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

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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, 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.

403 pages, Hardcover

First published January 1, 2013

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Noson S. Yanofsky

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Profile Image for BlackOxford.
1,081 reviews67.8k followers
March 8, 2022
Whatever Passes For Reason

“With understanding comes ambivalence.” This is an opening line that rates in concision and anticipatory depth with Dickens and Kafka. It also rings a suggestive note of intellectual honesty and reserve that gives the reader hope that what follows is tentative and provocative not an attempt to grind an intellectual axe. I was sympathetic with this Yarn0fsky from the start. Unfortunately, I found that as I began to understand Yanofsky, ambivalence about Yanofsky grew apace.

The book is really about logic despite its title and its stated ambitions about reason. It’s not a bad exposition of logical contradictions and how they occur. If the book were pitched as a popular introduction to this topic in logic, without pretending it had something to say about the philosophy of science, it might sit well in a certain niche for university undergraduates. But Yarnofsky and his publisher are after bigger game apparently. And there it fails to score.

“My goal is clarity,” Yanofsky says. And I think he succeeds. But at the cost of pedantry that is remarkable in its casual expression. He breezes from arbitrary definition to spectacularly prejudicial conclusion to sweeping implication with ease, confidence, and yes, great clarity. He is a university teacher after all, and apparently a teacher who takes no prisoners among his dissenting students. Neither nuance nor dialectic stands a chance in his class based on the contents of this book.

According to Yanofsky, “Rationality and reason are the methodologies used by a society to advance.” This is obviously meant to get the reader’s attention. But seriously? Does he mean by this broad claim to include those ancient societies of Asia, Middle, South and North America and Africa whose primary spurs to advance were often rather irrational and unreasonable (by today’s standards) religious beliefs. Or is the claim just a plug for scientistic boosterism, a nod to technology and material progress perhaps? In any case, whatever the “methodologies” employed by these societies, they were profoundly different from each other. So his claim that reason is the “tool of civilization” is tendentious as well as insulting to the intellectual traditions of most of the world.

But as his story unfolds, it becomes increasingly clear that Yanofsky really has no idea what he means by reason. He begins his exposition, arbitrarily and with a certain puerile insouciance, by choosing one among sixteen (his count) OED definitions of the term. No historical review, no comparisons, no explanation whatsoever except that he likes the definition he’s chosen. Ultimately, after conducting us on a grand tour of his real subject, contradictions, he concludes with his own re-considered definition: “Reason is the set of processes or methodologies that do not lead to contradictions and falsehoods.” One shudders. The circularity of this statement is apparent. Even Yanofsky recognises that it may not have any meaning at all except as a tautology: “The point I am making is that there are no exact rules to determine when some idea or process is part of reason or beyond its boundaries.” I don’t disagree. But so much for civilisation and its foundation in reason.

After he has trashed reason as an indeterminate process, then backtracks on his initial claims, Yarnofsky comes to a very narrow, very specific point. Despite the wide diversity of what constitutes reason among different scientific disciplines and among philosophers of science, he believes that “there is a property that most thinkers agree reason has: one cannot use reason to derive contradictions and false facts.” So, he continues, “We must keep such mental constructs [presumably theories] and language contradiction-free so as to describe the contradiction-free physical universe.” No discussion of which thinkers he has in mind or what their critiques might have said, or for that matter why the universe doesn’t choose to be contradictory. But Yarnofsky is quite sure that this is the Holy Grail of method that the world has overlooked.

So, here’s a fact: numerous experiments in quantum mechanics demonstrate entanglement, that is instantaneous action at (even great) distance, between paired photons. This contradicts much of modern physics. Is the issue factual (we’re measuring the wrong things) or theoretical (this actually happens and we can’t explain it)? Could be either. Could be both. Could be a presumption buried deeply in our logic of either our theoretical language or our physical technique (it may be some other issue entirely, for example: https://apple.news/AAWuNbce6RUGuiPjoM...). Yarnofsky’s discussion doesn’t get us any closer to pinpointing its locus much less solving the problem of entanglement. Or any other scientific problem. The logical technique of reductio ad absurdum, contradiction, is well-known to scientists in fields from physics to evolutionary anthropology. But to my knowledge it has never been of use in distinguishing factual from theoretical error. I don’t see how such a distinction is possible. And Yarnofsky’s not saying.

