Gottfried Leibniz · Mathematics
Leibniz’s most audacious dream: a universal symbolic language (the <em>characteristica universalis</em>) and a calculus of reasoning (the <em>calculus ratiocinator</em>) that would let all thought be carried out, and all disputes settled, by computation.
Of all Leibniz’s projects, the grandest - and the one he pursued from his youth to his death - was the dream of a universal characteristic: a symbolic language in which every concept would be represented by a sign, every complex idea built transparently from the signs of its simple parts, and every act of reasoning carried out by manipulating these signs according to fixed rules. He imagined an ‘alphabet of human thoughts’, a small set of primitive concepts from which all other concepts could be composed, just as all words are composed from letters and all numbers from digits. With such a language, Leibniz believed, the confusions and ambiguities that plague ordinary speech would vanish, because the very form of a symbol would display the structure of the thought it expressed.
Paired with this language was a second project: a calculus ratiocinator, a calculus of reasoning. If concepts could be reduced to symbols, then valid inference - drawing correct conclusions from premises - could be reduced to calculation, the rule-governed transformation of those symbols, exactly as the differential calculus reduced the finding of slopes to a procedure. Together the two projects would make reasoning a branch of computation. Disputes would no longer be won by the loudest voice or the cleverest rhetoric; they would be settled, the way an arithmetical disagreement is settled, by working out the answer. Leibniz returned to this vision again and again across his life, sketching fragments of it, never completing it. It was, perhaps, the most ambitious intellectual project ever conceived - nothing less than the mechanisation of human reason.
Leibniz did not merely gesture at this dream; he made concrete attempts to build it, and his most striking idea was to encode concepts as numbers. Suppose each primitive, indivisible concept is assigned a prime number - ‘animal’ might be 2, ‘rational’ might be 3. Then a composite concept is represented by the product of the primes of its components: ‘man’, being ‘rational animal’, would be 2 × 3 = 6. The logical relations among concepts would then become arithmetical relations among their numbers. To say ‘every man is an animal’ - that the concept ‘animal’ is contained in the concept ‘man’ - would be to say that the number for ‘animal’ (2) divides the number for ‘man’ (6) exactly. Checking whether one concept contains another would reduce to checking whether one number divides another - a purely mechanical arithmetical test.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned Leibniz’s twin projects: the characteristica universalis (a universal symbolic language of concepts) and the calculus ratiocinator (a mechanical calculus of inference). Explain the dream and how far it was eventually realised.
Leads to Gottlob Frege.
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