View source for Reverse Mathematics: Proofs from the Inside Out

{{short description|Book by John Stillwell}}
{{italic title}}
[[File:Reverse Mathematics.jpg|thumb|First edition]]
'''''Reverse Mathematics: Proofs from the Inside Out''''' is a book by [[Biography:John Stillwell|John Stillwell]] on [[Reverse mathematics|reverse mathematics]], the process of examining proofs in mathematics to determine which [[Axiom|axiom]]s are required by the proof. It was published in 2018 by the Princeton University Press.{{r|kahle|hirst|cohen|bultheel|kossak|case}}

==Topics==
The book begins with a historical overview of the long struggles with the [[Parallel postulate|parallel postulate]] in [[Euclidean geometry]],{{r|cohen}} and of the [[Philosophy:Foundations of mathematics#Foundational crisis|foundational crisis]] of the late 19th and early 20th centuries,{{r|case}} Then, after reviewing background material in [[Real analysis|real analysis]] and [[Computability theory|computability theory]],{{r|kahle}} the book concentrates on the reverse mathematics of theorems in real analysis,{{r|cohen}} including the [[Bolzano–Weierstrass theorem]], the [[Heine–Borel theorem]], the [[Intermediate value theorem|intermediate value theorem]] and [[Extreme value theorem|extreme value theorem]], the [[Heine–Cantor theorem]] on [[Uniform continuity|uniform continuity]],{{r|case}} the [[Hahn–Banach theorem]], and the [[Riemann mapping theorem]].{{r|kossak}}
These theorems are analyzed with respect to three of the [[Reverse mathematics#The big five subsystems of second-order arithmetic|"big five" subsystems of second-order arithmetic]], namely arithmetical comprehension, recursive comprehension, and the weak Kőnig's lemma.{{r|kahle}}

==Audience==
The book is aimed at a "general mathematical audience"{{r|kahle}} including undergraduate mathematics students with an introductory-level background in real analysis.{{r|hirst}} It is intended both to excite mathematicians, physicists, and computer scientists about the [[Philosophy:Foundations of mathematics|foundational]] issues in their fields,{{r|case}} and to provide an accessible introduction to the subject. However, it is not a textbook;{{r|cohen|bultheel}} for instance, it has no exercises. One theme of the book is that many theorems in this area require axioms in [[Second-order arithmetic|second-order arithmetic]] that encompass infinite processes and [[Computable function|uncomputable function]]s.{{r|cohen}}

==Reception and related reading==
Jeffry Hirst criticizes the book, writing that "if one is not too obsessive about the details, ''Proofs from the Inside Out'' is an interesting introduction," while finding details that he would prefer to be handled differently, in a topic for which details are important. In particular, in this area, there are multiple choices for how to build up the arithmetic on [[Real number|real number]]s from simpler data types such as the [[Natural number|natural number]]s, and while Stillwell discusses three of them ([[Decimal|decimal]] numerals, [[Dedekind cut]]s, and nested intervals), converting between them itself requires nontrivial axiomatic assumptions.{{r|hirst}}

However, James Case calls the book "very readable",{{r|case}} and Roman Kossak calls it "a stellar example of expository writing on mathematics".{{r|kossak}} Several other reviewers agree that this book could be helpful as a non-technical way to create interest in this topic in mathematicians who are not already familiar with it, and lead them to more in-depth material in this area.{{r|kahle|hirst|cohen}}

As additional reading on reverse mathematics in [[Combinatorics|combinatorics]], Hirst suggests ''[[Slicing the Truth]]'' by Denis Hirschfeldt.{{r|hirst}} Another book suggested by reviewer Reinhard Kahle is Stephen G. Simpson's ''Subsystems of Second Order Arithmetic''.{{r|kahle}}

==References==
{{reflist|refs=

<ref name=bultheel>{{citation
 | date = August 2018
 | journal = EMS Reviews
 | publisher = European Mathematical Society
 | title = Review
 | url = https://euro-math-soc.eu/review/reverse-mathematics-proofs-inside-out}}</ref>

<ref name=case>{{citation
 | last = Case | first = James
 | date = March 2019
 | journal = SIAM News
 | title = A new mathematical field answers old questions
 | url = https://sinews.siam.org/Details-Page/a-new-mathematical-field-answers-old-questions}}</ref>

<ref name=cohen>{{citation
 | date = October 2018
 | doi = 10.1080/00029890.2018.1502995
 | issue = 9
 | journal = [[American Mathematical Monthly]]
 | pages = 860–864
 | title = Review of ''Reverse Mathematics''
 | volume = 125| s2cid = 215791768 }}</ref>

<ref name=hirst>{{citation
 | last = Hirst | first = Jeffry L.
 | date = June 2018
 | doi = 10.1017/bsl.2018.19
 | issue = 2
 | journal = Bulletin of Symbolic Logic
 | jstor = 26473950
 | pages = 176–177
 | title = Review of ''Reverse Mathematics''
 | volume = 24| s2cid = 126256370
 }}</ref>

<ref name=kahle>{{citation
 | last = Kahle | first = Reinhard
 | journal = [[Mathematical Reviews]]
 | mr = 3729321
 | title = Review of ''Reverse Mathematics''}}</ref>

<ref name=kossak>{{citation
 | last = Kossak | first = Roman
 | date = November 2018
 | doi = 10.1007/s00283-018-9841-3
 | issue = 1
 | journal = [[The Mathematical Intelligencer]]
 | pages = 81–82
 | title = Review of ''Reverse Mathematics''
 | volume = 41| s2cid = 125295465
 }}</ref>

}}

[[Category:Mathematical logic]]
[[Category:Proof theory]]
[[Category:Computability theory]]
[[Category:Real analysis]]
[[Category:Mathematics books]]

{{Sourceattribution|Reverse Mathematics: Proofs from the Inside Out}}