Mark Bridger

Newtonian Supertasks: A Critical Analysis

In two recent papers Perez Laraudogoitia has described a variety of supertasks involving elastic collisions in Newtonian systems containing a denumerably infinite set of particles. He maintains that these various supertasks give examples of systems in which energy is not conserved, particles at rest begin to move spontaneously, particles disappear from a system, and particles are created ex nihilo. An analysis of these supertasks suggests that they involve systems that do not satisfy the mathematical conditions required of Newtonian systems at the time the supertask is due to be completed, or else they rely on the application of the time-reversal transformation to states which are not well-defined. Consequently, it is unjustified to conclude that the paradoxical results are arising from within the framework of Newtonian mechanics. In the last part of this article, we discuss various aspects of the physics of these supertasks.

Two ways of looking at a Newtonian supertask

A supertask is a process in which an infinite number of individuated actions are performed in a finite time. A Newtonian supertask is one that obeys Newtons laws of motion. Such supertasks can violate energy and momentum conservation and can exhibit indeterministic behavior. Perez Laraudogoitia, who proposed several Newtonian supertasks, uses a local, i.e., particle-by-particle, analysis to obtain these and other paradoxical properties of Newtonian supertasks. Alper and Bridger use a global analysis, embedding the system of particles in a Banach space, to determine the origin of the strange behavior. This paper provides a common framework for the discussion of both the local and global methods of analysis. Using this single framework, the areas of disagreement and agreement are made explicit. Further examples of supertasks are proposed to illuminate various aspects of the discussion.

Mathematics, models and Zeno's paradoxes

A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zenos paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time.

Looking at the Mandelbrot Set

In this column, readers are encouraged to share their expertise and experiences with computers as they relate to college level mathematics. Articles that illustrate how computers can be used to enhance pedagogy, solve problems, and model real-life situations are especially welcome. The Algorithm of the Bi Month will feature algorithms which solve mathematical problems through skillful computer use. It will also illustrate how the mathematical point of view may change in order to implement algorithms on a computer. New material for this column should be sent to John Blattner, Department of Mathematics, California State University, Northridge, California 91330.