Category: …
1 Introduction
Nicomachus of Gerasa (modern-day Jerash, Jordan) was a Pythagorean philosopher-mathematician who, while little respected in modern times, had a significant influence on the history of mathematics by popularizing arithmetic, which had long been neglected in favor of geometry. The reason his popularity has waned is that this mathematical project was embedded in a philosophical and religious framework which contemporary mathematicians tend to see as little more than superstition. It is thus compared unfavorably to the supposed rationalism of earlier Greek mathematicians, although of course it is the “superstitious” attachment of late antique Pythagorean and Platonic philosophers to mathematics which we have to thank for the survival of those earlier mathematical works.
But enough about the historiography of mathematics. The work at hand is Photius’ 9th-century summary of what was once a substantial work, the Arithmetical Theologumena, made some use of by Iamblichus (a follower of Nicomachus in many ways) in his own work of the same name. The meaning of the title is difficult to entangle, and the current English convention of translating the phrase as Theology of Arithmetic does little to clarify it. Theologumena are, properly speaking, things said in a mythological form of discourse, especially about the gods. Nicomachus used this term to describe a Pythagorean tradition of assigning the names of the gods to the numbers from one to ten. In this summary, it does not become entirely clear what that means metaphysically, and we cannot project the more explicit Neoplatonic theory backwards, so I will leave it there.
What I am most interested in is that this text contains a fairly extensive list of gods, organized in what seems at first a random way. But upon repeat reading, and especially in light of other, less summary texts in the same genre, it will be seen that there is a logic to the arrangement, and that its importance does not fall far short of more conventional taxonomies.
2 Photius’ introduction
I read the two books of Arithmetical Theologumena of Nicomachus of Gerasa. Now, the title is apt to arouse wonder and a keen desire, but the substance (pónos) – not to call it a work of unfounded and vain reasonings – falls far short of the title. For he treats the numbers, from the monad (one) to the decad (ten), not as in his Arithmetic* and the Introduction (Isagoge) before it, where he goes through the natural properties of the numbers and what serious investigation shows about them, but instead for the most part (gives forth) the fictions of a mind not clean from harm, one that does not adjust the reasonings to the nature of things, but instead strives to transform the nature of things in accordance to his private imaginings.
*Now lost, whereas the Introduction to Arithmetic is extant.
[…]
Now, to speak to the title, the theology of Nicomachus of Gerase is as follows.
3 Nicomachus on the Monad (the number one)
He says not a few inappropriate inventions about the monad, mixing the truth about it with related physical properties,
- that it is Noûs [‘intellect’, as a metaphysical entity],
- and further Arsenóthēlys [‘masculo-feminine’],
- and Theós [‘god’, here meaning the creator god],
- and somehow Hýlē [‘matter’], truly mixing up all things,
- and further Pandokheús [‘all-receptive’, another term for matter],
- Khōrētikḗ [‘containing’, another term for matter]
- Kháos [‘chaos’, probably intended as another synonym for matter as that which is still mixed up and not differentiated],
- Sýnkhysis [‘confusion, commingling’],
- Sýnkrasis [‘commixture’],
- Alampía [‘lack of illumination’; this and the other seem to be associated with Chaos as primordial darkness],
- Skotōdía [‘shadowiness’],
- Khásma [‘gap’]
- Tártaros [the lowest part of the underworld, sometimes poetically called Chaos].
They also monstriloquently call (teratologoûsi) it
- Stýx [one of the rivers of the underworld],
- Phrikōdía [‘horribleness’],
- Amixía [‘unmixedness’],
- Bárathron Hypokhthónion [‘subterranean pit’, the underworld],
- Lḗthē [one of the rivers of the underworld],
- Stiphrá Párthenos [‘firm virgin’],
- Átlas,
- And for them, he is Áxōn [celestial ‘axis’, equivalent to Atlas],
- Hḗlios [‘sun’],
- Pyrálios [‘fire-sun’],
- Morphṓ […]
- Zanòs pýrgos [‘tower of Zeus’, …]
- Spermatítēs lógos [‘seminal pattern’, a Stoic concept somewhat analogous to the Platonic forms]
- Apóllōn [as if ‘not-many’, a-polloí]
- Prophḗtēs [‘proclaimer, diviner’, title of Apollon]
- Lógios [‘of speech’, or here, ‘of oracles’, usually a title of Hermes]
And the cause for these names is, on the one hand, that his mind (phrḗn) is unstable and superstitious, on the other, that he has not escaped childish impressions. And this is the way that the monad is theologized and violated by Nicomachus and his teachers.
