So. I didn’t get through as much reading as I thought I would. Two chapters isn’t bad, but nothing more than I would have done. There are a couple of interesting claims here. I’m still completely confused about how they are rejecting Platonism. I think that they are quasi-Platonists at best, maybe full out Platonists at worst. They beginning of chapter 5 talks about forms and essence. I kind of want to skip to the end and read their summary first just to get the big picture.
This is about the essence of algebra. They claim there is a general algebraic essence metaphor. The most interesting part of these two chapters is the claim that algebraic structure is not inherently in what we use like we naively think. The integers under addition does not have the group structure in it. We impose that metaphor onto the set in order to make more sense of it. This seems to make sense. It is just nothing I would have ever guessed or thought of on my own.
Next comes a partial answer to my Platonism question. I don’t feel like explaining the term folk theory (since I’ve used it so much in 20th Century Philosophy). We have a western folk theory of essence in mathematics. This is the reason we feel that we should try to characterize all of mathematics in the fewest number of independent axioms possible. We believe these axioms to describe the forms of mathematics. Then when we do math we are just discovering things about these essences. (Look up these terms as well if you are fuzzy on “form” and “essence”).
It turns out that the folk theory of essences does not describe mathematical cognition. When we talk about rotations of the triangle (cyclic group of order 3), the rotations and group structure are conceptualized independently. The thing that gets us from one to the other is the idea of “linking metaphors” (as opposed to the previously encountered grounding metaphors).
Hmm…I don’t remember coming across anything particularly interesting in chapter 6, which is about Boolean logic. It talks about the Classes as Containers metaphor as a basis for Boolean logic. This was already touched upon though. It just goes one step further to the symbolic-logic mapping that mathematicians use. This is how we get the feeling of “blind manipulation.” All of this traces back to the embodied container schema, though. The closing thought is that no matter how advanced our system of logic becomes, it will never be able to capture true human reason. Thus if we solely use logic for reasoning we will miss some things.