To be fair, Yarnofsky knows he’s got a problem with the self-referentiality of his preferred “tools of reason,” mathematics and logic. Neither of these is entirely rational. That is to say, their internal consistency cannot be demonstrated. All we (or at least he) can say is that no one has yet shown them to be inconsistent. He strengthens this empirical statement with one of a quasi-religious plea. “Do not worry,” he says as if he were a preacher addressing a room full of Sunday worshippers, “that modern mathematics or even basic arithmetic is inconsistent. I assure you that they are consistent.” Trust me, I’m a Doctor, is not really an acceptable piece of scientific evidence or reasonable argument.

Yarnofsky glosses over Gödel’s second incompleteness theorem, the one that really scuppers self-referentiality, as if it contributes to his thesis about intellectual limits: “one of the central themes of this book: any self-referential system, no matter how powerful, is somewhat limited.” Yet he rests his whole case - about contradiction, about method, about reason, about civilisation itself - on the integrity of mathematics and logic? The inability to demonstrate that integrity is fatal for his project from the start. Rephrase his project: “I want to demonstrate to you the limits of reason by using tools that are themselves limited in their reasonability,” to understand as my growing ambivalence turned into annoyance.

So, after building the case for reason and rationality, or at least a case for the usefulness of logical contradiction, and then retracting it, Yarnofsky goes into a sort of general intellectual meltdown and abjures what he has built entirely:
“Since we do not know what will and will not cause one to err, our definition of reason is somewhat time dependent. What was once considered reasonable, could, in the future, be shown to cause contradictions. In fact, throughout history, there have been many times that something was considered part of science and only later turned out to be false.”
So much for logic and method and reason, and civilisation.

, as the philosopher, Charles Taylor, has said with pointed sarcasm about current thought patterns, “is whatever passes for reason.” Yarnofsky apparently agrees - without any sarcasm. If either of them could simply accept that the search for better methods of inquiry is a permanent part of inquiry, the world might be better off. It would certainly require less paper.
Profile Image for Liviu.
2,228 reviews625 followers
July 23, 2014
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
Profile Image for Emre Sevinç.
143 reviews269 followers
November 19, 2017
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.
Profile Image for Muhammad Arqum.
104 reviews68 followers
February 17, 2015
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*
Profile Image for يوسف بوحايك.
Author 1 book114 followers
July 13, 2016
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.
Profile Image for Chris.
383 reviews139 followers
September 3, 2014
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.
Profile Image for hayatem.
656 reviews170 followers
July 4, 2022
كتاب يحدثنا عن حدود العقل وإمكاناته، و في ما لا يمكن التنبؤ به أو وصفه أو معرفته، عن عجز قدراتنا على تجاوز قيوده، في ما لا يمكن معرفته. حتى أن العلم والمعارف البشرية على امتدادها منذ آلاف السنين عاجزة عن إخبارنا بأشياء نرغب في معرفتها، وما ذالك إلا قيد من هذه القيود.

يناقش يانوفسكي عبر تقصياته المختلفة في فصول هذا الكتاب موضوعاً مختلفًا كما يناقش القيود المفروضة عليه، كما ينظر فيما وراء أرسان العقل، إذا كان ثمة ما وراءه.

استكشاف القيود المختلفة لمعرفتنا يمكننا من فهم بنية العقل وحدوده بشكل أفضل : قيود مادية، قيود التشكيلات الذهنية( الرياضيات، ونظرية الفئات، والمنطق.)، قيود عملية، قيود الحدس؛ وفي جميع هذه المجالات المتنوعة إحالات ذاتية، وأوضاع مفارقية وقيود. أذكر عدداً من المواضيع على سبيل المثال لا الحصر ( اللغة، اللاتناهي، مفارقة رسل، مبرهنة غودل الأولى في اللاتمام، الاعتقادات.)؛ أهمية نظرية النسبية. أسباب نظرية السواش، عالم نظرية ميكانيكا الكم المتشابك.

أثبت يانوفسكي عبر عدة مسائل ومناهج وآليات أن المنطق، والرياضيات، والحاسوب، عاجزة عن حل مسائل بعينها بالوسائل العادية ؛ بيانات صحيحة ولكن لا يمكن إثباتها. ك المهام البسيطة التي قد تستغرق تريليونات القرون من أجهزة الكمبيوتر لإكمالها ومشكلات أخرى لا تستطيع أجهزة الكمبيوتر حلها أبدًا.

