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Yanofsky takes on this mindboggling subject with confidence and impressive clarity. He eases the reader into the subject matter, ending each chapter with further readings. His book is a fascinating resource for anyone who seeks a better understanding of the world through the strangeness of its own limitations and a must-read for anyone studying information science. * Publishers Weekly, (starred review) * Yanofsky provides an entertaining and informative whirlwind trip through limits on reason in language, formal logic, mathematics -- and in science, the culmination of humankind's attempts to reason about the world. * The New Scientist * In my view, Outer Limits is an extraordinary, and extraordinarily interesting, book. It is a cornucopia of mind-bending ideas. -- Raymond S. Nickerson * PsycCRITIQUES * The scope of the material covered is so wide, and the writing so clear and intuitive, that all readers will learn something new and stimulating. -- Thomas Colin * Leonardo Reviews *

About the Author

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

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I liked the way the author discusses various fields of science and philosophy. Recommended for students who would like to pursue sciences. Thought provoking and informative.

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.

The review by Featherless Biped is the only really useful one so I add here my explanation of incompleteness, paraconsistency and mathematical truth which are at the core of many of the issues Yanofsky raises. Regarding Godel and "incompleteness", since our psychology as expressed in symbolic systems such as math and language is "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 it-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 writings in the 30's and 40's are definitive. Although Shanker, Mancosu, Floyd, Marion, Rodych, Gefwert, Wright and others have done insightful work, it is only recently that W's uniquely penetrating analysis of the language games being played in mathematics have been clarified by Floyd (e.g., `Wittgenstein's Diagonal Argument-a Variation on Cantor and Turing'), Berto (e.g., `Godel's Paradox and Wittgenstein's Reasons , and `Wittgenstein on Incompleteness makes Paraconsistent Sense' and the book `There's Something about Godel ` , and Rodych ( e.g., Wittgenstein and Godel: the Newly Published Remarks', `Misunderstanding Gödel :New Arguments about Wittgenstein', `New Remarks by Wittgenstein' and his article in the online Stanford Encyclopedia of Philosophy `Wittgenstein's Philosophy of Mathematics' ). Berto is one of the best recent philosophers, and those with time might wish to consult his many other articles and books including the volume he co-edited on paraconsistency (2013). Rodych's work is indispensable, but only two of a dozen or so papers are free online. Berto notes that W also denied the coherence of metamathematics-i.e., the use by Godel of a metatheorem to prove his theorem, likely accounting for his "notorious" interpretation of Godel's theorem as a paradox, and if we accept his argument, I think we are forced to deny the intelligibility of metalanguages, metatheories and meta anything else. How can it be that such concepts (words) as metamathematics and incompleteness, accepted by millions ( and even claimed by no less than Penrose, Hawking, Dyson et al to reveal fundamental truths about our mind or the universe) are just simple misunderstandings about how language works? Isn't the proof in this pudding that, like so many "revelatory" philosophical notions (e.g., mind and will as illusions -Dennett, Carruthers, the Churchlands etc.), they have no practical impact whatsoever? Berto sums it up nicely: "Within this framework, it is not possible that the very same sentence...turns out to be expressible, but undecidable, in a formal system... and demonstrably true (under the aforementioned consistency hypothesis) in a different system (the meta-system). If, as Wittgenstein maintained, the proof establishes the very meaning of the proved sentence, then it is not possible for the same sentence (that is, for a sentence with the same meaning) to be undecidable in a formal system, but decided in a different system (the meta-system)... Wittgenstein had to reject both the idea that a formal system can be syntactically incomplete, and the Platonic consequence that no formal system proving only arithmetical truths can prove all arithmetical truths. If proofs establish the meaning of arithmetical sentences, then there cannot be incomplete systems, just as there cannot be incomplete meanings." And further "Inconsistent arithmetics, i.e., nonclassical arithmetics based on a paraconsistent logic, are nowadays a reality. What is more important,the theoretical features of such theories match precisely with some of the aforementioned Wittgensteinian intuitions...Their inconsistency allows them also to escape from Godel's First Theorem, and