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

Much of what science focuses on is reductionism and in particular the goal of reducing the world to the bare minimum of assumptions and postulates that can then be built up into all the complexity we see and feel. The outer limits of reason looks at the limits of what we can know and what we can reduce. It looks at lots of different ideas separately and analyzes lots of different forms of limitations that we must deal with. I'll give a quick overview of the subject matter.The author starts out by looking at limits to logical consistency in our evolved language. He looks at some repurcussions of paradox's on self referencing sentences and shows how such systems can be looked at symbolically. The author then starts discussing how there are many ideas which do not have a platonic ideal and that their meaning is effectively subjective. The author discusses an eroding artifact that is refurbished and questions at one point is the original not considered the same as the remaining. There is no right answer to such a question. The author discusses Zeno and some of the issues we face when considering infinity. The author ends the chapter by considering logic methods that can accomodate that we humans dont define every object or word precisely before using, in particular systems like fuzzy logic. Where truth values can be indeterminate. The author then moves onto set theory and looks at Cantor's analysis of infinity. Countable infinity is considered as well as uncountable infinity and the ladders above infinity created by power sets. These treatments are self contained in the book though some outside knowledge wouldnt hurt. The author then moves onto the limits of computation in particular polynomial time problems as well as NP problems. The author considers the problems associated with figuring out the travelling salesman problem and then frames other problems relative to one another in terms of difficulty and shows how many are in fact equivalent. The author's goal is to show how some problems which are easy to state currently only have solutions which can be found in times that are longer than the age of the universe and that quantum computation would not be the answer to solving such problems. The author then moves onto basic computation theory and Turing's halting problem. He shows how there can be no program which knows whether a given input program with a given input will compute in finite time. He uses counting arguments from a previous argument to show how this works and as such starts to refer to earlier material. The author then discusses the hierarchy of larger and larger systems for which one cannot solve the halting problem even with the help of divine oracles. The author moves on to the limits of what can know physically. He discusses both chaos and quantum mechanics and discusses how the precision needed to know initial conditions is a fundamental constraint in being able to predict a clockwork universe and discusses how separately quantum mechanics makes particle dynamics fundamentally uncertain as a separate limitation. The author then discusses science and philosophy and how science has philosophical limits imposed on it. He discusses how induction is used as a principal to "prove" our scientific postulates which then form the bedrock of the emergent theory dependent on those postulates. We have no proof that at some point experiments might not repeat themselves as they have in the past. The author discusses the relationship between science and math and is very well measured about why and how there are relationships and what to make of them. He discusses the anthropic principle and its scientific worth. The author then focuses on some more purely mathematical ideas like the real numbers, rational, irrational and within that algebraic and transcendental numbers, he discusses group theory and goes back to computation and then covers some pure logic and arithmetical systems and some of Godel's ideas. The author goes into how mathematical systems cannot deduce whether all statements about a given system can be shown to be true or false and that there are limits to knowledge about any given system irrespective of how many assumptions/axioms are included.There are a lot of different ideas the author discusses when familiarizing the reader with many of the limits of logic and reason we are fundamentally faced with. The author discusses grammar systems where terms are not all well defined, he discusses the problems associated with continuous space and motion on that space, he discusses chaos and quantum mechanics as well as the difficulty of computability of many easy to state problems- some due to time constraints others more fundamentally by the limits of what we can know. The author covers the limits of logic being able to answer all questions posed within any given mathematical system. A lot of information and thought has gone into this and one can learn a lot from this book. Each topic can be further researched and all are interesting overviews of where our knowledge fundamentally hits ceilings. I enjoyed reading much in this book.

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 placeYanofsky 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 entanglementDon’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.

Gran libro para todo aquél que tenga interés por la ciencia. Dada la naturaleza de la misma, hay conocimiento que está más allá de su alcance.

Amazing book! I have probably read the book thrice and continue to return to specific chapters for ready-reckoning. Prof. Yanofsky is able to explain extremely tricky subjects (minefields of paradoxes that relativity, logic and quantum mechanics are) with spectacular simplicity! Such simplicity, I am convinced, can come only to one who understands the subject completely! Other books on these subjects (Penrose, Hofstadter, Hawking) tend to digress, prevaricate, and over simplify easy things while expecting the reader to intuit the hard ones (that the authors invariantly gloss over). What makes this book delectable is that it ties disparate subjects -- physics, sciences and math -- with a common thread, namely the limits of reason they all stumble upon. To appreciate that limitation, one needs to know the foundations of relativity, quantum mechanics and logic as would a grad student! Yonofsky straddles all these subjects and guides the reader gently, by hand, to the edge of the precipice from where the view of the cosmos is just breathtaking! If the reader is philosophical minded, he will not be the same person after reading this book!This is among the top ten best books of my life!P.S. I read this book to learn about the implications of Godel's theorems. At the end of the book, Yonofsky warns that the theorems should not be carelessly employed to make pronouncements about the sciences, universe, whatever. Ha ha, that was funny, but a welcome reprimand for I was guilty of it!

Excellent BookAuthor has a talent for clearly explaining complex concepts

A must-read for thoughtful people, and very well written.

Dies ist eines der besten Bücher die ich in den letzten Jahren gelesen habe. Wunderbar einfach und klar dargelegt werden die Grenzen der komplexesten Fachgebiet der Wissenschaft wie auch die Philosophie der Wissenschaft selbst aufgezeigt. Es liest sich wie ein Krimi und ist doch einfach "nur" Mathematik und Physik sowie Philosophie.