifests in expression it is unable to distinguish between things for (or from) Itself. It is unable to evaluate one thing against another since it is at the same time both things, and all the rest besides. Each holds the same value of existence within the 1 as any other. This is the great equality of the One. What does 1 show us in its eccentric behaviour? It points to 2. And what is the lesson of 2? That all things are equal. But the process in 1 of multiplication demonstrates that through the fecund process of multiplication it restores itself. It returns to itself.
Why is multiplication fecund and what is addition? Multiplication is fecund because in combining two terms by multiplication we assemble consecutive rows of one number against the ranks of the other. In this way it is possible to identify the sources of the number generated. Strictly speaking this is only the case where simple numbers are multiplied together to give an answer that is incapable of further division. For example 12 may be derived from either 2 x 6 or 3 x 4. while 6 can only be derived from 3 x 2 unless we include the sum 1 x 6. Similarly with 10 it can only be 2 x 5 while 20 maybe 2 x 10 or 4 x 5 and so on. You are familiar with the idea now.
However in the case of addition there is no indication as to which of the numbers or modes of delivery have been chosen since in addition things are merely amassed in a disorderly fashion. They are, as it were, squashed together. 5 for example may be 1 +4 or 2 + 3 or 11 may be one of a number of origins from 1 + 10 to 5 + 6 and everything in between. Addition is therefore an unholy union, a mess, a scrabbling of bits to make a whole, where multiplication is a very orderly and thoroughly structured and pure a process.
But what of 3? Three stands at the head of the set of numbers. It is the first enclosure of space whether as a triangle or as circle. The centre, the radius and the circumference enclose the space of the circle. Each of the three lines and three angles which make up the triangle similarly enclose space. This brings us then to orders of Triangles, but we will leave those for another place.
Where numbers divide into odd and even – a duality, each of these series are then divided further into three different types. Of even numbers there are those which can be divided into equal parts until they return to 1. These are called Evenly Even and represent the series 2, 4, 8, 16 and so on. They are a relatively small group. Next there are those numbers which are capable of equal division into two parts but on making such a division are not capable of further equal division since they end in odd numbers. Such numbers as 6 or 10 or 14 are all capable of division into two equal parts but no further. These are Evenly Odd numbers. Unevenly Even numbers are those which are capable of equal division into two parts and capable of further division into equal parts but return eventually to an odd number, not making the division back to unity itself. Such numbers as 12, 20, 24, 28.
In the case of odd numbers the orders are not so simple, indeed it seems to me that one collection is a random and haphazard set which does not stand the same rigours as the other orders of odd numbers or the regularity of the even numbers. The first collection of odd numbers are those capable of division by themselves and the monad, 1, alone. These we call commonly Prime numbers. Taylor calls them ‘first and incomposite numbers’. The second collection he calls ‘second and composite’, giving such numbers as 9, 15, 21, 25, 27 and so on, as composed of two other odd numbers multiplied together. This series includes the squares. The third category I have trouble with. It is called by Taylor ‘the number which is second and composite of itself but with reference to another is first and incomposite’.
“ These numbers therefore, i.e. the first and incomposite and the second and composite being separated from each other by a natural diversity, another number presents itself to the view in the middle of these, which is indeed itself composite and second, receives the measure of another, and is therefore capable of a part with a foreign appellation, but when it is compared with another number of the same kind, is conjoined with it by no common measure; nor will these numbers have equivocal parts. Numbers of this description, are such as 9 and 25; for these have no common measure, except unity, which is the common measure of all numbers. They likewise have no equivocal parts. For that which in 9 is the third part is not in 25, and that which in 25 is the fifth part is not in 9. Hence both these numbers are naturally second and composite, but when compared with each other, they are rendered first and incomposite, because each has no other measure than unity, which is denominated from each; for in 9 it is the ninth, and in 25, the twentyfifth part.”
While Taylor gives only 9 and 25 as his examples we might question whether this particular collection might not equally well include the numbers 27 and 35 since no part of 27 is to be found in 35. It is for this reason I am unhappy with the collection. It seems an artificial devise to create symmetry. Unless, that is, we say that the squares alone belong to this collection. But yet Taylor does not say this.