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The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (MIT Press) von [Yanofsky, Noson S.]
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The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (MIT Press) Kindle Edition

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In my view, "Outer Limits" is an extraordinary, and extraordinarily interesting, book. It is a cornucopia of mind-bending ideas.--Raymond S. Nickerson "PsycCRITIQUES "

The scope of the material covered is so wide, and the writing so clear and intuitive, that all readers will learn something new and stimulating.--Thomas Colin "Leonardo Reviews "

In my view, Outer Limits is an extraordinary, and extraordinarily interesting, book. It is a cornucopia of mind-bending ideas.

--Raymond S. Nickerson "PsycCRITIQUES "

The scope of the material covered is so wide, and the writing so clear and intuitive, that all readers will learn something new and stimulating.

--Thomas Colin "Leonardo Reviews "


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.


  • Format: Kindle Edition
  • Dateigröße: 6698 KB
  • Seitenzahl der Print-Ausgabe: 419 Seiten
  • Verlag: The MIT Press; Auflage: Reprint (23. August 2013)
  • Verkauf durch: Amazon Media EU S.à r.l.
  • Sprache: Englisch
  • ASIN: B00F9MI2N6
  • Text-to-Speech (Vorlesemodus): Aktiviert
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  • Word Wise: Aktiviert
  • Verbesserter Schriftsatz: Aktiviert
  • Durchschnittliche Kundenbewertung: 5.0 von 5 Sternen 1 Kundenrezension
  • Amazon Bestseller-Rang: #186.655 Bezahlt in Kindle-Shop (Siehe Top 100 Bezahlt in Kindle-Shop)

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Format: Gebundene Ausgabe Verifizierter Kauf
Dies ist eines der besten Bücher die ich in den letzten Jahren gelesen habe. Wunderbar einfach und klar dargelegt werden die Grenzen der komplexesten Fachgebiet der Wissenschaft wie auch die Philosophie der Wissenschaft selbst aufgezeigt. Es liest sich wie ein Krimi und ist doch einfach "nur" Mathematik und Physik sowie Philosophie.
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Die hilfreichsten Kundenrezensionen auf (beta) 4.1 von 5 Sternen 48 Rezensionen
59 von 61 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Good Book 9. November 2013
Von jerryb - Veröffentlicht auf
Format: Kindle Edition Verifizierter Kauf
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.
40 von 40 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Fascinating exploration of the intersection of science and philosophy 28. Januar 2014
Von R. D Johnson - Veröffentlicht auf
Format: Gebundene Ausgabe Verifizierter Kauf
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.
14 von 16 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen A good overview of the limits of what math and logical reasoning can tell us 26. Februar 2014
Von A. Menon - Veröffentlicht auf
Format: Gebundene Ausgabe Verifizierter Kauf
Much of what science focuses on is reductionism and in particular the goal of reducing the world to the bare minimum of assumptions and postulates that can then be built up into all the complexity we see and feel. The outer limits of reason looks at the limits of what we can know and what we can reduce. It looks at lots of different ideas separately and analyzes lots of different forms of limitations that we must deal with. I'll give a quick overview of the subject matter.