from Church's undecidability result: there are, that is, demonstrably complete and decidable. They therefore fulfil precisely Wittgenstein's request, according to which there cannot be mathematical problems that can be meaningfully formulated within the system, but which the rules of the system cannot decide. Hence, the decidability of paraconsistent arithematics harmonizes with an opinion Wittgenstein maintained thoughout his philosophical career." W also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical `system,' rather than as a motley of pieces assembled by the random processes of natural selection. "Godel shows us an unclarity in the concept of `mathematics', which is indicated by the fact that mathematics is taken to be a system" and we can say (contra nearly everyone) that is all that Godel and Chaitin show. W commented many times that `truth' in math means axioms or the theorems derived from axioms, and `false' means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. W often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel's Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the `rest' of PA it cannot be used in the real world either. As Rodych notes "...Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra-systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics)..." Another way to say this is that one needs a warrant to apply our normal use of words like `proof', `proposition', `true', `incomplete', `number', and `mathematics' to a result in the tangle of games created with `numbers' and `plus' and `minus' signs etc., and with `Incompleteness' this warrant is lacking. Rodych sums it up admirably. " On Wittgenstein's account, there is no such thing as an incomplete mathematical calculus because `in mathematics, everything is algorithm [and syntax]and nothing is meaning[semantics]..." W has much the same to say of Cantor's diagonalization and set theory. "Consideration of the diagonal procedure shews you that the concept of `real number' has much less analogy with the concept `cardinal number' than we, being misled by certain analogies, are inclined to believe" and many other comments (see Rodych and Floyd). As Rodych, Berto and Priest (another pioneer in paraconsistency) have noted, W was the first (by several decades) to insist on the unavoidability and utility of inconsistency (and debated this issue with Turing during his classes on the Foundations of Mathematics). We now see that the disparaging comments about W's remarks on math made by Godel, Kreisel, Dummett and many others were misconceived. As usual, it is a very bad idea to bet against W. Some may feel we have strayed off the path here-after all in "The Limits of Reason" we only want to understand science and math and why these paradoxes and inconsistencies arise and how to dispose of them. But I claim that is exactly what I have done by pointing to the work of W and his intellectual heirs. Our symbolic systems (language, math, logic, computation) have a clear use in the narrow confines of everyday life, of what we can loosely call the mesoscopic realm--the space and time of normal events we can observe unaided and with certainty (the innate axiomatic bedrock or background ). But we leave coherence behind when we enter the realms of particle physics or the cosmos, relativity, math beyond simple addition and subtraction with whole numbers, and language used out of the immediate context of everyday events. The words or whole sentences may be the same, but the meaning is lost. It looks to me like the best way to understand philosophy is enter it via Berto , Rodych and Floyd's work on W, so as to understand the subtleties of language as it is used in math and thereafter "metaphysical" issues of all kinds may be dissolved. As Floyd notes "In a sense, Wittgenstein is literalizing Turing's model, bringing it back down to the everyday and drawing out the anthropomorphic command-aspect of Turing's metaphors." W pointed out how in math, we are caught in more LG's (Language Games) where it is not clear what "true", "complete", "follows from", "provable", "number", "infinite", etc. mean (i.e., what are their COS or truthmakers in THIS context), and hence what significance to attach to `incompleteness' and likewise for Chaitin's "algorithmic randomness". As W noted frequently, do the " inconsistencies" of math or the counterintuitive results of metaphysics cause any real problems in math, physics or life? The apparently more serious cases 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, but he does not really understand their basis, and fails to make clear that self-referencing is involved in the "incompleteness" and "inconsistency" (groups of complex LG's) of mathematics as well. Another interesting work is "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 (see below) since they wrote on universal computation and among his many accomplishments, Da Costa is a pioneer on paraconsistency. Chaitin also contributes to `Causality, Meaningful Complexity