[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
On Sat, Mar 15, 2008 at 3:12 PM, Dan Asimov <dasimov@earthlink.net> wrote:
In particular, there exists a strategy which, for an arbitrary function f from the reals into some set,
Maybe it only works for f : R -> {0} or some other one element set? I can do pretty well predicting that function's future. I would be surprised by not find it ludicrous if the image set had two elements. If the image set is the reals, it does indeed seem ludicrous. Maybe there's something in there about "describable functions" so there's only countably many functions possible? (E.g. functions that can be described with at most so many words, or functions containing finitely much information for some suitable definition of information). --Joshua Zucker
----- 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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participants (3)
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Dan Asimov -
Joshua Zucker -
Loren and Liz Larson