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Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a mathematician has grown in the last fifty years as some of his philosophically oriented mathematical ideas have connected with modern mathematicians and mathematics. That because of Leibniz's philosophical reflections, he foresaw aspects or parts of modern mathematics. Can anyone elaborate on these connections and recommend any references?

EDIT, Will Jagy. Editing mostly to bump this to the front of active. It is evident that Jacques and Sergey have good, substantial answers in mind. Please do not answer unless you have read Leibniz at length. I kind of liked philosophy in high school and college, or thought I did. Recently, I read one page of Spinoza and gave up.

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    $\begingroup$ It's as with Nostradamus: the more vague your predictions are, the more can be read into them. $\endgroup$
    – Franz Lemmermeyer
    Commented Nov 8, 2011 at 12:20
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    $\begingroup$ To be clearer, I think a less subjective question would be : In what ways did Leibniz's mathematics foresee modern aspects of mathematics ? $\endgroup$
    – François Brunault
    Commented Nov 8, 2011 at 13:56
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    $\begingroup$ For what its worth, Leibniz has an extensive Wikipedia entry: en.wikipedia.org/wiki/Gottfried_Leibniz $\endgroup$
    – Joel David Hamkins
    Commented Nov 8, 2011 at 14:34
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    $\begingroup$ @Daniel Geisler: Maybe you could say which readings of yours have led to this perception, or which of Leibniz's philosophical ideas you are referring to, or which parts of modern mathematics were foreseen in his philosophy. This question is way too vague as it stands. $\endgroup$
    – Todd Trimble
    Commented Nov 8, 2011 at 14:53
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    $\begingroup$ @Pete: Will do. see tea.mathoverflow.net/discussion/1204/… $\endgroup$
    – Jacques Carette
    Commented Nov 8, 2011 at 20:17

4 Answers 4

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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...

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    $\begingroup$ +1 for mentioning Candide (and I would +1 as well for the actual content of the answer, but I can't). $\endgroup$
    – Jacques Carette
    Commented Nov 8, 2011 at 18:11
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    $\begingroup$ Are you actually claiming that Leibniz's infinitesimals was a philosophical idea? Without wanting to fully raise the (of course very thorny) question of what exactly constitutes a "philosophical idea", let just express a little skepticism on this point. I would rather say that Leibniz introduced infinitesimals as a purely mathematical idea, albeit one that he could not adequately formalize or justify. Moreover I would summarize Robinson's great (and certainly wholly mathematical!) achievement as demonstrating that Leibniz's tentative mathematical idea was mathematically sound. $\endgroup$
    – Pete L. Clark
    Commented Nov 8, 2011 at 18:53
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    $\begingroup$ @Jacques: My pleasure. @Pete: Some would indeed claim philosophical origin of infinitesimals and relate them to monads. This may be a stretch, but at any rate Leibniz commented on them extensively, so they became a part of his philosophy. He may have held different views at different times, though; see e.g. muse.jhu.edu/journals/perspectives_on_science/v006/… $\endgroup$
    – Margaret Friedland
    Commented Nov 8, 2011 at 19:14
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    $\begingroup$ @Pete: I have read a lot of Leibniz in the last year, and yes, just about everything he did was done within a philosophical context. While we can discern and extract purely mathematical ideas out of his writings, he did not. $\endgroup$
    – Jacques Carette
    Commented Nov 8, 2011 at 19:26
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    $\begingroup$ It should be added that back in those days, there weren't very sharp distinctions made between mathematics, physics, and philosophy of the sort we would recognize today. Indeed, 'natural philosophy' was often what we might call physics today. So it's hard to say that Leibniz himself would have thought of infinitesimals as purely mathematical, and it wouldn't be surprising if they figured in his presumably more philosophical tracts. $\endgroup$
    – Todd Trimble
    Commented Nov 8, 2011 at 19:34
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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.

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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.

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To answer the title question, "In what ways did Leibniz's philosophy foresee modern mathematics?" one could mention the distinction between assignable and inassignable number that closely parallels the distinction between standard and nonstandard number in Abraham Robinson's (or Edward Nelson's) framework. Furthermore, Leibniz's notion of a generalized relation of equality closely parallels the modern notion of shadow (or standard part). Leibniz's law of continuity finds a close procedural proxy in the transfer principle of nonstandard analysis. The remainder of this answer will explain how one can make such claims without falling into the trap of presentism.

