#9. Does the phrase "mildly interesting exercise" suggest anything?
The quoted remark is
from Jean Dieudonne, the French mathematician and well known
Bourbakiste, in his dismissal of the Riemann integral as a suitable
object of study for undergraduate mathematicians.
Actually I find
the history itself more than mildly interesting and, with your
indulgence, will give a fractured account of it here. What happened to
Riemann follows a pattern that I can describe this way:
Suppose that you (Jones) discover an old theorem of Smith that you think you can improve on.
Theorem [Smith] Every object Y has the property Z.
You
(Jones) decide that it might be worthwhile to characterize this
property Z, especially since Smith seems to have gotten some fame from
it. You succeed and write it up like this:
Definition [Jones] We say that any object possessing the property Z is a zamboni.
Theorem [Jones] A necessary and sufficient condition for an object to be
a zamboni is [insert your characterization].
Corollary [Smith] Every object Y is a zamboni.
Thus you (Jones) have, with a stroke of the pen, transformed Smith's theorem into a trivial corollary of Jone's theorem.
The
success of Jone's manoever here is whether zambonis are going to have
any lasting interest. If they do and prove to be a significant concept,
then your fame exceeds Smith's, even though Smith had the original
insight. If zambonis fall flat on the mathematical world then move on to
something else.
Well, Riemann's zamboni is his integral. He took
a theorem of Cauchy asserting that every continuous function on a
compact interval [a,b] had a certain property with regard to
its integral. He then characterized that property and gave it a name.
Hence the definition of the Riemann integral, Riemann's characterization
of integrability, and Cauchy's theorem dropping down to the status of a
corollary. In fact the sums that Cauchy had used are now called
"Riemann sums," so history has taken all the credit from Cauchy and
shifted it to Riemann.
Well, Riemann doesn't need that credit
since he did far better things. Even the Riemann integral was just a
throwaway in a 1854 paper about trigonometric series and not anything
that he likely spent too much time thinking about. If he had seriously
directed his enormous intellect at the problem of integration itself he
would certainly have discovered the correct integral.
Unfortunately
for Riemann, however, is the fact that this particular zamboni was
misguided. Certainly it is a worthwhile professional project to
characterize the property that Cauchy had discovered, but it was a sad
mistake that future generations employed Riemann's integral as the
central tool of integration theory.
For the next fifty years or
so (18541901) mathematicians took Riemann's integral as if it were the
correct one for bounded functions, and spent their time on the problem
of integrating unbounded functions. Perhaps they thought of their
program as "extending the integral to unbounded functions." Too bad. If
you look at the problem this way you fail. If, instead, you tackle the
problem of how best to integrate bounded functions, then the unbounded
case takes care of itself. Enter the 20th century and Lebesgue.
Lebesgue's thesis swept away all the previous theories.
At that
point the history gets a bit strange. Lebesgue's theory is considered
too difficult for some to learn and for many to teach. So many did not
teach it, and many did not learn it. In the last 100 years there have
developed many different ways to teach integration theory, some not any
more difficult than teaching the Riemann integral itself. But, even so,
we still don't teach it until graduate school at many places.
Posted by BST at 6:02 PM January 19, 2008.
Top ten reasons for dumping the Riemann integral: #10, #9, #8, #7, #6, #5, #4, #3, #2, #1





 
pro quo hic escorol. Olypian quarrels et gorilla congolium sic ad nauseum. Souvlaki ignitus carborundum e pluribus unum. Defacto lingo est igpay atinlay. Marquee selectus non provisio incongruous feline nolo contendre. Gratuitous octopus niacin, sodium glutimate.
  

Olypian quarrels 
03/05/07
Pro quo hic escorol. Olypian quarrels et gorilla congolium sic ad nauseum. 
03/05/07
Pro quo hic escorol. Olypian quarrels et gorilla congolium sic ad nauseum. 
03/05/07
Pro quo hic escorol. Olypian quarrels et gorilla congolium sic ad nauseum. 
 