rhersh at math.unm.edu
Thu Jan 15 18:26:44 EST 1998
Sol suggests that math exists in the imagination.
That's right. When you are working on a math problem,
you imagine it.
Say your professor assigns a problem,
how many k-simplices are there in the boundary of an n-simiplex?
You start with n= 2 or 3, a triangle or a tetrahedron. Then you
think what happens when you go one dimension higher, what new
simplces do you get? So you get a recursion formula, and solve
it, and then you recognize the answer. The problem is not really
geometric, its combinatorial, how many k+1-tuples in an n+1-tuple?
All the way you are using your imaginations, visua and combinatorial,
along with some possibly relevant bits of knowledge from class.
But--whose imagination? That of some ideal Creative Intelligence,
a la Brouwer? No, I don't think that's what Sol means.
His own imagination? No, for there's also the teachers imagination,
and everybody else in the class. We all have imaginations.
Imagination is a form of thinking. So the simplices are just our
But what about Frege? Remember, in the Grundlagen, talking not about
simplices but about 2, he jeered at Berkeley and Schlomilch. If 2
is mental, then we would have millions, no billions of different twos,
all different!!jeered Frege.
How to answer Frege? These different twos or simplices are not unrelated or
disconnected. They match each other!
They are connected by a common schooling and communication,
which constrantly wipe out differences and keep your imaginary simplex
and my imaginary simplex congruent to each other. That's how we can
come to common understandings about simplices or polyhedra or whatever,
Now, there is a word to depict thoughts and understanding that
people have in common.
Unfortunately, in some quarters this has become a no-no, a bad word.
When Sol says "in the imagination," meaning I am sure in the
congruent and well-matched imaginations of the mathematical
community, and when I say "a social-cultural entity," we are
just using different words to say the same thing!!!!
Thank you for your kind attention.
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