Several reviews have appeared since the book's publication in April 2003, including one in a German daily newspaper - Frankfurter Allgemeine Zeitung (04.11.03) Der heiße Tip fürs Übersetzungsbüro/ David Corfield revolutioniert die Philosophie der Mathematik by Dietmar Dath.
I'll be using this page to respond to some of them.
Julian Cole MathSciNet MR1996199 (2004f:00005)
"I found much that resonated with my experiences as a research mathematician. Indeed the majority of problems that I had with the book were more a matter of overreaction to the status quo than outright errors. Unfortunately, however, not all philosophers of mathematics will find the book easy going. The author's development of the mathematics used in his case studies is often terse. And at a number of places where a diagram would have been extremely useful the author fails to include one. I do recommend plowing through however, perhaps with the aid of an introductory text in algebraic topology, for this book points the way towards a philosophical goldmine waiting to be plundered."
The only way it can be consistent to criticise someone of overreacting when they've located a goldmine, if your common goal is to find gold, is to believe that existing mining activity is already sufficiently productive. This is where Cole and I differ.
Here and elsewhere (Eduard Glas, Philosophia Mathematica XII: 65) some people reviewing my book have come away with an impression of my views on the value of Bayesian thinking for the philosophy of mathematics which I did not intend to give. I certainly don't believe it tells us much about what I consider to be the key issues of the philosophy of mathematics, such as how to characterise the rationality of the continuing quest for more adequate notions of space, quantity, dimension or symmetry. On the other hand, it's a very useful exercise to think about issues concerning plausibility through its lens. I invite a reconstruction of the following:
...it is my view that before Thurston's work on hyperbolic 3-manifolds and his formulation of the general Geometrization Conjecture there was no consensus amongst experts as to whether the Poincare Conjecture was true or false. After Thurston's work, notwithstanding the fact that it has no direct bearing on the Poincare Conjecture, a consensus developed that the Poincare Conjecture (and the Geometrization Conjecture) were true. Paradoxically, subsuming the Poincare Conjecture into a broader conjecture and then giving evidence, independent from the Poincare Conjecture, for the broader conjecture led to a firmer belief in the Poincare Conjecture.(John W. Morgan, 'Recent Progress on the Poincare Conjecture and the Classification of 3-Manifolds', Bulletin of the American Mathematical Society 2004, 42(1): 57-78)
It doesn't sound at all paradoxical to me, if you take Polya's "hope for a common ground" into account, see chapter 5. Also, I think I showed in chapter 6 how to deal with the famous problem of old evidence, which has taxed philosophers of science over the years. One reviewer, Joseph Melia (Metascience Volume 13, Number 3, December 2004) couldn't quite see it, so I'll spell it out.
Let's rehearse the argument. The problem runs: imagine you have a piece of firmly established evidence e, which you believe with total confidence, Pr(e) = 1. You devise some theory T, and rate how likely it is to be true, Pr(T). You then discover that T accounts for e. What should this do to your degree of belief in T? Well applying Bayes' theorem:
Pr(T|e) = Pr(e|T).Pr(T)/Pr(e) = 1.Pr(T)/1 = Pr(T) .
Apparently, there should be no boost to your degree of belief in T. This seems odd because scientists often do get encouraged when their theories turn out to explain some already observed phenomenon.
After Polya, I suggested the analogy with someone doing some mathematics. You've worked out as best you can a formula for the curved surface area of a frustrum of a cone (a cone with its nose chopped off), in terms of the height, h, and the radii of the upper and lower circles, r and R. You're not too sure you've got it right, but are encouraged to find the formula works for the area of a cylinder when you set R = r. Let's imagine you're now 50% certain that your formula is correct. A friend now says to you, "How would you feel about your formula, if I told you that if you set h = 0, you'll find your formula gives you the correct answer for the area of an annulus?". You say,"Much more confident, thank you", although you already know the area of an annulus. The key point is that you've learned something new, that your formula works in a special case. So, let F = "formula is correct", and A = "formula gives right result for annulus". Then Pr(F|A) = Pr(A|F).Pr(F)/Pr(A). Pr(A|F) = 1, but now Pr(A) = Pr(A and F) + Pr(A and not F). The first of the summands is 0.5, and the second your belief that the formula is wrong and yet still gives the right answer for the annulus, let's say 0.1. So you update your belief in your formula to around 83%.
Other reviews include:
Timothy Bays, Notre Dame Philosophical Reviews; James Robert Brown - Studia Logica 81(2), Nov 2005: 285-289; Andrew Arana - Mathematical Intelligencer (forthcoming); James Page - Philosophical Books 45(3): 277; John Baez - This Week's Finds in Mathematical Physics (Week 198); Reuben Hersh - SIAM Reviews 46(2): 365-367; Alexander Paseau - Studies in History and Philosophy of Science Part A, vol 36(1): 191-201; Christopher Pincock Philosophy of Science (forthcoming); Fernando Zalamea, Theoria 2006; Denis Lomas MAA Reviews.