4 Nicomachus on the Dyad (two)
[…]
- Zeús,
- Díkē [‘justice’],
- Îsis [similar to Greek ísos, ‘equal, even’],
- Phýsis [‘nature’],
- Rhéa [the Mother-of-Gods],
- Diomátēr [‘mother of Zeus’, transforming the name Demeter into a title of the Mother-of-Gods],
- […]
- Rhéa (a doublet),
- Phrygía [i.e., Phrygian Mother = Mother-of-Gods],
- Lydía [i.e., Lydian Mother = Mother-of-Gods],
- Dindymḗnē [i.e., Mother of Dindyme = Mother-of-Gods],
- Dḗmētēr [also identified with the Mother-of-Gods],
- Eleusinía [Demeter of Eleusis],
- Ártemis,
- Hímeros [‘yearning’],
- Díktynna [a byname or companion Nymph of Artemis],
- Aëría [‘aerial’, feminine],
- Astería [‘astral’, feminine],
- Dísamos
- Estṓ
- Mḗn
- Aphrodítē,
- Diṓnē
- Mykhaía
- Kythéreia [Aphrodite of Cythera],
- Ágnoia [‘ignorance’],
- Ágnōstía [‘ignorance’],
- Pseûdos [‘falsehood’],
- Adi(h)oristía (‘indefiniteness’],
- Heterótēs [‘otherness, difference’],
- Neîkos [‘strife’],
- Dikhostasía [‘dissension’],
- Móros [‘lot, death’].
- Thánatos [‘death’].
And in this manner, the dyad is theologized by them.
[…]
Notes
…
5 Nicomachus on the Triad (three)
[…]
- Kronía [‘pertaining to Kronos’, feminine]
- Latṓ [Leto, Doric dialect]
- Amaltheías kéras [‘horn of Amaltheia’, the so-called ‘horn of plenty’],
[…]
- Ophíōn
- Thétis
- Harmonía
- Hekáta [Hekate, Doric dialect],
- Erána
- Kharitía [‘pertaining to the Graces’, feminine]
- and of the Muses, Pol(h)ymnía
- Aḯdas [Hades, Doric dialect],
- Loxías [title of Apollon as oracle-giver],
- Árktos [‘dear’],
- Hélix
- […]
- Damatrámē
- Dioskoría (‘pertaining to the Dioscuri’),
- Mêtis
- Tridýmē
- Trítōn
- Thalassoûkhos
- Tritogéneia (‘triple-born’, title of Athena)
- Akhelôos
- Nástis
- Agyiópeza Kourêtis
- Krataḯs
- Harmonía
- Symbēnía
- Gámos
- Gorgonía
- Phorkía
- Trísamos
- Lýdios
[…]
Notes
…
6 Nicomachus on the Tetrad (four)
The tetrad (f.), again, is a very great marvel to them, another God (Theós, f.), Many-God (Polýtheos), or rather All-God (Pántheos). For to them, it is the fount of physical perfections, and the key-holder of Nature.¹ But it is also the forerunner and cause of the constitution and stability in accordance with mathematics.
But it is also, for them,
- another Phýsis
- Aióla
- Hēraklês
- Éxarma (
- Alkimōtátē (‘most brave’, f.),
- Arrhenikḗ (‘masculine’, f.),
- Hermês,
- Hḗphaistos
- Diónysos
- Sōrítas
- Maiadéus and Maiádēs (‘of Maia’, m.), because the tetrad is the son of Maia, that is, the dyad,
- Erioúnios
- Sôkos
- Diáktoros
- Bassareús
- Dimátōr (), as being born from the mother dyad,
- Thēlýmorphos (‘feminine-shaped
- Épandros (‘manly
- Arrhenourgós (‘masculinity-producing
- bakkhasmòn anegeírōn (‘arousing bacchasm
- Harmónita or Harmonía
- and of the Muses, Ouranía.
And this is how the tetrad is theologized by these monstriloquent people, and the first book of the theologized numbers (arithma) is brought to an end by Nicomachus.
Notes
…
7 Nicomachus on the Pentad (five)
In the second book, after the preface, the pentad fills his effort.
[…]
For the pentad is
- Aneikía
- Alloíōsis
- Pháos
- Dikaiosýnē
- the lowest summit of vitality
- Némesis
- Boubástia
- Díkēsis
- Aphrodítē
- Gamēlía
- Androgynía
- Kythéreia
- Zōnaía
- Kyklioûkhos
- Hāmítheos
- Zanòs pýrgos
- Didymaía
- Áxōn hedraía
They also give it the lofty name
- Ámbrotos
- Pallás
- Kradeâtis
- Hāgemonía
- Akreôtis
- Atálantos
- Ázyga
- Orthiâtis
- and of the Muses, Melpoménē
[….]
[…]
Notes
…
8 Nicomachus on the Hexad (six)
[…]
- Harmonía (‘’),
- Ouloméleia (‘’),
- and properly it is rather theologized as Aphrodítē Zygía (‘of the yoke’ of marriage)
- and Gamēlía (‘of marriage’, title of Aphrodite and other deities),
- and Androgynía (‘’).