The author starts out by looking at limits to logical consistency in our evolved language. He looks at some repurcussions of paradox's on self referencing sentences and shows how such systems can be looked at symbolically. The author then starts discussing how there are many ideas which do not have a platonic ideal and that their meaning is effectively subjective. The author discusses an eroding artifact that is refurbished and questions at one point is the original not considered the same as the remaining. There is no right answer to such a question. The author discusses Zeno and some of the issues we face when considering infinity. The author ends the chapter by considering logic methods that can accomodate that we humans dont define every object or word precisely before using, in particular systems like fuzzy logic. Where truth values can be indeterminate. The author then moves onto set theory and looks at Cantor's analysis of infinity. Countable infinity is considered as well as uncountable infinity and the ladders above infinity created by power sets. These treatments are self contained in the book though some outside knowledge wouldnt hurt. The author then moves onto the limits of computation in particular polynomial time problems as well as NP problems. The author considers the problems associated with figuring out the travelling salesman problem and then frames other problems relative to one another in terms of difficulty and shows how many are in fact equivalent. The author's goal is to show how some problems which are easy to state currently only have solutions which can be found in times that are longer than the age of the universe and that quantum computation would not be the answer to solving such problems. The author then moves onto basic computation theory and Turing's halting problem. He shows how there can be no program which knows whether a given input program with a given input will compute in finite time. He uses counting arguments from a previous argument to show how this works and as such starts to refer to earlier material. The author then discusses the hierarchy of larger and larger systems for which one cannot solve the halting problem even with the help of divine oracles. The author moves on to the limits of what can know physically. He discusses both chaos and quantum mechanics and discusses how the precision needed to know initial conditions is a fundamental constraint in being able to predict a clockwork universe and discusses how separately quantum mechanics makes particle dynamics fundamentally uncertain as a separate limitation. The author then discusses science and philosophy and how science has philosophical limits imposed on it. He discusses how induction is used as a principal to "prove" our scientific postulates which then form the bedrock of the emergent theory dependent on those postulates. We have no proof that at some point experiments might not repeat themselves as they have in the past. The author discusses the relationship between science and math and is very well measured about why and how there are relationships and what to make of them. He discusses the anthropic principle and its scientific worth. The author then focuses on some more purely mathematical ideas like the real numbers, rational, irrational and within that algebraic and transcendental numbers, he discusses group theory and goes back to computation and then covers some pure logic and arithmetical systems and some of Godel's ideas. The author goes into how mathematical systems cannot deduce whether all statements about a given system can be shown to be true or false and that there are limits to knowledge about any given system irrespective of how many assumptions/axioms are included.

There are a lot of different ideas the author discusses when familiarizing the reader with many of the limits of logic and reason we are fundamentally faced with. The author discusses grammar systems where terms are not all well defined, he discusses the problems associated with continuous space and motion on that space, he discusses chaos and quantum mechanics as well as the difficulty of computability of many easy to state problems- some due to time constraints others more fundamentally by the limits of what we can know. The author covers the limits of logic being able to answer all questions posed within any given mathematical system. A lot of information and thought has gone into this and one can learn a lot from this book. Each topic can be further researched and all are interesting overviews of where our knowledge fundamentally hits ceilings. I enjoyed reading much in this book.
12 von 13 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen An important book for any person who considers themselves educated 11. November 2013
Von Juana Cusi - Veröffentlicht auf
Format: Kindle Edition Verifizierter Kauf
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.
9 von 10 Kunden fanden die folgende Rezension hilfreich
4.0 von 5 Sternen fascinating 14. April 2014
Von Sy - Veröffentlicht auf
Format: Kindle Edition Verifizierter Kauf
While I was familiar with much of the contents of the book, it gave me the opportunity to really appreciate the implications. The book is fairly technical, so familiarity with logic, set theory and computer science is recommended. Nevertheless there were a few points where I got completely lost (diagonal proofs are not my best subject) yet I didn't feel I got any less out of the book even though I couldn't follow those parts.
I have only one quip with this book, which prevented me from giving it 5 stars and it regards the part on Quantum Physics. The author falls for the oldest pitfall about QP when he says that only conscious observation of a quantum phenomenon collapses the wave function. This is incorrect. First of all, we can't possibly physically observe quantum phenomena as they happen at an invisible scale, so no, we do not actually watch them happen. Moreover many particle experiments are automated and happen far from human eyes. Think of the LHC where the particle beams are secure and confined far from human beings, and results are recorded by computer systems and reviewed by human beings often days or weeks after the experiment is run. The term "observation" in QP refers to interaction with another system, and shouldn't be equivocated with the common use of the word. The double slit experiment yields the same results whether the human experimenter is in the room or not at the time it is run. A human being need not be in the same room for the wave function to collapse. Unfortunately this equivocation of the word "observation" has fueled much pseudoscience and religious thinking that this kind of book should not perpetuate.
Aside from this, I very much enjoyed the book and recommend it.
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