Leibniz as a Mathematician

Editor's Remarks

Leibniz is generally recognized as a master of notation, but is not so easy to substantiate this impression without getting lost in the labyrinthine technicalities of the development of the differential and integral calculus. Joseph Hoffmann and A. Rupert Hall, supplied a good basis for a convincing account of the story. In spite of our shared misgivings early on, we saw that the story of the priority conflict between Newton and Leibniz, though forbidding both in terms of the quantity of ink spilled on it, the animosities it once engendered, and its apparent triviality, actually deserves some attention for the light it casts, if approached judiciously, on scientific matters. Fortunately both Hoffmann and Hall, though not quite taking the same side, approach it in this spirit. As Hall points out, prior to his own work it had not actually been dealt with as a subject in its own right; we noticed that Hofmann's intention to set out his own views in a monograph on the subject was thwarted by his death in an accident.

Early on you make the following claim about Leibniz: "And yet although he became acquainted quite late in his life with the mathematical achievements of his generation, it will always be his innovations in this field that put him to the forefront of the enlightened thinkers of his era." I had my doubts about this, but they are diminishing. It's a bit difficult for me to see a talented mathematician in the man who was so callow on his arrival in Paris, but he really does seem to have accomplished a great deal by systematizing the calculus. This seems to be a case where good methods and a sound intelligence produced results that would otherwise have required genius.

Leibniz certainly had a good instinct for arranging mathematics so that demanding calculations could be carried out by men of only moderate gifts. Newton professed, late in life and in the midst of polemics (about 1712), not to see the utility of this, and perhaps he could not envision the day when hordes of engineers would routinely carry out calculations of the sort that he had himself devised. But I suspect Newton was not entirely sincere at this point.

"It was precisely for this reason that Leibniz had so much success in the field, in that he was unhampered by much of the dogma that might have hindered its progress." I'll part company with you about the "dogma", but it does seem that after devising methods that suited himself, he found as he expected that they would suit others very well indeed. The next question would be what exactly the Bernoulli's learned from him - one imagines it is exactly this "calculus" as an algorithm that impressed them.

"... through his own lens" ( or possibly the lens of Cavalieri, which in spite of its logical inadequacies had much to recommend it as the basis of an algorithmic formalism.

It seems to me at the moment that Leibniz was drastically ahead of his time in at least one respect: his unbridled optimism regarding the potential power of formal systems, an optimism which seems more reasonable, though still somewhat problematic, to our own age. Ultimately what interested him, it seems to me, is what we now refer to as artificial intelligence. Though he didn't phrase it in those terms, the result of combining his characteristica universalis with existing computers would be something very like artificial intelligence.

On the other hand, one can also relate the ideas of Leibniz to the work of the philosopher Frege, who in the late 19th century began to devise formal systems for reasoning that correspond quite well to some of Leibniz' ideas; though still restricted to t he field of mathematics, they encompass the whole of that field, rather than limited areas. There may yet be applications of these ideas to law and other aspects of civil society, though it will take at least one more crusader with the spirit, and abilities, of Leibniz to bring it about.

In any case, under the impact of your essay and some of your references, I find my picture of Leibniz evolving.

The Hofmann and the two books by A. Rupert Hall (one found by Anand, the other suggested by Professor Kosinski) are enormously helpful. The Hofmann is something I'd always meant to read, and was even better than I anticipated. Philosophers at War on the other hand was wholly unknown to me, and gives a very clear account of the whole controversy, with a useful chronology at the outset. As Hall tells the story, it casts considerable light on the history of the subject as well (at least until 1710, when it heats up: at that point my first impression is that it just becomes depressing).

Newton always attached great weight to his method of infinite series, and Leibniz was obsessed by Cavalieri's infinitesimals, which go back to the ancient Greeks (Democritos). Then as Hall stresses, Newton moved away from infinitesimals and in the direct ion of "flows" (fluxions), which amounts to replacing a rather primitive "atomic" theory of geometry by a more sophisticated physical theory of velocity, slightly closer to the rigorous theory of limites. Hall makes it plain that Newton claimed that the theory of infinite series is a central feature of the calculus. I hadn't realized he took that point of view, and again I wonder if it was really sincere. If so, he could have laid out that claim earlier, and more explicitly, since it ought to have been clear that the Continentals didn't see things that way.

These are interesting and subtle issues, and frequently the arguments associated with the priority fight touch on serious matters.

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