Revision date: 19 January 2004
This class is about reason and the limits of reason. We will let the notion of reason remain primitive, but reason par excellence we will assume is the practice of logic and mathematics. The kinds of questions we will consider include: Can we make reason explicit and even automatic? Can we ever be certain of our reasoning? What kinds of things can reason tell us about, and what kinds of things can it not tell us about? Is there anything beyond the reach of reason? What is the nature of mathematical knowledge? What can the nature of reason tell us about such things as knowledge, language, and the nature of mind?
This is a vast topic, and each of these questions deserves years of study. We will only have time and the ability to discuss a very few aspects of this issue, and trace a few historical developments. Our focus will be primarily on the goal to understand and to formalize (that is, to make explicit and verifiable) reason. We will trace just a few historical illustrations of this goal, and then focus upon some of the limits of formal reason that have been discovered in the 20th Century.
The course has two parts. In the first part, we will examine a few historical examples of attempts to find and make explicit an ideal language. We will also use as an example of the kind of challenge that demands of us more refined reasoning a single mathematical conundrum: the nature of infinity. We will begin with Galileo, who both considered geometry a privileged tool for the pursuit of science, and struggled with and exposed the problematic nature of reasoning about infinitesimals and infinity. Next we will discuss Leibniz and his dream of a Characteristica Universalis. Leibniz also makes an important contrast with Kant and provides the setting for one of the most important debates in the philosophy of mathematics. Galileo provides a pleasant introduction to infinity, which can serve to motivate our look at next at the logic of infinity developed by Cantor (Cantor's motivation to solve certain problems of analysis, instead, would be much less accessible to the non-specialist). Leibniz and Cantor provide two examples that set the background for our the intertwined development of set theory and of formal axiomatic methods, leading to the Turing machine. They also lead to the set theory paradoxes, Godel's incompleteness Theorems, and the Halting Problem. This very superficial review of some of the history of reasoning about reason will therefore lead from some early examples of the goal of formal reasoning, through several important developments, ultimately both to the partial realization of this dream in the Church-Turing thesis and also the realization of the limits of formal reason as shown by incompleteness results.
Other than Hilbert and those who follow him, these historical examples are not essential to the development of our reasoning about reason. They are illustrative, and accessible, and interesting. That was enough for them to be chosen as examples.
In the second part of the course, we will explore the limits of reason, and some consequences of these limits, by way of the powerful general framework, algorithmic information theory. We will also consider, as an interesting extension of the limits of reason, automata theory, and the related notion of the world as a deterministic but unpredictable system.