Mathematics at the Edge of the
Rational Universe
Mathematics is the art of logical
story-telling and its creations exist only in the mind. Yet it's amazing how
useful scientists have found these imaginative constructs in trying to
understand the material universe.
Nobody has ever seen a perfectly round circle or an infinitely
long line of zero width. They’re pure figments of the mathematical
imagination. As for imaginary square roots of 1, ideal points where
parallel lines meet, and 6-dimensional space! What fantasies can be dreamt
up by the fertile mind of a mathematician!
Stories, parables, fables, myths
and legends can carry profound truths that have a powerful impact on the lives
we lead. Mathematical stories are no exception. This gossamer web we
mathematicians spin might be pure fancy. But it's the best tool we have to
understand and predict the material universe. And it reaches far beyond.
This book will take you on a
journey to the extreme regions, just before the point where logic breaks
down. It discusses the impossible, the infinite, the unimaginable, the uncomputable and the undecidable.
Our motivation will be that of an
explorer. We simply want to know what's out there. Whether any
practical use can be made of what we find there is not our prime
concern. This book is not written for the practitioner in logic or
mathematics or computing science.
I'm certainly not the first to
have attempted to bring deep ideas of logic and mathematics to a wider
audience. Lewis Carroll was one of the first in Alice in Wonderland
— a book which delightfully introduces many ideas from logic. I have
also been influenced by Abbott's Flatland and the writings of Martin
Gardener and Douglas Hofstadter.
The final chapter goes beyond
transcendental mathematics to consider the philosophical/theological question
of the existence of something beyond the material world and proofs of the
existence of God. This isn’t a technical discussion of epistemology
but rather a drawing together of some of the ideas in the earlier chapters.
After each chapter there’s a story reflecting the ideas
developed in that chapter. These may or may not aid the understanding of
the chapter but at least they provide some breathing space before the next one
and hopefully maintain the whimsical frame of mind in which this material can
best be appreciated.
While a little high-school
algebra wouldn’t go astray, the emphasis will be more on the imaginative
and philosophical aspects than on the computational.
This book isn’t for
everybody. Is it for you? Here’s a check list. If you can
answer “yes” to some or all of them then go ahead and read this
book.
- Are you intrigued by the logical reflexiveness of the sentence “this sentence
is false"?
- Have you read and enjoyed Alice's
Adventures in Wonderland?
- Can you cope with the symbols in the
following?
Let P denote a computer program and let D denote some data on which it acts. Suppose we denote the output by P[D]. So if P is a program for duplicating data then P[D] = DD. And if such a program is given its own description to duplicate, we have the equation P[P] = PP. - Would it interest you if one could prove the
existence of God?
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