Rating: 
Summary: Mental masturbation at its highest form
Review: According to Rucker, set theory is the mathematician's religion. All I can say is, this certainly seems to be true for Rucker. Whether the REST of the mathematical world shares his quest for set-theoretical spiritual redemption is another matter. Myself, I don't think of set theory as a religion. To me, it's just a particular bunch of thought patterns that provide a working platform for day-to-day mathematical expression and communication. For me, set theory is like government, taxes, getting enough vitamins, washing your clothes, brushing your teeth, etc., -- processes which have to go on for daily existence, but hardly anything which gives my life ultimate meaning, much less spiritual salvation. So I find it hard to relate when Rucker keeps coming back to "cap-omega", (the proper class of all ordinals), trying to endow it with some kind of religious or mystical significance.This book is heavy on philosophy and metaphysical jibberish and short on math. Anyone familiar with the basics of ordinal and cardinal arithmetic will find little new material here. What you WILL find, though, is endless philosophical pontification: "Infinity commonly inspires feelings of awe, futility, and fear." [Only if you let it...] "I think of consciousness as a point, an 'eye', that moves about in a sort of mental space. All thoughts are already there in this multidimensional space, which we might as well call the Mindscape." [This just sounds like applying the idea of a "state-space" to consciousness...hmmmmm.] "God, the Cosmos, the Mindscape, and the class V of all sets [ooh, the proper class of all sets...I'm so SCARED!!!] -- all of these are versions of what philosophers call the Absolute" [Here is where Rucker leaves the world of mathematics and ventures into theology.] "Cantor proposed that each chunk of aether is made up of aleph-one aether-monads." [Cantor made many great contributions to mathematics, but this was NOT one of them.] "Imagine. Pursued by a tiny girl in a yellow dress, white ducks move deliberately across a green sward sloping down to stagnant brown water. How many ducks are there?" [How much LSD have I taken?] "As a person develops, he moves out to higher and higher transfinite levels. Although one cannot think of each natural number, to grasp the general idea of natural numbers is to jump out to V_omega. In terms of Love, or mutual knowledge, we could say that if A and B have a perfect understanding of each other...then they too are moving out past level omega." [Kum-ba-yah, my cardinal, kum-ba-yah...kum-ba-yah, my ordinal, kum-ba-yah...oh, Cantor, kum-ba-yah.] "Once again, think of the ordinals as an endless mountain you are climbing. Say that you have climbed as far as the first extendible cardinal, usually called lambda....Far below, you can make out the first inaccessible and first measurable cardinals...the part of the cliff nearest to you is rough with cardinalities...a large eagle floats nearby on the mountain drafts, flexing the finger-like tips of his wings as he circles." [Next, on National Geographic Explorer...] "The Way of Unity and the Inward Way have the same goal. Nothing is the same as Everything." [Join our cult. We worship every Tuesday night to Cantor and Aleph.] "It would seem, in particular, that God should be able to form a precise mental image of Himself." [Then why don't you just ask him?] "Infinity and the Mind is a work of transmission. I dedicate it with love and respect to everyone in the channel." [I SO did not make this up.] In short, if you like playing with your weiner or ruminating on how God expresses his spiritual essence in the form of transfinite cardinals, then this book is for you. If you really want to learn more about set theory, however, buy J. Donald Monk's "Introduction to Set Theory" instead.
Rating: 
Summary: A book that may encourage to read more about set/logic
Review: I have read the book in connection with AI. I find the chpater four ( Robots and Souls) is well written. Excursion 2 gives more details about Godel's incompeteness theorem. Chapter one gives the excitment. Chapter three ` the unnameable' was very diffcult for me. A very good book for philosophically minded.
Rating: 
Summary: A perfect book for someone like me
Review: 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.
Rating:
Summary: transcending the finite
Review: I recommend this book when I am engaged in dialogues about infinity on discussion boards. I especially like the presentation of ordinal numbers and the progressive infinitization of the number concept. I also appreciate the introduction to Robinson's Non-standard real and complex numbers and the background this book provides. Conway's surreal numbers are also described, but I find these more a math game than a useful tool. I agree with Rucker's characterization of Cantor's absolute infinity. I think the term 'infinity' should apply to it as a closer on the number concept and not a true number. The term 'transfinite' should be applied to the others, but it is probably too late to change common vocabulary now. The principle that says every specification of infinity however large is reflected in some transfinite number that is not absolutely infinite has application in philosophy.
