Discovering the Transfinite
The genius mathematician Georg Cantor (creator of set theory, 1845 - 1918) recognised a property of the number “field”, that through its smallest numbers generates harmony[1]. The whole of the number field is uncountable and therefore expresses the basic concept of Infinity. Cantor realised that where there are a large number of objects in relationship to each other, as the numbers themselves are, and as all the objects in the Universe must also be, then all these relationships far exceed the number of objects themselves. These relationships then become new but different uncountables that belong to a set of infinites.
Thus, some new types of number sets, greater than infinity, must exist and also some way of signifying and even operating on them. The basic type of Infinity he called Aleph Zero, which is created by counting an infinite set.
If all the different ways of grouping numbers were to be considered, then these would be uncountable and the result he called Aleph One, since it was itself made up of many uncountable infinites. These new types of super numerical relationships Cantor called Transfinite and within these there is some possibility of understanding angels as unique operators or archetypes, with a unique mathematical core for each. [2]
This idea of grouping number is already familiar. The Fibonacci series is one of many that approximate the golden mean between successive elements[3], and the bodies of living organisms often hold this relationship between their key dimensions. The golden mean therefore would make a very good candidate as an archangel of life whilst we know that, as a proportion, the Sun and Moon operate to generate it within the sky calendar, in harmony with the synod of Venus lasting 1.6 years[4] and Venus is associated well with the Virgin Mother and her precedents.
Locating the World of Spirits
J G Bennett became interested in the transfinite mathematics of Cantor because it explained a difference between two different worlds of experience, called by the Sufis The World of Bodies (aleph zero) and the World of Spirits (aleph one). In other words, the spirit world could be made up of relationships within and between the whole systems, giving them transfinite characteristics.
Using again the example of the golden mean, wherever this relationship/ proportion/ ratio exists, the spirit of the golden mean could be said to exist. Whilst there are an infinite number of ways of achieving this ratio, there is only one manifestation, the ratio itself. This could have inspired the thinking behind angels, especially when building monuments and temples that should have an appropriate connection to objective reality within the numerical world-view.
In fact our acts of knowledge are themselves composite perceptions, so that similar relationships tend to have a name as in “that is a metaphor” meaning that all metaphorical statements belong to a single class that are identifiable and uniquely named as “metaphor”. In this sense the World of Spirits is ever-present in the process of making meaning itself. But this can be lost, just as the dimensions of a building can have subliminal proportions that influence the aesthetic irrespective of any understanding such as “this is a golden mean proportion”.
This type of effect could well lie behind the idea of numerical enchantment at a given sacred site. If relationships and proportions are manifestations of a spirit world then the structuring of the landscape by ancient cultures is comprehensible as a spiritual technique. This provides important clues for the study of monuments, their measures and the intended meaning of a given site.
Is it a co-incidence for instance that the interrelation of number and function is especially clear within ancient metrology and musical harmony? When we look at the proportions used, they employ low prime number relationships as if these were somehow the primary gods of proportion, within the numbers below Twelve[5]. However, musical proportions are different to the golden mean; musical intervals express the cosmic order found in pure musical tones and this is seen clearly in the outer planet relationships to our Moon of (a) a whole tone of 8:9 to Jupiter’s synodic period and (b) a half tone of
The golden mean belongs to a different class of numbers that are called “irrational”, which means that they cannot be represented by any kind of simple fraction, involving one whole number over another whole number – no matter how large the numbers used are. Such numbers are generally only ever approximated to, either geometrically or through an infinite mathematical series. Thus, such important irrationals represent not so much a number as a proportion or a function which meets the criterion of belonging to Aleph One or the World of Spirits. In this respect they seem to naturally complement the ratios of musical harmony, rather like the demons that complement the gods in the Indian cosmological picture, The Churning of the Ocean of Milk.
[1] see chapter 2
[2] Discussed in The Dramatic Universe, vol 2, p244, by J G Bennett who went on to use an even higher transfinite number, Aleph Two, as a means to think of the divine world as an unfathomable source of relatedness.
[3] In fact, any series where Xn+1 = Xn + Xn-1 will tend towards the golden mean between successive members after a number of iterations and the Fibonacci series is just the exemplar for this property of blended sums.
[4] Matrix of Creation by Richard Heath, Inner Traditions,
[5] explained in chapter 2
