Lakoff and Núñez "Metaphorical Structure of Mathematics" (1997)

Lakoff, G. & Núñez, R. (1997). Metaphorical structure of
mathematics: sketching out cognitive foundations for a mind-based mathematics.
In English, L (Ed.) Mathematical reasoning: analogies, metaphors, and images.
(pp 21-89) Mahwah, N.J.: L. Erlbaum Associates.

Lakoff and Núñez start this article with a warning -- that the
cognitive science of mathematics is a new field of study that turns on its
head the 20th century idea of mind-free mathematics. This in turn
relies on http://www.wku.edu/~jan.garrett/120/newphil.htm#Meta the
Folk Theory of Kinds and the Metaphors of Essence</a>, which states (briefly
here) that science seeks the essential properties of things. But math is only "a
lens through which they see the world" (p.26) "... mathematics has
not been properly understood as a product of inspired human imagination, and
has been taught ... [as] a mere grasping of transcendental truths, something
lesser and less interesting (p. 29). Math is a commonplace conceptual metaphor
that needs to be stable (p. 30).

They continue by giving examples of math grounding metaphors: arithmetic is
Object Collection (sets); arithmetic is Object Construction (+ - x ÷);
arithmetic is motion/location (number line) (p. 35-38). And: functions are
machines (p. 47). And it goes on and on through all sorts of math concepts
I'd never learned including some fun ones like "monsters" (p. 72-79),
until we get to the Morals of the chapter (p. 83-85):

1. mathematical ideas cannot be expressed in formalisms with mathematical
   rigor.


2. mind-free math is a myth


3. there can be no ultimate foundations for math within math itself


4. the formal foundations program is constituted by a set of ideas (this is
   pretty cool, actually)


5. the study of ideas need not be vague or hazy


6. math is ultimately grounded in the human body


7. most interesting math comes from linking metaphors (that's true of poetry
   to, btw)


8. because metaphors preserve inference, proofs using metaphorical ideas stay
   proved


9. not all ='s mean the same thing, and most ='s are metaphorical


10. the continuous cannot be literally defined in terms of its opposite, the
    discrete.


11. one should neither expect, not seek absolute, literal, definitive foundations
    for math (that bears repeating, and applies to everything I've experienced
    in life).


12. a cognitively based philosophy of math is needed (and let's try to guess
    who should do it? Um... Lakoff and Nunez?)


13. math should be taught in terms of math ideas (vs. techniques of proof or
    calculation -- basically, though, it's a pragmatic thing, no?)

My take: I'm glad someone is looking at this; and I suspect
that I could probably learn to like math as much as I respect it. I recall
a high school conversation with my father about "Math vs. Other subjects" --
he liked math because the answers were either right or wrong, whereas English,
History, etc. could be argued, or subjectively disputed. It was exactly my
argument for disliking math. I saw it as rigid and dull.

But that's Math, and not Embodiment, and I need to wrestle with embodiment
here. So I come up with moral #6: math (like all language-based human ideas)
is grounded in the body, where "embodiment" means rooted in physical
experience. Can a computer be embodied? Sure, since it's a representation/manifestation
of a human idea, rooted in human metaphor, and since we're the one deciding,
we can call its data/knowledge embodied. And we can tell the machine to call
it embodied too. But some folks will probably object. We're not as good at
obeying.