Consistency without completeness

A good friend, whom I will call "Stewart," responded by email to the previous blog post, and pointed me to a column by Charles Krauthammer, as reprinted in his book, Things That Matter: Three Decades of Passions, Pastimes and Politics, pp. 64-66, Crown Publishing Group. The column is entitled, "The Central Axiom of Partisan Politics". (I would have liked to link to this column, but cannot.*)

As for the axiom itself, it is, "Conservatives think liberals are stupid. Liberals think conservatives are evil."

Where this gets interesting, to me, is his further claim that conservatives think liberals are stupid because they believe that everyone is basically good. Adding this reason to his axiom can be used to prove a contradiction, as follows (expressed from the point of view of liberals (according to conservatives according to Krauthammer)).

Every person is basically good. Conservatives are people. Therefore, conservatives are basically good. Oops. Contradiction, because, according to his axiom, liberals also believe that conservatives are evil.

A contradiction in a system of formal logic would not be a good thing. But (according to Nicholas Rescher**) people don't reason using formal logic, and so we can be perfectly happy with Krauthammer's axiom. That is, if we are inclined to agree with his views on politics. I will leave that up to you, dear reader, if you are inclined to look for, find, and read the entire column***.

Well, this post isn't meant to be about politics. Instead, the mathematician in me followed the path of least resistance in response to the word "axiom," and this post is about the journey along that path. As a mathematician, I love formal logic systems, which are based on a small number of axioms, and the things that can be proven from them. If a logical contradiction can be proven from the axioms, then they are considered to be inconsistent, which is not a good thing for an axiom system.

An ideal formal system is both consistent and complete. Unfortunately, early in the twentieth century, Kurt Gödel proved that any sufficiently powerful formal system is either incomplete or inconsistent. This undid the grand project of mathematicians at that time, who had hoped to find a set of axioms from which all true statements could be proven.

The cognitive dissonance of this post's title would exist in the minds of mathematicians, for whom consistency without completeness is not good for a formal axiomatic system. It is resolved, sadly, by Gödel's theorem demonstrating that the desired resolution is, in fact, impossible. And we're just going to have to live with that.

Mathematicians seemed to me, in the 1970's, when I was an undergraduate minoring in mathematics, to be still (forty years later) in denial about this result, for it was not taught in the mathematics department. I had to take a course from the philosophy department to learn more about it.

*The Washington Post, in which the column originally appeared on July 26, 2002, has a web site, but it requires payment to view articles older than 2005. Very interesting. This makes it impossible for me to do a proper attribution. Yet, at the same time, I understand that newspapers have been hard-hit by the Internet, and are desperately scrambling to find ways to monetize their work product. Thus, I am quoting from the book, rather than the original source, which is hereby at least acknowledged..

**Rescher, N. (1982). The Coherence Theory of Truth. University Press of America.

***Try the "Look inside" option of amazon.com once you have found his book there.