Spinning Yarns

The first time I read about symbolic logic, I thought it was the coolest thing since dry ice. Even today, I think symbolic logic is a pretty nifty idea, which is probably good, since the manipulation of logical formulae comprises a fairly substantial component of my Ph.D. dissertation, and it would be somewhat unfortunate if I had grown to hate it. Nevertheless, my perspective on logic has changed a lot, in the years since I first encountered the idea.

Put simply, the fundamental insight behind symbolic logic is that you can capture important propositions about “truth”, “belief”, “obligation”, “necessity”, “time”, and so forth, by writing them in a precisely-defined language of formulae, stripping away the supposedly inessential components of natural language, retaining only the “interesting” parts. Furthermore, you can manipulate these formulae according to simple, unambiguous rules, which we choose in such a way that if you start from one formula, and apply a rule to obtain a second formula, the second formula has all the same properties that the first one had, except that it looks different. Or, at least, that is the intent.

On the face of it, there is no harm in such an activity. I find it fun for much the same reason I enjoyed playing with wooden blocks as a child: All the shapes are simple, and you can combine them in a variety of easily-understood ways to obtain a variety of new shapes that are useful, pleasant to look at, or stimulating to the imagination. On the other hand, there is very little reason to suppose that the game of re-arranging this formula using those rules results in a formula that has any necessary relationship to objective reality. For the logician, there seem to be three possible responses to this criticism:

1. “Who cares? Since we cannot even be certain there is any ‘objective reality,’ one that exists independently of individual experience, it does not matter whether logical formulae have any relationship to such a thing. We are happy with the results, as long as they are consistent with our observations.”

2. “On the contrary! If we all agree that the rules are reasonable, and that they consistent with each other, then we can use the results to infer things about objective reality, based on our experience.”

3. “Go to Hell. I’m having fun playing with formulae.”

Despite the fact that it’s difficult to prove any real connection between the logic and the real world, a lot of people — myself included — have a great deal of faith in its power as a tool for the discovery and expansion of knowledge. And why shouldn’t we? All of the elaborate mathematics used by engineers, chemists, physicists, economists — and all the other architects of the modern world — has been based on logic. Even mathematicians sometimes use it — although for the most part, logic is a fairly esoteric branch of the discipline, and most mathematicians just use the simple parts, just like the rest of us. Anyway, it hasn’t failed us yet (right?), so why shouldn’t we trust it?

I see no reason we should not trust logic, but I worry that blind faith in its results is unwarranted, and probably dangerous. We should trust logic the way we trust an old, dear friend, or a faithful mechanic whose services we have come to depend upon: Logic is familiar, and reliable, and it tries to serve our best interests, but it can also be dangerous if we accept its advice without question. When you study logic closely, you quickly discover that while it is very easy to insure that it is internally consistent, it is very difficult to verify its relationship to reality. If this sounds like a description that could be applied equally well to religion, you are probably on the right track — for, as physicist John Barrow once observed,

“If we define religion to be a system of thought which contains unprovable statements, so it contains an element of faith, then Gödel has taught us that not only is mathematics a religion but it is the only religion able to prove itself to be one.”*

* This refers indirectly to Kurt Gödel’s famous “Incompleteness Theorem,” in which he proved, in essence, that any system of logic sufficiently powerful to be “interesting” contains formulae which are valid, but which cannot be proven.

Personally, I use logic all the time, and I don’t tend to question whether it is right or wrong when I’m doing it. It is difficult enough coming up with a logical proof at all, without miring myself in the philosophical intricacies of whether or not my has any value outside the boundaries of the logical framework I wrote it in. On the other hand, in my more reflective moments, I consider it extremely important that I, not only as a scientist, but as a curious and inquisitive human being, should be able to feel confident that the tools I am using are good ones, and worthy of my trust. In other words, I believe it is essential to question even the most basic truth-functional axioms of the very system of logic the rest of my knowledge is predicated upon. This is a scary business! What should happen to me, if I were suddenly to discover an inconsistency, whose explanation bespoke the collapse of my entire understanding? Would I go mad? Would the world end? Would my head catch fire?

On the other hand, what happens if I do not question these things?

