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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

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The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (MIT Press) by Noson S. Yanofsky Book Review

Name: Featherless Biped
Rating: 4.0 out of 5 stars
Title: .0 out of 5 stars The Universe According to Brooklyn — good science, not such good philosophy
Date: Reviewed in the United States 🇺🇸 on August 11, 2014
Review: Alvy’s Mom responding to his being depressed because the universe is expanding — “What has the universe got to do with it? You’re here in Brooklyn! Brooklyn is not expanding!”
This famous Woody Allen joke makes a profound point about the context sensitivity of language that applies throughout philosophy and science. It’s funny because it is obvious that the meaning of “expanding” in the two cases is quite different. Brooklyn might expand if the population increases or the city annexes outlying land, but the universe is said to expand due to cosmic telescopes that show a red shift indicating that stars are receding from each other or to measurements of matter density etc. Different meanings (language games)(LG’s) were famously characterized by the Austrian-English philosopher Ludwig Wittgenstein (W) as the central problem of philosophy and shown to be a universal default of our psychology. Though he did this beginning with the Blue and Brown Books (BBB) in the early 30’s, left a 20,000 page nachlass, and is the most widely discussed philosopher of modern times, few understand him. To Yanofsky’s (Y’s) credit, he has given much attention to philosophy and even quotes W a few times but without any real grasp of the issues. It is the norm among scientists and philosophers to mix the scientific questions of fact with the philosophical questions of how language is being used and, as W noted, — ‘Problem and answer pass one another by’. Yanofsky (a Brooklyn resident like many of his friends and teachers) has read widely and does a good job of surveying the bleeding edges of physics, mathematics and computer science in a clear and authoritative manner, but then we come to the limits of scientific explanation and it’s not clear what to say, so we turn to philosophy. Philosophy can be seen as the descriptive psychology of higher order thought or as the study of the contextual variations of language used to describe cognition or intentionality (my characterizations), or the study of the logical structure of rationality (Searle). Berkeley philosopher John Searle (S) is the best since W and his work can be seen as an extension of W. I have reviewed many books by them and others and together these reviews constitute a skeletal outline of higher order thought or intentionality, and so of the foundations of science.
It is common for books to betray their limitations in their titles and that is the case here. “Reason” and “limits” are whole complexes of language games. So I should stop here and spend the whole review showing how Y’s title reveals the deep misunderstanding of what the real issues are here. I knew we were in for a rough time by p5 where we are told that our normal conceptions of time, space etc., are mistaken and this was known even to the Greeks. This brings to mind W: “People say again and again that philosophy doesn’t really progress, that we are still occupied with the same philosophical problems as were the Greeks… at something which no explanation seems capable of clearing up…And what’s more, this satisfies a longing for the transcendent, because in so far as people think they can see the ‘limits of human understanding’, they believe of course that they can see beyond these. — CV (1931)” and also “The limit of language is shown by its being impossible to describe a fact which corresponds to (is the translation of) a sentence without simply repeating the sentence…” So I would say we just have to analyze the different types of language games. Looking deeper is essential but surrendering our prior use is incoherent.
Think about what is implied by “The Outer Limits of Reason”. “Outer”, “Limits” and “Reason” all have common uses , but they will be frequently used here in different ways and they will seem “quite innocent”, but this can only be discussed in some specific context.
We are using the word “question” (or “assertion”, “statement” etc.) with utterly different senses if we ask “Does 777 occur in the decimal expansion of Pi?” than if we ask “Does 777 occur in the first 1000 digits of the decimal expansion of Pi? (W)” In the latter case it’s clear what counts as a true or false answer but in the former it has only the form of a question but it is not clear if it can be answered. On p10 we find a group of “statements” which have quite different meanings. The first three are definitions one could understand them without knowing any facts about their use — e.g., X cannot be Y and not Y.
