Leibniz makes an excellent figure to use as an inspirational start,
since his ideas of the characteristica universalis, and of the
rational calculus, are prescient notions of reason made clear,
precise, and powerful.
Background for Leibniz (1646-1716) and Galileo (1564-1642)
Medieval philosophy was in several ways quite stagnant
- Simplistic metaphysics that followed
- Little explanatory power to philosophy
- Theology pollutes proto-science
Leibniz is a significant figure for several reasons
- Talented polymath (mathematician, philosopher,
diplomat, sinologist, legal scholar....)
- He recognized the utility of applying mathematics
to natural philosophy
- Unlike Galileo, however, he is eager to retain
some of neo-Aristotlean philosophy, especially the
Gottlobb Leibniz was born in 1646 in a noble family.
Both parents came from academic backgrounds.
He developed a vast body of work, which it is not easy to organize.
Much of it is also in correspondence (e.g., Leibniz had a long debate
with Newton's defender Clark about the nature of space).
His accomplishments include
- Development (he claims independently of Newton) of
- Development of a complex metaphysical theory that
included the rudiments of a modal logic
- Relevant to our interests: the concept of a
Universal Characteristic (or, Characteristica
Universalis) and of a Rational Calculus (or,
Universal Characteristic: a language that is purged of
ambiguity and vagueness, and where the rules of composition reflect
the nature of the universe. Communicating in this language, we will
be able to do science with much less error, and communicate our
knowledge with absolute clarity.
It is unclear whether Leibniz thought that this should be a revision
of a natural language (Latin, of course), or a newly invented
Rational Calculus: a universal characteristic would also have
the potential to be a rational calculus, which allowed us to settle
disputes by performing some kinds of agreed-upon operations in our
logical system. This appearas to be the first articulation of a
notion of developing complete algorithms for reasoning.
These are strikingly original notions. If we are to see progress in
knowledge and agreement in our disputes, some progress towards the
development, clarification, and universalization of reason will be
essential. Leibniz sees this as a task for developing a formal
language and reasoning system.
Leibniz's dream will be our guiding thread this semester. (And the
problems of reasoning about Infinity will be our primary challenge
to such a dream.)
Leibniz's Two Kinds of Knowledge
According to Leibniz, all mathematics is truths of reason. To deny a
mathematical claim is to contradict yourself.
- Truths of reason (necessary truths)
- Matters of fact (contingent truths)
This actually matters a great deal to the issue of whether we can
develop a universal characteristic and rational calculus. For, if
Leibniz is right, we can found all of mathematics and other forms of
reasoning on necessary propositions.
Contrast with Kant: the philosopher Kant argued that much of
mathematics is given through synthetic a priori intuitions or
judgments. This means that these are not just matters of meaning
(that's what "synthetic" means: there is more here than just
definition) but that we know these things without having to learn
about them from experience (that's what a priori means).
If some of reasoning (including especially mathematics, which is
reasoning par excellence) is given in this way, then it becomes
unclear how we could ever create something like the universal
characteristic with its rational calculus. We need to discern
what is given to reason,
There are still many mathematicians who share some of Kant's view,
as we will see.