Slicing the Truth: On the Computable and Reverse Mathematics of Combinatorial Principles (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore) by Denis R Hirschfeldt (PDF)

Description

This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey’s Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions.

User’s Reviews

Editorial Reviews: From the Back Cover This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey’s Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a nice and much needed introductory text to the topic of reverse maths, where one investigates the strength of various theorems, in the sense of the principles needed to prove them and conversely the facts they imply (under a minimal formal framework). The writing style is conversational and personal. The readers that are familiar with the previous monograph of the author (joint with Rod Downey) will find that this is a very different type of book. It does not aim to be exhaustive (and it is far from it). Instead, it tries to tell a story of reverse maths in a rather personalized and often slightly poetic style. You will find sentences like “…the ashes of Hilbert’s Program have proved a fine fertilizer” as well as quotations like “Nothing is built on stone, all on sand, but our duty is to build as if the sand were stone” in discussions describing the development of the subject and the underlying philosophy.There is something to be said about encyclopedic monographs. Have you tried to read such books from beginning to end? When I subject myself to such exercises (misguided by the idea that I should know everything in an area before I start working in it) I quickly give up with the same feeling you get when you read a dictionary or an encyclopedia, chapter-by-chapter.It is always futile to set out to exhaust a subject, before you exhaust your readers. Denis is well aware of that, and opted to telling a story instead.On the other hand, intuitive ideas are almost always followed by rigorous mathematical arguments and facts, which is to say that this is a solid mathematical textbook. The organization of the material is also very attractive, starting with motivation, a description of the mathematical tools used, and gradually building up to the main topics that the authors chose to cover, and in particular the study of combinatorial principles like Ramsey’s theorem. The book is full of interesting digressions, and the reader receives a warning for this early in the introduction. One of them that stands out and is particularly interesting, is Chapter 5 which is entitled “In Defense of Disarray”, where the author steps back and discusses the reverse maths programme under his epistemological views against certain types of unnecessary organization. Nice contrast as, in a way, reverse maths is an effort to organize mathematics. Oh well, that’s all for now – this is not meant to be a thorough review. It is worth noting that the text is written by a computability theorist, as this subject can be approached from all the different areas of mathematical logic, and different disciplines have vastly different styles in presenting the same theory. Not much background is needed, except basic understanding of mathematical logic and some computability theory. Wow I haven’t said anything negative! This is definitely a well written book. If I find anything annoying I will come back and write about it.

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