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on August 9, 2000
I've read a few of Rucker's other nonfiction books (his fiction is another topic entirely), and I think this one is still his best. I bought and read it when it was new and I'm about to buy a replacement copy.

The "book description" on this page touches briefly on one of Rucker's key points: "the transcendent implications of Platonic realism." This is well put, and the remarks above correctly relate this point to Rucker's "conversations with Godel." Godel was a mathematical Platonist -- that is, he believed that mathematical objects are real in their own right and that the mind has the power to grasp them directly in some way.

Rucker gets this right, unlike some other better-known interpreters of Godel who have co-opted his famous Theorems in the service of strong AI. Rucker, too, thinks artificial intelligence is possible, but for a different reason which he also here explores: he takes the idealistic/mystic view that _everything_ is conscious in at least a rudimentary [no pun intended] way, and so there's no reason to deny consciousness to computers and robots. Heck, even rocks are conscious -- just not very :-). (I don't know whether Rucker would still defend this idea today or not. At any rate, for interested readers, a more elaborate version of panpsychism is developed and defended in Timothy Sprigge's _The Vindication of Absolute Idealism_.)

These and other speculations are the jewels in a setting of solid exposition. Rucker is powerful in general on the topic of set theory, which he takes to be the mathematician's version of theology. And his discussions are a fine introductory overview of the various sorts of infinity, including but not limited to mathematical infinities. He is remarkably familiar with the literature of the infinite both inside and outside of mathematics, e.g. calling attention to certain neglected works by Josiah Royce (who discusses infinities in an appendix to _The World and the Individual_). He also discusses, quite accessibly, some of the paradoxes that arise from treating the set-theoretic "universe" as a completed, all-there-at-once set in its own right.

Rucker, a descendant of G.W.F. Hegel in both body and spirit, could be read profitably on this topic by a pretty wide audience. In particular he is a good cure, or at least the beginning of a cure, for certain philosophers who (more or less following Aristotle) would deny the real existence of actual infinities in particular and mathematical objects in general. (Also for interested readers: another, more technical defense of realism with regard to mathematical objects can be found in Jerrold Katz's _Realistic Rationalism_.)

My original copy of this book was published, with some justification, in Bantam's "New Age" series. I am glad to see the new edition is published by Princeton University Press.
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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!
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on June 28, 2002
I know very little about any of the subjects discussed in this book, although I do have a degree in philosophy of science, and I liked this book a lot.
I can't believe I made it through 7 years of senior school and 2 years of degree level maths and nobody ever bothered to tell me about infinity, transfinite numbers, set theory and its relationships with, and underpinning of other branches of mathematics in a way I could understand rather than simply regurgitate. Rucker on the other hand manages to do this in 362 pages.
I slso found the stuff about Godel and the impossibility of complete formulisms very useful, not only philosophically, but also just for my own peace of mind.
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on January 2, 1999
Yeah, its a book on math and a book on philosophy. Actually, it's a bit more. It's a book that uses mathematics as an approach to philosophy, but certainly not in a mechanist or reductionist fashion; after all, what's the last book by a mathematician that treated mysticism as a serious philosophy? Inside you learn about: what sorts of infinity there might be, why the ancients and medievals were uncomfortable with infinity, truth, randomness, transfinite arithmetic, Hilbert's Hotel, robots and souls, and quite a bit more. Bottom line-the mathematical discussions can be tough slogging at times, but are explained thoroughly and in fine detail, with wit and charm, and the whole constitutes one of the richest scource of ideas I have ever come across. A bargain at twice the price!
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on November 3, 2010
Firstly, please ignore the silly one and two star reviews. They seems to be written by
cranks. For reasons unknown to me Set Theory seems to attract cranks who get very upset
by the work of Georg Cantor and the "diagonal method" in particular.

This book should be on all bookshelves of popular accounts of mathematics.

Infinity and the Mind is a superb popular account of Set Theory as an approach to
exploring the concept of infinity. I know of no other book like it. It is quite accessible
to anyone who has studied high school mathematics. The book tackles deep issues of
mathematics, logic and philosophy. Rudy Rucker explores topics such as Godel's Theorems,
paradoxes of Set Theory, orders of infinity (large cardinals), artificial intelligence,
the logical foundations of mathematics and much more. The maths is pretty rigorous too
for a popular work.

