Logic and Word Games

Over at Rotten Chestnuts, a post on Hegel and Marx that underlines the giant Problem of Philosophy.

One reason “underpants gnome metaphysics” appeals, of course, is that Hegel et al had a point. Classical logic has some huge gaps, as the Classical Greeks — i.e. the guys who developed it in the first place — well knew. Consider the famous “Achilles” paradox of Zeno of Elea (c. 490–430 BC). Achilles and a tortoise are running a race. The tortoise gets a ten foot head start. Can Achilles catch up?

In reality, of course, Achilles blows by the tortoise, but consider it from the “logical” perspective. In order for Achilles to catch the tortoise, he has to cut the distance in half. Now he’s five feet away. But to bridge that gap, he has to cut the remaining distance in half. Which he does, and now he’s 2.5 feet away. To bridge that gap, he has to halve the remaining distance again, and now he’s 1.25 feet away, then 0.625 feet away, then 0.3125 feet away, and so on, out to infinity. According to “logic,” at least, Achilles never catches up.

Severian, “Final Sample“, rottenchestnuts.com

As Severian mentions, this is probably an analogy for a high-level mathematical concept, and NOT an argument that motion does not exist (much the same way Schrodinger’s Cat is an analogy on the difficulty of observing sub-nuclear particles, and NOT a denial of the Law of Non-Contradiction). Because the minor premise of this “logic” (the infinite halves) is, shall we say, entirely questionable. All Achilles needs to do to catch the tortoise is run faster than the tortoise does. If the tortoise is ten feet ahead, and moves at .5 feet per second (a generous estimate) and Achilles runs at 4 feet per second, then at the end of 3 seconds, Achilles has gone 12 feet from the starting line, and the tortoise is 11.5 feet from the same starting line. Boom. It’s over. Simple mathematics, which, I’m told, is entirely logical.

{But first he has to go HALF! Yes, and at a constant rate, Achilles will cover 2 feet in half as much time as 4 feet, and 1 foot in a quarter as much time, 1/2 foot in an eighth, etc. Achilles could be a dolt at times, but nobody’s stupid enough to slow their speed by half each second of a foot race. Stop being such a nerd.}

The next paradox he mentions is even dumber:

Consider an equally puzzling Ancient Greek problem, the sorites paradox. How many grains of sand make a heap? Or, since this is the Internet, how many hairs must Jean-Luc Picard, the best captain of the starship Enterprise, lose before he’s considered bald?


There’s actually a formal fallacy under this name: the Continuum Fallacy. But that’s less important than the deep and abiding idiocy of expecting that “bald” to be a precisely defined term. It isn’t, any more than “heap” is. And this is the Problem of Philosophy I mention earlier: it’s lost in an endless race to the bottom of granular defining. It’s all word games.

The Zen Master holds up a staff. He says to his pupils: “If you call this a staff, you deny its eternal life. If you do not call it a staff, you deny it’s present fact. Tell me, just what do you propose to call it?”

Like all Zen Koans, there isn’t one answer, but the one I find most useful is to look at this as a meditation on the limitations of language as such. No matter what word you use, you’re focusing on a specific aspect of the thing’s existence. No word exists that describes the fact that the staff as once part of a tree, which was one a seed, which was once part of another tree, and will shortly be dust to be sucked up by the roots of another tree, while also describing the fact that it’s an inanimate object you can smack your stupider students on the butt with.

Translation: words communicate ideas, and can so be very very vague while still being effective. Humans use language as a tool, language is not fixed. So demanding absolute precision in words outside of a highly technical context isn’t just nerdery, angels-on-pinheads; it’s a sisyphean nightmare, on the order of the infinite cyle-of-halves. The aforementioned Schrodinger may be right about subatomic particles (for all I know — I took Physics for Non-Science Majors), but that doesn’t translate up to an inability to determine if a cat is dead or not. Much of modern and post-modern philosophy is an exercise in playing word games in a fruitless quest to provide the metaphysical underpinning that used to come to us via religion. It won’t work, because words are too flexible and incomplete to meet these needs. Whatever conceptual structure we create, (say “gender”), we can uncreate just as fast, as soon as we find the limits of it. It’s nothing but Imaginary Wack-a-Mole.

Happy New Year.


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