and Embodied Cognition' (2010), replete with articles having the usual mixture of insight and incoherence and as usual, nobody is aware that W can be regarded as the originator of the position current as Embodied Cognition or Enactivism. Many will find the articles and especially the group discussion with Chaitin, Fredkin, Wolfram et al at the end of Zenil H. (ed.) `Randomness through computation' (2011) a stimulating continuation of many of the topics here, but lacking awareness of the philosophical issues. Again cf. Floyd on W:"He is articulating in other words a generalized form of diagonalization. The argument is thus generally applicable, not only to decimal expansions, but to any purported listing or rule-governed expression of them; it does not rely on any particular notational device or preferred spatial arrangements of signs. In that sense, Wittgenstein's argument appeals to no picture and it is not essentially diagrammatical or representational, though it may be diagrammed and insofar as it is a logical argument, its logic may be represented formally). Like Turing's arguments, it is free of a direct tie to any particular formalism. [The parallels to Wolpert are obvious.] Unlike Turing's arguments, it explicitly invokes the notion of a language-game and applies to (and presupposes) an everyday conception of the notions of rules and of the humans who follow them. Every line in the diagonal presentation above is conceived as an instruction or command, analogous to an order given to a human being..." W's prescient grasp of these issues including his embrace of strict finitism and paraconsistency is finally spreading through math, logic and computer science (though rarely with any acknowledgement ). Bremer has recently suggested the necessity of a Paraconsistent Lowenheim-Skolem Theorem. "Any mathematical theory presented in first order logic has a finite paraconsistent model." Berto continues: "Of course strict finitism and the insistence on the decidability of any meaningful mathematical question go hand in hand. As Rodych has remarked, the intermediate Wittgenstein's view is dominated by his `finitism and his view [...] of mathematical meaningfulness as algorithmic decidability' according to which `[only] finite logical sums and products (containing only decidable arithmetic predicates) are meaningful because they are algorithmically decidable.'" In modern terms this means they have public conditions of satisfaction-i.e., can be stated as a proposition that is true or false. And this brings us to W's view that ultimately everything in math and logic rests on our innate (though of course extensible) ability to recognize a valid proof. Berto again: "Wittgenstein believed that the naïve (i.e., the working mathematicians) notion of proof had to be decidable, for lack of decidability meant to him simply lack of mathematical meaning: Wittgenstein believed that everything had to be decidable in mathematics...Of course one can speak against the decidability of the naïve notion of truth on the basis of Godel's results themselves. But one may argue that, in the context, this would beg the question against paraconsistentists-- and against Wittgenstein too. Both Wittgenstein and the paraconsistentists on one side, and the followers of the standard view on the other, agree on the following thesis: the decidability of the notion of proof and its inconsistency are incompatible. But to infer from this that the naïve notion of proof is not decidable invokes the indispensability of consistency, which is exactly what Wittgenstein and the paraconsistent argument call into question...for as Victor Rodych has forcefully argued, the consistency of the relevant system is precisely what is called into question by Wittgenstein's reasoning." And so: "Therefore the Inconsistent arithmetic avoids Godel's First Incompleteness Theorem. It also avoids the Second Theorem in the sense that its non-triviality can be established within the theory: and Tarski's Theorem too-including its own predicate is not a problem for an inconsistent theory "[As Priest noted over 20 years ago]. Prof Rodych thinks my comments reasonably represent his views but notes that the issues are quite complex and there are many differences between he, Berto and Floyd. And again, `decidability' comes down to the ability to recognize a valid proof, which rests on our innate axiomatic psychology, which math and logic have in common with language. And this is not just a remote historical issue but is totally current. I have read much of Chaitin and never seen a hint that he has considered these matters. The work of Douglas Hofstadter also comes to mind. His Godel, Escher, Bach won a Pulitzer prize and a National Book Award for Science, sold millions of copies and continues to get good reviews (e.g. almost 400 mostly 5 star reviews on Amazon to date) but he has no clue about the real issues and repeats the classical philosophical mistakes. So with these additional thoughts this book is quite useful.

