Thursday, June 17, 2010

Logic and Faithfulness

One question I'm fascinated by is whether or not the rules of logic are necessarily right. That is, can we imagine a universe where the rules of logic are different? I realize this is an ill-posed problem. It is very easy to point out that the word necessarily is rather ill-defined, and therefore one can go around in circles all day arguing both sides of the question to no avail. But the question still intrigues me, especially since it has to do with the very foundations of my work as a mathematician.

Where do the laws of logic come from? Do we simply take them as axiomatic and then move on? Why do we attempt to constrain our use of language to a set of precise rules that can never be broken? I am speaking, of course, only of subjects such as mathematics where logical precision is a defining characteristic. Venture into poetry, speculative philosophy, or mystical theology and I'll grant that logic does not need to constrain our use of language.

As I grow accustomed to reading mathematical papers, and especially now that I have been introduced to the task of reviewing a paper (which involves the daunting task of checking it for correctness) I have noticed just how essential it is that words are, in some sense, faithful. For one thing, every proof depends on the fact that symbols don't change their meaning from page to page. For another, proofs of theorems constantly rely on lemmas that must in turn be based on earlier knowledge. This, too, demands some sort of faithfulness on the part of words and their corresponding concepts--I need to know that every time a certain set of hypotheses is satisfied, I get a desired conclusion.

When a mathematician defines a term, he is really establishing certain relationships between words. He is building a certain kind of trust. Whenever you see these symbols, he says, you can count on these other symbols being connected in this way. Theorems are maps of those relationships that take you much longer distances than the bare definitions. Think of a family tree. Each child is the child of two parents--this corresponds to a mere definition. But every child can be connected to myriads of other people by tracing his ancestry back to common ancestors and then down through some other line of descent. This is kind of what theorems are. Definitions are immediate connections that we create; theorems are broad connections that establish some sort of common bond that may have been unexpected.

How could there be any logic in the universe if there were no faithful relationships? If we had never encountered the possibility of trusting in someone or something, it's hard to imagine developing logical reasoning. If our minds never retained connections, never established relationships between things, then all hope of doing mathematics in particular and logic in general would be lost.

Whatever codified rules of logic we may have inherited from the Greeks on down, it seems to me that logic is really the simple manifestation of the faithfulness of the Creator. In some sense, logic is faithfulness. Language becomes illogical when words are used in an untrustworthy manner. You say one thing but you mean another; you use the same word to mean multiple things; you follow one sentence with another sentence without remaining faithful to the former.

There is a moral element to logic. It isn't just some abstract, impersonal idea. We hear this constantly if we listen to scholars talk about their interpretations of historical documents, particularly in biblical studies. They say, "I have tried to remain faithful to the text." There must always be a moral commitment in order for there to be any logic. Hence the Proverb, "The fear of the Lord is the beginning of knowledge."

So do other things such as poetry deviate from this faithfulness? I don't think so. As long as the reader understands that words may not always mean exactly one thing, and that words may even contradict themselves to create something beautiful, then no faithfulness is lost. Like I said, our codified laws of logic are no substitute for the real heart of the matter: logic is faithfulness in words. Poetry and music have their own logic, for they have their own inner faithfulness. And when that faithfulness is not present, a good listener will notice. It is no different from a trained mathematician being able to spot a mistake in a mathematical research article.

Thus whenever we hear of a modern society based on reason and logic as opposed to faith in God, we must ask ourselves, on what are reason and logic based? They are always based on a relationship of some kind. Throw out any God save ourselves, and what are we left with? To what, exactly, are we to be faithful to?