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The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us [Hardcover]

Noson S. Yanofsky
4.3 out of 5 stars  See all reviews (3 customer reviews)
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Book Description

20 Sep 2013
Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.

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The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us + Our Mathematical Universe: My Quest for the Ultimate Nature of Reality + Nothing: From absolute zero to cosmic oblivion - amazing insights into nothingness
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Product details

  • Hardcover: 328 pages
  • Publisher: MIT Press (20 Sep 2013)
  • Language: English
  • ISBN-10: 0262019353
  • ISBN-13: 978-0262019354
  • Product Dimensions: 22.9 x 15.5 x 4.1 cm
  • Average Customer Review: 4.3 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: 11,606 in Books (See Top 100 in Books)

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Yanofsky takes on this mindboggling subject with confidence and impressive clarity. He eases the reader into the subject matter, ending each chapter with further readings. His book is a fascinating resource for anyone who seeks a better understanding of the world through the strangeness of its own limitations and a must-read for anyone studying information science. Publishers Weekly, (starred review) Yanofsky provides an entertaining and informative whirlwind trip through limits on reason in language, formal logic, mathematics -- and in science, the culmination of humankind's attempts to reason about the world. The New Scientist

About the Author

Noson S. Yanofsky is Professor in the Department of Computer and Information Science at Brooklyn College and The Graduate Center of the City University of New York. He is a coauthor of Quantum Computing for Computer Scientists.

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Most Helpful Customer Reviews
2 of 2 people found the following review helpful
4.0 out of 5 stars Erudite, stimulating and informative 4 Jan 2014
Format:Hardcover|Verified Purchase
This is an excellent book for anyone with an interest in epistemology, science and mathematics. Yanofsky delivers on his title and then some. Using worked examples, anecdotes and historical analysis, Yanofsky demonstrates that there are and always will be limits to what we can possibly know. What's more, what we don't know will always infinitely outweigh what we do know. Along the way, he discusses classical and quantum physics, determinism and free will, Turing's halting problem, Godel's Incompleteness Theorem, the P-NP problem, the structure of the Universe and the anthropic principle (not an exhaustive list). Neither does Yanofsky avoid philosophical issues like mathematical Platonism and why did the Universe create an intelligence capable of reflecting and understanding its own origins and structure. This is a deep book on many levels, intellectually stimulating, informative and reflective. A combination often attempted but rarely achieved with such aplomb. Easy to read, given its esotericism.
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5 of 6 people found the following review helpful
By Mr R
Format:Hardcover|Verified Purchase
This book, despite some defects, is well worth buying if only for the few magnificent, lucidly written sections, some of which are much better than I've seen elsewhere. For example, the section on Zeno's paradoxes (pp41-49)makes the point that history has sold us short, by having the paradoxes handed down to us by people who wanted to prove him wrong (Zeno's original writings having been lost). Yanofsky does a truly brilliant job of presenting and examining their true, and astounding, import. Another very impressive section is that on quantum entanglement and Bell's inequalities (pp194-201) - all so convincing and elegant. And the geometrical proof of the irrationality of the square root of 2 (pp300-301)is the most concise and magical I've seen.
Unfortunately, there are some irritations and errors. Who, for instance, could possibly be interested in ploughing through the detailed argument (p116) of converting a 31-digit number of seconds to a number of centuries (this given to 15 digit accuracy!)? "Divide by 60...." blah blah blah.
On page 214 we read that the special theory of relativity "deals with the universe without gravity or acceleration"; then on page 226 we read about acceleration in special relativity, which of course is perfectly ok. Indeed, relativity doesn't get very careful treatment: page 221 tells us, on the subject of length contraction, "it must be stressed that it is not the case that the moving space shuttle appears to shrink or seems like it is shrinking. Rather, it shrinks". But no mention is made here of frames of reference. In its rest frame, the shuttle most emphatically does NOT shrink. Further on, 4 dimensional spacetime is attributed to Einstein, with no mention of Minkowski. And on page 225, we read the word 'pressure' where 'force' would be appropriate.
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0 of 1 people found the following review helpful
5.0 out of 5 stars Great 22 Dec 2013
Format:Hardcover|Verified Purchase
Bought this as a christmas present and it looks good and arrived on time. Excellent thank you. My husband will love it.
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Most Helpful Customer Reviews on (beta) 4.0 out of 5 stars  28 reviews
30 of 31 people found the following review helpful
5.0 out of 5 stars Good Book 9 Nov 2013
By jerryb - Published on
Format:Kindle Edition|Verified Purchase
I really enjoyed this book; it made think about subjects I hadn't thought about in years. The bibliography is excellent and the author's comments and examples are often surprising but to the point. I think it could be better organized. Self reference keeps popping up, I wish it were all in one place. The book is organized into areas where reason presumably fails. I wish it were organized by types of failure. Also I think he casts too wide a net. I object to his inclusion of some mathematics examples. For instance the inclusion of the ancients' problems demonstrates as he points out later that their problems were not using the correct tools, not that reason was at fault. Similarly for solving quintic equations. It's as if we were out on a starry night and he said "look up. See those moons of Jupiter." I would say that it's not a fault of science or reason that I can't see them. I don't have the proper tool. Give me a telescope and I'll be able to see them. I have the same basic criticisms of including quantum theory. If there is not a final theory why say that there is a limit of science or reason to understand quantum phenomena? He may acknowledge that there are several theories or maybe a theory that no one has ever thought of which might finally be right. But that doesn't prove that there is a limit to what science might conclude in the future. Overall I think the author's best suit is logic and computing and I think it would have been a better book to have stuck to those subjects.

