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An exploration of the scientific limits of knowledge that challenges our deep-seated beliefs about our universe, our rationality, and ourselves. 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, ma An exploration of the scientific limits of knowledge that challenges our deep-seated beliefs about our universe, our rationality, and ourselves. 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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An exploration of the scientific limits of knowledge that challenges our deep-seated beliefs about our universe, our rationality, and ourselves. 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, ma An exploration of the scientific limits of knowledge that challenges our deep-seated beliefs about our universe, our rationality, and ourselves. 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.

30 review for The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

  1. 5 out of 5

    Liviu

    excellent drive through the paradoxes and limits of science - thankfully free of useless speculations and very moderate in the authorial bias towards this or that position, I would say that this book is a great example of how natural philosophy books should be written. Very clear presentation - I knew most of the stuff there and read and thought about lots of it across time, but I still immensely enjoyed the book and there was new thing for me too, the Goodestein sequence example (of a more natur excellent drive through the paradoxes and limits of science - thankfully free of useless speculations and very moderate in the authorial bias towards this or that position, I would say that this book is a great example of how natural philosophy books should be written. Very clear presentation - I knew most of the stuff there and read and thought about lots of it across time, but I still immensely enjoyed the book and there was new thing for me too, the Goodestein sequence example (of a more natural Godel proposition - true, not provable in Peano but provable in a system that has some features of Peano but limits induction and adds some infinitary stuff about ordinals) more highlights - Zeno paradoxes clearly explained; NP vs P clearly explained; excellent introduction to naive set theory and a discussion of ZF, ZFC, and as mentioned Peano, Godel etc; not much on uncountable cardinals and the continuum problem but hey this is not advanced set theory so that was fine great stuff about physical limitations and while the pilot wave stuff was a bit rushed with more weight on Copehangen and multiverse (eiether many worlds or more modern stuff from string theory), the hidden variable approach was explained superbly lots of historical vignettes and lots of extra reading in each chapter with some of the best stuff I am aware of mentioned - useful notes too Highly, highly recommended

  2. 4 out of 5

    Muhammad Arqum

    What a book! What a gem of a book! Initially I was a tad disappointed because I thought this would be a debate on metaphysics and most probably on existence of God or case of religion and that was the reason why I picked it up in the first place. It did not start that way, it did not even give an impression that it would eventually lead to that. The first few chapters introduce what logic is, how can we define reason and so fourth. It had plenty of problems that I had studied in my Programming a What a book! What a gem of a book! Initially I was a tad disappointed because I thought this would be a debate on metaphysics and most probably on existence of God or case of religion and that was the reason why I picked it up in the first place. It did not start that way, it did not even give an impression that it would eventually lead to that. The first few chapters introduce what logic is, how can we define reason and so fourth. It had plenty of problems that I had studied in my Programming and Algorithms classes, but it was fun nevertheless to revisit those tricky problems. Then the book got interesting as I entered into the realm of the unknowable! Problems that computation can't solve. Problems that mathematics can't solve. And then finally I saw what the author was trying to do, slowly unfolding areas and then building upon them, started to prove how stupid and helpless we can be at times. Eventually you do reach a point where you can contemplate on your own, about those metaphysical questions that I mentioned in the beginning. The author does not explicitly make those questions as his topic, however it makes you do the assessment in your mind. Plenty of mind numbing paradoxes, theorems, quantum superpositioning, philosophy, metaphysics and a whole world of amazing thought provoking ideas combined in one book. Amazing, interesting as hell read. Highly recommended resource for the knowledge hungry. *Goes back to understanding Godel's theorem*

  3. 5 out of 5

    Emre Sevinç

    Back in the day, sometime in 1930, David Hilbert, a very smart mathematician said "Wir müssen wissen. Wir werden wissen" (We must know. We will know. (It sounds way more authoritative in German, try it.)) to which another smart guy, Mr. Gödel had already replied: "Hold my beer..." The rest is history. And what a history it's been. Among the stellar achievements of humankind are the ones that sound like "oops, seems like there's no way we can solve this, here's the proof. Sorry." Because, we wouldn' Back in the day, sometime in 1930, David Hilbert, a very smart mathematician said "Wir müssen wissen. Wir werden wissen" (We must know. We will know. (It sounds way more authoritative in German, try it.)) to which another smart guy, Mr. Gödel had already replied: "Hold my beer..." The rest is history. And what a history it's been. Among the stellar achievements of humankind are the ones that sound like "oops, seems like there's no way we can solve this, here's the proof. Sorry." Because, we wouldn't be considered that smart if we didn't know the limits of our methods, right? A man's got to know his limitations, as Dirty Harry said once. This book, in a sense, is a very brief introduction into the wonderful world of philosophical, mathematical, and physical limitations, as well as all sorts of paradoxes you'll find if you force your way into some realms. The limitations are not there because we're not smart enough or for the lack of computers that are sophisticated enough; on the contrary, the book's topic is the limitations that are fundamental, that arise from the logic, from the foundations of mathematics, as well as physics. Those limitations are described mathematically, and proved. They constitute a domain into which you can look, and ponder about the mysteries of universe, and the whole human reason. (The author tried to do that, without falling into much mysticism, though one could be forgiven to do just that when the issues are so fundamental and proved to be beyond reason). I think this book can serve as a nice introduction for undergraduate students that are just getting started with logic, mathematics, physics, philosophy of science, computer science, and linguistics. The discussion and presentation in this book serve as a good starting point for many fresh minds, as well as a warning to the practitioners in those fields: E.g. if you're a programmer, and someone says something can't happen because, you know, Halting Problem, you won't try to object (that is, if the claim can really be reduced to the Halting Problem). And also when you're faced with a problem that can be reduced to a Traveling Salesman Problem, you won't throw in the towel, but will try to find some approximate algorithm, knowing full well that the perfect solution will stay out of your reach. In other words, this book is in the same category as What We Cannot Know, and both of the volumes give nice introductions. I found this book more fluid, and I've especially liked the notes at the end because all of them had very nice pointers to more detailed books and articles. If you're curious about the impossibility results in mathematics, logic, physics, linguistics, and computer science, and you haven't read much about these before, then I can easily recommend this book. If on the other hand you had basic mathematics, engineering, physics, logic, computer science, linguistics, and philosophy of science classes, then probably you'll not find anything new in this volume. Still it can act as a concise refresher, and you might even learn a thing or two such as Yablo's paradox.

