Status Updates From Where Mathematics Come From...

using a suite of metaphors, Lakoff and Nunez intuitively explain Euler's magic formula and how an exponential e raised to an imaginary power i times Pi equals negative one. It is amazing that with normal human brains, we can grasp something so abstract and amazing.

going over the metaphors to Euler's magic formula e^i(pi)-1=0

I agree that embodied cognition is a good gateway to math, but I still think Math exists independently of the mind. These concepts kick back like when you kick tires. So, l like embodied cognition for pedagogical purposes. I still think math is discovered, not just metaphor play.

Dedekind cuts and metaphors for the continuum, and Weierstrass and taming monster functions.

Points and the continuum and space-filling curves, and the way we build them from the concrete.

Transfinite diagonalization arguments and the specks or dots of infinitesimals.

extending this infinity to infinite numbers, polynomials, and sequences. And the concept of a least upper bound to get Dedekind cuts and the real numbers filling the number line.

Infinity and indefinite iterative processes like breathing and day-night cycles. Iterative processes happen all over the place, and many have no definite end and are good metaphors for the infinity of natural numbers.

ordered pairs and the metaphor of set theory in terms of ordinality and Cardinality.

essences and numbers and sets and classes and symbolic logic of Boolean algebra.

Taking numbers as things and the collections schema, we get natural numbers by mapping them out to directions, we get positive and negative and imaginary numbers via a directional schema.

Goes into how we take simple concrete schemas and metaphorically generalize them to math concepts, like say set theory concepts of number, and how to manipulate through common operations gets more to the nuts and bolts, and names the types of schemas involved.

One of the surprising things about cognition is that a lot of it is unconscious. Some of these unconscious frameworks or schemas are involved in cognition, which are applied concretely and, by metaphor, abstractly. In math, some are the above-below schema, a succession schema, a container schema, a path goal schema, and others that get used for daily needs but can be adapted via metaphor to abstraction.

inherent subitizing abilities of newborns and some animals. This is a kernel of numbers in the brains of humans and some animals.

How we build abstract mathematical concepts via concrete bodily and sensory metaphors and use a via odd form of metaphor to abstract and generalize. I disagree, I think Plato's Plenum of mathematical forms is real, but manifests in patterns in the physical reality. I will say, however, that the cognitive body/sensory approach is the right one for pedagogy.

Just starting to reread.

Promising but the book was stolen before it's could finish it :(

I’m growing convinced of the basic point that (some) mathematics is developed according to conceptual models that have metaphoric expression and that changes in metaphoric expression can precede and drive changes in mathematical practice. I’m getting a little itchy for some implications at this point.

The book’s purpose is a bit muddled so far. I’m not yet convinced that math is metaphorically grounded in embodied experience. At best the authors have shown that bodily experience coherently overlaps math, but in a way that accounts more for rational numbers and arithmetic. Even there causal directionality is hard to show. Does math derive from embodied experience or does embodied sense derive from math?