- Leibniz invented the first digital mechanical calculator capable of addition, subtraction, multiplication, and division. Called the "Stepped Reckoner", it was an improvement over Schickard's earlier model.
- He might have been the first to document the base 2 (binary) system.
- He developed formal logic, not as an abstract idea, but rather in connection with his work on calculators.
- Bertrand Russell was so impressed by Leibniz that Russell's first book was about Leibniz's philosophy (The Philosophy of Leibniz). Russell (with Whitehead) of course went on to write Principia Mathematica, which influenced Gödel and Turing who actually began the age of programmable computing.
- In Leibniz's metaphysics, relational properties reduce to monistic properties. A relational property is expressed in the proposition "The earth and the sun are 93 million miles from one another." A monistic property is expressed in the proposition "Socrates is mortal." Monistic properties follow from the essence of the thing itself. So it's in the nature of Socrates to be mortal. It's not in the nature of the earth or the sun to be such and such distance from one another—or so it seems. According to Leibniz, an infinite intellect, like that possessed by God, could derive all relational properties just by having insight into the nature of the substances themselves. If one knew everything contained in the concept "Caesar", one could analytically derive that he would cross the Rubicon in 49 B.C. Human beings, possessed of finite, not infinite, intellect, must go out and discover things empirically, using the methods of natural science. But brute empirical facts, like the distance between the sun and the earth, or the date Caesar crossed the Rubicon, are only apparently part of the furniture of the universe. In fact, there are only monads and their intrinsic properties, and they are related to one another, not by brute relations of space and time, but rather by their concepts. To put it in philosophical parlance, natural science gives us an ideal picture of the world, but pure thought expresses what is absolutely real. (Compare with Newton and Kant who basically thought the opposite.)
What does this have to do with information and computation? One way of describing Leibniz's metaphysics is to say that the world is souls all the way down. Every existing thing is a windowless monad which has no real relationship to anything outside of itself. It might have an ideal or made-up relationship to other monads when viewed from the human perspective, but that's a mere artifact of our finite intellect that would go away as soon as one adopted a divine, infinite perspective.
But imagine one were to adopt that infinite perspective. Imagine one had a complete account of the world as it really is, and that all of that information were contained in a book. Because our intellects are finite, we can't know what information is contained in that book, but we can know the general form it would take (assuming Leibniz is correct). It would contain a list of substances, things like "Caesar", "Socrates", "Sun", etc., and for each substance there would be given a derivation, using symbols of formal logic which Leibniz devised, of everything contained in the substance's concept.
So basically what Russell and Whitehead were trying to do with mathematics—ground the whole thing in logic—Leibniz was trying to do with the whole of reality, right down to its very core. Another way to put it: every substance does in space and time what is programmed into the concept which defines what it is. The reason I see you walking toward me is not because either of us is really moving through space and time. It's because the substance that I am is programmed to see you moving toward me just at the same time the substance you are is programmed to walk toward me. In fact, there is nothing more to me being me and you being you to be programmed exactly like this. To see it any other way is merely an illusion. This is what Leibniz means by "preestablished harmony".
Saturday, March 20, 2010
Leibniz and information theory
Posted by der Augenblick at 7:21 AM