  4. 5 out of 5

    يوسف بوحايك

    Oh I finished this awesome book ! The first time I read the book’s title, I thought it's gonna be a boring deep philosophical book fulling of complex philosophers’ discussions, but instead I found a simple book discussing the different limitations and contradictions that our thinking might lead us to, the book raised these limitations by presenting several paradoxes and its explication using proof by contradiction, to clear its impossibility as a limitation to our logic. In other chapters, the boo Oh I finished this awesome book ! The first time I read the book’s title, I thought it's gonna be a boring deep philosophical book fulling of complex philosophers’ discussions, but instead I found a simple book discussing the different limitations and contradictions that our thinking might lead us to, the book raised these limitations by presenting several paradoxes and its explication using proof by contradiction, to clear its impossibility as a limitation to our logic. In other chapters, the book presented the limitation of reason in scientific fields, presenting the Quantum mechanics theory and its chocking principle of Superposition, and also the Einstein’s Special and General Relativity and its challenges to our intuition (Distance relativity), the author summarized that by: “Bad intuitions are not feared. Only contradictions are feared”. In the last part, the author tried to discuss the relation between Reason, Mathematics and Science, and summarized the different philosophers’ interpretations for the coherence between mathematics and reality. This book is simply a pleasure, and one of the books that make your mind clear and well-organized.

  5. 4 out of 5

    Chris

    From its suggestive title, many might expect this to be a book (by a scientist, nonetheless) arguing for the relevancy of religious faith as an explanation for reality when science itself is unable to do so. Well, sorry to disappoint, but the book, thankfully, has nothing to do with religion vs. science. Rather, it's about the structure of reason itself and is an intellectual tour of paradoxes, limits to observation in physics, computing impossibilities, and the incompleteness of mathematical lo From its suggestive title, many might expect this to be a book (by a scientist, nonetheless) arguing for the relevancy of religious faith as an explanation for reality when science itself is unable to do so. Well, sorry to disappoint, but the book, thankfully, has nothing to do with religion vs. science. Rather, it's about the structure of reason itself and is an intellectual tour of paradoxes, limits to observation in physics, computing impossibilities, and the incompleteness of mathematical logic. All of this material is common in college-level courses of one type or another—for example, the NP-P limit is covered in all computer science curricula. In fact, this book is essentially a "lightweight" textbook: not good enough for a class, but perfect as an overview of the whole topic of "limits of reason." Yanofsky includes plenty of suggested reading for more thorough material. Four-star instead of five only because there is no new thought presented.

  6. 5 out of 5

    Bjørnar Tuftin

    Who would I recommend this book too ... On the one hand it would be someone who knew less about the many topics in this book, as I found a lot of it tedious and overly detailed. But at the same time some of it went over my head. I definitely know the second to last chapter did. It was like revisiting a class I almost failed in university, and it was still incomprehensible. So perhaps someone who doesn't mind only enjoying part of a book? Then again I also have issues with some of the contents, wh Who would I recommend this book too ... On the one hand it would be someone who knew less about the many topics in this book, as I found a lot of it tedious and overly detailed. But at the same time some of it went over my head. I definitely know the second to last chapter did. It was like revisiting a class I almost failed in university, and it was still incomprehensible. So perhaps someone who doesn't mind only enjoying part of a book? Then again I also have issues with some of the contents, which makes me doubt the author's ability elsewhere. He feeds the uninformed reader with a tiny spoon when it comes to subjects where that is possible, such as the basic logic and math "puzzles" in the early chapters, subjects that he's also apparently an expert on, but only skims the surface on the really tricky bits like quantum physics, relativity and metaphysics where he seems to be a very enthusiastic and well read dabbler, but a dabbler nevertheless. Now maybe I'm wrong, I'm definitely not as well-read, but here are the major flaws that turned me off recommending this book: 1. Quantum physics requires a conscious observer. This is definitely not the agreed-upon truth that the author presents it as, and as he goes into detail about the basics interpretations of QP he could at the very least have spent some time presenting conflicting views here. I see this as either ignorant or misleading. 2. The symmetry of special relativity effects is not explicitly mentioned. Maybe I'm nitpicking here, but with the detail he goes into regarding relativity I find it a serious issue that "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." isn't accompanied by mention of the symmetry of the situation. The lack of preferred reference frames is sort of explained further on on the page, but I think it's a grave mistake to not mention that the observer on the space shuttle will see the so called stationary observer as moving-shrinking-time dilated. 3. Occam's razor is only about simplicity Yanofsky gives relativity vs. Newton and similar developments as examples where Occam's razor fails. This is a nonsense definition of the razor. Unless the criterion for the competing hypotheses is that they have identical explanatory powers is included, Occam's razor becomes just a question of esthetics, which is not how the book presents it. Considering the space the book devotes to Occam's razor this seems a major flaw. 4. The weak anthropic principle is mangled. Yanofsky writes "The weak anthropic principle says that the observed universe must be of a form that would permit the existence of intelligent human observers. In other words, not all universes are possible." and thereby does a terrible job at distinguishing the weak anthropic principle from the strong anthropic principle. It doesn't get better when right after explaining the strong AP comes a paragraph where "the anthropic principle", no qualifier, is said to restore intelligent humans beings (and any other intelligent beings in the universe) to the spot in the center of the universe Copernicus, Darwin and Freud displaced us from. This seems bizarre to me who sees the weak AP as the complete opposite. It shows us we can't even take our existence in the universe as special, since our observation of the universe can only tell us that the universe is suitable for us to at some point observe it. Now if you're still reading this review, do pick up and browse this book. If you find parts of it interesting, read them. If it's the math and information theory stuff, you might learn a lot or at least a little, but if it's the more difficult physics and metaphysics stuff I recommend reading some more competent authors before sharing what you've learned.

