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Under The Microscope


Reviewed by Russell Dear

There was once a cartoon beetle who told his friend he was setting off to search for infinity. On his return the friend asked him how he'd got on. "I almost made it", he replied, "I'll try again tomorrow". I guess that's the nature of infinity. It's certainly the nature of this book.

The author investigated every reference he could find on the subject and yet infinity remains as elusive as ever. Apart from two long appendices (73 pages) on transfinite cardinals and Gödel's incompleteness theorems, the book is, as the author puts it, digestible to the average reader providing it is well chewed. It opens with a comprehensive review of peoples' understanding of infinity throughout history from Aristotle to the present day, via St Thomas Aquinas and Galileo, and follows with a discussion of the problems associated with infinite number sets as investigated by mathematicians from Pythagoras to Cantor.

The book continues by looking in detail at some of the famous logical paradoxes, including Berry's on the largest number that can be described. The author then gets entangled in a discussion of whether truth can be defined -- mostly revolving around the time-worn statement This sentence is false -- and concludes that no finite system can define truth. The fourth section discusses Gödel's Incompleteness Theorem and makes speculations about mechanical intelligence and the nature of intelligence.

It includes, for me, the high point of the book which could be called conversations with Gödel. Here the author describes discussions he had with Kurt Gödel over a series of informal meetings earlier in his career; such questions as whether the mind is localised in the brains of people or out there everywhere. Gdel also clearly believes that the future is already here, that the passage of time is an illusion. The final section, apart from the appendices, considers the classical one-many problem -- can everything be regarded as a unity, as a single definite thing? Here, I believe, the author loses his way when he confuses absolute (non-subjective) issues with infinity and argues that infinity can't be discussed without some notion of God.

Besides being a well-known writer of science fiction, Rudy Rucker is Professor of Mathematics and Computer Science at San Jose University specialising in set theory, so is ideally placed to write on the subject of infinity. The book is amusingly illustrated with Rucker's own artwork and each chapter ends with puzzles and paradoxes giving the reader the opportunity of testing his/her understanding of what they have just read.

Russell Dear is a Mathematician living in Invercargill