Y recommends the documentary “Into the Infinite” but actually it cannot be viewed unless you are in the UK. I found it free on the net shortly after it came out and was greatly disappointed. Among other things it suggests Godel and Cantor went mad due to working on problems of infinity — for which there is not a shred of evidence — and it spends much time with Chaitin, who, though a superb mathematician, has only a hazy notion about the various philosophical issues discussed here. If you want a lovely whirlwind “deep science” documentary I suggest “Are We Real?” on Youtube , though it makes some of the same mistakes.
W noted that when we reach the end of scientific commentary, the problem becomes a philosophical one-i.e., one of how language can be used intelligibly. Yanofsky, like virtually all scientists and most philosophers, does not get that there are two distinct kinds of “questions” or “assertions” (LG’s) here. There are those that are matters of fact about how the world is — that is, they are publicly observable propositional (True or False ) states of affairs having clear meanings (Conditions of Satisfaction — COS) in Searle’s terminology — i.e., scientific statements, and then there are those that are issues about how language can coherently be used to describe these states of affairs, and these can be answered by any sane, intelligent, literate person with little or no resort to the facts of science. Another poorly understood but critical fact is that, although the thinking, representing, inferring, understanding, intuiting etc. (i.e., the dispositional psychology) of a true or false statement is a function of the higher order cognition of our slow, conscious System 2 (S2), the decision as to whether particles are entangled, the star shows a red shift, a theorem has been proven (i.e., the part that involves seeing that the symbols are used correctly in each line of the proof), is always made by the fast, automatic, unconscious System 1 (S1) via seeing, hearing, touching etc. in which there is no information processing, no representation (i.e., no COS) and no decisions in the sense in which these happen in S2 ( which receives its inputs from S1). This two systems approach is now the standard way to view reasoning or rationality and is a crucial heuristic in the description of behavior, of which science, math and philosophy are special cases. There is a huge and rapidly growing literature on reasoning that is indispensable to the study of behavior or science. A recent book that digs into the details of how we actually reason (i.e., use language to carry out actions — see W and S) is ‘Human Reasoning and Cognitive Science’ by Stenning and Van Lambalgen (2008), which, in spite of its limitations (e.g., limited understanding of W/S and the broad structure of intentional psychology), is (as of mid 2014) the best single source I know.
Regarding “incompleteness” or “randomness” in math, Y’s failure to mention the work of Gregory Chaitin is truly amazing, as he must know of his work, and Chaitin’s proof of the algorithmic randomness of math (of which Godel’s results are a corollary) and the Omega number are some of the most famous results in the last 50 years.
Likewise one sees nothing about hypercomputation and other topics. The best way to get articles on the cutting edge is to visit ArXiv.org where there are tens of thousands of free preprints on every topic here (be warned this may use up all your spare time for the rest of your life!).
Regarding Godel and “incompleteness”, since our psychology as expressed in symbolic systems such as math and language are “random” or “incomplete” and full of tasks or situations (“problems”) that have been proven impossible (i.e., they have no solution-see below) or whose nature is unclear, it seems unavoidable that everything derived from them — e.g. physics and math) will be “incomplete” also. Afaik the first of these in what is now called Social Choice Theory or Decision Theory (which are continuous with the study of logic and reasoning and philosophy) was the famous theorem of Kenneth Arrow 63 years ago, and there have been many since. Y notes a recent impossibility or incompleteness proof in two person game theory. In these cases a proof shows that what looks like a simple choice stated in plain English has no solution. Although one cannot write a book about everything I would have liked Y to at least mention such famous “paradoxes” as Sleeping Beauty(dissolved by Read), Newcomb’s problem(dissolved by Wolpert) and Doomsday, where what seems to be a very simple problem either has no one clear answer or it proves exceptionally hard to find one. A mountain of literature exists on Godel’s two “incompleteness” theorems and Chaitin’s more recent work, but I think that W’s comments made in the 30’s are quite substantive and will not be excelled. W’s comments have been debated frequently by Floyd, Putnam and others. Viktor Rodych has written the most incisive series of papers on W’s mathematical views but they have been almost totally ignored to date. In any case W points out the fact that we are here caught in another LG (Language Game) where it is not clear what “true”, “complete”, “follows from” and “provable” mean (i.e,, what are their COS in THIS context) and hence what significance to attach to ‘incompleteness’ and now likewise for Chaitin’s algorithmic “randomness”. As W noted frequently, do the odd formulas or inconsistencies that don’t fit the pattern cause any issues for math, physics or life? The even more serious case of contradictory statements –e.g., in set theory — -have long been known but math goes on anyway. Likewise for the countless liar (self referencing) paradoxes in language which Y discusses.