  7. 5 out of 5

    Kaśyap

    An interesting overview of the practical and theoretical limits of logic, mathematics and science. Deals with the limits of knowability, determinism, computation and predictability in science and mathematics. This is a clear and concise study of various topics from classical logic to modern physics, but lacks a quest for deeper understanding.

  8. 4 out of 5

    Vasil Kolev

    This was horrible. Even the things I knew were explained so bad that I was unable to understand them, and as a whole the book tries to dazzle idiots instead of explaining the (actually pretty interesting) problems.

  9. 5 out of 5

    Harold Struik

    Brilliant, in depth but not exhausting.

  10. 5 out of 5

    Alexandra!

    And here I thought quantum computing would be kind of a revolution...

  11. 5 out of 5

    Nv

    This is such a thought provoking and beautiful book, it’s really made me wonder why there are so few poetic evocative works of popular science to combat the one reason religion always beats rational thought, the romance and the emotional content. Maybe I just haven’t read enough. Alan Lightman and Carl Sagan come to mind, and sure there are beautiful passages in many of the really serious books, but the popular science overviews made for filthy casuals like me are usually so prosaic. Passages of This is such a thought provoking and beautiful book, it’s really made me wonder why there are so few poetic evocative works of popular science to combat the one reason religion always beats rational thought, the romance and the emotional content. Maybe I just haven’t read enough. Alan Lightman and Carl Sagan come to mind, and sure there are beautiful passages in many of the really serious books, but the popular science overviews made for filthy casuals like me are usually so prosaic. Passages of beauty, mystery and wonder seem tailor-made for the most abstract of our knowledge, and yet they’re so few and far between that their presence only calls attention to their scarcity. No surprise then that the battle for hearts and minds is being won spectacularly by religion despite bringing a pin to a gunfight. A pin with infinite angels should still be no match for the weapons of profundity at the disposal of science. All this from a book about what science doesn't know. Speaking of what it doesn't know, I picked this up after watching the Vsauce video on Banach-Tarski which blew my mind. I don’t know what to make of the fact that there is nothing about B-T in this book. Neither is there anything in the other book he recommended in the video, Why Beliefs Matter. I guess those are the limits of the video’s reason too. Notes Chessboard: place 2-square dominos. If you put the Q on diametrically opposite squares it is impossible to place the dominos, because both squares are same color, now there are 32 B and 30 W, whereas a 2-square domino will need to be on 1 B 1 W square. If you make assumptions that leads to a paradox, you need to change assumptions. That’s the message a paradox sends. Linguistic paradoxes (liar’s paradox, heterological (french is not french, monosyllabic is not monosyllabic) is heterological or not) explained away saying language full of inconsistencies. Real-world paradoxes like barber paradox explained away saying doesn’t exist in reality. But what about Russell’s paradox, it’s a mathematical construct which follows axiomatic rules, can’t be explained away. But math is also a language? The problem of self-reference. Yablo’s paradox of saying everything k+1 is false. No self-reference. All-encompassing truths invalidate themselves. Democratic party in 1790s supporting state rights over federal. Opposite now. What if you preserve planks of Theseus ship. Real number line with series of points zero thickness.. field generated by each point created thickness? Xenos paradox: time is discrete. But stadium paradox, time is not discrete (relativity). If time travel is a paradox, what's the assumption that's wrong. The universe itself will not allow contradictions? How many grains is a heap. 1.2.n. epistemic vagueness: a binary exists that we don't know. Ontologic vagueness: there is no clear boundary. Fuzzy logic. Paraconsistent logics, deriving meaningful statements about vague terms Monty hall 25 host variant. He's giving you Information. Continuum hypothesis: Hilbert, no set exists that is more than set of natural numbers but less than irrational numbers (0,1). Platonists say we haven't found it. Formalists/nominalists say it's irrelevant whether it exists in reality, we are free to say set theory A assumes it's true, and set theory B assumes it's false, and then proceed with both till we get a contradiction How to go from brute force problem solving to algorithmic thinking. Need to solve problems and reflect on the solutions. Reduce one np to another. Solve one. Solve the other. Np complete is one that any np can reduce to. Solve it, solve all. Traveling salesman. Approximation algorithms, heuristics. You will never know how I feel when I solve a problem. Alan Turing proved no program would decide the halting problem: whether input X on program y would halt. Liars paradox, a program that halts only if it doesn't halt. Mandelbrot? Geometric intuition? Harder still: check which inputs will make it halt. Set of programs that always result in zero. Rice theorem. No interesting property of a program can be determined by a program Goldbach conjecture. Nxn matrix of products, unroll into linear number line. Gaps are primes. Distance keeps growing. Cantor dust. Matter is a combination of actual matter (nucleus electrons) + space between them. More than just matter. Is mind the combination of brain waves + space between them, more than brain? Electric field? Amplitude of electric signals creates different fields, not just about the pathway? Interaction of fields forms a different variable altogether. Shainberg memories of amnesia Chaos in epidemiology: single individual can have large scale impact Turing found chaos in morphogenesis of zygote which takes same DNA and differentiates, based on signal of position within the organism as well as in relation with outside. N body problem is deterministic but chaotic and unsolvable. Lunar month is off by+-15hrs. 1) sensitive dependence of initial conditions. 2) large number of components. Statistical mechanics. Crocodile dilemma. I'll return child if u guess what I'll do. You'll keep it. Uncertainty is more like noncommutativity: measuring X then y is different set of values than y then X. Is double slit only weird because our Aristotelian language of logic isn't equipped? Distributive law doesn't hold in quantum logic. Interference X and (top b or bottom c) not equal to X and b or X and c. Train lightning thought experiment destroys simultaneity. Therefore destroys causality too. All is frame of reference. Ravens paradox. All ravens are black. Non black objects are nonravens. Paradox: green sweater provides confirmatory evidence for both. Resolution: each evidence is weighted, so if million ravens, each black raven provides 1/1000000. But infinitely many black/nonblack objects so green sweater, while evidence, is close to 0. What a nice way to argue against reasoning that feels wrong. Assign everything a weight. Simplicity of hypotheses works against simplicity of ontology. Multiverse. Popper was amazed that a single falsificagion of Newton by Eddington was enough for science to discard Newton. Contrast with politics, religion, morality Math revealing reality. Herschel Neptune. Dirac positron. Wigner saying math unreasonable effectiveness of math in natural sciences. Gelfand: unreasonable ineffectiveness of math in biology. Latitude longitude are parallel grid in 2d map. Think of empty space in same way. Now to convert to globe, need to make ends meet. Think of spacetime curvature that way Natural selection of universes. Universes with more black holes leave more copies. Bigger stars have black holes. Evolve towards the perfect universe with conditions for intelligent life. Max tegmark. All mathematical structure that is coherent exist and describe some universe. Participatory anthropic principle. Superposition of all possible universes collapsed by the one that developed consciousness. John wheeler. Postulating a multiverse is just delaying the question. What explains the superlaws or metalaws that governs the multiverse. What differentiates belief in a multiverse from God. Both are unprovable untestable unobservable Symmetry used by Einstein: law should look the same even if we flip perspective. Any conservation law has symmetry and vice versa. Momentum: symmetry of place. Angular momentum: orientation. Energy: time (experiment done at t=1 Vs t+1) Hippasus killed by Pythagoreans for pointing out rt2 is irrational, against their pure view of the world Beautiful proof for sqrt2 as irrational. Remember? Missed out fact of godel: unprovable within that system but procwgle within the larger system. For instance arithmetic by zermelo Frankel set theory. Are there true statements unprovable in any system? Ptolemy’s geocentric model actually worked better than Copernicus’ heliocentric, until Kepler used Apollonius’ conic sections to model orbits as elliptical not circular. Euclid’s 10 theorems were all obvious. One stood out. No.5 that talked about parallel lines that would never intersect infinitely. He was never comfortable with this. Noone could prove it. Gauss shows up and says it is both true and untrue. True = Euclidian geometry. False = Non-euclidian geometry. Physicists ignored it until Einstein used this to formulate relativity. Cardano proposed sqrt(-1) as i. Mathematicians developed rules of complex numbers. Physicists ignored it until superposition perfectly described by them Hamilton extended i to ijk, quaternions. Found multiplication is noncommutative. Physicists ignored until found Heisenberg uncertainty perfectly described by noncommutative xy <> yx Group theory to find roots of polynomials. Eventually used to describe subatomic particles.

