On 19 Sep 2012, at 13:16, Roger Clough wrote:


Leibniz's characteristic numbers

I apologize if this is just a can of worms, but it seems related to comp:

All of his life, Leibniz sought to be able to develop a universal calculus,
in which (as I understand it) ideas could be discussed and debated
unambiguously. Issues could be settled arithmetically or algebraically.

Here is a comment on Leibniz' presumed goal:

http://www.rbjones.com/rbjpub/philos/classics/leibniz/meth_math.htm

"Towards a Universal Characteristic
An ancient saying has it that God created everything according to weight, measure, and number. However, there are many things which cannot be weighed, namely, whatever is not affected by force or power; and anything which is not divisible into parts escapes measurement. On the other hand, there is nothing which is not subsumable under number. Number is therefore, so to speak, a fundamental metaphysical form, and arithmetic a sort of statics of the universe,
in which the powers of things are revealed.'


OK. But the idea of a universal language to settle all problems is also close to Hilbert idea of settling the math foundation in such a language.

That very dream of Hilbert is what Gödel, but I would say more deeply just the existence of the universal numbers or machines refutes, even effectively, that dream.

Hilbert's dream had been qualified as Hilbert's nightmare and in that sense, it is a relief that it is inconsistent, as it prevents normative theory for the possible beings (notably the Löbian one and above).

There is a universal language for computability, (CT), and a universal machine understanding it, but this entails, in two diagonalizations, that there are no general theories in which we can settle all questions, notably due to the presence of all those universal entities.

Was Leibniz closer than Hilbert to the universal machine? Question for future historians.

The universal machine needed Boole, Frege, Post, Turing, Church, Markov, Kleene, ... Babbage got it, I think, from the study of an amazing book of 1936 by Jacques Lafitte "La Science des machines". He got the concept and understood that the snake eats its own tail.

Bruno







I appologize it this is just a can of worms, but in relation to that project
Leibniz developed a set of "characteristic numbers", (put


leibniz characteristic numbers

into Google.)

which to me, a non-mathematician, appears to be a form of set theory,
such as, if I got it right,

all a is b
some a is b
no a is b
some a is not b

so each number a is actually a pair of numbers (or IMO terms)
logically related by Aristotle's logic.

It has been applied (apparently with some problems) to
cognitivism by Dreyfus:

http://www.springerlink.com/content/gk5115u747772604/








Roger Clough, rclo...@verizon.net
9/19/2012
"Forever is a long time, especially near the end." -Woody Allen

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