Rating:
Summary: transcending the finite
Review: I recommend this book when I am engaged in dialogues about infinity on discussion boards. I especially like the presentation of ordinal numbers and the progressive infinitization of the number concept. I also appreciate the introduction to Robinson's Non-standard real and complex numbers and the background this book provides. Conway's surreal numbers are also described, but I find these more a math game than a useful tool. I agree with Rucker's characterization of Cantor's absolute infinity. I think the term 'infinity' should apply to it as a closer on the number concept and not a true number. The term 'transfinite' should be applied to the others, but it is probably too late to change common vocabulary now. The principle that says every specification of infinity however large is reflected in some transfinite number that is not absolutely infinite has application in philosophy.
Rating:
Summary: Rucker's best.
Review: 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.
Rating:
Summary: Rucker's best.
Review: 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.
Rating:
Summary: Infinity made simple and understandable
Review: In many ways, infinity is the most abstract concept of all. Many of the advances in understanding how to manipulate infinities had unpleasant consequences. As the legend goes, the first one to announce that there are infinite non-repeating decimals was rewarded by being drowned. Georg Cantor, the first to prove that there are different levels of infinity, faced extreme criticism and ultimately went mad. Fortunately, Rudy Rucker provides a gentle introduction to this concept, one that can be read by most with the only consequence being enlightenment.
The entire range of infinities (what a phrase!) is covered in this book. From the simplest infinity (omega), to the multi-universe theories of quantum theory. All are put forward in a very readable style, although there are times when one must slow down and read very carefully if one is to understand. Rucker's encounters with Kurt Godel is a welcome contrast with the common depiction that he was a dry, humorless man. It is refreshing to hear that he laughed and had a sense of humor.
Many different test scenarios have been put forward to determine if a computer is indeed intelligent. At this time, I would propose that any machine that can understand the concept of infinity must be considered intelligent. Any human wishing to pass that test need only read this book. It should be required reading in all undergraduate mathematics programs.
Published in Journal of Recreational Mathematics, reprinted with permission.
Rating:
Summary: Infinity made simple and understandable
Review: In many ways, infinity is the most abstract concept of all. Many of the advances in understanding how to manipulate infinities had unpleasant consequences. As the legend goes, the first one to announce that there are infinite non-repeating decimals was rewarded by being drowned. Georg Cantor, the first to prove that there are different levels of infinity, faced extreme criticism and ultimately went mad. Fortunately, Rudy Rucker provides a gentle introduction to this concept, one that can be read by most with the only consequence being enlightenment.
The entire range of infinities (what a phrase!) is covered in this book. From the simplest infinity (omega), to the multi-universe theories of quantum theory. All are put forward in a very readable style, although there are times when one must slow down and read very carefully if one is to understand. Rucker's encounters with Kurt Godel is a welcome contrast with the common depiction that he was a dry, humorless man. It is refreshing to hear that he laughed and had a sense of humor.
Many different test scenarios have been put forward to determine if a computer is indeed intelligent. At this time, I would propose that any machine that can understand the concept of infinity must be considered intelligent. Any human wishing to pass that test need only read this book. It should be required reading in all undergraduate mathematics programs.
Published in Journal of Recreational Mathematics, reprinted with permission.
Rating:
Summary: Ouch! Ouch ! POW! Damn, there goes another brain gasket...
Review: It's not often a first-rate mathematician like Rucker (who is descended from Hegel) takes mysticism as a serious subject. (But then, Godel did, too.) Rucker seems to think mathematics supports the idea of what he calls a "Mindscape," the whole of reality through which our consciousness moves, much like a bubble in the ocean. This is pure mystical Idealism. This is a hard book. The math is beyond me but I understood what he was saying. Rucker obviously is having a good time romping through all of reality. And he does his best to show his readers a good one also.
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