Coming back to the B.Berkeley critics, there is a common denominator of all known getarounds, both the two mainstream ones (Wstrass and NSA) and exotic ones like the SDG interpretation.
That is, one considers an extension - call it $R^+$ - of the true reals R and a map
$R^+ \to R\cup \{\infty\}$ - call it the valuation map. For instance,
1) $R^+$ consists of all convergent infinite real sequences and the valuation map is the
`` taking the limit'' map
2) $R^+$ is a nonstandard extension of $R$ and the valuation map is the ``standard part'' map ($\infty$ for infinitely large objects)
3) Nilpotent or any other applicable exotics.
It occurs that the evaluation map cannot be a homomorphism, it always lacks something. For instance the value of a non-0 infinitesimal is 0, the value of its inverse is $\infty$, but $0\cdot \infty=1$ makes little sense in $R$.
This is I believe the only sound way to view the medieval controversies around infinitesimals. That is, accept that a non-0 infinitesimal is not equal to the real number 0, it just has the value 0. Maybe, a devoted scholar of Leibnizz, Euler, etc. (although there is no much of etc. after Euler!) can find a support of this point of view.
Obviously, a modern mathematician would ask for either a concrete mathematically defined model of both $R^+$ and the valuation map - and the two mainstream such models are listed above, with perhaps more yet to come under category 3 - or at least to set it up in the form of calculus of propositions, with rigorous rules of inference albeit w/o a fixed interpretation of objects.
Vladimir Kanovei
Best Answer
The famous paper of Dedekind and Weber:
R. Dedekind, H. Weber: Theorie der algebraischen Functionen einer Veränderlichen, J. Reine Angew. Math 92 (1882) 181-290.
is the first place where the points of a Riemann surface are described in terms of ideals of the ring of functions. To put this into context, Dedekind had only invented the notion of ideal a few years earlier. They also give an algebraic proof of the Riemann-Roch theorem.
I think the analogy between function fields and number fields started here.