The author is well qualified to write on the subject. As well as doing research in
Set Theory he has, incredibly, interviewed Kurt Goedel himself shortly before he died.
An account of his discussions with Goedel is included in the book.
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on May 10, 2011
As a university professor I read zillions of, frankly often quite dreadful, mathematics and physics and astronomy books. I read this one when it first came out in 1982. It really bowled [or Booled?] me over even then, and every time I've picked it up since I find something new and clever I hadn't given full thought to before. It is a MUCH better introduction to transfinite numbers than the also still quite good book by the late David Wallace Foster [if I've ordered names those correctly]. When I say it's one of the five best science books ever written, I am not exaggerating, provided, of course, that you have a basic grasp of Calculus and a touch of "naive set theory" and basic analysis under your belt, which if you got through high school or college you probably do. Rucker's non-fiction books are always excellent. His fiction doesn't interest me as much, but some of the stories have interesting conceptual leaps etc.
Now what are the four OTHER TOP FIVE SCIENCE BOOKS? Well, George Gamow's "Mr. Tompkins" books are pretty darn good but a bit dated in presentations. Stan Ulam's auto?-biography is good as well, As are the two books on the making of the A and H bombs. It was Ulam BTW who really figured most of it out. But those are not top fiver. "A Primer of Real Functions" by Ralph Boaz is great, as is the old ? Thomas Very Complete "Calculus" book. [In fact it was so complete you couldn't really get through the work even teach a three quarter series of courses with it. [Ah, such a noble task!] I remember they used it at Macalester College when I was there in the early 70s. They were very proud of using such an advanced text too, as they should have been. Try using the same books these days, and the students would probably immediately haul you off to the Provost and try to get you fired for giving them "thinking headaches." ... but I'm not getting starting to get bitter after 30+ years of teaching ... am I? Oops, one last thing, as great as Rucker's book is, even he doesn't provide several intuitively helpful conceptual images of the deeply mysterious "measurable cardinal" first discovered by Ulam BTW. It still all "infinite intersections and unions of sets, ultra-filters etc., if he even goes that far down the road. Though he does say something mind-expanding like "they are so much larger than all the cardinals that come before them that they sort of stand to Aleph infinity as Aleph Nought does to a large finite," or something like that. And, even in the 28 years since my first read, I still haven't found anyone, online or off, who can conjure up an intuitive picture of measurable cardinal for me ... oh why, oh why, do I bother to go on? "Man by nature seeks to know" is all Aristotle would say.
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on May 20, 2000
The book mentiones : Infinity commenly inspires feelings of awe, futility and fear. Reading of the book makes one agree to it. The book is written for a reader who is philosophically curious and patient in reading. After introducting the various context ( spatial, temporal , physical) where one encounter the issue of infinity, the author explain clearly the debate of potential vs actual infinity. Here author points out about the Greek philosophical tendencies. Chapter two discusses the revolution brought by Cantor's works. He explains the concept using a lot of symbols, diagrams and illustrations. The reader is made to understand the notion of transfinite number. The chapter ends with an extract from his novel White Light which deals with the idea of the chapter. Next chapter discusses the kind of paradoxes one encounter in thinking the theme of infinity within modern mathematical logical framework. Chapter four discusses the implications of Godel's theorems in question of Robot consciousness. He gives details about his personal interactions with Godel. He mentiones about his dream about Godel the day before Godel's death. This is most humanistic chapter. Last chapter discusses the abstract philosophical reflections. There are two well written excursion chapters : one on Cantor's set theory and one on Godel's Incompleteness theorems. Every chapter has well thought puzzles and paradoxes section.
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on July 2, 2011
Rucker does a good job in Chapters 2, 3 & 5, on transfinite numbers, paradoxes, the one/many problem, truth, nameability... And in exploring the many routes towards the unattainable/inconceivable absolute...

His treatment of transfinite & large cardinals, in Excursion 1, is more complete than Stillwell's.

His treatment of Gödel's incompleteness theorems, in Excursion 2, is more detailed but totally unstructured as compared to Stillwell's exposition (via Emil Post's & Gentzen's discoveries...).

BUT, but, if you haven't been exposed to books such as Smullyan's, Stillwell's on various topics such as sets, ordinals, cardinals, ZFC, NBG, class/set differences, finite/transfinite induction & recursion, foundation, rank, constructible sets, independence of the continuum hypothesis... then Rucker's skimming over those concepts will appear confusing.