  12. 5 out of 5

    Sam Eccleston

    This is a pretty interesting read- it covers a lot of information while managing to keep the discussion concise and to the point. The author tries, and mostly manages, to be fair to the different positions he outlines, even those he disagrees with, and is refreshingly honest about the difficulty of finding certain answers to difficult questions. Having said this, he sometimes does not express positions he disagrees with in quite the detail or forcefulness that they deserve; on occasion he skirts This is a pretty interesting read- it covers a lot of information while managing to keep the discussion concise and to the point. The author tries, and mostly manages, to be fair to the different positions he outlines, even those he disagrees with, and is refreshingly honest about the difficulty of finding certain answers to difficult questions. Having said this, he sometimes does not express positions he disagrees with in quite the detail or forcefulness that they deserve; on occasion he skirts around arguments which are major strengths of, for example, Platonism, or another position he finds unconvincing. He also has a somewhat confusing tendency of changing the perspective from which he is writing without indicating that he has done so. Thus, he ends up treating the ontologies implied by both quantum mechanics and relativity as factual, even though they are contradictory. He does occasionally acknowledge this, but does not really do justice to the mystery here; both theories are able to give extremely accurate predictions about the world, but they imply very different things about what it is like. Additionally, like many authors who are also professional scientists, he often explains problems which require quite precise description in a rather offhand way which renders the meaning of what he is saying somewhat ambiguous, particularly to those who do not have any formal scientific or mathematical training. Throughout the book, he insists that there cannot be a contradiction in reality, but oddly, he does not explore whether there actually are any real contradictions; for example could super-position be described as a contradiction? Is it not contradictory to say that an object is both in place a and place b at the same time and in the same sense (i.e. spatially) when it does not continually occupy the space between those places? In summary, while much of the content of this work is very interesting, it somehow manages to miss out on what could have been even more fascinating discussions about the already fascinating subject matter.

