12 August 2007

first let's kill all the lawyers

Language Log writes about the semantics of pork, inspired by this New York Times article. A current health care bill includes statements like

"any hospital that is co-located in Marinette, Wis., and Menominee, Mich., is deemed to be located in Chicago"

and other similar references that sound innocuous -- they're just defining a metropolitan area -- until you think, wait a minute, is that anywhere near Chicago? (I checked a map; it's 259 miles away.) And how many hospitals like that could there be? (As it turns out, exactly one.) Apparently hospitals in metropolitan areas get bigger reimbursements from Medicare for various procedures, on the theory that the cost of living for their employees is greater than that for rural hospitals.

Language Log goes on to point out that is an example of how there are two ways to define any particular set: by listing its elements, or by specifying a set of constraints. A mathematical example that immediately comes to mind is "the set of even primes". Or, somewhat more innocuously, the statement "let p be an even prime". You don't see this that often, because it's easier to say "let p equal 2". But I have seen it, in proofs of the form: "Theorem: All primes p have property x. Proof: let p be an odd prime. Then (proof). Alternatively, let p be an even prime. Then (simpler proof)."

Of course, when one specifies a set by giving constraints, there's always the problem that the set might be empty. What would happen if the health care bill in question said, say, "$100,000 should be distributed evenly between all hospitals within a quarter-mile of Isabel's apartment?" There are none. Who gets the money? (I suspect there is some conventional interpretation for this sort of thing. I don't think you'd see it in this sort of bill, but I can imagine, for example, a doting grandparent writing in their will "my grandchildren shall equally split my [large sum of money]" and then the grandchildren tragically die before the grandparent.) I've heard the apocryphal story of a student who goes into his PhD defense and says that he will be presenting results on a certain sort of algebraic structure satisfying the following eight conditions. One member of the committee interrupts and says that he can prove there are no such groups. The student doesn't get the PhD. Proving things about something that doesn't exist is considered worthless, no matter how ingenious the proofs might be.

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