Daniel Geisler speculates that "because of Leibniz's philosophical reflections, he foresaw aspects or parts of modern mathematics" and asks: "Can anyone elaborate on these connections and recommend any references?"

Several responders mentioned the connection to Robinson's theory. On the other hand, François Brunault rightly cautioned: "The statement that someone (even Leibniz) foresaw parts of modern mathematics is potentially controversial because of its subjectivity. I think most historians of mathematics now insist on the fact that the works by earlier mathematicians should also be studied from the point of view of that time, before extrapolating possible connections."

François Brunault is correct in suggesting that there is resistance among historians of mathematics to the idea of seeing continuity between Leibniz and Robinson. Indeed, the prevalent interpretation of Leibnizian infinitesimals is a so-called syncategorematic interpretation, pursued notably by R. Arthur and many other Leibniz scholars. On this view, Leibnizian infinitesimals are merely shorthand for ordinary ("real") values, assorted with a (hidden) quantifier, viewed as a kind of a pre-Weierstrassian anticipation. These scholars rely on evidence drawn from various quotes from Leibniz where he refers to infinitesimals as "useful fictions", and explains that arguments involving infinitesimals can be paraphrased a l'ancienne using exhaustion. In this spirit, they interpret the Leibnizian "useful fictions" as LOGICAL fictions, denoting what would be described in modern terminology is a quantified formula in first-order logic.

For example, Levey writes:

"The syncategorematic analysis of the infinitely small is ... fashioned around the order of quantifiers so that only finite quantities figure as values for the variables. Thus,

(3) the difference $|a-b|$ is infinitesimal

does not assert that there is an infinitely small positive value which measures the difference between~$a$ and~$b$. Instead it reports,

($3^*$) For every finite positive value $\varepsilon$, the difference $|a-b|$ is less than $\varepsilon$.

Elaborating this sort of analysis carefully allows one to express the now-usual epsilon-delta style definitions, etc."

This comment appears in the article

Levey, S. (2008): Archimedes, Infinitesimals and the Law of Continuity: On Leibniz's Fictionalism. In Goldenbaum et al., pp.~107--134. The book is

Goldenbaum U.; Jesseph D. (Eds.): Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Berlin-New York: Walter de Gruyter, 2008, see http://www.google.co.il/books?id=tWWuQ9PHCusC&q=

I personally find it hard to believe Levey is talking about Leibniz, but there you have it. Whether or not Levey's analysis stems from a "Desire To Preserve The Orthodoxy of Epsilontics Against The Heresy of Infinitesimals", as Yemon likes to put it, is anybody's guess.

What the "syncategorematic" view tends to overlook is the presence of DUAL methodologies in Leibniz: both an Archimedean one, and one involving genuine "fictional" infinitesimals. On this view, Leibnizian infinitesimals are PURE fictions (rather than logical ones). Such a reading is akin to Robinson's formalist view, and sees continuity not merely between Leibniz's and Robinson's mathematics, but also their philosophy. This view is elaborated in a text entitled "Infinitesimals, imaginaries, ideals, and fictions" by David Sherry and myself, to appear in Studia Leibnitiana, and accessible at http://arxiv.org/abs/1304.2137

HOPOS (Journal of the International Society for the History of Philosophy of Science) just published our rebuttal of syncategorematist theories that seek to sweep Leibnizian infinitesimals under a Weierstrassian rug.

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    $\begingroup$ I think you know this already, but what Francois refers to is common to many if not most historians of anything - one should avoid reading the present into the past, and beware telological narratives. In my view, that general philosophy is completely separate from any supposed Desire To Preserve The Orthodoxy of Epsilontics Against The Heresy of Infinitesimals $\endgroup$
    – Yemon Choi
    Commented Apr 18, 2013 at 9:36
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    $\begingroup$ @YemonChoi, my point was precisely to note that traditional historians tend to read the Weierstrassian present in the Leibniz/Euler/Cauchy past as if accepting a principle that Weierstrass (and infinitesimal calculus minus infinitesimals) was the inevitable result of progress. In a way consonant with Francois' comment, our published research tends to steer clear of such assumptions. $\endgroup$
    – Mikhail Katz
    Commented Mar 22, 2016 at 12:09

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