Source and meaning of ἀεί ὁ ἄνθρωπος ἀριθμητίζει?

Question

In the preface to the first edition of Essays on the Theory of Numbers, Dedekind writes:

"In this sense which I wish to express by the word formed after a well-known saying ἀεί ὁ ἄνθρωπος ἀριθμητίζει, I hope that the following pages, as an attempt to establish the science of numbers upon a uniform foundation will find a generous welcome..."
_{Source: Hawking, God Created the Integers, reprinted courtesy of Dover Publications}

I'm taking the Greek to mean something like "Man always mathematizes", which in the context Dedekind is using it, relates to his idea that: "Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things."

The phrase is quite similar to ἀεὶ ὁ θεὸς γεωμετρεῖ, attributed to Plato, translated as "God always geometrizes."

Three related questions, answers to any of which are welcome:

• Is my translation accurate?

• What's going on with ἀριθμητίζει? I can't find that specific form, although the root and meaning seem clear.

• Is there a source for this phrase? Does it come from the Classical period, or is it a later play on Plato's phrase?

Answer 1

My first inclination was that the verb means "count", not "mathematize", as the simple ἀριθμός means "number". This translation is supported by translator Robin Mackay in a pre-print draft translation of Alain Badiou's book, Number and Numbers.

Thus, I would translate the phrase (as Badiou does): "Man always counts."

Mackay explains the history in a footnote.

[aei o anthropos arithmetizei - "man always counts". Plutarch (Convivialium disputationum, liber 8,2) reports that : "Plato said God geometrizes continually". Kepler’s repetition of the statement in his Mysterium Cosmographicum (either placatory or ironic, given the decidedly non-platonic nature of his proposed coelestis machina) was followed by Gauss’s modification: o theos arithmetizei, god arithmetizes, counts or calculates. Dedekind’s ‘copernican revolution’ consisted of transforming this once again into: aei o anthropos arithmetizei - man is always counting; completing the transformation

So it started with Plato's phrase, ἀεὶ ὁ θεὸς γεωμετρεῖ. Then Gauss modified it to ἀεὶ ὁ θεὸς ἀριθμητίζει. Then Dedekind modified it to ἀεὶ ὁ ἄνθρωπος ἀριθμητίζει.

Indeed, the verb is hard to look up, and I will edit this post if I find an online entry.