Really enjoyed the 75% or so that I understood and I just skimmed the rest. ( )

The Outer Limits of Reason is as far as I remember one of the best research essay that I have read so far. I definitely recommend the book to all inquiring minds around and to anyone who is too much confident about him/her-self knowledge.
Noson S. Yanofsky confirmed his knowledge page by page and with astonishing clarity explain hard topic such as Chaos, Relativity Theory and Quantum Mechanics. As a graduate student of Computer Science and Engineering I have found really well written explanations about theoretical computer science. Not for nothing 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.
Awards
Winner, 2013 American Publishers Award for Professional and Scholarly Excellence (PROSE Award) in Popular Science & Popular Mathematics, presented by the Professional and Scholarly Publishing Division of the Association of American Publishers
Quotes
Science is a human activity. It is created by finite, flawed human beings attempting to search for the ultimate truth.

The mathematics becomes abstract and about nothing in particular. Because these concepts are about nothing, they are about everything.

Rather than asking why the laws of physics follow mathematics, ask why there are any laws at all.

When we talk about the limits of scientific reasoning, we must keep in mind how we are observing the universe…the way we look at the universe is the way it will present itself to us.

Do not mistake the metaphor for reality.

SPOILER: Yanofsky conclusion thought
We human beings already live beyond reason. Real life has importance only when it includes ethics, values, and beauty. Reason is a powerful but nevertheless limited tool.
Table Of Contents

• Introduction 1


• Language Pradoxes 15
  
  
  
  • Liar! Liar! 15
  
  
  • Self-Referential Paradoxes 19
  
  
  • Naming Numbers 26
  
  


• Philosophical Conundrums 31
  
  
  
  • Ships, People, and Other Objects 31
  
  
  • Hangin’ with Zeno and Gödel 41
  
  
  • Bald Men, Heaps, and Vagueness 50
  
  
  • Knowing about Knowing 57
  
  


• Infinity puzzles 65
  
  
  
  • Sets and Sizes 66
  
  
  • Infinite Sets 69
  
  
  • Anything Larger? 76
  
  
  • Knowable and Unknowable 85
  
  


• Computing Complexities 97
  
  
  
  • Some Easy Problems 98
  
  
  • Some Hard Problems 109
  
  
  • They’re All Connected 121
  
  
  • Almost Solving Hard Problems 129
  
  
  • Even Harder Problems 131
  
  


• Computing Impossibilities 135
  
  
  
  • Algorithms, Computers, Machines, and Programs 136
  
  
  • To Halt or Not to Halt? 139
  
  
  • They’re All Connected 146
  
  
  • A Hierarchy of the Unknown 152
  
  
  • Minds, Brains, and Computers 157
  
  


• Scientific Limitations 161
  
  
  
  • Chaos and Cosmos 161
  
  
  • Quantum Mechanics 175
  
  
  • Relativity Theory 214
  
  


• Metascientific Perplexities 235
  
  
  
  • Philosophical Limitations of Science 235
  
  
  • Science and Methamatics 252
  
  
  • The Origin of Reason 272
  
  


• Mathematical Obstructions 297
  
  
  
  • Classical limits 298
  
  
  • Galois Theory 304
  
  
  • Harder Than Halting 309
  
  
  • Logic 320
  
  
  • Axioms and Independence 331
  
  


• Beyond Reason 339
  
  
  
  • Summing Up 339
  
  
  • Defining Reason 345
  
  
  • Peering Beyond 349
  
  


• Notes 355


• Bibliography 379


• Index 393

( )

A wide and shallow review of various logical conundrums, language paradoxes, mathematical impossibilities, and quantum confusion. Yanovsky presents all this as representing the limits of ‘reason,’ but the kind of assumptions that he makes let you know that he is a computer scientist and not a philosopher.

I read a lot of popular mathematics books, so there is little that is new here, but it's interesting to find all these paradoxes in a single place. A good synthesis of the limits of logic and the strangeness of contemporary physics. ( )

Back in the day, sometime in 1930, David Hilbert, a very smart mathematician said

"Wir müssen wissen. Wir werden wissen" (We must know. We will know. (It sounds way more authoritative in German, try it.))

to which another smart guy, Mr. Gödel had already replied:

"Hold my beer..."

The rest is history. And what a history it's been. Along the stellar achievements of humankind are the ones that sound like "oops, seems like there's no way we can solve this, here's the proof. Sorry." Because, we wouldn't be considered that smart if we didn't know the limits of our methods, right? A man's got to know his limitations, as Dirty Harry said once.

This book, in a sense, is a very brief introduction into the wonderful world of philosophical, mathematical, and physical limitations, as well as all sorts of paradoxes you'll find if you force your way into some realms. The limitations are not there because we're not smart enough or for the lack of computers that are sophisticated enough; on the contrary, the book's topic is the limitations that are fundamental, that arise from the logic, from the foundations of mathematics, as well as physics. Those limitations are described mathematically, and proved. They constitute a domain into which you can look, and ponder about the mysteries of universe, and the whole human reason. (The author tried to do that, without falling into much mysticism, though one could be forgiven to do just that when the issues are so fundamental and proved to be beyond reason).

I think this book can serve as a nice introduction for undergraduate students that are just getting started with logic, mathematics, physics, philosophy of science, computer science, and linguistics. The discussion and presentation in this book serve as a good starting point for many fresh minds, as well as a warning to the practitioners in those fields: E.g. if you're a programmer, and someone says something can't happen because, you know, Halting Problem, you won't try to object (that is, if the claim can really be reduced to the Halting Problem). And also when you're faced with a problem that can be reduced to a Traveling Salesman Problem, you won't throw in the towel, but will try to find some approximate algorithm, knowing full well that the perfect solution will stay out of your reach.

In other words, this book is in the same category as "What We Cannot Know" by Marcus du Satoy, and both of the volumes give nice introductions. I found this book more fluid, and I've especially liked the notes at the end because all of them had very nice pointers to more detailed books and articles.

If you're curious about the impossibility results in mathematics, logic, physics, linguistics, and computer science, and you haven't read much about these before, then I can easily recommend this book. If on the other hand you had basic mathematics, engineering, physics, logic, computer science, linguistics, and philosophy of science classes, then probably you'll not find anything new in this volume. Still it can act as a concise refresher, and you might even learn a thing or two such as Yablo's paradox. ( )