NB. I was sent this book as a review copy.
I’m not sure I’ve read a mathematics book which was so hard to review, not because of the quality of the book (which is superb), but because the way of thinking is in some senses so different to the way we normally think about mathematics. This, indeed, is also the book’s best feature. This book gets you thinking about mathematics in ways which I have never explored before, and which have definitely given me a new, and I think, improved perspective on formal mathematics.
In general in mathematics we start with a set of assumptions (axioms), and explore the consequences of them. Within Euclidean geometry we start with ideas about lines, and points, and circles, and then see what other theorems can be proved from these. Within set theory too, we start with a set of ideas about equalities of sets, existence, pairings, unions etc which we hold to be true and then see what can be said of other properties of sets, which are not straightforwardly stated in the axioms. Within the construction of the natural numbers, we take the existence of 0, the existence of a successor for any given number, and a number of other properties to be true, and then see what we can prove about natural numbers from there. Number theory is essentially the outcome of these axioms.
It took until the mid 19th century for the real numbers to be put on any sort of firm footing, with Cantor, Dedekind, Cauchy and many others providing the reasoning which allowed us to define the real numbers with the same confidence with which we could describe the natural numbers. Having ‘constructed’ the reals, the ZF axioms can then be used to provide the way forward into the subject of real analysis, from which calculus, as well as many other topics emerge. The idea then is to take a set of axioms and explore what we can proved with them.
Reverse mathematics takes an alternative view. Reverse mathematics asks, given a statement that we want to prove, what is the minimum set of axioms needed which can be used to prove it? Or, given a subset of axioms with which we are unable to prove a theorem, can we at least show, using that subset, that the theorem is equivalent to another theorem? It turns out that in analysis, a relatively weak subset of axioms (The or Recursive Comprehension Axiom system) can be used to prove that the Bolzano Weierstrass theorem is equivalent to the least upper bound property which is equivalent to the Könnig’s Lemma, amongst others. None of these theorems can be proved in , but should one either include the Bolzano Weierstrass theorem as an extra axiom, or an equivalent, then all of these can necessarily be proved (as proved under ).
Reverse Mathematics takes us on a journey through analysis, computability and on to Elementary Formal Systems which provides a deep understanding of the connection of theorems within mathematics. It is rather like taking mathematics as a two dimensional landscape of ideas, and seeing that in fact these ideas are connected by tunnels which come from subsets of axioms. Higher dimensions are built up with stronger and stronger axiomatic systems, but to understand the fundamental ideas, scraping back the ‘superfluous’, stronger statements allows for a deeper appreciation of what we can say within any formal system.
The book is a fascinating tour of this subject which I knew nothing of before reading the book, and came away feeling enlightened, not only of the power of reverse mathematics, but I think with a deeper appreciation of mathematics performed the ‘normal’ way around.
The level of this book is such that a strong student who has studied real analysis should be able to get a good deal from it, though having a foundation in formal systems and computability will certainly make the reader have a deeper appreciation of the mental gymnastics which are used throughout it.