In the Gorgias (451 AC), Socrates defines arithmetics : "It belongs to that
knowledge which deals with the even and the odd, with reference to how much
either happens to be"(hoti toon peri to artion te kai peritton (gnoosis),
hosa an hekatera tugchanei onta"). Even if there is a theoretical
arithmetics for Plato, the Greek sense is closer to "art of counting" than
to "number theory" in the modern sense.
Meanwhile, Plato puts the accent on geometry : "The knowledge of which
geometry aims is the knowledge of the eternal"(Resp, VII, 52) Over the
entrance to Plato's Mouseion was the inscription "Let not one destitute of
geometry enter my doors"(Ageometretos medeis eisito) (cf Phlp. in de An.
cvii 26-7; Olymp., Proll. viii, 39-ix 1)
As for the famous saying "God geometrizes", it is not to be found in Plato's
own writings, but in a quote from Plutarch (Convivialium disputationum,
liber 8,2) : "Plato said God geometrizes continually"
In the modern times, Gauss uses the sentence again, but changes it : "God is
doing arithmetics"
Gauss changes the sentence into : "o theos arithmetizei", "god computes,
calculates", or, better, "is using arithmetics". Gauss certainly wanted to
express the importance given to arithmetics in the modern times, while the
Greeks, especially from Plato's viewpoint, saw geometry as the foremost
mathematical discipline.
However, there is no verb "arithmetidzo" in Greek, as far as I can see from
my Greek-French Lexicon Bailly. There is, however, a verb "arithmeo^-o^",
meaning, "to count", while "arithme\sis" refers to the act of counting.
Dedekind changed the saying one more time, transforming it into : "aei o
anthropos arithmetizei", because he clearly wanted to show how it was
possible to build arithmetics with the means of the sole human thought,
without reference to transcendent realities, much in the vein of
Weierstrass' arithmetization of analysis.