Leibniz’s Universal Characteristic: The Algebra of Thought

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Egyptian and Chinese hieroglyphics and the symbols of astronomers and chemists belong to the first category, but Leibniz declared them to be imperfect, and desired the second category of characters for what he called his “universal characteristic”.2 It was not in the form of an algebra that Leibniz first conceived his characteristic, probably because he was then a novice in mathematics, but in the form of a universal language or script.3 It was in 1676 that he first dreamed of a kind of algebra of thought,4 and it was the algebraic notation which then served as model for the characteristic.5Ibid., p. 81.Ibid., pp. 51, 78Ibid., p. 61.Ibid., p. 83.Ibid., p. 84.Ibid., p. 84.Leibniz attached so much importance to the invention of proper symbols that he attributed to this alone the whole of his discoveries in mathematics.6 And, in fact, his infinitesimal calculus affords a most brilliant example of the importance of, and Leibniz’ s skill in devising, a suitable notation.7Ibid., p. 84.Now, it must be remembered that what is usually understood by the name “symbolic logic”, and which–though not its name–is chiefly due to Boole, is what Leibniz called a _Calculus ratiocinator_, and is only a part of the Universal Characteristic. In symbolic logic Leibniz enunciated the principal properties of what we now call logical multiplication, addition, negation, identity, class-inclusion, and the null-class; but the aim of Leibniz’s researches was, as he said, to create “a kind of general system of notation in which all the truths of reason should be reduced to a calculus. This could be, at the same time, a kind of universal written language, very different from all those which have been projected hitherto; for the characters and even the words would direct the reason, and the errors–excepting those of fact–would only be errors of calculation. It would be very difficult to invent this language or characteristic, but very easy to learn it without any dictionaries”. He fixed the time necessary to form it: “I think that some chosen men could finish the matter within five years”; and finally remarked: “And so I repeat, what I have often said, that a man who is neither a prophet nor a prince can never undertake any thing more conducive to the good of the human race and the glory of God”.In his last letters he remarked: “If I had been less busy, or if I were younger or helped by well-intentioned young people, I would have hoped to have evolved a characteristic of this kind”; and: “I have spoken of my general characteristic to the Marquis de l’Hopital and others; but they paid no more attention than if I had been telling them a dream. It would be necessary to support it by some obvious use; but, for this purpose, it would be necessary to construct a part at least of my characteristic;–and this is not easy, above all to one situated as I am”.Leibniz thus formed projects of both what he called a _characteristic universalis_, and what he called a _calculus mitocinator_; it is not hard to see that these projects are interconnected, since a perfect universal characteristic would comprise, it seems, a logical calculus. Leibniz did not publish the incomplete results which he had obtained, and consequently his ideas had no continuators, with the exception of Lambert and some others, up to the time when Boole, De Morgan, Schroder, MacColl, and others rediscovered his theorems. But when the investigations of the principles of mathematics became the chief task of logical symbolism, the aspect of symbolic logic as a calculus ceased to be of such importance, as we see in the work of Frege and Russell. Frege’s symbolism, though far better for logical analysis than Boole’s or the more modern Peano’s, for instance, is far inferior to Peano’s –symbolism in which the merits of internationality and power of expressing mathematical theorems are very satisfactorily attained–in practical convenience. Russell, especially in his later works, has used the ideas of Frege, many of which he discovered subsequently to, but independently of, Frege, and modified the symbolism of Peano as little as possible. Still, the complications thus introduced take away that simple character which seems necessary to a calculus, and which Boole and others reached by passing over certain distinctions which a subtler logic has shown us must ultimately be made.Let us dwell a little longer on the distinction pointed out by Leibniz between a _calculus ratiocinator_ and a _characteristic universalis_ or _lingua characteristic_. The ambiguities of ordinary language are too well known for it to be necessary for us to give instances

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