FOM: Mathematics in the imagination.

[Solomon Feferman]

To avoid misunderstandings about my use of the word 'imagination' in my 
Thesis 1 let me call on a book that develops related
ideas (completely independently).  This is by the philosopher Mark
Johnson, and it is called *The Body in the Mind: The bodily basis of
meaning, imagination and reason*.  Johnson draws his ideas from work that
he did with the cognitive linguist, George Lakoff, in their earlier book,
*Metaphors that We Live By*.  People interested in following up their
direction of thought should also look at the collection (L.D.English, ed.)
*Mathematical Reasoning: Analogies, metaphors and images*, and in
particular the article by Lakoff and R. Nunez therein, "The metaphorical
structure of mathematics: Sketching out cognitive foundations for a
mind-based mathematics".  My reference to these sources should not be
considered a general endorsement, but there is a certain overlap in our
views and they have a lot of interesting ideas and evidence.  At any rate,
here is a quotation from Johnson's *Body in the Mind* (p. 198), which
pretty much expresses well what I have in mind in my own use of the word
'imagination' in Thesis 1:

"We have seen that human rationality is imaginative through and through,
insofar as it involves image-schematic structures that can be
metaphorically projected from concrete to more abstract domains of
understanding.  Obviously, this is not imagination in the Romantic sense
of unfettered creative fancy; rather, [it is] an extended Kantian view of
imagination as a capacity for ordering mental representations into
unified, coherent, meaningful wholes that we can understand and reason
about.  Imagination, in this sense, mediates between sense perception and
our more abstractive conceptualizing capacities; it makes it possible to
conceptualize various structural aspects of our experience and to
formulate propositional descriptions of them."

More to come.  In particular, I owe John Steel an explanation of how it is
we can teach our students about the real numbers without ever having seen
one.  

Sol Feferman