I'll buy that: the particular model space may not have to be a single
one. And our readings hopefully will lead to the good ones.
A model does, however, have to satisfy Timothy Cowers's notion of
abstraction: that after the intuition drives you to an abstraction,
you can cut the cord to to the intuition and live entirely within the
abstraction.
The 10^-1 example: what in the hell does it mean to multiply something
by itself -1 times! The abstraction simply says that exponents are
added during multiplication and subtracted during division. That lets
us also make sense of 10^0 = 1, which drives non-math folks mad!
And the abstraction does also have to be constructive: i.e. one can
use the abstractions to create new entities within the abstraction.
TC's example of a 5-space cube being simply lists of 5 numbers. His
example of a unit 5-cube was great: (0 1 0 0 1) as an example vertex.
That counting the number of possible 5-tuples of this sort (2^5=32)
tells you the number of nodes the 5-cube has. And so on for the edges
(all tuples differing by a single digit).
This all biases me toward computational abstractions. Besides, its
way fun to write programs!
BTW: Gowers Mathematics: A Very Short Introduction has gotten great
reviews on Amazon. And, sigh, has made me have much more respect for
philosophy (blush!). If Russell and Wittgenstein helped Tim to arrive
at a Fields medal and deliver the Millennium keynote, it can't be all
bad.
Hey, maybe our next seminar should use Gowers's two books, both the
small and the large (Princeton Companion to Mathematics)! When the
PCM gets to its second printing (thus reducing the errata
considerably), I definitely will buy it. And there is a digital
version, making it easier to work with in a seminar.
-- Owen
On Oct 10, 2009, at 5:40 PM, Nicholas Thompson wrote:
Great!
We seem to agree that models are important. You are keener on
mathematical
models ... that is models that are accompanied by a mathematical
formalization ... I am keener than you about models like "natural
selection"., where the model space is some phenomenon one feels one
understands better than the phenomenon under examination, but in both
cases, the procedure is the same ... commit your self to a domain that
represents the phenomenon of interest, work within that domain, and
then
return to the phenomenon to see where you have gotten.
I think we need to think hard about the process by which the model
comes
about in the first place ... the eureka moment, or as popper called
it, The
Bold Conjecture." Clearly some models are crap and others are very
useful.
What I think we are doing now is assembling the equipment to
generate a
good model as a opposed to a crappy one..
Nick
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