Chapter 4 has lots of repeats. It basically just gives more evidence that the four grounding metaphors are true plus some examples of them. There have been some really interesting new points brought up, though.
1. ERF’s (i.e. Equivalent Result Frames). This is a set of three things: a desired result, essential actions and entities, and a list of alternative ways of performing those actions with those entities to achieve the result. From the embodied mind standpoint, ERF’s are how we get associativity of arithmetic. Thus, according to this theory, associativity is not an axiom, but a consequence. I sort of like this. We’ll see how this stands up later (e.g. when we throw out associativity in non-associative algebras or loops).
2. The four grounding metaphors are isomorphic as mappings over the natural numbers. Well, I’d hope so. Also, the “proof” of isomorphism is a little sketchy. Also, as structures they aren’t isomorphic, since some of them produce things like fractions, irrationals, and negatives while others don’t. This isomorphism is only with respect to how they map to the naturals.
3. Uh. Why does Reuben Hersh endorse this book again? Numbers are things in the world, thus “Since things in the world have an existence independent of human minds, numbers have an existence independent of human minds. Hence, they are not creations of human beings and they and their properties can be ‘discovered.'” Well, I guess that settles the discovered vs invented question. Hersh is a hard-core invented proclaimer, though. Also, I’m not sure how much I buy the “things in the world have an existence independent of human minds.” I guess that depends on how you interpret the necessity of an observer in quantum mechanics.
Also, is this really what embodied mind is saying? Embodied mind is the idea that how we interact with these objects is how we perceive them and interpret them. Isn’t it true that if another species of intelligent beings formed somewhere, and they interacted with objects, in say a non-associative way, associativity would not be a consequence? In the same section we have, “There is only one true arithmetic, since things in the world have determinate properties.” I disagree precisely because of what I just said. There are many true arithmetics. In fact, in physics non-associative structures are rearing their ugly heads. If we had developed such that we saw and interacted with those structures before interacting with macro-structures, non-associativity could be the norm.
4. They are heavily pushing the metaphor idea. Good idea, but do all these metaphors they are referencing exist in all languages? This seems to be an English-centric argument. Maybe a certain person who has studied the cultural influence of mathematics can tell us if there are cultures that didn’t use, say, the motion metaphor. (i.e. 5 is close to 6 since we interpret motion from one natural number to the next as a short path).
5. Extension of subitizing to closure. I don’t remember if I mentioned this. The way we tell the difference between small numbers of things (up to about 4) is to subitize. We know whether something has 1,2,3, or 4 objects instantly. Groups of larger size get harder. We need to group into these smaller groups that we know to figure out how many are there. So this is not a closed operation. 2+3=5. We can subitize 2 and 3, but not necessarily 5. Why is it natural that we extend this to a closed operation? It seems unnatural actually, since the natural innate thing is not closed.
6. Probably the best thing about this chapter is the idea that (I can’t actually find it right now) symbols are important in the sense that the symbol representing the number (or whatever) is not the thing itself. A proper choice of notation is necessary to make calculating and manipulating as easy as possible. This has been my biggest soapbox argument for the past year. Let’s get rid of some of this outdated crap. String diagrams are amazing, box notation is amazing for algebra, categorification is amazingly compact, etc.
Now, don’t get me wrong. I love what these people are trying to do, but I think they aren’t even interpreting their own results properly. Maybe I’ll take my most cogent points from these posts and write a more well-founded and thought out essay to send to them.