Lou van den Dries
vddries at math.uiuc.edu
Tue Oct 28 17:16:51 EST 1997
The rumors of my "selling out" have been greatly exaggerated.
Anyway, what is of "general intellectual interest" is I think
a very difficult issue to discuss sensibly. In listing 2. and 3.
I just had a very modest goal in mind, showing that some key
theorems in diophantine geometry do have consequences on
the "high school algebra level". In fact, I now realize that
these theorems themselves do have equivalent forms that can be
stated on that level. (And any number theorist could have
probably told us this right away.) But I do have to look this up
a little so I don't make some stupid mistake.
By the way, the distinction "high school algebra" (language of
polynomials, and algorithms about them, like Euclidean alg. ),
"college algebra" (language of fields and rings and ideals)
and "university algebra" (cohomology, etc.) has been introduced
by Abhyankar a long time ago, and it is a well known phenomeneon
that many results in algebraic geometry can be stated on all
these three levels. (And this can be done for Siegel's theorem
and for Falting's theorem as well, and probably has been done,
but I"ll come up with something like this when I have a little
time. Many other things need to be done.) By the way, I do like
this kind of exercise to some extent, and I am happy to oblige,
Steve and Harvey, who'd like to see things stated on that level.
But I 'd be careful to draw any kind of "dramatic conclusions"
from such things, as foundationalists seem so eager and willing
to do. (By the way, the general notion of algorithm does not seem to
me on the high school math level, so any result that uses this notion
essentially seems to me not "of general intellectual interest"
according to some of the definitions I have heard bandied about,
like "you can tell it to your neighbour", etc. And if you are
willing to make a case that Turing's or Church's analysis can be
made accesssible on the high school level, you could make just as
good a case (probably a better case) that complex numbers,
and algebraic curves and their genus can be made accessible on
Sorry for the hasty way this is written. -Lou van den Dries-
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