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Noson S. Yanofsky.
The Outer Limits of Reason: what science, mathematics, and logic cannot tell us.
MIT Press. 2013

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes.

Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world—our ideas about space, time, and motion, and the complex relationship between the knower and the known.

Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.

Noson S. Yanofsky.
Theoretical Computer Science for the Working Category Theorist.
CUP. 2022

rating : 2 : great stuff
review : 11 April 2022

Using basic category theory (category, functor, natural transformation, etc.), this Element describes all the central concepts and proves the main theorems of theoretical computer science. Category theory, which works with functions, processes, and structures, is uniquely qualified to present the fundamental results of theoretical computer science. In this text, we meet some of the deepest ideas and theorems of modern computers and mathematics, e.g., Turing machines, unsolvable problems, the P=NP question, Kurt Godel’s incompleteness theorem, intractable problems, Turing’s Halting problem, and much more. The concepts come alive with many examples and exercises. This short text covers the usual material taught in a year-long course.

I have been wanting to dip my toe in the sea of category theory for a while now, but have not found a suitable entry point. The books written for mathematicians are impenetrable. Those written for computer scientists lose me in the detail. When I came across this one, a book about computer science, written for category theorists, I was intrigued. I was reminded of a colleague of mine who collected foreign languages: he would learn them by reading a text he knew well in the language he wished to learn (memory offers up the thought it was Macbeth, but it may have been a different Shakespeare play). Maybe I could similarly learn the foreign language of category theory by reading about a topic I know, when written in that language? Especially since the book is a brisk 130 pages.

So, I am definitely not the target audience. And yet I feel I have learned more about category theory from this book than from any of the others I have tried. Not enough to be fluent, but enough that I can see what the concepts are, how they are put together, and how they can help illuminate a subject.

This was initially available as a PDF from the publishers site. I downloaded, and started active reading, which involved annotating the PDF with questions, thoughts, and ideas. (I come from a generation where annotating an actual physical book is Not Done.) I did skim a few of the more technical parts, but did read the whole thing. I then bought a hard copy.

As well as having a better understanding of Category Theory, I have a new perspective on theoretical computer science: the subset of functions that comprise computable functions is somewhat ugly to formalise.

This is the first in a series of Cambridge Elements: shortish works on Applied Category Theory. This one is highly recommended: I will be looking out for the others.