ويختم يانوفسكي حديثه بقدرتنا رغم كل هذه القيود المحكومة بحدود العقل والمعقولية على تجاوز أرسان العقل" العالم الذي يقطنه البشر ليس عالم العقل، المنطق، والرياضيات، والعلم البارد قاسي القلب. أذهاننا لا تعيش في عالم الأحجار، وصور الحياة المؤسسة على الكربون، والجزيئات التي تتمثل لقوانين الفيزياء المعتادة، بل لدينا مشاعر وعواطف لا يمليها العقل والمنطق. لدينا حس بالجمال، والدهشة، والأخلاق، والقيم التي تتجاوز العقل وتتحدى التفسير المعقول.
،…….. قراراتنا لا تتخذ تأسيساً على المنطق والعقل، بل تعول على الإستاطيقا ، والخبرة العلمية، والميول الأخلاقية، والنزوات، والعواطف، والأحداس، والمشاعر. وبهذا المعنى ، كل منا يتجاوز أصلاً أرسان العقل."
Profile Image for Bjørnar Tuftin.
208 reviews15 followers
July 1, 2016
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.
Profile Image for Kaśyap.
271 reviews123 followers
January 8, 2018
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.
Profile Image for Vasil Kolev.
1,036 reviews159 followers
January 22, 2016
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.
443 reviews2 followers
April 11, 2020
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.

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.
Profile Image for Testunde.
3 reviews
December 26, 2022
It is fine. It gets to the point at some stage in the book (last chapters) but until then has lengthy explanations of examples, which especially computer scientists will start to recognize. I found it hard to get a clear line of what the author wanted to explain. Also, the depth of details of the posed problems are relatively shallow. I understand, however, that this book is written for the popular science audience.
Profile Image for Alexandra.
89 reviews26 followers
March 26, 2018
And here I thought quantum computing would be kind of a revolution...
150 reviews12 followers
November 4, 2016
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.
Profile Image for Doctor Moss.
461 reviews17 followers
May 31, 2020
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.
Profile Image for Janice.
462 reviews9 followers
January 20, 2014
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!!!
57 reviews1 follower
November 11, 2013
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.
Profile Image for Tim Sharp.
22 reviews
March 7, 2016
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.
Profile Image for Steve.
922 reviews39 followers
March 8, 2014
Really enjoyed the 75% or so that I understood and I just skimmed the rest.
Profile Image for Giacomo Mantani.
68 reviews1 follower
October 6, 2017

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.


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


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

Profile Image for Gary Schroeder.
126 reviews9 followers
March 16, 2021
This is an entertaining review of the concept of knowability. Some authors out there (Like David Deutch, for instance) have written books that attempt to prove that there are no limits to knowledge, that all problems are solvable. Noson Yanofsky takes a different approach: he tries to show all of the problems for which a solution does not exist and all of the aspects of the universe which may be forever unknowable.

Some unknowbale things are so because of the limitations of language. Some things are theoretically knowable but the execution of the necessary calculations would exceed the lifetime of the universe. Many aspects of mathematics are known to be true and work flawlessly, but can nevertheless never be formally "proven" as unassailably true.

A lot of time is dedicated to classic mathematical paradoxes and conundrums...like figuring out the shortest route for a traveling salesman who lives in a city with five bridges, or whether or not it's theoretically possible for a machine to compute the ability of geometrical shapes to be used for floor tiling without leaving any gaps. These puzzles are intellectual curiosities and may have some esoteric application to the real world...but it's often hard to see them as anything more than pure intellectual challenges for mathematicians.

The more interesting parts of the book are dedicated to the physical world, particularly at the subatomic quantum level. There, Yanofsky gets into some fascinating territory by exploring the same sorts of weird conundrums...but with real-world consequences. One of the most interesting of which is "entanglement." At quantum scales, there are observable physical phenomena in which what's happening in front of you can result in a physical change halfway across the world...or across the galaxy. Why? Becasue the universe itself is part of the system being observed and can't be separated in the way that we intuitively think. Mind-bending. Don't ask me to explain it further.

Another interesting excursion is into the concept of superposition in which information in a closed system exists in multiple states...until we look at it. It's not a game; physicists are actually telling us that physical things exist in multiple states at once and the deciding factor for which state it is when we open the closed system (like a box) is the act of our looking at it...because there's something about "consciousness" which collapses the superposition. Again, don't ask me to explain further.