The more zealous ought to get “Godel’s Way”(2012) by Chaitin, Da Costa and Doria. In spite of its many failings — really a series of notes rather than a finished book — it is a unique source of the work of these three famous scholars who have been working at the bleeding edges of physics, math and philosophy for over half a century. Da Costa and Doria are cited by Wolpert since they wrote on universal computation and among other things, Da Costa is a pioneer on paraconsistency. Chaitin also contributes to an excellent volume(though with many of its own confusions) covering some of the territory of this one ‘Causality,Meaningful Complexity and Embodied Cognition’ (2010).
Different contexts mean different LG’s (meanings, COS) for “time”, “space”, “particle” “object” , ”inside”, “outside”, “next”, “simultaneous”, ”occur”, “happen”, “event” ,”question”, “answer” ,“infinite”, “past”, “future”, “problem”, “logic”,“ontology”, “epistemology”, “solution”, “paradox”, “prove”, “strange”, “normal”, “experiment”, ”complete”, “uncountable”, “decidable”, “dimension”,“complete”,“formula”, “process”, “algorithm”, “axiom”, ”mathematics”, “physics”, “cause”, “place”, “same”, “moving”, “limit”, “reason”, “still”, “real” “assumption”, “belief”, ‘know”, “event”, ”recursive”, “meta — “, “self referential” “continue”, “particle”, “wave”,, “sentence” and even (in some contexts) “and”, “or”, “also”, “add” , “divide”, “if…then”, “follows” etc.
To paraphrase W, most of what people (including many philosophers and most scientists) have to say when philosophizing is not philosophy but its raw material. Yanofsky joins Hume, Quine, Dummett, Kripke , Dennett, Churchland, Carruthers,Wheeler etc. in repeating the mistakes of the Greeks with elegant philosopical jargon mixed with science. Perhaps the quickest way to dispel the fog is to read some my reviews and as much of Rupert Read as possible — but at least some of his articles in A WITTGENSTEINIAN WAY WITH PARADOXES and WITTGENSTEIN AMONG THE SCIENCES or go to academia.edu and get his articles , especially ‘Kripke’s Conjuring Trick’ and ‘Against Time Slices’ and then as much of S as feasible but at least his most recent such as ‘Philosophy in a New Century’, ‘Searle’s Philosophy and Chinese Philosophy’, ‘Making the Social World’ and ‘Thinking About the Real World’ and perhaps his forthcoming volume on perception.
Y does not make clear the major overlap that now exists (and is expanding rapidly) between game theorists, physicists, economists, mathematicians, philosophers, decision theorists and others, all of whom have been publishing for decades closely related proofs of undecidability, impossibility, uncomputability, and incompleteness. Godel was first but there have been an avalanche of others. As noted, one of the earliest in decision theory was the famous General Impossibility Theorem discovered by Kenneth Arrow in 1951 (for which he got the Nobel Prize in economics in 1972 — and five of his students are now Nobel laureates so this is not fringe science). It states roughly that no reasonably consistent and fair voting system (i.e., no method of aggregating individuals’ preferences into group preferences) can result in sensible results. The group is either dominated by one person and so is often called the “dictator theorem”, or has intransitive preferences Arrow’s original paper was titled “A Difficulty in the Concept of Social Welfare”. It is impossible to formulate a social preference ordering that satisfies all of the following conditions: Nondictatorship; Individual Sovereignty; Unanimity; Freedom From Irrelevant Alternatives; Uniqueness of Group Rank. Those familiar with modern decision theory accept this and the many related constraining theorems as their starting points. Those who are not may find it (and all these theorems) incredible and in that case they need to find a career path that has nothing to do with any of the above disciplines. See” The Arrow Impossibility Theorem”(2014) or “Decision Making and Imperfection”(2013).
