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.
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Top reviews from the United States
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- Reviewed in the United States on February 25, 2014
- Reviewed in the United States on May 31, 2020
Top reviews from other countries
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Alan Rodriguez GonzalezReviewed in Mexico on December 5, 2019
Excelente libro de divugación científica
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- A CustomerReviewed in the United Kingdom on September 21, 2014
If you think this is going to be the familiar popular science tropes, please be assured this is a fairly original work
- Kindle CustomerReviewed in India on November 27, 2018
The most simple & lucid explanations of arguably the most abstruse principles in Sciemce & Logic!
