In what ways did Leibniz's philosophy foresee modern mathematics?

Abraham Robinson explicitly referred to Leibniz's idea of infinitesimal quantities when developing non-standard analysis in 1960's. Wikipedia article has a quotation from his book Robinson, Abraham (1996). Non-standard analysis (Revised edition ed.). Princeton University Press. ISBN 0-691-04490-2.

Added: the idea of expressing logic in an algebraic way is credited to Leibniz; see e.g. the following article in Stanford Encyclopedia of Philosophy:

http://plato.stanford.edu/entries/leibniz-logic-influence/#DisLeiMatLog

Added: Saul Kripke introduced a semantics of possible worlds (really, relational semantics) for modal logic. http://en.wikipedia.org/wiki/Modal_logic#Semantics

The idea of possible worlds precedes Leibniz, but he devoted a lot of consideration to it. Ironically, his claim that our existing world is the best out of possible ones is perhaps most known from the ridicule it received in Voltaire's "Candide". Oh wait, this is Math Overflow...

My version, quickly, would be that he envisaged "points" that were abstractions. Whence "logical space" as came in first around 1900 (long discussion) as implied by Boolean algebra, which he also anticipated. Also "extensionality", still a scary concept for mathematics even post-Grothendieck. Sadly MO is hardly the place: the recent book by Daniel Garber on Leibniz makes the good point that his thought is a moving target, often distorted by later authors.

Edit: Since this question has survived closure, some more. If you look at the April 2004 version of the article "Sheaf (mathematics)" on Wikipedia. it says that some aspects of sheaf theory trace back to Leibniz. I put that in; no doubt it was rightly taken out. I just think it shows how far a serious discussion might lead. The codification of four "laws of thought" from Leibniz is probably an example of distortion, if hugely influential. It broke down around 1910 (Bertrand Russell round then wrote up three laws), and the extensionality implied by A = B if (and only if but that is trivial) A and B have the same attributes had to come back into mathematics by the back door, really. Parts of this question would be fruitful as new questions.

Practically, Leibniz preceded computer science by inventing the Stepped Reckoner, a mechanical computer which was the first to be able to compute addition, subtraction, multiplication, and division.

More abstractly, he sought after a "calculus ratiocinator", a framework for dealing with logical statements. You can think of this as sort of a primitive formal language, although I doubt Leibniz had in mind as heavy restrictions that we use for formal grammars today.