Y mentions the famous impossibility result of Brandenburger and Keisler(2006) for two person games (but of course not limited to “games” but like all these impossibility results it applies broadly to decisions of any kind) which shows that any belief model of a certain kind leads to contradictions. One interpretation of the result is that if the decision analyst’s tools (basically just logic) are available to the players in a game, then there are statements or beliefs that the players can write down or think about but cannot actually hold. “Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong” seems unexceptionable and has been assumed in argumentation, linguistics, philosophy etc., for a century at least, but they showed that it is impossible for Ann and Bob to assume these beliefs. And there is a rapidly growing body of such impossibility results for 1 or multiplayer decision situations(e.g., it grades into Arrow, Wolpert, Koppel and Rosser etc). For a good technical paper from among the avalanch on the B&K paradox, get Abramsky and Zvesper’s paper from arXiv which takes us back to the liar paradox and Cantor’s infinity (as its title notes it is about “interactive forms of diagonalization and self-reference”). Many of these papers quote Y’s paper “A universal approach to self-referential paradoxes and fixed points. Bulletin of Symbolic Logic, 9(3):362–386, 2003. Abramsky(a polymath who is among other things a pioneer in quantum computing) is a friend of Y’s and so Y contributes a paper to the recent Festschrift to him ‘Computation, Logic, Games and Quantum Foundations’ (2103). For maybe the best recent(2013) commentary on the BK and related paradoxes see the 165p powerpoint lecture free on the net by Wes Holliday and Eric Pacuit ’Ten Puzzles and Paradoxes about Knowledge and Belief’.
For a good multi-author survey see ’Collective Decision Making(2010).
One of the major omissions from all such books is the amazing work of polymath physicist and decision theorist David Wolpert, who proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, and even independent of the laws of physics, so they apply across computers, physics, and human behavior, which he summarized thusly: “One cannot build a physical computer that can be assured of correctly processing information faster than the universe does. The results also mean that there cannot exist an infallible, general-purpose observation apparatus, and that there cannot be an infallible, general-purpose control apparatus. These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one uses an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine.” He also published what seems to be the first serious work on team or collective intelligence (COIN) which he says puts this subject on a sound scientific footing. Although he has published various versions of these over two decades in some of the most prestigious peer reviewed physics journals (e.g., Physica D 237: 257–81(2008)) as well as in NASA journals and has gotten news items in major science journals, few seem to have noticed and I have looked in dozens of recent books on physics, math , decision theory and computation without finding a reference.
It is most unfortunate that Yanofsky and others have no awareness of Wolpert, since his work is the ultimate extension of computing, thinking, inference, incompleteness, and undecidability, which he achieves (like many proofs in Turing machine theory) by extending the liar paradox and Cantors diagonalization to include all possible universes and all beings or mechanisms and thus may be seen as the last word not only on computation, but on cosmology or even deities. He achieves this extreme generality by partitioning the inferring universe using worldlines (i.e., in terms of what it does and not how it does it) so that his mathematical proofs are independent of any particular physical laws or computational structures in establishing the physical limits of inference for past, present and future and all possible calculation, observation and control. He notes that even in a classical universe Laplace was wrong about being able to perfectly predict the future (or even depict the past or present) and that his impossibility results can be viewed as a “non-quantum mechanical uncertainty principle”(i.e., there cannot be an infallible observation or control device). Any universal physical device must be infinite, it can only be so at one moment in time, and no reality can have more than one(the “monotheism theorem”). Since space and time do not appear in the definition, the device can even be the entire universe across all time. It can be viewed as a physical analog of incompleteness with two inference devices rather than one self referential device. As he says, “either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique(unlike algorithmic information complexity) in that there is one and only one version of it that can be applicable throughout our universe.” Another way to say this is that one cannot have two physical inference devices (computers) both capable of being asked arbitrary questions about the output of the other, or that the universe cannot contain a computer to which one can pose any arbitrary computational task, or that for any pair of physical inference engines, there are always binary valued questions about the state of the universe that cannot even be posed to at least one of them. One cannot build a computer that can predict an arbitrary future condition of a physical system before it occurs, even if the condition is from a restricted set of tasks that can be posed to it — that is, it cannot process information(though this is a vexed phrase as S and Read and others note) faster than the universe. The computer and the arbitrary physical system it is computing do not have to be physically coupled and it holds regardless of the laws of physics, chaos, quantum mechanics, causality or light cones and even for an infinite speed of light. The inference device does not have to be spatially localized but can be nonlocal dynamical processes occurring across the entire universe. He is well aware that this puts the speculations of Wolfram, Landauer, Fredkin, Lloyd etc., concerning the universe as computer or the limits of information processing, in a new light (though the indices of their writings make no reference to him and none of them are mentioned by Yanofsky either). He says it shows that the universe cannot contain an inference device that can process information as fast as it can, and since he shows you cannot have a perfect memory nor perfect control, its past, present or future state can never be perfectly or completely depicted, characterized, known or copied. He also proved that no combination of computers with error correcting codes can overcome these limitations.
However once again note that “infinite”, “compute”, “information” etc., only have meaning in specific human contexts — that is, as Searle has emphasized, they are all observer relative or ascribed vs intrinsically intentional (i.e., not like animals). The universe apart from our psychology is neither finite nor infinite and cannot compute nor process anything. Only in our language games are our laptop or the universe able to compute. Wolpert also notes the critical importance of the observer(“the liar”) and this connects us to the familiar conundrums of physics, math and language that concern Y.
On p140 we might note that 1936 was not actually “long” before computers since Zeus in Germany and Berry and Atanasoff in Iowa both made primitive machines in the 30’s and Wittgenstein discussed the philosophical aspects of computers before they existed.
However not everyone is oblivious to Wolpert. Well known econometricians Koppl and Rosser in their famous 2002 paper “All that I have to say has already crossed your mind” give three theorems on the limits to rationality, prediction and control in economics. The first uses Wolpert’s theorem on the limits to computability to show some logical limits to forecasting the future. Wolpert notes that it can be viewed as the physical analog of Godel’s incompleteness theorem and K and R say that their variant can be viewed as its social science analog, though Wolpert is well aware of the social implications. Since Godel’s are corollaries of Chaitin’s theorem showing algorithmic randomness (incompleteness) throughout math (which is just another of our many symbolic systems), it seems inescapable that thinking (behavior) is full of impossible, random or incomplete statements and situations. I have never seen anyone try to “explain” (or as W says we really ought to say “describe”) this but since we can view each of these domains as symbolic systems evolved by chance to make our psychology work, perhaps it should be regarded as unsurprising that they are not “complete”. For math, Chaitin says this randomness shows there are limitless theorems that are true but unprovable — i.e., true for no reason. One should then be able to say that there are limitless statements that make perfect “grammatical” sense that do not describe actual situations attainable in that domain. I suggest one might find this much less problematic if one considers W’s views. He wrote many notes on the issue of Godel’s Theorems, and the whole of his work concerns the plasticity, “incompleteness” and extreme context sensitivity of our symbolic systems.
K and R ‘s second theorem shows possible nonconvergence for Bayesian (probabilistic) forecasting in infinite-dimensional space. The third shows the impossibility of a computer perfectly forecasting an economy with agents knowing its forecasting program. The astute will notice that these theorems can be seen as versions of the liar paradox and the fact that we are caught in impossibilities when we try to calculate a system that includes ourselves has been noted by Wolpert, Koppl, Rosser and others in these contexts and which has now circled back to the puzzles of physics when the observer is involved. K&R conclude “Thus, economic order is partly the product of something other than calculative rationality”. Bounded rationality is now a major field in itself, the subject of thousands of papers and hundreds of books.
