Saturday, April 11, 2009

So What Is It That's Missing, Then?

This is about Kurt Gödel, and what he did and did not prove back in the '30's. He is one of the names in mathematics or mathematical physics that is most regularly abused by those who do not understand what the Incompleteness Theorems mean. Words like Incompleteness, Relativity, Uncertainty, Chaos, Entropy and Evolution are bandied about, not just by blokes in bars who think that they give a sense of authority to their arguments, but by politicians, journalists and academics who have made no real attempt to understand these concepts, but for some reason assume they just do.

Now this makes a certain amount of sense, knowing what we do of the human mind's attitude to itself- when I was 15 I knew everything; at 25 I was confident of filling in the gaps by the time I was 30; now in my 40's I often despair at the extent of my ignorance; when I am 60 I shall scarcely dare to open my mouth; similarly, in normal circumstances I can manage to make general remarks about a number of subjects, speculate about others, and perhaps raise what may or may not be pertinent questions about a few more, but around the fourth pint of Abbot's I percieve and express the great questions and answers of the moment and of all time with a clarity and lucidy that Einstein himself surely never experienced.

In other words, there is nothing like learning a little about something to appreciate your ignorance of it. Gödel's Incompleteness Theorem is not something most people bother to understand, because oddly enough, many of them think they already do. It is an assumption of breathtaking arrogance, in fact, to imagine you not only understand the theorem just from having heard a couple of third-hand references to it, but that you can also discern previously unnoticed applications of it to other fields that you don't know anything about either.

It is often said that Gödel's theorem is simple to understand. This is not true. It is not so hard to express in simple terms what it proves about mathematical systems, and it is not hard to give an idea of the approach that Gödel used (in terms of the Liar Paradox, or something similar), but in order to prove it true, rather than to engage in anecdotal discussion about the limits of self-referential sentences, it must be expressed in purely mathematical terms, and that is not at all easy to understand. That is, it is not hard to get a grasp of what the Incompleteness Theorem says (though most people fail to do so) but to know that it is true is a very different matter.

Gödel's first incompleteness theorem proved (in the mathematical sense) that in any sufficiently complex axiomatic system there must exist true statements which are not provable within the system. This does not mean that no result of the system is provable (most are, and all true statements can be proven true by expanding the system), or that there is no such thing as truth.

It does have some interesting consequences for mathematics, but none whatsoever for other types of systems, or collections of statements, nor for philosophy, psychology, cultural studies, or the humanities/waffle-blathery crowd in general. This is because it is a statement about a particular type of entity in mathematics, and must be understood in those terms before any possible relevance to anything else can be derived. It is also true (in the mathematical sense), not just a theory, an idea, or a set of general remarks. This is why, I think, those without a science background think they can play with a few of the words used to decribe the theorem, and expect to arrive at something useful- because their very notion of what truth is is completely different.

Looking for specific, significant examples of misuse of the name of Gödel or of incompleteness, I found this page, maintained by Torkel Franzen of Lulea University (that's his story, anyway). He tabulates and discusses the common misuses, saving me the trouble of doing it, and causing me the kind of intellectual envy I didn't have to experience when I was 20. So I shall let him do the rest of the work. I also found this, by Rebecca Goldstein, a philosopher who actually does seem to understand both the relevant mathematics and the limits of its applicability to other fields.

In other news, Johann Hari discusses a common type of fallacious argumentative gambit, a form of false relevance, or misdirection. Hardly new, and it is mostly a self-centred and self-justifying piece, but this blog is interested in the search for truth, and here it is discussed in a national newspaper.

And finally, The Times suggests that the only qualifications required for being a newspaper columnist are being blonde and having nothing at all worth saying.

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