I would like to explore the origin of the name 'place' in number theoretic sense. A place of an algebraic number field $K$ is defined as an equivalence class of absolute value functions on $K$. There are two types of places: the infinite and infinite places.
It is well known that the 'finite' places of a number field are indeed in one-to-one correspondence with the (maximal) prime ideals of $K$'s ring of integers $R$. See for example this discussion. If we think about the variety/ scheme $V= \operatorname{Spec}(R)$ associated to $R$ then the finite places can be regarded as 'points' in Zariski topology and this seems at least for me to provide a justification for the geometrically afflicted notation 'place'.
First of all, is this exactly the original reason why (finite) places are called 'places'? i.e. that they are sitting somewhere in the associated topological space $V$. The reason why I doubt and think that this is not the proper origin or the name 'place' is simply because the Grothendieck's fondation of algebraic geometry in language of schemes arises later then the study of absolute values of algebraic number fields.
Does anybody know the true origin and motivation of the name 'place'?
In addition, there are also infinite places which cannot be associated with prime ideals of ring of integers $R$. Is there also any reason why they carry geometrically afflicted name 'places'?
I checked first if the language of places appears in the Artin-Whaples paper in 1945 on axiomatizing number fields and function fields over finite fields in terms of a set of absolute values satisfying a product formula. However, I did not find the technical term "place" there. It also is not in the Artin-Tate notes on class field theory (they refer to "primes" or "canonical absolute values") or Tate's thesis (he writes about "prime divisors"), which brings us to 1950.
In Edwin Weiss's Algebraic Number Theory, which was published in 1963 but was based on a course he gave at Harvard in 1958, he refers to an equivalence class of absolute values as a "prime divisor" and uses "place" to refer to the residue class field mapping $\mathcal O \to \mathcal O/\wp$ associated to a nonarchimedean prime divisor.
Bachman's 1964 book Introduction to $p$-adic Numbers and Valuation Theory also essentially uses "place" in the same sense as Weiss: the residue field mapping for an absolute value (same as mapping for any equivalence absolute value, since they all have the same valuation ring).
The terminology of places in its meaning today appears in Weil's Basic Number Theory, which was published in 1967 but was based on a course he gave at Princeton in 1961-1962. See Definition 3 in Chapter III. Weil defines a place as an equivalence class of completions rather than an equivalence class of absolute values, but that is morally the same idea, at least closer than the usage of Weiss and Bachman.
So if I had to bet on who finally gave the term "place" its standard meaning today in valuation theory, I'd go with Weil. But keep in mind that the term "place" was used earlier for related ideas in valuation theory. The analogy with points on (compact) Riemann surfaces giving rise to the discrete valuations on their function fields (normalized to have value group $\mathbf Z$) was certainly in the back of the minds of people developing all this mathematics