Here’s how I see it: Suppose you have a best friend, whose advice you greatly trust. She has never ever steered you wrong, even once, in all the many long years you have known her. When she picks lottery numbers for you, you win. When you take her advice on career choices, you advance. When she tells you to take an alternate route home, you avoid being part of a fifteen-car pile-up on the expressway. After a while, you stop even wondering how it is that she could possibly know these things, and simply accept them for what they are: Good, practical advice.

And then, one day, you ask her for advice on what vegetables to buy for a party you’re having next week, and she tells you, “I advise you to climb up to the top of the Brooklyn Bridge, and jump off.” On the face of it, this is patently ridiculous — it makes no sense, and any fool can see that doing this is going to result in death or serious injury. And yet, on the other hand, she’s never been wrong, and you are probably thinking to yourself, “now, wait a minute, maybe there’s something to this. Shouldn’t I just follow her advice, the way I always have?” If you’re incautious, you might assume she’s right, simply because she always has been, and wind up leaping to your death — certainly not an auspicious end to a beautiful summer’s day.

Belief in the results of logic should be approached with the same skeptical mind. If she advises you to do something that makes sense, there is probably no harm in going along with what she says. On the other hand, if her advice baldly contradicts your intuitive understanding of reality, you should at least be prepared to seek a second opinion. After all, there are lots of things about our world which are true, but seem quite contrary to what we call “common sense”.** On the other hand, you probably shouldn’t trust a map that tells you to walk five hundred feet due north from the south lip of the Grand Canyon, even if that map was made with advanced satellite-based mapping technology (or perhaps, especially not then).

** Albert Einstein once defined common sense as “the collection of prejudices acquired by age eighteen.”

So, in short, while logic is a great tool, and a lot of fun to play with, you should be careful not to hurt yourself by using it improperly. And, well, is that really such a revelation? My father taught me the same thing about the tools in his workshop, when I was growing up. Maybe the real take-home lesson here is that we should all listen carefully to our fathers. But, not too carefully.

Filed by Michael at 21:10 under Philosophical
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Although we have not yet been greeted with morning frost, you can definitely tell that the summer has passed us by. When I went out this morning, both air and ground were chilly, not in the gentle, friendly way of a warm July morning, but with a sharp nip to it, that bespeaks the decline of the year. Even by the end of July, you could tell that the days were getting shorter, but now the sun is setting at 6:36pm, and you know it’s only going to get worse. But, once the sun comes up, and the morning fog has risen up like the ghosts of a forgotten army out of the fields, it’s still quite pleasant to be outdoors, and you can still get away with wearing short-sleeved shirts and sandals, if you are not afraid of a little cool weather at dawn and at dusk.

The contrast in the length of days was made even more apparent to me at the end of July, when I went to Sweden for the wedding of my friends David and Annica. With Stockholm being about fifteen-and-a-half degrees further north in latitude than good old Hanover, you could get up with the sun before 6am, and still be outside reading by sunlight at a quarter to eleven in the evening. At first, I figured I was sleepy simply because of jet-lag, but eventually I figured out it probably also had something to do with being awake for eighteen hours at a stretch. Even so, Sweden in the summertime is quite beautiful, and well worth being awake for. The wedding was pretty good, too.

Anyway, now it’s the Fall term, and after a long summer of research, I’m back to teaching. I am quite happy with how the summer came out, both in regard to the work I accomplished, and my own personal happiness. It’s felt a little bit odd, at times, getting back into the swing of the regular term, with lectures and grading and scheduling and all that, but so far I think it is going pretty well. Hopefully my students will wind up agreeing, at the end of the term. Either way, however, I’m enjoying the course so far, and that’s a good start.

Filed by Michael at 14:49 under Personal
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Well, I’ve gone and done it. I’ve accepted my fate, and started a real web-log. For the time being, I will leave my existing News and Thoughts page where it is, but my longer-term intention is to import all of the older postings from that system into this new framework.

After all the ranting and raving I have done about web logs, I suppose one could argue I’m being a hypocrite, but if this is the worst sin I commit in my life, I think I will be accounted a good man. So, damn the torpedoes, mates — make revolutions for twenty knots, and may god save us all. Meanwhile, feel free to register yourselves, and post comments, if you wish.

Filed by Michael at 12:47 under Miscellaneous
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