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Reverse Mathematics: Proofs from the Inside Out
The first book surveying the history and ideas behind reverse mathematics
Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. In Reverse Mathematics , John Stillwell offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. 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.
Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. In Reverse Mathematics , John Stillwell offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. 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.
200 pages, Paperback
Published September 24, 2019
21 people are currently reading
329 people want to read
About the author
John Stillwell
51 books60 followersJohn Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University
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Displaying 1 - 6 of 6 reviews
December 15, 2024
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.
November 3, 2018
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.
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.
May 24, 2018
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.
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.
March 26, 2024
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.
March 21, 2023
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.
Worth reading for anyone interested in the philosophy of mathematics or generally skeptical about the real numbers and analysis.
March 21, 2023
This book covers some extremely interesting topics and ideas.
Displaying 1 - 6 of 6 reviews