Quelltext der Seite Werk:The Theology of Arithmetic/On the Μοnad

[[Datei:On the Monad.png|zentriert|100x100px]]
[1] The monad is the non-spatial source of number. It is called 'monad' because of its stability,<ref>The author follows a traditional etymology of ''monas'' and derives it from ''menein'' (to be stable).</ref> since it preserves the specific identity of any number with which it is conjoined. For instance, 3 x 1 = 3, 4 x 1 = 4: see how the approach of the monad to these numbers preserved the same identity and did not produce a different number.

Everything has been organized by the monad, because it contains everything potentially: for even if they are not yet actual, nevertheless the monad holds seminally the principles which are within all numbers, including those which are within the dyad. For the monad is even and odd and even-odd;<ref>For these and other terms, see the Glossary. The monad is called both even and odd because if it is added to an even number, the result is odd, while if it is added to an odd number, the result is even. It was therefore held to have the properties of both evenness and oddness.</ref> linear and plane and solid (cubical and spherical and in the form of pyramids from those with four angles to those with an indefinite number of angles); perfect and over-perfect and defective; proportionate and harmonic; prime and incomposite, and secondary; diagonal and side; and it is the source of every relation, whether one of equality or inequality, as has been proved in the Introduction.<ref>The reference is probably to Nicomachus, ''Introduction to Arithmetic'', 1.23.4-17 and II.2.1-2. Assuming that the monad is the source of equality because the first manifestation of equality is 1=1, then those passages are relevant, because they argue that all forms of inequality (see p. 79, n. 14) are derivable from equality and reducible back to it.</ref> Moreover, it is demonstrably both point and angle (with all forms of angle), and beginning, middle and end of all things, since, if you [2] decrease it, it limits the infinite dissection of what is continuous, and if you increase it, it defines the increase as being the same as the dividends (and this is due to the disposition of divine, not human, nature.)<ref>The text of this sentence is extremely difficult and there may be a lacuna before 'since,' but it seems to mean, first, if you take 1/n, then however many parts 1 is divided into, each part is still a monad in its own right; second, if you take n/1, then n is still 1 + 1 + 1 … n.</ref>

At any rate (as was demonstrated at the beginning of the ''Arithmetic'' in the lambda-shaped diagram),<ref>The familiarity of the reference makes it seem as though it is again to Nicomachus' ''Introduction'' (see p. 35, n. 3), but there is no lambda-shaped diagram near the start of that. However, in his commentary on Nicomachus' book, Iamblichus derives from it a lambda diagram which demonstrates the 'natural contrariety' of integers and fractions. So the reference is to Nicomachus, but as commented on by Iamblichus (14.3 ff.). The application of Iamblichus' diagram which is relevant to our text is as follows: [[Datei:lambda diagram natural contrariety of integers and fractions a.png|150x150px]]or [[Datei:lambda_diagram_natural_contrariety_of_integers_and_fractions_b.png|150x150px]]</ref> each of the parts within the monad correspond to and offset the integers. Hence, just as if ''x'' is double ''y'', then ''x''<sup>2</sup> is four times ''y''<sup>2</sup>, and ''x''<sup>3</sup> is eight times ''y''<sup>3</sup>, and if ''x'' is triple ''y'', then ''x''<sup>2</sup> is nine times ''y''<sup>2</sup>, and ''x''<sup>3</sup> is twenty-seven time ''y''<sup>3</sup>, in the orderly arrangement of all numbers; so also in the orderly arrangement of parts, if x is half ''y'', then ''x''<sup>2</sup> is one quarter ''y''<sup>2</sup>, and ''x''<sup>3</sup> is one eighth ''y''<sup>3</sup>, and if ''x'' is one third ''y'', then ''x''<sup>2</sup> is one ninth ''y''<sup>2</sup>, and ''x''<sup>3</sup> is one twenty-seventh ''y''<sup>3</sup>.

Every compound of plurality or every subdivision is given form by the monad; for the decad is one and the chiliad is one, and again one tenth is one and one thousandth is one, and so on for all the subdivisions ad ''infinitum''.

