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What is infinity? Does it even exist? How can there be more than one infinity? Using vivid imagery Rucker shows that throughout the ages man has pondered these questions. Ranging from Plato to Cantor, infinity has inspired awe, fear and feelings of futility. Through a fun exercise of 'name the largest natural number' the book starts exploring the concept of different levels of infinity. From the 'smallest' countable infinity ω (omega) to the whole succession of Cantor's א (aleph) numbers. Discussions are accompanied by drawings that are sometimes enlightening and other times mind-boggling. Every chapter concludes with puzzles and paradoxes for the reader. Fortunately the answers are included in the back, since many questions left me scratching my head.
Absolute Infinity and the Mindscape
It doesn't take long for Rucker to mix philosophy into his - so far mostly mathematical - discussion of infinity and transfinite numbers. Starting with the concept of Absolute Infinity, or Ω (Omega), the inconceivable infinity that contains all ordinals. Now, if you're asking yourself how can you sensibly talk about something inconceivable, read Rucker's response:
A skeptical reader could, rightly, demand to know how it is possible to discourse rationally about an inconceivable object like Ω. I would respond that Ω is a given, an object of our immediate pre-rational experience. And to use the tools of symbolic logic to investigate an empirically existing phenomenon is not to commit a category mistake, anymore than it is to look at living cells through the inanimate lenses of a microscope. We have a primitive concept of infinity. This concept is inspired, I suspect, by the same deep substrate of mind that conditions religious thought. Set theory could even be viewed as a form of exact theology.
One other philosophical notion I especially enjoyed is the Mindscape. Here Rucker conjectures that, like our bodies all occupy part of the same physical universe, our thoughts occupy part of a shared Mindscape. Your own thoughts are like a private room, and you can expand your view of the Mindscape by exploring it. Like in the physical universe, a single 'point' in the Mindscape can only be occupied by a single consciousness at the same time. But at the same time thoughts can be observed by many. Every possible thought is in the Mindscape, whether someone has thought of it or not. Of course the Mindscape is discussed in tandem with the Absolute Infinity, using set theory to formalize the notion.
After discussing the theory and philosophy of infinities, the book continues exploring some famous paradoxes. The Berry paradox, Richard's paradox and the Liar paradox (think: 'this sentence is false') are treated in great detail. However, the goal is not to bask in the glory of these paradoxes. No, Rucker wants to persuade the reader that the infinite regresses that underly these paradoxes point to mental concepts defying any exact formalization. Yet we as humans can understand these paradoxes and reason about them anyway.
Thinking machines and Gödel
If the last sentence left you wondering what this means for artificial intelligence and whether machines can ever think like humans, you're thinking in the right direction. The book introduces Gödel's incompleteness theorems in a chapter titled 'Robots & souls'. Ultimately, we are lead to the conclusion that we cannot possibly formalize all of our mathematical intuitions in a finite way. Because to formulate our intuitions is to create a formal system. This formal system must be incomplete and its consistency (even if we know it is consistent) cannot be proven by the system itself according to Gödel's theorems. However, this does not mean that such a 'thinking' machine for all our mathematical intuitions cannot exist. It just means we won't be able construct or verify it. Rucker goes as far as stating that:
… rationality can never penetrate to the final, ultimate truth.
What I loved about this book is that Rucker shares his first-hand conversations with Kurt Gödel. It turns out Gödel had a mystic turn of thought, which Rucker shares throughout this book. It's less fuzzy than it sounds at first:
… "mysticism" does have a precise meaning [..] not to be confused with occultism, strange rites and so on. Mysticism is just the simple awareness of the direct identity of the individual soul and the Absolute. Too much rationality quickly becomes inane and boring. What is needed is some kind of bridge between the two.
And that is exactly what the book sets out to achieve. First using the pure mathematics of infinity, set in its historic background. But also by acknowledging that mathematics and science cannot answer all interesting questions posed by the existence of Absolute Infinity. As a Christian I recognize many of the conclusions Rucker draws while exploring the boundaries of our knowledge.
Now, I'm a programmer. In my case that means I have some affinity for math, but clearly I'm out of my depth in several of the more rigorous mathematical interludes. But that's ok. The surrounding narrative is easy to follow, even if some of the finer points escape me. Especially if the alternative is for Rucker to hand-wave his way through important details, I prefer rigor. I might even have learned a thing or two along the way. What made this book most interesting is the fact that it goes beyond just the math. It tries to derive meaning from notions like (absolute) infinity, relating it concepts like consciousness and the One/Many problem.
This is not a book for casual reading. You really have to make an effort, but when you do it is a very mind-expanding book. Which reminds me, I really should finish Gödel, Escher and Bach as well some day... Since I only have finite space for this review, let me conclude by saying that I thoroughly enjoyed reading this book. If you have any interest in mathematics, philosophy and the intersection of these two, I highly recommend reading 'Infinity and the Mind'.