The book in purely a scientific way catalogues limits in different mental constructs like science, language, mathematics and computer science. While human imaginations are limitless the models we create have limitations.

5.0 out of 5 starsFascinating exploration of the intersection of science and philosophy

28 January 2014 - Published on Amazon.com

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This is a book that is targeted to those who like to think. Although written for a general audience, it would be helpful to have some background in math and science beyond a few long-ago semesters in high school before diving in. You should have some comfort level with set theory, exponents, real and imaginary numbers, and the basic tenets of calculus and probability. The same is true for general physics, such as polarization, particle spin, and the like. Nothing deep, mind you, as the book isn't expecting you to solve equations or anything of the sort. However, the 'outer limits' involve travel beyond where science is now, so knowing something about where science currently is helps.

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

I do have one complaint with this book. There are copious footnotes in each chapter, some which are simple references, but many others which are additional explanatory material. These are all grouped together in a 'Notes' section in the back of the book. This required me to flip continually back and forth from each chapter to the 'Notes' section to read the additional material. It would have been better to present this material as true footnotes on each page; doing so would have eliminated a lot of tedious page-flipping.

I really enjoyed this book; it made think about subjects I hadn't thought about in years. The bibliography is excellent and the author's comments and examples are often surprising but to the point. I think it could be better organized. Self reference keeps popping up, I wish it were all in one place. The book is organized into areas where reason presumably fails. I wish it were organized by types of failure. Also I think he casts too wide a net. I object to his inclusion of some mathematics examples. For instance the inclusion of the ancients' problems demonstrates as he points out later that their problems were not using the correct tools, not that reason was at fault. Similarly for solving quintic equations. It's as if we were out on a starry night and he said "look up. See those moons of Jupiter." I would say that it's not a fault of science or reason that I can't see them. I don't have the proper tool. Give me a telescope and I'll be able to see them. I have the same basic criticisms of including quantum theory. If there is not a final theory why say that there is a limit of science or reason to understand quantum phenomena? He may acknowledge that there are several theories or maybe a theory that no one has ever thought of which might finally be right. But that doesn't prove that there is a limit to what science might conclude in the future. Overall I think the author's best suit is logic and computing and I think it would have been a better book to have stuck to those subjects.

One shouldn't be dissuaded by my criticisms. Read the book, you'll enjoy it. I wouldn't have gone to the trouble of writing what I did write if I didn't like the book.

This book is a real tour de force. It covers difficult topics in the areas of philosophy, science and mathematics but writes in a way that makes these topics accessible to an intelligent layperson who need not have studied advanced math or physics. I do think that professional philosophers, mathematicians, and physicists will also enjoy the book. The book starts out with paradoxes involving self reference in ordinary language, eg. the liars paradox, and proceeds to cover Godel's theorem, Turing's theorem and the work of Cantor. The book goes on to cover problems in quantum mechanics and relativity theory in a very lucid way.

The author has a doctorate in mathematics and is a computer science professor. Given the way he writes, he could be an English professor as well. The great mathematician David Hilbert said that a mathematician should be able to explain what he is working on to an intelligent person off the street; this author has put this advice into action with this book. I have a Ph.D. in philosophy from Princeton. I wish this book had been written when I was an undergraduate or graduate student as it gets a reader to the edge of research in these areas very quickly

A unifying theme of the book is to point out our limitations of knowledge in the various areas ordinary language, mathematics and science. But it treats the various areas rigorously on their own terms without trying to make loose analogies at the outset. At the end, after doing the hard, detailed work in the various areas, the author offers remarks tying the various areas together. It is much better than a book like "Godel, Bach, and Escher" which rather strains to force analogies from different areas. This work treats the different areas of language, computer science, mathematics and physics on their terms before offering insightful comments at the end on what the results from the different areas show us.

It is the best book I have read in the last 10 years and the best of its kind that I have seen.

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