One shouldn't be dissuaded by my criticisms. Read the book, you'll enjoy it. I wouldn't have gone to the trouble of writing what I did write if I didn't like the book.
8 of 8 people found the following review helpful
5.0 out of 5 stars Fascinating exploration of the intersection of science and philosophy 28 Jan 2014
By R. D Johnson - Published on
Format:Hardcover|Verified Purchase
This is a book that is targeted to those who like to think. Although written for a general audience, it would be helpful to have some background in math and science beyond a few long-ago semesters in high school before diving in. You should have some comfort level with set theory, exponents, real and imaginary numbers, and the basic tenets of calculus and probability. The same is true for general physics, such as polarization, particle spin, and the like. Nothing deep, mind you, as the book isn't expecting you to solve equations or anything of the sort. However, the 'outer limits' involve travel beyond where science is now, so knowing something about where science currently is helps.

As an engineer whose math and science education hasn't totally faded away yet, I found this book fascinating. It explains the huge difference between 'countable infinity' and 'uncountable infinity' (something I had never been taught in school), and how the infinite number of solvable problems are dwarfed by an infinitely greater number of unsolvable ones. It goes over the P-NP and Halting problems in Computer Science with far more clarity than any CS textbook I've ever read. It covers chaos theory, the strange quantum world, and the equally curious world of general relativity and the mysteries therein that science has yet to (and in some cases never can) solve. It will also expose you to the philosophical debate about the curious relationship between math, science and consciousness, without having to plow through a course in philosophy. This book is a wonderful antidote to those (far-too-many) books that present science and math as always settled fact and incontrovertible truth. It shows you why intuition often fails, why the scientific dogma of one era is often debunked by the next, and explains how some knowledge of our universe will always remain forever beyond our grasp simply because we cannot 'step outside' our own self-referential existence. In some ways we're like the inhabitants of 2-D Flatland (another excellent book btw) trying to understand a wider 3-D world.

I do have one complaint with this book. There are copious footnotes in each chapter, some which are simple references, but many others which are additional explanatory material. These are all grouped together in a 'Notes' section in the back of the book. This required me to flip continually back and forth from each chapter to the 'Notes' section to read the additional material. It would have been better to present this material as true footnotes on each page; doing so would have eliminated a lot of tedious page-flipping.
8 of 8 people found the following review helpful
5.0 out of 5 stars An important book for any person who considers themselves educated 11 Nov 2013
By Amazon Customer - Published on
Format:Kindle Edition|Verified Purchase
Excellent explanation of tough concepts of higher math. The chapter on the infinite is particularly good, putting the understanding of quite complex concepts within reach of reasonable educated people. Dr. Yanofsky not only explores the mathematics world, but also the philosophical implications of the limits of math and logic.
6 of 6 people found the following review helpful
5.0 out of 5 stars Incredible compilation of diverse topics related to the limits of knowledge and science in general 24 Dec 2013
By Santiago Herrera - Published on
Format:Hardcover|Verified Purchase
Very well written, highly enjoyable. If you like authors/thinkers as Hofstadter, Penrose, John D. Barrow, Godel, etc., who stand in the limits of knowledge, then you will love this book. It brings together philosophy, mathematics, computation, physics, in one marvelous exposition without breakups or interruptions.
12 of 15 people found the following review helpful
5.0 out of 5 stars Excellent survey 13 Oct 2013
By Rohit Parikh - Published on
Format:Hardcover|Verified Purchase
The book provides an excellent survey - in a very informal manner - of a lot of topics which are "in the news" but a mystery to many.

It runs the gamut from puzzles in set theory, in computation, to the ship of Theseus.

Reading this book could be a mini education!

Something I WOULD like to see in the second edition is a discussion of Arrow's theorem and the Gibbard Satterthwaite theorem. These are paradoxical results about elections which Americans would be well off knowing. Many Americans think that "the more choices the better" but in fact more choices mean more chaos and more chance of something going wrong. The paradoxical No Trade theorem of Milgrom and Stokey based on Aumann's "Agreeing to Disagree" could also stand a treatment.
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