This book is a lot more hesitant, more exploratory, and more firmly based in experiment than *WMCF*, and this is probably why it is both more interesting and less famous. There’s a real, tangible basis in cognitive science, and one that is not used just to form a base to extrapolate but also to test the ideas it puts forward at various points along the way.

We start with some really nice experiments to assess the numerical capacities of animals and children. There’s a lot about Piaget.

According to the theory first set forth some fifty years ago by Jean Piaget, the founder of constructivism, logical and mathematical abilities are progressively constructed in the baby’s mind by observing, internalising, and abstracting regularities about the external world. p. 42

There’s an interesting discussion of some of the experiments and how effective they are at testing understanding of number rather than of expectation.p. 47. p. 58 talks about babies whose expectations are based more on trajectory than number.

Then there are some interesting questions about the relative differentiation between numbers (p. 74) and discussion of a number line (p. 79) – this is stuff I’ve seen before in Giaquinto but this is a really nice comprehensive survey and could be very useful.

P. 84 has some very interesting stuff about synaesthesia and number, and even of mapping personal number lines, which have particular curves and forms. I don’t get any of this at all, though I have a friend who attaches very particular colours. I’d love to find somebody who gives the number line a form, and try to map it. p. 86 suggests that non-integers etc. are difficult to grasp because there is no immediate way to intuitively connect them to a category in the brain – intuitionism is very present in this book.

p. 108-9 talks about round and sharp numbers – we’re likely to talk about something being ‘around twenty’, but not ‘around seventeen point eight five’. Before this there are some other comments about linguistic usage of one, two, three and higher numbers – how much we use these ideas in our lives. The idea of number as a technology is in here.

Chapter 5 – 118 onwards – starts looking at calculation and how this relates to our understanding of numbers. 3-year-old children understand that you count the numbers in order, but objects don’t need to be counted in order – but don’t associate this with ‘how many’ questions p. 121. Children reach the principle of commutativity long before they would possibly know its logical foundations. p. 123. See: Wittgenstein!

Dehaene notes the difficulty of multiplication tables given the associative way that our memories work – drawing connections between many similar things, unlike a computer, so all those similar-looking sums are difficult to distinguish (p. 128). He looks at various telling ‘bugs’ in calculation p. 133. There’s something interesting here about meaning vs. foundations, and on p. 142 he talks about getting kids to play with abacuses, thermometers, dice, board games, and gain that number sense from that experimentation – and that improving kids’ understandings of number quite dramatically.

Dehaene also relates mathematical sense to spatial awareness on p. 150, citing some experiments on gifted children.

p. 171 talks about mathematical creation, enthusiasm and creativity. Then there’s a whole Oliver Sacks-style section about the effect of various brain injuries, disorders etc on calculation; examples of the diverse effects of lesions backs up the idea of multiple ideas acting at once in maths p. 176. Numbers, after all, mean an awful lot of different things to us, and that affects how we treat them. No American will treat 911 exactly like 910.

On p. 238 there’s something nice about intuition vs. axioms – foundationalism – ends up supporting a Kantian view, Poincaré, intuitionist, and addresses that famous “Unreasonable Effectiveness”.