Tom Chavez
The pin-ball universe or, how useful is mathematics, really?
The usefulness of mathematics in science is a miracle with no rational explanation. It is a gift for which we should be grateful. — Eugene Wigner
Physicists give a higher degree of reality to mathematical models of the universe than they accord the ordinary world of sensation. — Steven Weinberg
Some scientists believe that mathematical laws apply to all phenomena of nature, and the discovery of such laws will lead to a complete understanding of the universe.
Richard Thompson’s thought experiment consists of an array of fixed 2-inch spheres, spaced four inches apart, extending in all directions. Consider the track of a moving sphere that rebounds elastically off the fixed spheres.
A slight variation in the initial direction of the moving sphere will be greatly magnified as it bounces off others. We need to know the sphere’s initial direction with great accuracy to predict its path correctly.
Thompson calculates that if the sphere moves at sixty miles per hour, to predict its path for one hour would require accuracy of roughly two million decimal places, which would take over 700 pages to write out.
To specify the sphere’s movements for one year, would require more than six million pages.
Generalizing the experiment by allowing all the spheres to move and interact by force laws of various kinds gives a 2D classical Newtonian model of nature.
Thompson concludes that the prospects for a simple universal mathematical theory are dismal, as it must account for almost unlimited detailed information.
Conclusion: the practical usefulness of mathematics is great, but limited.
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