Neil Tennant just reviewed my book (subscription to Philosophia Mathematica required for the link).
We adverted above to an appearance of slight strain between two assertions in the work. The first of these is confidently sweeping:
. . . arithmetical truths are conceptual truths; or, at least, enough arithmetical truths are conceptual truths to enable us to account for all of our a priori arithmetical knowledge once we add in knowledge secured by inference from other truths known in this way. (p. 123; emphasis added)
But, thirty pages later the author retracts:
To account for all of our knowledge of arithmetic is a tough call, even when we allow that much can be achieved by deduction from previously known arithmetical facts. Godel’s incompleteness results are a measure of how tough a call this is. (p. 153)
There are two problems with what Neil says here about my asserting then retracting some claim. (1) The first quoted passage is not an assertion. (2) The second quoted passage is not a retraction.
(1) The passage from p. 123 appears within the scope of an ‘On the view that I’m interested in’ operator, which Tennant carefully omits to reproduce. I never claim to have established that the view in question is true. Indeed, I repeatedly stress that this is not an aim of the book.
(2) The two passages are nonetheless obviously consistent. To make things even clearer, the discussion on the very next page (p. 154) explains why I would like to maintain my optimism that the proposal on offer can account for all our a priori arithmetical knowledge even given that this is a tough call in the way just described.
I am genuinely puzzled as to why Tennant presented these passages in the way he did. (This is just one example; I had similar thoughts about many of Tennant’s other criticisms.)
The foregoing notwithstanding, I’m feeling fortunate compared to Chris Peacocke, to date the only other recipient of a review by Neil.
UPDATE: Neil tells me he has, in fact, written other reviews. I had been assuming the list on his OSU ‘publications’ page, which mentions only the Peacocke review, was complete.