  13. 5 out of 5

    Doctor Moss

    I was hoping for two major discussions in Yanofsky’s book. First, a survey of paradoxes and other conundrums, frustrations, etc. having to do with the limits of “reason” as a tool for understanding the world. And then a probably very speculative analysis to find themes and maybe some theoretical conjectures about how we might tie together and understand those limits. We get much more of the first than the second. Yanofsky takes us through a fascinating survey of paradoxes and other types of limit I was hoping for two major discussions in Yanofsky’s book. First, a survey of paradoxes and other conundrums, frustrations, etc. having to do with the limits of “reason” as a tool for understanding the world. And then a probably very speculative analysis to find themes and maybe some theoretical conjectures about how we might tie together and understand those limits. We get much more of the first than the second. Yanofsky takes us through a fascinating survey of paradoxes and other types of limitations. He starts with the simple liar’s paradox — “I am lying”. The statement is true if false and false if true. The liar’s paradox is one example of problems we run into with self-reference, when we speak about speaking, calculate about calculating, compute about computing, . . . Yanofsky’s survey is organized into chapters on language, philosophy, infinity, computing, science, metascience, and math. As he takes us through those different domains, it’s interesting to try to find your own common themes cutting across them. You can categorize and reflect on them in many different ways, such as: - There are paradoxes, like the liar’s paradox, which is false if true and true if false (i.e., implies a contradiction) - Limits to knowledge, like deterministic but unpredictable phenomena, like the three body problem in physics - Things that just aren’t the way we normally think about them, like quantum mechanical reality — superpositions and entanglement — or, a very different example, the Monty Hall problem - Limits to the feasibility of calculation or computability, like the traveling salesman problem - Limits to calculability itself, like the Halting Problem - Vagueness that defies reduction to precision, such as the sorites or heap paradox - And there is reason’s reliance on apparently unreasonable principles, such as the problem of induction - On the positive side, there are uncanny successes of reason, such as the reliability of induction, the success of mathematics as a language in which to describe physical reality, and the precise but seemingly fragile suitability of the universe for the evolution of intelligent, reasoning creatures in the first place Yanofsky offers his own categorization of all of these paradoxes and problems in his final chapter, Beyond Reason: - Physical Limitations, like time travel (which I’m sure some readers would want to dispute, whether successfully or not) - Mental-Construct Limitations, like Zeno’s paradoxes - Practical Limitations, like the traveling salesman problem - Limitations of Intuition, like quantum indeterminacy or entanglement Don’t worry if you don’t know what these problems and paradoxes are. Yanofsky provides short, usually very clear, explanations of each. The survey itself is entertaining and edifying, and it’s probably the best part of the book. What I don’t think Yanofsky really does is tie all of this into a theoretical statement about the limitations of reason. He gives something of a prescription for how to stay safely within the bounds of reason, by not following reason down a path of implication toward contradictions or “false facts.” And, also in that final chapter, he reminds us that reasoning, or science (since much of what he means by “reason” is bound to science), isn’t the only way we have of relating to the world and of coping with its mysteries. He excludes art, morality, religion, and others from the discussion of the bounds of reason. It’s a little beside the point of Yanofsky’s book, but one remark about the place of morality (and art) vis a vis reason left me jaw-dropped. In fact his blithe treatments of philosophical problems like nominalism, realism, and naive realism surprised me, given that Yanofsky actually seems well-read in the history of those problems. But here's the zinger, where he distinguishes problems like the Halting Problem in computability as something having to do with objective features of reality, as opposed to “some subjective, wishy-washy idea like artistic taste or morality.” Morality is “subjective” and “wishy-washy?” Like I said, that’s kind of beside the point of the book, but I was so gobsmacked by the remark that I couldn’t let it go. Back to the core concern of the book. I can’t shake the feeling that by focusing with Yanofsky on what we might call “formal” reasoning, we are missing something in a more mundane sense of “reason” and “reasoning.” For example, in an everyday use of “reasons,” you may or may not have reasons for what you do or what you say or what you believe. That’s not quite the same sense of “reasons” as Yanofsky’s more formal sense, as employed in scientific, logical, philosophical, or mathematical reasoning. And in that ordinary sense, we aren’t especially surprised, much less dismayed, that our reasoning isn’t perfect or that it doesn’t lead to optimal outcomes. Of course, reasoning isn’t perfect. We may have reasons for what we do, say, or believe, but we’re often wrong. That’s life. We are only human. That perspective, as distinct from Yanofsky’s focus on more formal reasoning, seems to accord with a remark by David Hume, quoted by Yanofsky — “What peculiar privilege has this little agitation of the brain which we call thought, that we must thus make it the model of the whole universe?” Yanofsky’s examples tend to gather around theoretical contexts — physics, mathematics, logic, . . . Nothing wrong with that, but a more rounded diet of what we call “reasoning” would, I think, include examples in which we would have more of the “Well of course” reaction to the limitations of reason than a reaction of surprise. In fact, it may be worth thinking about whether or not the more formal senses of “reasoning” aren’t an extension of the more mundane senses , but now with unrealistic expectations. It’s a good book, and I hope i’ve demonstrated that it is thought-provoking. I’d also recommend a couple of other books on themes I’ve touched on a little. One is Dan Ariely’s The Upside of Irrationality, which concerns the more practical sense of “reason.” His focus is on systematically irrational decision-making, especially in consumer behavior. His discussion there is entertaining and demonstrates how our decision-making sometimes has only the appearance and not the substance of rationality. I also recommend something very different — Paul Feyerabend’s The Tyranny of Science — on that final point Yanofsky only briefly touches, that science is just one way of relating to and making sense of the world. Feyerabend is a notorious opponent of science as the one and only, or the superior way of understanding reality. And that book is particularly focused on opening minds to both the limitations of scientific reasoning and the alternatives that often compete in its shadow.

  14. 4 out of 5

    Janice

    I wanted some unique insight into reason but did not find it. I was happy that Douglas Adams and Yogi Berra were each quoted once, though. The one insightful take-away was from the Norwegian Boy Scout Handbook: "If the terrain differs from the map, believe the terrain." Are you listening GPS users!!! I wanted some unique insight into reason but did not find it. I was happy that Douglas Adams and Yogi Berra were each quoted once, though. The one insightful take-away was from the Norwegian Boy Scout Handbook: "If the terrain differs from the map, believe the terrain." Are you listening GPS users!!!

