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, [email protected]
9/19/2012
"Forever is a long time, especially near the end." -Woody Allen
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