THE MATHEMATICS AT THE EDGE OF THE RATIONAL UNIVERSE
Mathematics is the art of story-telling. 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 minus 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.
In this book we'll go on a journey to the edge of the rational universe. 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.
Material which had hitherto remained locked up in courses with such intimidating names as Advanced Algebra, Axiomatic Set Theory and Theory of Computation is too fascinating to leave there. All it needs is a little less emphasis on symbolic formality and a little more imaginative presentation.
That's not to say that having read this book you'll be on a par with the students who graduate from my courses. I like to think that what I've done is to build a road into a national park that has hitherto only been accessible on foot.
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 Adventures in Wonderland, a book which delightfully introduces many ideas of logic. I have also been influenced by Abbott's Flatland and the writings of Martin Gardener and Douglas Hofstadter.
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.
(1) Are you intrigued by the logical reflexiveness of the sentence "this sentence is false"?
(2) Have you read and enjoyed Alice's Adventures in Wonderland?
(3) 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.
(4) Would it interest you if one could prove the existence of God?