Wednesday, March 21, 2007

Theorem and Non-sequitir

I recently had occasion to ask myself what exactly constitutes thinking. Since I was trained as a mathematician, I generally regard thinking as something akin to the process of stepping through a sequence of formal steps: starting with premises, then stepping logically to intermediate results and finally arriving at a conclusion.

If reason is reasonable, the conclusion will be as true as the premises and the validity of the steps to the conclusion.

There's a similar process in forensic procedures where cases are made tallying up the arguments in favor of and counter to the point of dispute. When making cases, facts are marshaled to serve as the premises of a reasoned process leading to the advocate's position.

In the forensic setting, we can't count on the neat set-piece logic of the mathematical proof. So, the arguments tend to be more inductive and the logical steps of the argument somewhat less rigorous.

Now, what has this to do with thinking? There's a cartoon that came out when I was in college. I'm wearing it on a t-shirt as I write this. That was the constant temptation when I was getting my Masters' in Mathematics: I'd be in the middle of a proof and realize the next step I needed to take, but I didn't know how to do it. I dearly wished I could say, "then a miracle occurs," to make the missing step. They don't let you do that in Mathematics.

In Law, I think it's a matter of getting away with it. If you try it in law school, it depends on how closely the prof is watching, or in court, how closely the judge is watching. If they catch you, you'll be handed your head.

There's a term-of-art for the unsubstantiated step in an argument: the non sequitir. The step does not follow. It should be cut and dried, but that's not necessarily so.

When I was taking Calculus, the prof would write up proofs on the board and I would not understand the step that he took. My dullness of mind could not perceive how one step followed from another. The ultimate math-prof cop-out is to say, "this is left as an exercise for the student."

So then, it requires some subtlety to distinguish between the theorem one does not understand and the non sequitir. Charity demands that when someone throws something that appears to be a mismash of non sequitir and gratuitous assertions, that it not be dismissed out of hand. But how hard must one work before throwing up one's hands and saying, "I'm not this stupid," and decide that whatever you're looking at is no theorem, but non-sequitir. Is this from a deep thinker or from a braggart dilettante?

One strategy suggests itself. If your interlocutor has demonstrated mastery of a broad spectrum of difficult subjects and familiarity therewith, it suggests the former. Conversely, if he clutches a small handful of facts that he has difficulty integrating with other ideas, it suggests the latter.

Which am I? I suppose it's safest for me to regard myself a dilettante until proven otherwise.

No comments: