Sunday, 16 December 2018

abstract algebra - Is there an algebraic-geometric solution to the problem of the Leibnizian formalism?

The precise question appears at the end of this entry.



With all the recent advances in understanding infinitesimals, we still don't fully understand why Leibniz's definition of dydx as literally a ratio works the way it does and seems to explain numerous facts including chain rule.




Note that Robinson modified Leibniz's approach as follows. Suppose we have a function y=f(x). Let Δx be an infinitesimal hyperreal x-increment. Consider the corresponding y-increment Δy=f(x+Δx)f(x). The ratio of hyperreals ΔyΔx is not quite the derivative. Rather, we must round off the ratio to the nearest real number (its standard part) and so we set f(x)=st(ΔyΔx). To be consistent with the traditional Leibnizian notation one then defines new variables dx=Δx and dy=f(x)dx so as to get f(x)=dydx but of course here dy is not the y-increment corresponding to the x-increment. Thus the Leibnizian notation is not made fully operational.



Leibniz himself handled the problem (of which he was certainly aware, contrary to Bishop Berkeley's allegations) by explaining that he was working with a more general relation of equality "up to" negligible terms, in a suitable sense to be determined. Thus if y=x2 then the equality sign in dydx=2x does not mean, to Leibniz, exactly what we think it means.



Another approach to dy=f(x)dx is smooth infinitesimal analysis where infinitesimals are nilsquare so you get equality on the nose though you can't form the ratio. On the other hand, Leibniz worked with arbitrary nonzero orders of infinitesimals dxn so this doesn't fully capture the Leibnizian framework either.




Question: In an algebraic-geometric or other algebraic or analytic context (with suitable limitations on f), is there a way of assigning a precise sense to the Leibnizian generalized equality using global considerations?





Note. Related material can be found at this MO post.

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