  15. 4 out of 5

    Zac

    Quite interesting, covers a pretty broad range of topics in terms of their limitations. The book is largely equation free, but I'd recommend learning a bit about logic and its notation if you want to get the most out of it. Quite interesting, covers a pretty broad range of topics in terms of their limitations. The book is largely equation free, but I'd recommend learning a bit about logic and its notation if you want to get the most out of it.

  16. 5 out of 5

    Tim Sharp

    A grand tour of various fallacies, logic problems, linguistic contradictions and mathematical/philosophical issues with reason. Very tied down to rationality, as one would expect. This wasn't quite the grandslam against reason I expected, but a generally entertaining read nonetheless. A grand tour of various fallacies, logic problems, linguistic contradictions and mathematical/philosophical issues with reason. Very tied down to rationality, as one would expect. This wasn't quite the grandslam against reason I expected, but a generally entertaining read nonetheless.

  17. 4 out of 5

    Mills College Library

    001.01 Y24 2013

  18. 4 out of 5

    Giacomo Mantani

    The Outer Limits of Reason is as far as I remember one of the best research essay that I have read so far. I definitely recommend the book to all inquiring minds around and to anyone who is too much confident about him/her-self knowledge. Noson S. Yanofsky confirmed his knowledge page by page and with astonishing clarity explain hard topic such as Chaos, Relativity Theory and Quantum Mechanics. As a graduate student of Computer Science and Engineering I have found really well written explanations The Outer Limits of Reason is as far as I remember one of the best research essay that I have read so far. I definitely recommend the book to all inquiring minds around and to anyone who is too much confident about him/her-self knowledge. Noson S. Yanofsky confirmed his knowledge page by page and with astonishing clarity explain hard topic such as Chaos, Relativity Theory and Quantum Mechanics. As a graduate student of Computer Science and Engineering I have found really well written explanations about theoretical computer science. Not for nothing 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. Awards Winner, 2013 American Publishers Award for Professional and Scholarly Excellence (PROSE Award) in Popular Science & Popular Mathematics, presented by the Professional and Scholarly Publishing Division of the Association of American Publishers Quotes Science is a human activity. It is created by finite, flawed human beings attempting to search for the ultimate truth. The mathematics becomes abstract and about nothing in particular. Because these concepts are about nothing, they are about everything. Rather than asking why the laws of physics follow mathematics, ask why there are any laws at all. When we talk about the limits of scientific reasoning, we must keep in mind how we are observing the universe…the way we look at the universe is the way it will present itself to us. Do not mistake the metaphor for reality. SPOILER: Yanofsky conclusion thought We human beings already live beyond reason. Real life has importance only when it includes ethics, values, and beauty. Reason is a powerful but nevertheless limited tool. Table Of Contents Introduction 1 Language Pradoxes 15 Liar! Liar! 15 Self-Referential Paradoxes 19 Naming Numbers 26 Philosophical Conundrums 31 Ships, People, and Other Objects 31 Hangin’ with Zeno and Gödel 41 Bald Men, Heaps, and Vagueness 50 Knowing about Knowing 57 Infinity puzzles 65 Sets and Sizes 66 Infinite Sets 69 Anything Larger? 76 Knowable and Unknowable 85 Computing Complexities 97 Some Easy Problems 98 Some Hard Problems 109 They’re All Connected 121 Almost Solving Hard Problems 129 Even Harder Problems 131 Computing Impossibilities 135 Algorithms, Computers, Machines, and Programs 136 To Halt or Not to Halt? 139 They’re All Connected 146 A Hierarchy of the Unknown 152 Minds, Brains, and Computers 157 Scientific Limitations 161 Chaos and Cosmos 161 Quantum Mechanics 175 Relativity Theory 214 Metascientific Perplexities 235 Philosophical Limitations of Science 235 Science and Methamatics 252 The Origin of Reason 272 Mathematical Obstructions 297 Classical limits 298 Galois Theory 304 Harder Than Halting 309 Logic 320 Axioms and Independence 331 Beyond Reason 339 Summing Up 339 Defining Reason 345 Peering Beyond 349 Notes 355 Bibliography 379 Index 393

  19. 4 out of 5

    Horatio

    Very interesting read, although it was quite the effort (a lot of re-reading of explanations and passages) to even get a basic grasp of a lot of the concepts that were being discussed (that being said, the models and diagrams really helped). My biggest gripe with the book is the interpretation of quantum mechanics that the author describes in a few passages. I'm an absolute layman, but as Mr. Yanofsky stated that consciousness causes wave function collapse (during the exploration of the Schröding Very interesting read, although it was quite the effort (a lot of re-reading of explanations and passages) to even get a basic grasp of a lot of the concepts that were being discussed (that being said, the models and diagrams really helped). My biggest gripe with the book is the interpretation of quantum mechanics that the author describes in a few passages. I'm an absolute layman, but as Mr. Yanofsky stated that consciousness causes wave function collapse (during the exploration of the Schrödinger's Cat thought experiment) I couldn't help being in disbelief. There's no explanation for why that has to be, and that's because it's just an interpretation among many others (the "Von Neumann-Wigner interpretation" to be exact), and I personally find it a large leap of faith to think that humans have some sort of resolving effect on the elementary particles of the universe (and where does consciousness even begin? Shouldn't the cat collapse the superposition by observing the poison vial breaking or not breaking? Or do humans change everything by virtue of our brains being more complex?). Isn't it far easier and more along the lines of Ockham's razor to assume that the measurement apparatus collapses the wave function (measurement at that scale is bound to influence the particle in question), or that the state of superposition for some reason cannot be transferred to macro-objects (i.e. a wire which conducts electricity for the mechanism that breaks the poison vial)?

