Jonathan Aronson alerted me to the relevance of Howard Gardner’s work on Multiple Intelligences to my “hard intangibles” project.
Gardner argues that intelligence isn’t a one-dimensional capacity that can be measured by (say) an IQ test. He defines it as “the ability to solve problems, or to fashion products, that are valued in one or more cultural or community settings.” He argues that there are seven distinct intelligences: linguistic; logical-mathematical; spatial; musical; bodily-kinesthetic; interpersonal; and intrapersonal. Each person has a different mix of skills. 
He applies this theory to education. In ref.  he gives the example of a child that’s having trouble learning mathematics because the principle to be learned (the content) exists only in the logical-mathematical world and it ought to be communicated through mathematics (the medium); however, the child struggles with math. A good teacher finds a way around this problem by translating the principle into another domain, e.g. through a story or a spatial model. Gardner observes that this alternative route to understanding “is at best a metaphor or translation. It is not mathematics itself. And at some point, the learner must translate back into the domain of mathematics.”
This raises a question about Lakoff’s work  about the underlying sensory-motor metaphors in mathematics. If Gardner is correct that mathematics is a domain with its own intelligence, and if there’s a distinct basis for each intelligence in brain physiology, then his claim that mathematics is based on spatial and bodily-kinesthetic metaphors may be nothing more than a way to make math intelligible to people with good spatial and bodily-kinesthetic intelligence. Those who are proficient at mathematics use their “mathematical faculty”, and don’t have to fall back on sensory-motor metaphors.
From my reading of their work, Lakoff makes a more persuasive case than Gardner, and I’m therefore inclined to doubt that spatial models in mathematics are simply crutches.
Still – it’s notable that two of Gardner’s intelligences (logical-mathematical, and musical) do seem more remote from the sensory-social experiences of childhood, which Lakoff argues shapes our cognitive abilities, than the others. It suggests that one might expect another collaboration from Lakoff, on “Where Music Comes From”.
 Howard Gardner (1993), “Multiple Intelligences: The Theory in Practice”
 George Lakoff and Rafael Núñez (2000), “Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being”