As for the philosophy side, Rucker's book is packed with interesting details and it would have taken a Hofstadter to structure the whole lot, into what could have become a fascinating exposition... Alas, not everyone is a Hofstadter .

Finally, concerning presentation and readability : sections are not numbered, let alone paragraphs which are overpacked ; figures are "thrown" in the text, unrelated ; proofs are half achieved, are not even stated as proofs ; conclusions are loosely tied to proofs and theorems, when they are...

If that's how Rucker teaches, then I feel really sorry for his students !
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on August 28, 2008
I was first introduced to this book by a mathematical philosopher friend in 2001. Immediately I was drawn into the book, because it dealt with many subjects I'd been thinking, such as how there are more real numbers than natural numbers, how infinity comes in different sizes, and how the mere existence of infinity is to be questioned. Soon I got lost in all the numbers and had to put the book down a few times until the summer of 2006.

Rucker's writing was more like personal notes he wrote for himself than a well-constructed thesis on the subject. And here are some of my own personal notes about this book.

Chapter one reviews the history of infinity, and introduces the concept of mindscape. Years ago I was excited about the idea of mindscape, but after I had the fortune to see the Reality as a whole, I found this idea rather intuitive and basic. I was happy to see the mention of the Absolute as part of the discussion of Infinity.

Chapter two is about all the numbers. Again soon I became confused with the names of different infinities. Unless one can tightly grab onto the endless symbols Rucker introduced incessantly throughout the chapter (and the book) one would have a difficult time follow the text. Also his figures are ill-labeled. I don't think I am missing much by skipping some of the paragraphs. I also skipped the two excursions because they are even more technical.

Chapter three is titled "The Unnameable", and Rucker discussed the Berry Paradox and discussed the reality of Truth, among other subjects. It's interesting to see how systematically and detailed he talks about the logic of "This sentence is false", and even distinguishes it from "This sentence is not true". I skipped the more technical section of Richard's Paradox, assuming it is along the similar line of the truth discussion. I was glad to find out that Rucker is also a Borges's fan (I only wish I could write reviews of books and movies as clearly and originally as Borges). From Borges's story about the Library of Babel--the library of all possible books, Rucker introduced a clever tool--to code each book into a natural number. Furthermore, the whole universe can be coded into a natural number, and thus we can think about the infinity nature of the universe the way we think about numbers.

Chapter four is about robots and souls, but the more interesting part is the three conversations Rucker had with Godel. I was happy to know that Godel is a mystic, partly because I am becoming more and more identified with the label mystic.... Godel has found. Rucker is still seeking.

Chapter five is on the One and the Many, the most philosophical chapter of the book. Rucker probably does not have the One figured out, but it's interesting to see how he compares the One and the Many in a rational way.

Rudy Rucker wrote this book in 1982. Perhaps he has reached another stage in his search. Despite of many inadequacies I found in this book, it nonetheless has showed me fascinating new ways of thinking about the universe. For this I am grateful. I would rate this book 4 out of 5 stars.

p.s. I feel I am much slower than 5 years ago. I don't seem to be able to comprehend complex systems as effortlessly as I used to--perhaps a sign that my brain power is declining?
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Rudy Rucker's book is one of the best introductory texts into the problems of infinity and understanding the problems inherent in dealing with an infinity. He covers quantum mechanics to some extend and the problem of an infinite number of parallel worlds which are created and collapsed on a near constant basis. There are many examples of how when dealing with infinities most of our logic deserts us and the very basis on which we understand things may not hold true. For example if you have two sets of numbers and one set has all integers and the other has the squares of all numbers possible then which set is larger? The set with the squares of all numbers is missing an infinite number of numbers (for example, 3, 5, 6, 7, 8, 10, etc) and so the one with all possible integers is larger since it contains all numbers. But since all numbers have a square then they are the same size. How can the be they same size and one infinitely larger than the other at the same time? When dealing with infinities those rules don't apply and there is no contradiction.
Mr. Rucker leads us on a wonderful trail of discovering how these things work on a philosophical level and how they all relate together. What sort of infinities exist and how does understanding infinities affect our concepts of philosophy are questions that this book attempts to work through. A thoroughly enjoyable read for those who would like to expand their way of thinking or who enjoy delving into concepts that defy logic while being totally logical.
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