Rudy Rucker, son of a cleric and mathematics whiz kid, produced this book on `Infinity and the Mind' years ago, but reading and re-reading it, I continue to get insights and the chance to wrap my mind around strange concepts.
`This book discusses every kind of infinity: potential and actual, mathematical and physical, theological and mundane. Talking about infinity leads to many fascinating paradoxes. By closely examining these paradoxes we learn a great deal about the human mind, its powers, and its limitations.'
This book was intended to be accessible by those without graduate-level education in mathematics (i.e., most of us) while still being of interest to those even at the highest levels of mathematical expertise.
Even if the goal of infinity is never reached, there is value in the journey. Rucker provides a short overview of the history of 'infinity' thinking; how one thinks about divinity is closely related often, and how one thinks about mathematical and cosmological to-the-point-of-absurdities comes into play here. Quite often infinite thinking becomes circular thinking: Aquinas's Aristotelian thinking demonstrates the circularity in asking if an infinitely powerful God can make an infinitely powerful thing; can he make an unmade thing? (Of course, we must ask the grammatical and logical questions here--does this even make sense?)
Rucker explores physical infinities, spatial infinities, numerical infinities, and more. There are infinites of the large (the universe, and beyond?), infinities of the small (what is the smallest number you can think of, then take half, then take half, then take half...), infinities that are nonetheless limited (the number of divisions of a single glass of water can be infinite, yet never exceed the volume of water in the glass), and finally the Absolute.
`In terms of rational thoughts, the Absolute is unthinkable. There is no non-circular way to reach it from below. Any real knowledge of the Absolute must be mystical, if indeed such a thing as mystical knowledge is possible.'
At the end of each chapter, Rucker provides puzzles and paradoxes to tantalise and confuse.
* Consider a very durable ceiling lamp that has an on-off pull string. Say the string is to be pulled at noon every day, for the rest of time. If the lamp starts out off, will it be on or off after an infinite number of days have passed?
Rucker explores the philosophical points of infinity with wit and care. He explores the ideas behind and implications of Gödel's Incompleteness Theorem, and leads discussion and excursion into self-referential problems and set theory problems and solutions.
He also discusses, contrary to conventional wisdom, the non-mechanisability of mathematics. We tend to think in our day that mathematics is the one mechanical-prone discipline, unlike poetry or creative arts and more 'human' endeavours. But Rucker discusses the problems of situations which require decision-making and discernment in mathematical choices that no machine can (yet!) make.
* Consider the sentence S: This sentence can never be proved. Show that if S is meaningful, then S is not provable, and that therefore you can see that S must be true. But this constitutes a proof of S. How can the paradox be resolved?
This is a beautifully complex and intriguing book on the edges of mathematics and philosophical thinking, which is nonetheless accessible and intellectually inviting. You'll wonder why math class was never this fun!
