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

Leibniz is a significant figure for several reasons

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

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 one.

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.

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, and how.

There are still many mathematicians who share some of Kant's view, as we will see.