Saturday, January 15, 2011

Trade Off #17: Secular vs Sacred

Ever wanted to rid yourself of all assumptions and reason your way to a full, coherent philosophy? You're not alone.

Rene Descartes made the most famous attempt to do so, but by most contemporary accounts he failed miserably. And in truth, if someone ever did succeed in assuming nothing, we'd probably never hear from em again.

That's because assumptions are necessary for action. They are like the ground, without which you can't propel yourself forward. Assumptions come with a price, though, which is that they can distort reality. (The ground distorts reality, too. Relative to the earth's surface our position appears to be constant, even though the earth is actually hurtling through spacetime at ~ 30 km / s.)

So although you'd prefer to assume as little as possible, you must make some assumptions if you want to get anything done. The trade off is in deciding where precisely to draw this line. Examples:
  • Euclid's fifth assumption was the parallel postulate, that two perpendicular lines extended indefinitely will never meet. Making this assumption enabled him to prove many useful geometrical theorems, but it also obscured some other real ways the universe can behave, which later mathematicians would explore. (see here)
  • Psychologically, we humans all have a certain amount of anxiety, regarding our place in the universe and our identity. Some believe that everyone has a "dogma quota," which they must assign somewhere, to ethics or empirics or whatever, to overcome this anxiety and function in society. (see here)
  • In Kuhn's conception of science, the paradigm is the set of all assumptions that allows scientists to solve puzzles and make advances. But it comes at the cost of institutionalizing the assumptions. So even when the reality-distorting effects of the assumptions become obvious, the institutions supporting these assumptions are difficult to overthrow and require mini-revolutions. (see here)
Mathematicians often make broad assumptions when they first prove a theorem, and then someone else will come along later and "generalize" that theorem by coming to the same conclusions with fewer assumptions. Isn't this a violation of the trade off, because the later proof takes the same "action" with fewer assumptions?

Sort of, and it's true that most people make stronger assumptions than they have to. But at the base level certain assumptions are truly essential. For centuries bright minds tried to generalize Euclid's parallel postulate, to prove the fifth assumption from the first four, with no success. It is this "rate-limiting" case where assumptions are essential that this trade-off describes.

We're a bit worried about this trade off conflicting with precision vs simplicity. From a stats perspective, maybe it's best to think of precision vs simplicity as referring explicitly to the number of parameters in the model, while this trade off refers to the form of the model, including the act of modeling itself. But possibly they can and should be united.

(kudos to Denis Collette for the nice photo)