Philosophy 167: Class 4 - Part 15 - Varieties of Generalization: Exact, Essentially Exact, and Approximate.
Smith, George E. (George Edwin), 1938-
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This is a more subtle set of distinctions. I threw it out, the top one, last week. And I can reinforce it now. It's very, very much present in Kepler. Are the generalizations exact, essentially exact, merely approximate? Where essentially exact means, it's a nice subjunctive. Would be exact, were there not various extraneous factors at work.
And when Kepler started thinking that the Jupiter Saturn is not entirely well behaved. What he thought their magnetic fibers are interacting with one another. That's disturbing things. It's still essentially exact, but those fibers are disturbing the underlying physics enough that it would be exact if they weren't interacting.
That's the sort of mood. Where merely approximate, means all you've got is a curve that agrees with the data. No claim that's exact, or would be exact, under any specific circumstances. That's fairly straightforward. Now I'm gonna introduce a four fold distinction. A box of two and two. The first one says, suppose it's not exact, there are deviations from it.
Then the question becomes does it hold in the mean, or to the contrary, are the discrepancies skewed, they don't balance out across the generalization. Now a comment about that. Almost no principles of physics hold in the mean. My favorite example is the ideal gas law, every real gas deviates from it in a different way, but the ideal gas law is not in the mean of any of them.
They all depart systematically from it. So it doesnt hold in the mean. That's point one. Point two, you want 'em. I'll come right back if you have a question. Point two, if you want to make predictions, what do you want the generalization to do? Hold in the mean.
We can prove that the probable error is least when you've got something that holds in the mean. That's what least squares fitting does. Is it fits curves so they hold in the mean. Which tells you something else when I tell you physics has almost no principles that hold in the mean.
Physics is not in the business of trying to maximize predictions. I'm an engineer. Curves I use holding the mean, you better bloody well count on it, because I'm in the business of predicting. But that's a real distinction for anything that holds an approximation. Second distinction, generalizations that would hold exactly in certain specifiable circumstances.
Now I go back the way I did it initially cause I'm misleading. Idealizations, something that you simply have for one reason or another said it's near enough to this, I'm gonna idealize it. It's near enough to a circle, I'm gonna treat it as a circle. Near enough to an ellipse, I'm gonna treat it as an exact ellipse.
That's an idealization. Kepler's bisection of the eccentricity of the earthside orbit. That was an idealization. He couldn't establish it was exactly that. So was Ptolemy's. So was the ideal gas law. That's an idealization. Now that versus mere approximation, simply holds approximately to very high accuracy but we can't say it's an idealization.
Now I characterize, these are not the only two kind of idealizations, but they'll suffice for us till very late in the course. One is one that would hold exactly in specifiable circumstances. So the ellipse, I'll do it now. The ellipse is an idealization. It would hold if the planets were not interacting with one another.
It would hold exactly if there were only the Sun and one planet, and no other forces. I can do other examples like this. The idea gas law would hold exactly, if molecules were point masses with no forces between them. So the deviations are measuring the forces between them.
That versus pure approximation, which is what you get with a curve fit. Then the other kind of idealization are ones we do purely for mathematical reasons. And the classic example is linear elasticity. About the only thing you ever learn in a classroom, there are no linear elastic bodies.
You get a linear elasticity, I hope you people know what this means, from a Taylor series by truncating all but the first term. That's how you derive linear elastic theory, is write a Taylor series out and chop off all the higher terms. So, and we have other examples of that mathematically where what we're doing is the mathematical cractibility is so great, we idealize even without worrying whether there are any circumstances in which it holds exactly.
So that's a four fold distinction. Where do Kepler's generalizations fall in that? I'll come right back. He thought they didn't hold exactly, so he's ready to concede they hold approximately. He wanted them to hold centrally exactly. Not to be idealizations. I think he wanted them to hold in the mean.
Because that's how he's assessing their accuracy. But that doesn't mean that's not, may not mean what it is at all. So once again, any time you have a generalization you have another kind of evidence problem. What are we to make of this generalization? What's its status? What's its standing in this four fold, actually, it's a four fold division on top of a two fold division.
Exact, or if not exact, four fold. Fair enough? And these are again, you turn and ask what evidence is gonna help me sort that out.