Analogy: ἀναλογία

We ought to perhaps start our journey into the concept of the analogia entis by etymology, though the etymological source of the word analogia (analogy) does not by any means admit us into the real meaning of this concept. In fact, it may serve to confuse. If for no other reason than perhaps to understand how these words analogia entis have a life separate and apart from their origins we ought to start with etymology.

The word analogy comes to us from the Latin analogia, but the word is Greek in ultimate origin. The Greek term ἀναλογία (analogia) means proportion. It is a compound word formed from ana (meaning "upon" or "according to") and logos (mean "reason," "word," or "speech"). When used by, for example, Plato and Aristotle, the term ἀναλογία means a proportion of mathematical kind. We find it used by Plato in his Timaeus and by Aristotle in his Nicomachean Ethics. We might quote from Plato's Timaeus:

[31c] for there must needs be some intermediary bond to connect the two. And the fairest of bonds is that which most perfectly unites into one both itself and the things which it binds together; and to effect this in the fairest manner is the natural property of proportion. For whenever the middle term of any three numbers, cubic or square, [32a] is such that as the first term is to it, so is it to the last term, and again, conversely, as the last term is to the middle, so is the middle to the first,—then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity.

[31ξ] οὐ δυνατόν: δεσμὸν γὰρ ἐν μέσῳ δεῖ τινα ἀμφοῖν συναγωγὸν γίγνεσθαι. δεσμῶν δὲ κάλλιστος ὃς ἂν αὑτὸν καὶ τὰ συνδούμενα ὅτι μάλιστα ἓν ποιῇ, τοῦτο δὲ πέφυκεν ἀναλογία κάλλιστα ἀποτελεῖν. ὁπόταν γὰρ ἀριθμῶν τριῶν εἴτε ὄγκων [32α] εἴτε δυνάμεων ὡντινωνοῦν ᾖ τὸ μέσον, ὅτιπερ τὸ πρῶτον πρὸς αὐτό, τοῦτο αὐτὸ πρὸς τὸ ἔσχατον, καὶ πάλιν αὖθις, ὅτι τὸ ἔσχατον πρὸς τὸ μέσον, τὸ μέσον πρὸς τὸ πρῶτον, τότε τὸ μέσον μὲν πρῶτον καὶ ἔσχατον γιγνόμενον, τὸ δ᾽ ἔσχατον καὶ τὸ πρῶτον αὖ μέσα ἀμφότερα, πάνθ᾽ οὕτως ἐξ ἀνάγκης τὰ αὐτὰ εἶναι συμβήσεται, τὰ αὐτὰ δὲ γενόμενα ἀλλήλοις ἓν πάντα ἔσται.

Later in the same work, Plato again invokes the word analogia:

Thus it was that in the midst between fire and earth God set water and air, and having bestowed upon them so far as possible a like ratio one towards another—air being to water as fire to air, and water being to earth as air to water, —he joined together and constructed a Heaven visible and tangible. For these reasons [32c] and out of these materials, such in kind and four in number, the body of the Cosmos was harmonized by proportion and brought into existence.

οὕτω δὴ πυρός τε καὶ γῆς ὕδωρ ἀέρα τε ὁ θεὸς ἐν μέσῳ θείς, καὶ πρὸς ἄλληλα καθ᾽ ὅσον ἦν δυνατὸν ἀνὰ τὸν αὐτὸν λόγον ἀπεργασάμενος, ὅτιπερ πῦρ πρὸς ἀέρα, τοῦτο ἀέρα πρὸς ὕδωρ, καὶ ὅτι ἀὴρ πρὸς ὕδωρ, ὕδωρ πρὸς γῆν, συνέδησεν καὶ συνεστήσατο οὐρανὸν ὁρατὸν καὶ ἁπτόν. καὶ διὰ ταῦτα ἔκ τε δὴ τούτων τοιούτων [32ξ] καὶ τὸν ἀριθμὸν τεττάρων τὸ τοῦ κόσμου σῶμα ἐγεννήθη δι᾽ ἀναλογίας ὁμολογῆσαν . . . .

It is an error to start the journey of understanding the analogia entis with a concept of analogy that is based upon mathematical or geometric proportion. The biggest single impediment to understanding the term analogy (when used in the concept analogy of being) is failing to see the term analogy itself analogical, at least analogical relative to its original sense of proportion.

Plato and Aristotle both use the term analogia in the meaning of "sameness of ratio," so that A:B :: C:D (e.g., 2:4 :: 8:16). This analogia or proportion may be discontinuous (because there are no shared terms among the four components) or continuous if there is a shared term. For example, a continuous analogia would be A:B :: B:C (e.g., 2:4 :: 4:8). In the Timaeus, however, Plato appears to be using the term as a continuous geometric analogia.