On p19 Yanofsky says math is free of contradictions, yet it has been well known for over half a century that logic and math is full of them — just google inconsistency in math or see the works of Graham Priest or the article by Weber in the Internet Encyclopedia of Philosophy. In fact as Priest notes, W was the first to predict (and debated the point with his student and colleague Turing) inconsistency or paraconsistency, which is now a common feature and a major research program in Geometry, Set Theory, Arithmetic, Analysis and computer science. He returns to this issue other places such as on p346 where he says reason must be free of contradictions, but it is clear that “free of” has different uses. What he should say is that inconsistency is common but we have mechanisms to contain it.
Regarding time travel (p49),I suggest Rupert Read’s “Against Time Slices” in his free online papers or “Time Travel-the very idea” in his book “A Wittgensteinian Way with Paradoxes.”
Regarding the discussion of famous philosopher of science Thomas Kuhn on p248, those interested can see the work of Rupert Read and his colleagues, most recently in his book “Wittgenstein Among the Sciences” and while there, you may make a start at eliminating the hard problem of consciousness by reading “Dissolving the hard problem of consciousness back into ordinary life”(or his earlier essay on this which is free on the net).
It is in the last chapter “Beyond Reason” that philosophical failings are most acute as we return to the mistakes suggested by my comments on the title. Reasoning is another word for thinking, which is a disposition like knowing, understanding, judging etc. As Wittgenstein was the first to explain, these dispositional verbs describe propositions ( sentences which can be true or false) and thus have what Searle calls Conditions of Satisfaction (COS). That is, there are public states of affairs that we recognize as showing their truth or falsity. “Beyond reason” would mean a sentence whose truth conditions are not clear and the reason would be that it does not have a clear context. It is a matter of fact if we have clear COS (i.e., meaning) but we just cannot make the observation — this is not beyond reason but beyond our ability to achieve, but it’s a philosophical (linguistic) matter if we don’t know the COS. “Are the mind and the universe computers?” sounds like it needs scientific or mathematical investigation, but it is only necessary to clarify the context in which this language will be used since these are ordinary and unproblematic terms and it is only their context which is puzzling. E.G, the “self referential” paradoxes on p344 arise because the context and so the COS is unclear.
On p347, what we discovered about irrational numbers that gave them a meaning is that they can be given a use or clear COS in certain contexts and at the bottom of the page our “intuitions” about objects, places, times. length are not mistaken- rather we began using these words in new contexts where the COS of sentences in which they are used were utterly different. This may seem a small point to some but I suggest it is the whole point. Some “particle” which can “be in two places” at once is just not an object and/or is not “being in places” in the same sense as a soccer ball.
Regarding his reference on p366 to the famous experiments of Libet, which have been taken to show that acts occur before our awareness of them and hence negate will, this has been carefully debunked by many including Searle.
It is noteworthy that on the last page of the book he comments on the fact that many of the basic words he uses do not have clear definitions but does not say that this is because it requires much of our innate psychology to provide meaning, and here again is the fundamental mistake of philosophy. “Limit” or “exist” has many uses but the important point is — what is its use in this context. “Limit of reason” or “the world exists” do not(without further context) have a clear meaning (COS) but “speed limit on US 15” and “a life insurance policy exists for him” are perfectly clear.
Regarding solipsism on p369, this and most classical philosophical positions were shown by W to be incoherent.
And finally why exactly is it that quantum entanglement is more paradoxical than making a brain out of proteins and other goop and having it feel and see and remember and predict the future? Is it not just that the former is new and not directly present to our senses (i.e., we need subtle instruments to detect it) while animal nervous systems have been evolved to do the latter hundreds of millions of years ago and we find it natural since birth?
Overall an excellent book provided it is read with this review in mind.

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Name: Doctor Moss
Rating: 4.0 out of 5 stars
Title: .0 out of 5 stars Great Survey of Paradoxes and Problems, Not a Lot on a Theory of Reason’s Limitations
Date: Reviewed in the United States 🇺🇸 on May 31, 2020
Review: 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.