Reverse Mathematics: Proofs from the Inside Out

by John Stillwell

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis—finding the “right axioms” to prove fundamental theorems—and giving a novel approach to logic.Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the “right axiom” to prove it.By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

Genres: Mathematics, Science, Logic, Physics, Nonfiction

200 pages, Kindle Edition

Published January 1, 2018

About the author

John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University

Reviews

[J.]

With this text in hand, it's now possible to talk about RM with a general philosophical audience. Its main weakness, of course, is the lack of detailed treatment of the two more powerful principles of RM, namely Arithmetic Transfinite Recursion and Pi-1-1 Comprehension. But that just means that I eagerly await a sequel.

[William Schram]

With Reverse Mathematics John Stillwell demonstrates the ideas and properties of Reverse Mathematics. I know it is a bit of a tautology to speak of it in this manner, so I will try to explain. Stillwell shows that mathematics had always been establishing axioms and then finding the results of those axioms. Reverse Mathematics goes in the other direction by taking a theorem and finding the axioms needed to prove it.

It is an interesting approach to Analysis, but I don’t really know all that much about Mathematical Analysis. I never took it in school, since I only got up to Calculus II in college. Therefore, there are several things that I find annoying, but that is only because I wasn’t paying attention the first time through. For instance, there is a portion where ZF is mentioned and I didn’t know what that was. The book might use a lot of acronyms but it usually explains what they mean before the author dives into the gist. This is important for when Stillwell decides to drop stuff like ACA and WKL and other such ideas.

In writing this book, it is clear from the text that the author wanted to establish a solid foundation for analysis given some advances in Logic. This is mentioned in the text itself. It doesn’t have any problems to solve or questions to answer.

The book is quite interesting as I mentioned before. It is rather short, but it is densely packed with ideas. It doesn’t have a glossary, choosing instead to jump straight to the Bibliography and Index. In that sense, I could say that the author dropped the ball, but it is possible that the editor thought it unnecessary for the target audience. Then again, you could always search for what he means by using the Internet.

[Jared Davis]

I'm very much enjoying this book. It's been recomended to me before; I don't know why I didn't pick it up sooner.

As I read, I'm playing a game where I reconstitute coursework I had taken in college, high school, or independent studies later in life. The obvious touchpoints are things like Euclid's postulates, Reimann integrals, or epsilon/delta proofs. Newer (to me) ideas are König's lemmas or the recursive comprehension axiom. But even in these less familiar areas, Stillwell presents the concepts in such a way that I recognize that they were around before, lurking in coursework I didn't pay attention to or missed out on.

[Zak]

Sûrement le meilleur point d'entrée dans cette discipline peu connue et pas uniquement parce que c'est le seul existant.
L'ouvrage n'est pas bien long, mais nécessite tout de même d'y consacrer du "temps de concentration disponible" pour bien saisir les détails. L'auteur essaie de décrire ce que sont les mathématiques à rebours en laissant les aspects trop techniques et spécifiques de côté pour permettre aux plus grand nombre de mathématiciens de comprendre les grandes idées et principes de ce programme. C'est réussi.

L’aspect historique est d'abord couvert puis le livre se structure en différents chapitres permettant d'aborder différents fondamentaux: arithmétisation classique, analyse classique, calculabilité, arithmétisation de la calculabilité pour enfin arriver à la "compréhension arithmétique" ou les précédentes parties sont mises en applications pour exposer les "puissances" des systèmes axiomatiques les plus communs de cette discipline pour les théorèmes d'analyse. Un dernier chapitre porte enfin sur une discussion plus légère sur des sujets un peu plus profonds qui semble être touchés, au moins partiellement, par cette branche de la logique.

Le livre présente réellement des résultats de logiques mathématiques (et d'une façon un peu tordue d'informatique théorique) qui mêlent arithmétique, algèbre, analyse, axiomatique et calculabilité. Même sans plonger dans les détails trop techniques, les théorèmes sont présentés avec leurs démonstrations, et le livre est très accessible pour une personne avec des connaissances, même basiques, dans ces divers domaines. On en ressort avec beaucoup de nouvelles questions et une nouvelle façon d'appréhender ou de concevoir des résultats connus et inconnus jusqu'alors.

Je partais pourtant avec l'a priori que les mathématiques à rebours était une niche remplie de matheux aimant se compliqué la vie pour la beauté du sport. Je pense maintenant que cela pourrait jouer un grand rôle pour mieux comprendre les relations entre les différents domaines, ce qui à l'ère des mathématiques par IA pourrait amener à de nouvelles percées et une compréhension plus fine de la mathématique.

[Miguel Aguilar]

A very well written book about the interplay between logic and analysis. A good deal of mathematical knowledge is necessary to understand it fully, even though one can go trough it just having some decent undergrad studies in math. I bought it to motivate my self to write my master thesis and that aim has been accomplished.

[Julian]

I really enjoy Stillwell's accessible writing on mathematics; Mathematics and Its History is still one of my favorite books. This book was no exception, and although I had originally obtained it hoping for more details about where ideas for specific axioms and systems came from and the failures along the way (the "wrong axioms"), I enjoyed what it is, which is a bit of historical recap on the search for foundations, and then a building up of the Big Five systems of second order arithemetic that seem to be the key systems of reverse mathematics.

Worth reading for anyone interested in the philosophy of mathematics or generally skeptical about the real numbers and analysis.

[Tobias]

This book covers some extremely interesting topics and ideas.