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13 of 15 people found the following review helpful
4.0 out of 5 stars Stimulating and enticingly written, but flawed in places., 7 Dec. 2013
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This review is from: The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (Hardcover)
This book, despite some defects, is well worth buying if only for the few magnificent, lucidly written sections, some of which are much better than I've seen elsewhere. For example, the section on Zeno's paradoxes (pp41-49)makes the point that history has sold us short, by having the paradoxes handed down to us by people who wanted to prove him wrong (Zeno's original writings having been lost). Yanofsky does a truly brilliant job of presenting and examining their true, and astounding, import. Another very impressive section is that on quantum entanglement and Bell's inequalities (pp194-201) - all so convincing and elegant. And the geometrical proof of the irrationality of the square root of 2 (pp300-301)is the most concise and magical I've seen.
Unfortunately, there are some irritations and errors. Who, for instance, could possibly be interested in ploughing through the detailed argument (p116) of converting a 31-digit number of seconds to a number of centuries (this given to 15 digit accuracy!)? "Divide by 60...." blah blah blah.
On page 214 we read that the special theory of relativity "deals with the universe without gravity or acceleration"; then on page 226 we read about acceleration in special relativity, which of course is perfectly ok. Indeed, relativity doesn't get very careful treatment: page 221 tells us, on the subject of length contraction, "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". But no mention is made here of frames of reference. In its rest frame, the shuttle most emphatically does NOT shrink. Further on, 4 dimensional spacetime is attributed to Einstein, with no mention of Minkowski. And on page 225, we read the word 'pressure' where 'force' would be appropriate.
Yanofsky's mathematical remarks are suspect in places: either he is oversimplifying without saying so, or he is unaware of certain crucial details. One example: on page 335, mention is made of Galois Theory concerning the solvability (using the 4 basic operations, and taking roots) of certain equations(presumably polynomial with integer coefficients, but I'll take that as given). He states, "However, there are calculus-based methods that can always solve such equations". Really? In terms of radicals? I don't think so! The quintic can be solved in terms of elliptic functions, and some higher degree equations in terms of modular functions, but it's clear that Yanofsky has 'tried too hard' here, and made a hash of things.
Forget the defects though, because most of the book is magnificent, and eminently readable.
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3 of 4 people found the following review helpful
4.0 out of 5 stars Erudite, stimulating and informative, 4 Jan. 2014
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Paul P. Mealing (Melbourne, Australia) - See all my reviews
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This review is from: The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (Hardcover)
This is an excellent book for anyone with an interest in epistemology, science and mathematics. Yanofsky delivers on his title and then some. Using worked examples, anecdotes and historical analysis, Yanofsky demonstrates that there are and always will be limits to what we can possibly know. What's more, what we don't know will always infinitely outweigh what we do know. Along the way, he discusses classical and quantum physics, determinism and free will, Turing's halting problem, Godel's Incompleteness Theorem, the P-NP problem, the structure of the Universe and the anthropic principle (not an exhaustive list). Neither does Yanofsky avoid philosophical issues like mathematical Platonism and why did the Universe create an intelligence capable of reflecting and understanding its own origins and structure. This is a deep book on many levels, intellectually stimulating, informative and reflective. A combination often attempted but rarely achieved with such aplomb. Easy to read, given its esotericism.
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2 of 3 people found the following review helpful
5.0 out of 5 stars If you think this is going to be the familiar popular science tropes, please be assured this is a fairly original work, 21 Sept. 2014
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This review is from: The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (Hardcover)
Great book. Very well written with some very interesting points to be made. If you think this is going to be the familiar popular science tropes, please be assured this is a fairly original work well worth a weekend of your time.
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0 of 1 people found the following review helpful
4.0 out of 5 stars Stimulating look at how we get into trouble when we ignore the subtleties of language, 28 Dec. 2014
This review is from: The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (Hardcover)
The review by Featherless Biped is the only really useful one (see USA site) 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.
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0 of 3 people found the following review helpful
5.0 out of 5 stars Great, 22 Dec. 2013
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This review is from: The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (Hardcover)
Bought this as a christmas present and it looks good and arrived on time. Excellent thank you. My husband will love it.
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0 of 4 people found the following review helpful
4.0 out of 5 stars Mount Evariste, 30 July 2014
This review is from: The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (Hardcover)
The quintic equation can be solved with elliptic functions, which is outside the definition of solution by radicals that Galois worked within. I've heard there are also developments in differential geometry that do the same.
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