Werk:The Theology of Arithmetic/Foreword

Gespräch:Übersetzung Die Theologie der Arithmetik Vorwort

Werk:Die Theologie der Arithmetik/Vorwort

Kairos is privileged to present this timely translation of The Theology o f Arithmetic, and is most grateful to Robin Waterfield for his dedicated and scholarly translation. We offer the following observations to emphasize the inner nature of a document like this, which invites interpretation.

Aetius (first or second century A.D.) is one of the most important sources for the opinions of the earliest Greek philosophers, whose actual works have for the most part perished. He says of Pythagoras (I.3.8) that he was the first to call the search for wisdom "philosophy," and that he "assumed as first principles the numbers and the symmetries existing among them, which he calls harmonies, and the elements compounded of both, that are called geometrical. And," Aetius continues, "he says that the nature of Number is the Decad." However 'unreliable' we choose to treat this transmission, even this brief extract does contain some remarkably significant possibilities which call for interpretation.

That numbers were assumed as first principles is not unexpected, but the idea of "symmetries existing among them" is particularly evocative. Their harmonies are likewise said to be among them. Further, the geometricals are elements compounded of the numbers with their inherent symmetrical harmonies. The progression given by Aetius is from first principles to numbers, symmetries, harmonies and geometricals, and it seems to reflect the progression arithmetic, harmonic and geometric, which are the three primary means that Plato proposed that the Divine Artificer (demiurgos) used to proportion the 'world soul' (Timaeus 36).

Finally, that "the nature of Number is the Decad" could place quite a different perspective on the issues that took up so much of Aristotle's time, as he chewed over the paradox of the linear generation of the 'incomparable' ideal numbers. The indication is that the Decad is both complete and is the essential nature of all number—a monad of minimum ten. That ten is 'complete at four' is a well-known Pythagorean paradox based on the simple cumulative progression of 1 + 2 + 3 + 4 = 10; or, in the direct manner of those who had no separate number symbols:

Figure 1

However, there is obviously an allegorical reasoning which allows paradox to play a free part in the 'generation' or simultaneous 'being' of the first nine (or ten) ideal numbers. Given that Aetius' ultimate source is Theophrastus, who wrote in the fourth century B.C. the Doctrines of Natural Philosophers (Phusikon Doxai), then we have a most interesting Pythagorean idea to contend with—that ideal number is not necessarily subject to a sequential or causal progression from one through to ten, but is rather a unity with ten essential and potential qualities, simultaneously present in the Decad or Tetraktys.

Although it is well known that numbers traditionally come 'before' bodies, there is the possibility of understanding the shadow allegory of Plato's Cave in terms of the number points, which we might call 'dots,' representing 'incomparable sets' of one dot for oneness, two dots for duality, three dots for triplicity, four dots for quaternity, and so on. In this case, the 'shadows' would work in reverse, as it were. From the pure, indescribable light of unity, the first visual evidence, we might say, is for the units to become surface 'dots,' which eventually become matter as spheres, i.e. projecting into the dimensions.

In this way we can see a progression that embraces Plato's doctrine that 'images' in the human mind reflect the universal source in the mind of the Divine Artificer. So we get One reflecting as a single dot or shadow on a surface for us to 'imagine' or remember Oneness from. If we should take the Pythagorean tradition literally, the first and comprehensive evidence of 'ideal' or archetypal number is an array of 'dots' in triangular form:

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Figure 2	Figure 2a	Figure 2b
Figure 2. The Pythagorean Tetraktys as 'dots.' Figure 2a. The Pythagorean Tetraktys as close packed circles in triangular form. Figure 2b. The Tetraktys of ten 'dots' with the basic symmetries between them.

This pattern might be called, in current usage, a refraction pattern of a single triangle. It is important to remember that the ancient Greeks did not have an abstract system of number symbols, and used the letters of their alphabet as number symbols. They also commonly manipulated pebbles to leam arithmetic and used these small stones on calculating boards. In this case, number patterns were their common experience of arithmetic. From this use of pebbles, we have inherited the word 'calculation/ from the Latin calculus, which means 'pebble.'