  20. 5 out of 5

    Stewart

    What can we know and what can we not know? This is the subject of Noson S. Yanofsky’s thought-provoking book “The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us.” The branch of philosophy that deals with what we can know is called epistemology. Although this area of thought may seem far removed from our everyday lives and hopelessly intellectual, it is, in fact, of fundamental importance to understanding what it means to be a human being. This 2013 book enumerates, What can we know and what can we not know? This is the subject of Noson S. Yanofsky’s thought-provoking book “The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us.” The branch of philosophy that deals with what we can know is called epistemology. Although this area of thought may seem far removed from our everyday lives and hopelessly intellectual, it is, in fact, of fundamental importance to understanding what it means to be a human being. This 2013 book enumerates, in subjects such as language, mathematics, science, philosophy, and computing, the limits of our knowledge. Out of these 10 chapters, a few of the author’s insights especially intrigued me. Yanofsky, a professor of computer and information science at Brooklyn College, spends several pages on Einstein’s two theories of relativity and how they have changed the way humans look at the universe and understand the basic concepts of space and time. Einstein’s famous theories, in opposition to Newtonian physics, state that absolute space and time do not exist. Yanofsky has a great way of making these theories relate to our everyday life. “Although it seems like the Earth is not moving, it is, in fact, constantly moving in a wild pattern,” Yanofsky writes. “Remember that the Earth is spinning on its axis at about 1,000 miles per hour. It is rotating around the sun at about 67,000 miles per hour. Furthermore, our solar system is moving around our galaxy at about half a million miles per hour. Poke your finger into the air. Wait a second. Now poke your finger ‘in the same place.’ Realize that the two places where you poked your finger are hundreds, if not thousands, of miles apart. A stationary observer on Earth is far from stationary. There are no absolute observers, no absolute measurements, and no absolute space and time. All is relative.” The book examines induction, humans’ using small amounts of information to create generalizations, something we do every day of our lives. We turn on the shower expecting, because of past experience, that water will come out. We expect the sun to appear every morning. We assume many, many things in life based on limited information, sometimes amazing minute amounts. And we are right to do so; human life would be almost impossible if we were not able to discover patterns in our lives from small anecdotal evidence. Usually these inferences are right, but they can sometimes be wrong. Yanofsky writes, “Each action has in the past caused a particular effect that we assume is going to happen in the future. [philosopher David] Hume says that a person who is using induction is making an assumption that the universe is somehow uniform over time. There is no reason to believe this assumption.” Science also depends on induction. “The problem of induction is at the very core of science. Scientific laws are formulated by looking at phenomena and generalizing them to universal rules we call laws of nature. There are, however, no real reasons why we have the right to come up with such generalizations.” “Not only science, but our entire worldview is built from induction. We observe phenomena and formulate theories about the true nature of the world. Every time we close the refrigerator door, we are sure the light goes off, even though we do not see it go off. As [theoretical physicist John A.] Wheeler wrote, ‘What we call “reality” … consists of an elaborate papier-mache construction of imagination and theory fitted in between a few iron posts of observation.” There is a certain arrogance in our using the word “universe” and “universal” about our existence on that speck of dust on the edge of our galaxy called Earth. Not only is the pretentious name of the Miss Universe contest laughable, but so are the “universal” laws of science, which may not be applicable in other parts of our known universe, for instance in those violent destroyers of space-time called black holes. The book mentions Karl Popper and his idea that if a scientific law has the possibility of being shown to be incorrect but is not found to be incorrect, he would call that scientific knowledge. “All swans are white” is a statement that could be proven to be false. However, if all observations of swans show them to be white, then you have a scientific law. Yet, obviously just one new observation could in the future prove any scientific law or everyday generalization to be false, for instance the observation of a black swan. “For Popper, all scientific knowledge is provisional and not absolute.” I would say all knowledge is provisional, even this sentence. A crucial limitation to what we can know is that we do not see the universe “as it is.” Rather, we see a “mediated reality” created by our senses and our brain’s complex processes. Philosophy, in the person of Immanuel Kant, discovered this basic insight long before science. “For Kant, our view of the universe is influenced by our own mind and we cannot observe what is really out there ‘in itself’ without these built-in notions,” Yanofsky writes. Yanofsky says that we should not be too concerned that our reason and rationality have limits. He says, “We human beings already live beyond reason.” Humans have imagination, willpower, emotions, systems of ethics, concepts of beauty, values of all kinds that precede reason. After all, reason and logic are tools, and the motivation to use these tools comes first, before we actually use them. He ends his book with this paragraph: “Not only do we have this unreasonable part of our psyche, but this irrational component is our most important component. It is what gets us out of bed in the morning. It is our motivation and will. There is no logical reason to do anything. Reason and logic tell us what is and in some cases can tell us what will be. These tools can be used to help us get what we want. But they do not tell us what to want or what ought to be. Only will and desire tell us that. Unless love, desire, music, and art exist, our world has no meaning. Real life has importance only when it includes ethics, values, and beauty. Will and desire are fundamental, while reason is a tool for that will and desire. Reason is a powerful – but nonetheless limited – tool.”

  21. 5 out of 5

    Sayed H Fatimi

    An absolute must read for all those with an insatiable appetite for knowledge and understanding. Without delving too deep into each topic, this book does well to introduce the reader to a diverse, but well connected, web of information and problems that face human beings; our existence and the universe, not being the least. From elementary logical formalism, to introductory physics spanning a couple centuries, paradoxes that boggle the greatest minds, mathematics underpinning our very reality, p An absolute must read for all those with an insatiable appetite for knowledge and understanding. Without delving too deep into each topic, this book does well to introduce the reader to a diverse, but well connected, web of information and problems that face human beings; our existence and the universe, not being the least. From elementary logical formalism, to introductory physics spanning a couple centuries, paradoxes that boggle the greatest minds, mathematics underpinning our very reality, philosophical and metaphysical ideologies that just seem to want to be broken, and a look at information theory from an epistemological perspective, this book does very well to present so much. Yanofsky took on a great challenge, bringing the most grandest of theories to the hands of the masses; I would argue he did very well.

