According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective algebraic curves. This suggests that the archimedean places are infinite in the sense of being ‘points at infinity’ and not just because rational integers become arbitrarily ‘large’.
Is there any reason to expect this, i.e. was there some intuition that the archimedean places were the points needed to make $\operatorname{Spec} \mathscr{O}_K$ complete in some sense? I am aware of the construction of abstract smooth projective curves from function fields via DVRs (e.g. as described in Hartshorne [Ch. I, §6]), but the set $\{ x \in K : \left| x \right|_v \le 1 \}$ fails to be a subring, let alone a DVR, for any archimedean place $v$. (At any rate, archimedean absolute values do not correspond to valuations.)
Was the term ‘infinite prime’ used for an archimedean place before these analogies were known? If so, why?