Peter Smith

"Beta version" of forthcoming 2024 book. Corrected and expanded version of much-downloaded notes on category theory, which should provide entry-level, relatively gentle, preliminary reading for those going on to take an industrial-strength course. Should also be approachable by those with just a little maths who want to know something of what the categorial fuss is about.

An Appendix to the Beginning Mathematical Logic Study Guide -- the Guide is arranged by topic, while this Appendix looks book-by-book at a somewhat random selection of texts. The discussions range from half a page to more detailed three or four page notes. This could be of interest to (relatively) beginning math logic students who want to know a bit about what is covered where.

From the blurb: "Formal logic provides us with a powerful set of techniques for criticizing some arguments and showing others to be valid. These techniques are relevant to all of us with an interest in being skilful and accurate reasoners. In this very accessible book, extensively revised and rewritten for the second edition, Peter Smith presents a guide to the fundamental aims and basic elements of formal logic. He introduces the reader to the languages of propositional and predicate logic, and develops natural deduction systems for evaluating arguments translated into these languages. His discussion is richly illustrated with worked examples and exercises, and alongside the formal work there is illuminating philosophical commentary. This book will make an ideal text for a first logic course and will provide a firm basis for further work in formal and philosophical logic."

This corrected version of the second edition is free to download, but is also available as a low cost paperback by print on demand from Amazon

From the blurb: “In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter?  Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book -- extensively rewritten for its second edition -- will be accessible to philosophy students with a limited formal background. It will be of equal interest to mathematics students taking a first course in mathematical logic.”

Gödel's famous First Incompleteness Theorem shows that, for any sufficiently rich theory that contains enough arithmetic, there are some arithmetical truths the theory can express but cannot prove. How is this remarkable result established? This short book explains. It also discusses Gödel's Second Incompleteness Theorem. The aim is to make the Theorems available, clearly and accessibly, even to those with a quite limited formal background.

The book is available both as an inexpensive paperback and as a hardback (details on my website.)

John Earman wrote on the cover of my 1998 book Explaining Chaos “This book is a splendid achievement. With a minimum of technical apparatus, the author gives the reader a good feeling for the mathematics that underlies the collection of phenomena that collectively have come to be known as chaos. At the same time he provides a much needed debunking of the breathless claims about the revolutionary nature of chaos theory. There are also major positive contributions to our understanding of the nature of scientific methodology.”

I couldn't have put it better myself. If by some unhappy chance you've missed out on reading the book (it's quite short), then now you have no excuse not to rectify the omission. For by very kind permission of Cambridge University Press, you can now freely download a copy.

[March 28, 2024 version] An update to the early chapters of my earlier Gentle Introduction notes, requiring only modest mathematical background. There are chapters on categories, and on constructions like products, pullbacks, exponentials that can occur in different categories. There is also  a first encounter with functors.

Part II continues the story, talking about natural transformations between functors, the Yoneda lemma and adjunctions and we take an introductory look at the idea of an elementary topos

This book Part I is available as a cheap print-on-demand paperback from mid April, but I think of it as a beta version, still work in progress and all comments are still most welcome.