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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?

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    I'm guessing the phrase is Dedekind's own. "Formed after a well-known saying" implies as much, though this English translation is a bit unclear. – TKR Aug 26 '17 at 1:49
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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.

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    "Man always counts" has a nice double-entendre in that "man is the measure of all things", although I might favor "man always calculates", because Dedekind comes after Laplace, who developed a fully formed theory of probability, thus a different, "game theoretic" double-entendre. (I'm a little surprised "mathematize" turns out to be actual word—I suspect I may have chosen it because of the zeta;) Wonderful answer. Thank you so much! – DukeZhou Aug 26 '17 at 2:28

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