This Decad or pattern can be taken both as a graphic (differentiated) form of ten dots in four rows and/or as one triangular form outlined in nine dots with a center point. In terms of the Platonic formula for knowledge of any object (see the Platonic Seventh Letter), this would be the second stage of definition.

From the above pattern, we can see how the Tetraktys of the Pythagoreans may represent both a minimal oneness and a maximal tenness simultaneously. The Pythagoreans must have based their deep regard for this particular form (see the Pythagorean oath quoted on the dedication page) on the breadth and depth of symbolism it carried or could carry for them.

The doctrine of the essentiality of number, which has held so much inspirational power for so long (and is hardly far from modem concepts of mathematical atomism) is (a) unlikely to have been totally divulged in the literature and is (b) likely to have had more than one teaching related to it. Yet this multiplicity of teaching, that inevitably leads to paradox, is not necessarily a helpful ingredient during the time of nurturing dialectic and logic in a developing philosopher; there was, therefore, good reason for the controlled unfolding of the meaning of the symbol.

This concept of unfolding or unveiling was particularly developed by the 'Platonizing' school of Chartres in France, during the eleventh and twelfth centuries, most notably by Bernardus Silvestris in his commentary on Martianus Capella's Marriage of Mercury and Philologia. In it, the commentator defines the practice of integumentum, which literally means 'cover' or 'wrapping' and usually referred to clothing. However, here we have the word used to refer specially to symbolic narratives and their hidden meanings. "Figurative discourse," says our Chartrain author, "is a mode of discourse which is called a veil. Figurative discourse is twofold, for we divide it into allegory and integumentum." He goes on to say, "Allegory pertains to Holy Scripture, but integumentum to philosophical scripture." Hence he explains that his commentary on Capella uses this latter method because Capella "is unveiling the deification of human nature," and "speaks like a prudent theologian, because all his utterances contain a hidden truth." The work contains instruction and this instruction has been put into figurative form.

Figuration can also be one of the values or qualities that the Divine Artificer uses to decorate or adorn the invisible first principles of the creation (the word kosmos for 'universe' means 'adornment'). Thus arithmetic and geometric figuration, with the resultant harmonic (musical) figurations, are the most essential tools that Plato posits the Divine Craftsman uses to adom the 'likeness' of the perfect cosmos, which he is attempting to delineate in the psycho-cosmogony of the Timaeus.

There is not space here to develop all the interesting implications of how 'four' is 'ten' and so on. So what we propose to do is to look at how Plato took care to reveal only so much of what are called by Aristotle the 'unwritten doctrines.'

When Plato proposed the portioning by number of the Same, Other and Being in their unified mixture (Timaeus 35b-c), he set up what has subsequently been called his Lambda. It is a portioning arranged into two 'arms/ each with three intervals.

Figure 4	Figure 4
Figure 3. Plato's Lambda from the Timaeus. The first seven portions of the triadic mixture of Same, Other and Being corresponding to the seven planetary lights. Figure 4. The missing positions in the Platonic Lambda.

Starting from one at the top, the generation from 1 to 2 and 3 gives the eventual seven stages. Here we see the principle of 'twice' and 'thrice' as well as the progression from 2 into a plane 4 and a cubic 8, while on the opposite arm is 3 with its plane 9 and cubic 27. Now we have seven positions representing two progressions, which leaves us with the impression that there could be three more 'points' or positions within the open triangle, so that if filled the model would become like the sacred Tetraktys of the Pythagoreans or the triangular 'four' representing the full decad. Even if Plato himself did not suggest the lambda form in the Timaeus, yet because the convention of triangular numbering and the image of the Tetraktys in four lines of dots were completely familiar to the Pythagoreans of the day, it would be inevitable that they would make the comparison. The idea of doing so is not new: the pattern we are about to look into was, in fact, published by the Pythagorean Nicomachus of Gerasa in the second century A.D.