- But they also make it Zygîtis (‘’),
- Philotēsía (‘’),
- Eirḗnē (‘’),
- Philía (‘’),
- Hygeía (‘’),
- Ákmōn (‘’),
- Alḗtheia (‘’),
- and of the Moiraí (Fates), they make her Lákhesis (‘’),
- and the beginning (arkhḗ) and half of the universe (or ‘everything’, pân),
- Hekatēbelétis (‘’),
- Tri(h)odîtis (‘’),
- Dikhronía (‘’),
- Perseía (‘’),
- Trímorphos (‘’),
- Amphitrítē (‘’),
- Ankhidíka (‘’),
- and of the Muses, Tháleia (‘’),
- and Panákeia (‘’).
[…]
Notes
…
9 Nicomachus on the Heptad (seven)
[…]
- Týkhē (‘Fortune’),
- Kairós (‘’),
- Athēnâ,
- Árēs,
- Akreôtis (‘’),
- Ageleía (‘’),
- Atrytṓnē (‘’),
- Phylakîtis (‘’),
- Obrimopátra (‘’),
- Tritogéneia (‘triple-born’, title of Athena),
- Glaukôpis (‘grey-eyed’, title of Athena),
- Alalkoméneia (‘’),
- Panteukhía (‘in full armor’, title of Athena),
- Ergánē (‘’),
- Polyarḗtē (‘’),
- Ouloméleia (‘’),
- Amaltheías génos (‘’),
- Aigís (the aegis worn by Zeus and Athena),
- Ósiris,
- Óneiros (‘dream’),
- Phōnḗ (‘’),
- Audḗ (‘’),
- and of the Muses, Kleiṓ (‘’).
- And, if he likes, Krísis (‘’),
- Adrásteia (‘’),
and much nonsense of that kind. In this way is their all-reverend (pánseptos) heptad, by a rather great effor (pónos), hymned as a great and multiple (polýs) god.
Notes
…
10 Nicomachus on the Ogdoad (eight)
[…]
- Pan(h)armonía (‘universal Harmony’),
- Kadmeía Mḗtēr (‘Cadmean/Theban Mother’, unclear; Mother-of-Gods?),
- Rhéa (Mother-of-Gods),
- Thēlypoiós (‘producing feminity’),
- Kybélē (Mother-of-Gods),
- Kybḗbē (Mother-of-Gods),
- Dindýmē (Mother-of-Gods),
- Polioûkhos (‘keeper of the city’, title of various deities including the Mother-of-Gods),
- Érōs (‘desire, love’),
- Philía (‘friendship, love’),
- Mêtis,
- Epínoia,
- Oreía (‘of the mountain’, Mountain Mother = Mother-of-Gods),
- Thémis,
- Nómos,
- Ēlitom-?
- and of the Muses, Eutérpē.
Notes
…
11 Nicomachus on the Ennead (nine)
[…]
[…]
- Ōkeanós,
- and is hymned as Horízōn (‘’).
They give it the sacred names (hierologoûsin)
- Promētheús (‘’),
- Homónoia (‘’),
- Perseía (‘’),
- Hā́lios (‘’),
- Aneikía (‘’),
- Homoíōsis (‘’),
- Hḗphaistos,
- Hḗra, the sister and spouse of Zeus,
- Hekáergos (‘’),
- Paiân (‘’),
- Nyssēḯs (‘’),
- Agyieús (‘’),
- Enyálios (‘’),
- Ageleía (‘’),
- Tritogéneia (‘triple-born’, title of Athena),
- Homónoia (‘’),
- Peithṓ (‘’),
- Kourêtis (‘’),
- Kórē,
- Hyperíōn (‘’),
- and of the Muses, Terpsikhórē (‘’).
Notes
…
12 Nicomachus on the Decad (ten)
[…]
- Pân (‘everything, universe’),
- Theòs hypértheos (‘god above god[s]’),
- Theòs theôn (‘god of gods’),
[…]
But the decad, for them, is:
- Kósmos (‘order, cosmos, world’),
- Ouranós (‘Heaven’),
- Heimarménē (‘Fate’),
- Aiṓn (‘eternity’),
- Krátos (‘strength’),
- Pístis (‘faith, faithfulness’),
- Anánkē (‘necessity’),
- Átlas,
- Akámas Theós (‘untiring god’, used, e.g., of the Sun),
- Phánēs,
- Hḗlios (‘Sun’),
- Ouranía (‘celestial one’, a Muse or Aphrodite),
- Mnḗmē (‘Memory’, a Muse),
- Mnēmosýnē (‘Memory’, the mother of the Muses),
[…]
Notes
…
13 Photius’ epilogue
…
Notes
…
SARTRIX 