In each of these cases there is the same monad in terms of form, yet different monads in respect of quantity, because it produces itself out of itself, as well as producing them, just as if it were the principle of the universe and the nature of things; and because it maintains everything and forbids whatever it is present in to change, it alone of all numbers resembles the Providence which preserves everything, and is most particularly suited both to reflect the principle of God and to be likened to him, in so far as it  is closest to him.

It is in fact the form of forms, since it is creation thanks to its creativity and intellect thanks to its intelligence; this is adequately demonstrated in the [3] mutual opposition of oblongs and squares.<ref>Nicomachus, ''Introduction to Arithmetic'' n.19, argues that the whole universe is skilfully contrived by harmonizing the opposition of the sequences of squares and oblong numbers.</ref>

Nicomachus says that God coincides with the monad, since he is seminally everything which exists, just as the monad is in the case of number; and there are encompassed in it in potential things which, when actual, seem to be extremely opposed (in all the ways in which things may, generally speaking, be opposed), just as it is seen, throughout the ''Introduction to Arithmetic'', to be capable, thanks to its ineffable nature, of becoming all classes of things, and to have encompassed the beginning, middle and end of all things (whether we understand them to be composed by continuity or by juxtaposition),<ref>See e.g. pp. 55, 60, n. 13: all things are either continuous and have size, or are juxtaposed and have quantity.</ref> because the monad is the beginning, middle and end of quantity, of size and moreover of every quality.

Just as without the monad there is in general no composition of anything, so also without it there is no knowledge of anything whatsoever, since it is a pure light, most authoritative over everything in general, and it is sun-like and ruling, so that in each of these respects it resembles God, and especially because it has the power of making things cohere and combine, even when they are composed of many ingredients and are very different from one another, just as he made this universe harmonious and unified out of things which are likewise opposed.

Furthermore, the monad produces itself and is produced from itself, since it is self-sufficient and has no power set over it and is everlasting; and it is evidently the cause of permanence, just as God is thought to be in the case of actual physical things, and to be the preserver and maintainer of natures.

So they<ref>8. Any unattributed 'they' in this treatise means 'the Pythagoreans.'</ref> say that the monad is not only God, but also 'intellect' [4] and 'androgyne.' It is called 'intellect' because of that aspect of God which is the most authoritative both in the creation of the universe and in general in all skill and reason: even if this aspect of God were not to manifest itself as a whole in particular matters, yet in respect of its activity it is Intellect, since in respect of its knowledge it is sameness and unvarying. Just so, the monad, which even if differentiated in the different kinds of thing has conceptually encompassed everything within itself, is as it were a creative principle and resembles God, and does not alter from its own principle, and forbids anything else to alter, but is truly unchanging and is the Fate Atropos.<ref>One of three Fates: her name means 'not to be turned aside.'</ref>

That is why it is called 'artificer' and 'modeler,' since in its processions and recessions it takes thought for the mathematical natures, from which arise instances of corporeality, of propagation of creatures and of the composition of the universe. Hence they call it 'Prometheus,' the artificer of life, because, uniquely, it in no way outruns or departs from its own principle,<ref>The etymology of Prometheus is here derived from the Greek for 'not outrunning.'</ref> nor allows anything else to do so, since it shares out its own properties. For however far it is extended, or however many extensions it causes, it still prohibits outrunning and changing the fundamental principle of itself and of those extensions.

So, in short, they consider it to be the seed of all, and both male and female at once—not only because they think that what is odd is male in so far as it is [5] hard to divide and what is even is female in so far as it is easy to separate, and it alone is both even and odd, but also because it is taken to be father and mother, since it contains the principles of both matter and form, of craftsman and what is crafted; that is to say, when it is divided, it gives rise to the dyad. (For it is easier for a craftsman to procure matter for himself than for the reverse to happen—for matter to procure a craftsman.)<ref>Here the argument is that the monad take precedence over the dyad (see also p. 42), but on p. 46 there is an argument that they co-exist. Both passages are probably from Nicomachus, so they should be resolved into the idea that the monad is the most important of the primary and co-existent pair of sources, the monad and the dyad, or sameness and difference.</ref> And the seed which is, as far as its own nature is concerned, capable of producing both females and males, when scattered not only produces the nature of both without distinction, but also does so during pregnancy up to a certain point; but when it begins to be formed into a foetus and to grow, it then admits distinction and variation one way or the other, as it passes from potentiality to actuality.<ref>The relevant tenets of Greek embryology are that the father gives form, the mother matter; and that differentiation of gender does not occur for the first few days of pregnancy (cf. pp. 83-4, 93-4).</ref>