Before proceeding further, we will do well to observe that we are allocating numbers to each of the 'dots' of the pure Tetraktys model (figure 1 ); we are accordingly already moving into symbolic representation. Each dot becomes representative of a position in the process of generation—from undifferentiated unity into 'twiceness' and 'thriceness' in the first dimension, 'foumess' and 'nineness' as their respective planar reflections, and 'eightness' and 'twenty-sevenness' as their projections into the ultimate third dimension or bodily world. We have two squared and cubed, and three squared and cubed, giving us a dyad of forms of generation passing through the three dimensions on each arm.

The challenge that follows the allocation of three more points within the triangle (figure 4) is to discover what numbers they should be, if they are not part of the existing progressions of 2 or 3. The solution becomes a matter of the logic of the symmetry or 'angles' that the progressions of 'times 2' and 'times 3' follow. If we start from the monad or one at the top and follow down the righthand slope or angle, the fourfold progression is 1 x 3 = 3, x 3 = 9, x 3 = 27. If we now shift down from 1 to the position of 2 and follow a similar angle down through the pattern, we pass through the center point, which becomes 2x3=6, and arrive at the point on the base, which becomes 6 x 3 = 18. If we do the same, starting with the diagonal that moves down from 4, we have 4 x 3 = 12.

Figure 5	Figure 6
Figure 3. Plato's Lambda from the Timaeus. The first seven portions of the triadic mixture of Same, Other and Being corresponding to the seven planetary lights. Figure 4. The missing positions in the Platonic Lambda.

Now, to check that this allocation is valid, we have the symmetrical diagonals following down the other way and multiplying by 2. Starting from 3 this time, we would get 3 x 2 = 6 and 6 x 2 = 12: the pattern agrees with the times-three symmetry. Finally, starting from 9, we have 9 x 2 = 18, which is the same result as on the times-three slope. The pattern is consistent and valid.

Figure 7	Figure 8
Figure 7. The generating angle of the 'times two' symmetry, from 3 generating 6 and 12: 2 x 3 = 6, 2 x 6 = 12. Figure 8. The second 'times two' diagonal from 9 generating 18: 2x9-18.

Referring back to Aetius' statement of the Pythagorean position, that number has the Decad (or the Tetraktys, we might say) as its 'nature,' and that the numbers have 'symmetry' and produce 'harmony' within, then we now have a pattern with all those qualities simultaneously present—our numerically symbolic Tetraktys, developed from Plato's Timaean Lambda.

In summary, we might say that the point positions, one by one, represent the Decad; the sets of points represent the oneness, twoness, threeness and fourness of the constituent horizontal lines which we can call the first symmetry; the allocated number progressions represent 'states of being' of the refraction of one, we might say, and particularly of two and three respectively as they move from plane to cubic form and thereby provide a pattern that has within it the essential numerical harmonies. In particular, this Decad contains those numerical harmonies proposed by Plato for such a vital story as the generation of the 'world soul' in the Timaeus, as we shall see.

On the basis of simultaneous qualities interpenetrating and mutually present, we might take the point model as representative of the background archetypal 'isness' of the 'incomparable' ideal numbers—those complete at ten. In another sense, we could take the allocated soul-generating numbers to be representative of the numbers moving into bodiliness or cubic being and ready to facilitate the geometrical basis of materiality (Timaeus 36e, 53c-e). The geometrical basis of materiality consists of the tetrahedron or 'fourness' of fire or light, the octahedrality or 'sixness' of the gaseousness of air, the icosahedrality or 'twelveness' of the liquidity of water, and the cubic 'eightness' of the solid earthiness. These numbers are the points in space that define the solidity of the regular figures.

Each of the basic 'geometricals' can be generated out of a series of morphic points as three-dimensional reflections of the first monad or the 'original' singular spherical body—the sphere being the most perfect and most simple three-dimensional form, simultaneously representing the unique, unity and the unified.