Aristotle in his Nicomachean Ethics (1131a31) imports the term analogia into the concept of justice, using this mathematical proportion as a term for justice.

Justice is therefore a sort of proportion; for proportion is not a property of numerical quantity only, but of quantity in general, proportion being equality of ratios, and involving four terms at least.

ἔστιν ἄρα τὸ δίκαιον ἀνάλογόν τι. τὸ γὰρ ἀνάλογον οὐ μόνον ἐστὶ μοναδικοῦ ἀριθμοῦ ἴδιον, ἀλλ᾽ ὅλως ἀριθμοῦ: ἡ γὰρ ἀναλογία ἰσότης ἐστὶ λόγων, καὶ ἐν τέτταρσιν ἐλαχίστοις. ἡ μὲν οὖν διῃρημένη ὅτι ἐν τέτταρσι, δῆλον. ἀλλὰ καὶ ἡ συνεχής: τῷ γὰρ ἑνὶ ὡς δυσὶ χρῆται καὶ δὶς λέγει.

Aristotle later tells us (1131b13) that this analogia is borrowed from the geometers.

This kind of proportion is termed by mathematicians geometrical proportion; for a geometrical proportion is one in which the sum of the first and third terms will bear the same ratio to the sum of the second and fourth as one term of either pair bears to the other term. Distributive justice is not a continuous proportion, for its second and third terms, a person and a share, do not constitute a single term.

καλοῦσι δὲ τὴν τοιαύτην ἀναλογίαν γεωμετρικὴν οἱ μαθηματικοί: ἐν γὰρ τῇ γεωμετρικῇ συμβαίνει καὶ τὸ ὅλον πρὸς τὸ ὅλον ὅπερ ἑκάτερον πρὸς ἑκάτερον. ἔστι δ᾽ οὐ συνεχὴς αὕτη ἡ ἀναλογία: οὐ γὰρ γίνεται εἷς ἀριθμῷ ὅρος, ᾧ καὶ ὅ. τὸ μὲν οὖν δίκαιον τοῦτο, τὸ ἀνάλογον: τὸ δ᾽ ἄδικον τὸ παρὰ τὸ ἀνάλογον.

But Aristotle seems to use the term in other ways as well: as an arithmetic analogia, as distinguished from a geometric analogia.

For example, let 10 be many and 2 few; then one takes the mean with respect to the thing if one takes 6; since 6 —2 = 10 — 6, and this is the mean according to arithmetical proportion.

οἷον εἰ τὰ δέκα πολλὰ τὰ δὲ δύο ὀλίγα, τὰ ἓξ μέσα λαμβάνουσι κατὰ τὸ πρᾶγμα: ἴσῳ γὰρ ὑπερέχει τε καὶ ὑπερέχεται: τοῦτο δὲ μέσον ἐστὶ κατὰ τὴν ἀριθμητικὴν ἀναλογίαν.

N.E., 1106a36. As Carl A. Huffman interprets it,* "[i]t appears that from the more precise definition of 'equality of ratio' [proportion] there developed a looser sense in which any similarity, which could be defined in accordance with a mathematical account (ἀνα λόγον.), could constitute an analogia." "In its broadest sense," Huffman continues, "Aristotle uses analogia to refer to any similarity in the relationships between two pairs of things." It is in this broader sense that Aristotle uses the term analogia to compare the scales of a fish as feathers are to a bird. Aristotle, History of Animals, 486b17.

The notion of analogy that is used in the concept of analogy of being, however, is transmathematical. Not only is it transmathematical, it is not a protraction or extrapolation of the mathematical concept of proportion. We do not simply extend out or protract the meaning of the mathematical term "proportion," a term which is univocal, and in some sense derive a meaning of "super-proportion" to understand the use of the term. Multiply a univocal term by a million and you still have a univocal term. The term analogy as used in the concept of analogy of being is not some sort of "inflated univocal notion." Anderson (1967), 1.

The notion of analogy of being is, rather, metaphysical, ontological (it relates to being), beyond time, space, and quantity, and so it is not constrained or bounded by any dimensional qualities. "[O]ntological analogies cannot be mere extrapolations from the real of mathematical ratios." Anderson (1967), 1.

We do best then to forget the etymological roots of the word analogia, for it is sure to steer us wrong. It is a false friend.

___________________________
*Carl A. Huffman, Archytas of Tarentum: Pythagorean, Philosopher, and Mathematician King (Cambridge: Cambridge University Press 2005), 179-80.
*Anderson (1967), 1.