I found the early chapters a bit of a slog because they rest so heavily on exploring pure math problems, but the later chapters that delve into quantum phenomena are fascinating.
Profile Image for Marko Beljac.
9 reviews
December 28, 2022
Quite simply, this was fantastic and a pleasure to read. There’s so much to learn in this book. This is definitely one of the best books I’ve read on the relation of science, mathematics, and logic to reason. Yanofsky has the knack of explaining things exceptionally well, his renderings of Bell’s Theorem, Godel and the P NP problem were works of clarity and exposition for instance. Just a quibble, however. The subtitle of the book is; what science, mathematics and logic cannot tell us. You’ll find plenty here about that, but what I think Yanofsky shows is all three are not a consequence of reason. That is science, mathematics and logic go beyond reason so the boundary of reason is not coterminous with the boundaries of science, mathematics and logic. Whatever faculty, or faculties, of the mind enable us to form sciences and mathematics it cannot be a faculty of reason. This we’ve known since David Hume, and I think what Yanofsky does is show how the subsequent history of physics and mathematics bears out Hume’s deflating of reason. Also, Yanofsky says on a number of occasions that the universe is rational but he demonstrates quite clearly that classical logic and quantum mechanics don’t mix which suggests something else. Yanofsky also states that language is a tool to describe the world in which we live. That it certainly isn’t. Language is thought. For what it’s worth I think that reason doesn’t exist, that is reason as a central commanding faculty of the mind. Reason is normative.
29 reviews
September 20, 2020
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)?
Profile Image for Mike Lisanke.
145 reviews5 followers
August 2, 2022
There are no uninteresting topics in this book on the outer limits of reason... because if you find some section of this book uninteresting, that'd be interesting! :-p
Seriously, this is a very good book Because it stays on topic and doesn't wonder into any social science not related to the topic (so, none).
If you're a mathematician already then this book would be far too basic. If you're an Engineer (like I am) you may find you know much about most of the topics covered. If you're a "scientist" then there's no hope for you! :-p
So, the key target audience for this book are those who are intellectually stimulated by what's possible to reason about and why some things which we think are reasonable can often confound that conclusion. And if you like some simple stories to convey many powerful ideas you'll find them here. Just don't expect to learn the details without following the references.
There are several problems... for one far too many examples where long lists of numbers are presented which aren't easy to skip over in an auto-reader! Also, there are many other answers to the controversial limits of reason than those presented by the author which the readers should also explore... some presented in his notes (e.g. solipsism as answer to anthropomorphic principle). Obviously, this book can't mention everything on this interesting broad topic... but would need to in order to do the topic justice... call it the limits of reason incompleteness theory! :-p
Profile Image for Nader Saeed.
22 reviews10 followers
July 23, 2021
A thorough investigation of the epistemic and ontological limits of our logic and reason. The author examines logical and scientific questions whose answers lay beyond the faculties of our reason, language, mathematics, and physics. He navigates through the concepts of seemingly impenetrable philosophical and mathematical abstractions to demonstrate our limitation in dealing with them with a lucid and informative approach that greatly enriches their conceptualization.

This limitation is revealed to us everywhere we look; in the paradoxes of language and the mind; chaos theory and unsolvable math problems; the restricted power of proofs; our intuition’s ineptitude dealing with what relativity and quantum mechanics are telling us about reality; and the undecidability of some computational problems (The P vs NP problem was exceptionally well-presented).
Even so, if I wanted to nitpick, some elements were still a bit disconcerting in the way they were addressed, for example, the role consciousness plays in collapsing the wave-function in quantum mechanics.

I enjoyed the book throughout and would not hesitate to recommend it to anyone.

“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?” – David Hume
Profile Image for Stewart.
319 reviews13 followers
February 23, 2016
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.”
Profile Image for Kirjoittaessani.
10 reviews1 follower
January 28, 2023
The book turned out slightly different than expected: From the title, I had assumed to find a list of things that "we can never know", and -- being a follower of Hilbert -- I was more than sceptical.
Instead, the author delivers what in another genre would have been a collection of anecdotes: more or less disjointed facts, all in some way connected to limitations of understanding. There are classic paradoxes, Gödel's incompleteness theorem, NP-hardness, and the halting problem. There are some metaphysical questions as well as the more counterintuitive aspects of quantum mechanics. In the end, all of this is stuff that an educated person should know about, there's very little that,'s truly unknowable: the paradoxes conjure up situations that are actually impossible; quantum mechanics is, for the most part, well understood (it just doesn't agree with common sense): and undecidability and infeasibility in computer science pertains to general problems only -- deciding the halting problem for a given program can actually be quite doable. And even Gödel might be circumvented with extra axioms.
Profile Image for Sayed H Fatimi.
13 reviews
September 2, 2020
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.
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