  22. 4 out of 5

    Marcus Chee

    This review has been hidden because it contains spoilers. To view it, click here. Immense book. Yanofsky brings the reader on an incredible journey, carefully illustrating philosophical conundrums from all facets of science. The book also invokes child-hood memories of simple English self-reference paradoxes such as the Liar Paradox, yet bringing the reader to explore the famous P & NP problem. In the end, what is beyond Reason? Yanofsky satiates the curious minds and outlines that Humans already live beyond the boundaries of Reason. Feelings & Emotions exist outside the frame Immense book. Yanofsky brings the reader on an incredible journey, carefully illustrating philosophical conundrums from all facets of science. The book also invokes child-hood memories of simple English self-reference paradoxes such as the Liar Paradox, yet bringing the reader to explore the famous P & NP problem. In the end, what is beyond Reason? Yanofsky satiates the curious minds and outlines that Humans already live beyond the boundaries of Reason. Feelings & Emotions exist outside the framework of logical Reasoning and yet, it is these very irrational thoughts that allow us to appreciate the beauty of art and music; of love and morality.

  23. 5 out of 5

    Apurva

    The author explains how although we consider science and reason to be infallible and beyond doubts, in a lot of ways, these also have their own limitations. The very basis of science - induction, which is the ability to predict something based on few sample studies, is flawed. He doesn't downplay the importance of science though, simply opens our eyes to that fact that like everything, this too is not without its share of drawbacks and science can certainly not answer everything. He does use a lo The author explains how although we consider science and reason to be infallible and beyond doubts, in a lot of ways, these also have their own limitations. The very basis of science - induction, which is the ability to predict something based on few sample studies, is flawed. He doesn't downplay the importance of science though, simply opens our eyes to that fact that like everything, this too is not without its share of drawbacks and science can certainly not answer everything. He does use a lot of maths and reason to explain his arguments. But even if you hate equations, you will still find plenty to take away from this book.

  24. 4 out of 5

    Mario Krapp

    I had to put this down after the first chapter. I do not remember how this book ended on my „to read“ list but usually those books are recommendations I pick up from anywhere else (and usually they are good). I also had to read a few review los of this very book, just to see if I’ve missed a point or something. I haven’t. This book is simply not for me. I don’t like the style. It’s overly detailed, repetitive, and boring. Again, I’ve just read the introduction and the first chapter. But skimming t I had to put this down after the first chapter. I do not remember how this book ended on my „to read“ list but usually those books are recommendations I pick up from anywhere else (and usually they are good). I also had to read a few review los of this very book, just to see if I’ve missed a point or something. I haven’t. This book is simply not for me. I don’t like the style. It’s overly detailed, repetitive, and boring. Again, I’ve just read the introduction and the first chapter. But skimming through the rest of the book, I fell that it doesn’t look like there is something in for me. Still feeling bad for giving it just one star. But maybe it helps others not wasting their time.

  25. 5 out of 5

    Prahesa

    explain advanced topics using simple term, yet somehow fails to clearly explain them. what's the target reader of this book? somehow it fails to clearly define that. too much tedious yet half-baked proof and statements provided. non-stem readers will be overwhelmed by this book, yet more advanced reader will not be satisfied with its content as well (hello? can you please not say "it is too tedious to put the proof here" when you flashed the topic in early part of the book please?) explain advanced topics using simple term, yet somehow fails to clearly explain them. what's the target reader of this book? somehow it fails to clearly define that. too much tedious yet half-baked proof and statements provided. non-stem readers will be overwhelmed by this book, yet more advanced reader will not be satisfied with its content as well (hello? can you please not say "it is too tedious to put the proof here" when you flashed the topic in early part of the book please?)

  26. 5 out of 5

    Ogi Ogas

    My ratings of books on Goodreads are solely a crude ranking of their utility to me, and not an evaluation of literary merit, entertainment value, social importance, humor, insightfulness, scientific accuracy, creative vigor, suspensefulness of plot, depth of characters, vitality of theme, excitement of climax, satisfaction of ending, or any other combination of dimensions of value which we are expected to boil down through some fabulous alchemy into a single digit.

  27. 5 out of 5

    Mishehu

    There's a lot of very interesting material in this book, and it's generally well presented. Less one star because there's also some surface skimming, under-explanation, and repetition. And because the author is not the most sparkling of prose stylists. A good read, not a great one, that covers a lot of the same ground as other books on the subject. Still, I profited from reading it. There's a lot of very interesting material in this book, and it's generally well presented. Less one star because there's also some surface skimming, under-explanation, and repetition. And because the author is not the most sparkling of prose stylists. A good read, not a great one, that covers a lot of the same ground as other books on the subject. Still, I profited from reading it.

  28. 5 out of 5

    Howard H. Stevenson

    A great challenge to the accepted dominance of reason The book shows the difference between unknowable and irrational. Often academics and scientists presume that something is knowable and deny the irrationality of that assumption. Thank you!

  29. 4 out of 5

    Tony Gualtieri

    I read a lot of popular mathematics books, so there is little that is new here, but it's interesting to find all these paradoxes in a single place. A good synthesis of the limits of logic and the strangeness of contemporary physics. I read a lot of popular mathematics books, so there is little that is new here, but it's interesting to find all these paradoxes in a single place. A good synthesis of the limits of logic and the strangeness of contemporary physics.

  30. 5 out of 5

    Dana Paxson

    The author takes us on an amazing trail that skirts the cliff-edge of human comprehension, from the halting problem in computing to the hierarchy of infinite infinities, from the incompleteness of logic to the elusiveness of consciousness. A wonderful ride.

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