If the potential of every number is in the monad, then the monad would be intelligible number in the strict sense, since it is not yet manifesting anything actual, but everything is conceptually together in it.

There is a certain plausibility in their also calling it 'matter' and even 'receptacle of all,' since it is productive even of the dyad (which is matter, strictly speaking) and since it is capable of containing all principles; for it is in fact productive and disposed to share itself with everything.

Likewise, they call it 'Chaos,' which is Hesiod's first generator,<ref>Hesiod, ''Theogony'' 116.</ref> because Chaos gives rise to everything else, as the monad does. It is also thought to be both 'mixture' and 'blending,' 'obscurity' and 'darkness,' thanks to the lack of articulation and distinction of everything which ensues from it.
----Anatolius says that it is called 'matrix' and 'matter,' on the grounds that without it there is no number.
----The mark which signifies the monad is a symbol of the source of all things.<ref>Presumably a (alpha) is the 'mark which signifies the monad' (see next note). It starts the word ''arche'' (source).</ref> [6] And it reveals its kinship with the sun in the summation of its name: for the word 'monad' when added up yields 361, which are the degrees of the zodiacal circle.<ref>The letters of the Greek alphabet also served as numerical symbols: hence a system of gematria was obvious. The Greek ''monas'' is 40 + 70 + 50 + 1 + 200. Presumably 361, rather than 360, is given as the number of degrees because the first one is counted twice, to indicate a complete circle.</ref>
----The Pythagoreans called the monad 'intellect' because they thought that intellect was akin to the One; for among the virtues, they likened the monad to moral wisdom,- for what is correct is one. And they called it 'being,' 'cause of truth,' 'simple,' 'paradigm,' 'order,' 'concord,' 'what is equal among greater and lesser,' 'the mean between intensity and slackness,' 'moderation in plurality,' 'the instant now in time,' and moreover they called it 'ship,' 'chariot/ 'friend/ 'life/ 'happiness.'

Furthermore, they say that in the middle of the four elements there lies a certain monadic fiery cube, whose central position they say Homer was aware of when he said: "As far beneath as is Hades, so far above the Earth are the heavens."<ref>''Iliad'' 8.16. See p. 106, n. 15, for the four-element model of the universe.</ref> In this context, it looks as though the disciples of Empedocles and Parmenides and just about the majority of the sages of old followed the Pythagoreans and declared that the principle of the monad is situated in the middle in the manner of the Hearth, and keeps its location because of being equilibrated; and Euripides too, who was a disciple of Anaxagoras, mentions the Earth as follows: "Those among mortals who are wise consider you to be the Hearth."<ref>Euripides, fragment 938 (Nauck<sup>2</sup>).</ref> 

Moreover, [7] the Pythagoreans say that the right-angled triangle too was formed by Pythagoras when he regarded the numbers in the triangle monad by monad.<ref>See p. 82, perhaps.</ref>
----The Pythagoreans link matter closely with the dyad. For matter is the source of differentiation in Nature, while the dyad is the source of differentiation in number; and just as matter is indefinite and formless, so also, uniquely among all numbers, the dyad is incapable of receiving form. Not least for the following reason also the dyad can be called indefinite: shape is encompassed in actuality by means of at least and in the first instance three angles or lines, while the monad is in potential.<ref>1 am fairly sure that this paragraph is out of place and belongs somewhere in the section on the dyad.</ref>

Calling the monad 'Proteus/ as they do, is not implausible, since he was the demigod in Egypt who could assume any form and contained the properties of everything, as the monad is the factor of each number.