Figure 9. The Platonic Lambda filled out into the Pythagorean Tetraktys by the addition of6, 12 and 18.

We learn from Plato that 'soul' embraces both the mean proportionals as well as the uniting principle of opposites (Timaeus 31c, 35a-b). We now can find out how the particular means, which Timaeus proposes the Divine Artificer uses to fill in the intervals between the primary numbers, can be found 'in the symmetries/ or in other words how the 'harmonies' can be read.

It is good to remember that Plato describes the cosmogony as 'a likely story,' indicating the objective laws that lie behind the ordering principles that we find in our cosmos, because this cosmos is only a 'likeness' of the ideal [Timaeus 29c). This cosmogony or creation story is a rhythmic separating and uniting of forces which are given 'provisional' names to enable us to even conceive of what is involved. First, there is the complementarity of Sameness and Otherness with their (proportional) uniting principle of Being. When the Divine Craftsman "made of them one out of three, straightaway He began to distribute this whole into so many portions..." (Timaeus 35b).

There follow the seven portions multiplying unity by two and three to get the sequence 1, 2, 3, 4, 9, 8, 27. Having made this division and related it to the sevenness of the planetary system, Plato goes on to describe the filling in of the intervals. This is done by placing two means between each of the powers of 2 and powers of 3. These are the arithmetic and harmonic means which, with the geometric mean, complete the triad of means. The means set up proportional unions between extremes and are therefore in themselves the epitome, in mathematical terms, of the mediating principle—in common with the definition of psyche ('soul') as mediating between the metaphysical (intelligible) domain and the physical (sensible) domain. The means themselves are set in a hierarchical tendency, as one might call it. The geometrical is the most heavenward (metaphysical), the harmonic the most central (psychic and anthropological), and the arithmetic the most earthward (phvsical).

Figure 10	Figure 11
Figure 10. Tetraktys showing full display of musical ratios: the octave proportion of 2:1; the musical fifth proportion of 3:2; the musical fourth proportion of 4:3.; and the tone interval of 9:8. Figure 11. The arithmetic proportion with the three arithmetic 'means' of 3, 6 and 9.

If we take these three means—the arithmetic, the harmonic and the geometric—we find that the legs or diagonals to the right or left are in geometric proportion (i.e. times 2 or times 3). The pattern in figure 11 above gives the symmetry of the arithmetical proportions. The mean is the middle (uniting) term between the two extremes; so we have 2, 3, 4, and 4, 6, 8, and 6, 9, 12, which are all examples of arithmetic proportion; and 3,6 and 9 are the means or mid-terms. An arithmetic proportion, it should be noted, is one where the second term exceeds the first by the same amount as the third exceeds the second.

The harmonic proportion can be drawn as paths of symmetry within the pattern in the following way:

Figure 12. The harmonic proportions with the three harmonic means of 4, 8 and

These new interior patterns give us the harmonic proportions of 3, 4, 6, and 6, 8, 12, and 9,12,18. The harmonic proportion is defined as the proportion where the mean exceeds one extreme and is exceeded by the other by the same fraction of the extremes (see Timaeus 36). If we take as an example 6, 8, 12, then eight exceeds six by one third of six (i.e. by two), and eight is exceeded by twelve by one third of 12 (i.e. by four).

Figure 10	Figure 11
Figure 13. The central six as a key to the whole number intervals devised by Francesco Giorgi. Figure 14. The whole-number arithmetic and harmonic intervals between each of the seven original numbers of Plato's Lambda as proposed by Francesco Giorgi. HM = Harmonic Mean, AM = Arithmetic Mean. Note that this results in there being the non-repeating sequence of 6,8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 87, 108 and 162: sixteen numbers of which the central number 36 occurs three times and is the square of 6 or 6 to the second power I62).

