----- Original Message ----- From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Saturday, March 15, 2008 5:12 PM Subject: [math-fun] A very peculiar paper
In the Feb. '08 Monthly, a strange article appears. Here is the abstract:
<< A Peculiar Connection Between the Axiom of Choice and Predicting the Future
Authors: Hardin, Christopher S.; Taylor, Alan D.
Abstract:
We consider the problem of how one might try to guess values of a function based only on knowledge of the function on a subset of the domain, without any assumptions about the function being analytic or even continuous. At the level of a single point, this is a hopeless problem. When one considers a collection of many points, however, it is often possible, using the Axiom of Choice, to guess in such a way that most of the guesses are correct.
In particular, there exists a strategy which, for an arbitrary function f from the reals into some set, will for all but countably many t correctly guess the value of f on an interval [t, t + d), d > 0, given only the values of f(x) for x < t. If one interprets this function as the evolution of a system over time, this means that, in principle, one can almost always predict an interval of the system's future based only on its past, without any assumption of continuity.
I haven't read it, but this sounds so ridiculous on the face of it, it's tempting to just ignore it completely.
Can someone give a hint as to why the claim made in the abstract (that it is possible to predict something about a arbitrary function's values for x
= t from its values from x < t) should not be considered ludicrous?
--Dan
P.S. I checked the calendar, and indeed it is not April 1.
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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