The number six occupies the central point of our Tetraktys (figure 13). We can, as Francesco Giorgi did (inspired by Ficino's commentary on the Timaeus), multiply the peripheral original set of seven numbers to arrive at figure 14. We have constructed a new Tetraktys, based on unit 6. This results in the next insertions of harmonic and arithmetic means, following the Timaean cosmogony as we did above, giving rise to a total series as follows (given by F. Giorgi in his De Haimonia Mundi Totius, Venice, 1525): 6, 8, 9, 12, 16, 18, 24 , 27, 32, 36, 48, 54, 81, 108, 162. The original seven are underlined; the eight which are not underlined are harmonic or arithmetic means between the sets above and display musical ratios.

Francesco Giorgi gives another interesting veiled hint as to the meaning of the three means in relation to the three realms or domains. He calls the geometric mean the mean "par excellence," which we would propose refers to the heavenly or theological realm. He calls the arithmetic mean "excess," which we propose refers to the earthly or cosmological domain. Finally, he says of the harmonic mean that it is "the harmony of the other two," which in turn we would propose refers to the human, psychological or anthropological domain. This links the three domains through the metaphor of musical proportion, and with the pattern of the Sacred Decad or Tetraktys.

This is a deep subject that needs much more development than current space permits. Our intention is mainly to indicate that the 'literature' of the Pythagorean tradition tends to be a stimulus or an aid for the mind in its recall of the unwritten doctrines—or so it would appear.

Finally, we are drawn to another idea that results from adding one to each member of the first series:

Figure 15. The 'new' Tetraktys starting with two or the'indefinite dyad/resulting in cosmic numbers or cycles.

• 2 The Sun and Moon, day and night.
• 3 The geocentric pattern of Mercury loops.
• 4 The four seasons or quarters of the year.
• 5 The geocentric pattern of Venus loops.
• 7 The cycle of the Moon's quarter or a week.
• 9 The trine conjunctions of Saturn and Jupiter over a 60-year period or a lifetime.
• 10 The Tetraktys.
• 13 The lunar months of the year.
• 19 The Metonic cycle of coincidence between Sun and Moon every 19th year.
• 28 The full cycle of the Moon.

This new set of numbers has an interesting cosmological quality; it includes numbers which were not represented in the previous set, and numbers that appear in the cosmic cycles. The apex takes up the generating role of the dyad or principle of multiplicity and thereby the created order. The first pair are the three and the four. Four completes the Decad in the sense we have explained, yet this first pair of 3 and 4 also represents threeness and foumess as 'spirit' and 'matter' or principle and expression—aspects of cosmic duality. Further, the cosmological seven of the planetary systems and one quarter of the lunar cycle (one week) are produced by 3+4, and seven takes up the central position of the 'new' Tetraktys. Five and ten, the 'quintessence' and the 'decad,' flank the seven. The nine, which is the last archetypal number before returning to the unity of the decad or ten, begins the row of the four lowest positions. Thirteen, which is the number of the occurrences of lunar cycles in a single year, as well as the number of closest-packed spheres around their nuclear sphere, is followed by the very special number nineteen, which is not only the magic number of years it takes for the sun and moon to be in the same relationship, but is also a sacred number for both the Holy Qu'ran and the Christian Gospels. Finally, twenty-eight is the full cycle of the moon's phases and is also the sum of the central number 7, because 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

Taken arbitrarily, such facts can seem curiosities and usually do to those ignorant of arithmosophy, yet in the light of a wisdom teaching they demonstrate particular depths of symbolic value and meaning lying within the arrangement of numbers. These examples of interpretation are an indication that Plato put forward the Lambda pattern in the first instance as one of profound signification and yet left the inquiring mind to unfold the layers of meaning and the 'Pythagorean' source for themselves.

It has been said that written teachings are only ever partial (the Platonic Seventh Letter says as much), so we should expect the patterns given in the study of arithmosophy to appear simple and yet reveal more through sincere and genuine seeking. The truth, after all, is eventually one.

Both in the world and in m an the decad is all.

— Keith Critchlow