#8. What would Sir Isaac say?
Newton
died in 1727 so contacting him about Riemann (1826 – 1866) was not
easy. But, with some help from a single malt scotch and a ouija board, I
have managed to channel him. Here is the session, as best as I can
recall.
He first gave me his definition of the integral.
Classical Definition.
A function f has an integral on an interval [a,b] provided there is a
function F defined on the interval and a subset N of the interval for
which
* N is a finite set
* f is defined at all points of [a,b] except possibly points of N.
* F'(x)=f(x) at all points x of [a,b] expect possibly points of N.
* F is continuous at points of N.
In that case the value of the integral is assigned to be F(b)F(a).
Modern Definition.
A function f has an integral on an interval [a,b] provided there is a
function F defined on the interval and a subset N of the interval for
which
* N is a set of measure zero.
* f is defined at all points of [a,b] except possibly points of N.
* F'(x)=f(x) at all points x of [a,b] expect possibly points of N.
* F has zero variation on N.
In that case the value of the integral is assigned to be F(b)F(a).
[A brief explanation. Only the notion zero variation on a set may be unfamiliar. You can find it in our dripped analysis text and in THE CALCULUS INTEGRAL as well as in THEORY OF THE INTEGRAL.
N is a set of measure zero if the identity function F(x)=x has zero
variation on N. The connection between the two definitions is that F has
zero variation on a finite set N if and only if F is continuous at each
point of N.]
transcript of ouija board session
Me: Is this the correct definition of the integral?
Newton: Certainly. Integration on the real line is antidifferentiation. You can characterize that in numerous other ways, but that is what it is and that should be disclosed to students up front.
Me: Your definition has been called descriptive.
Newton: Indeed. It describes exactly
what an integral is. Don't get confused by the fact that there are
constructive versions. Start with this and get the constructions later.
Me: Why two versions?
Newton: For the 18th century the ``finite exceptions'' version
was adequate for all applications. It was only with Fourier's
introduction, in the 19th century, of series representations by
trigonometric series that the ``measure zero exceptions" version became needed. Teach the former to your integral calculus students and save the subtle version for later.
Me: You say "get the constructions later." Who did that?
Newton:
First there is Cauchy. He clarified the notion of continuity. We had a
vague idea before, but he nailed it and showed that continuous functions
have integrals. He also characterized the situation for functions with
finitely many discontinuities.
Me: And ...
Newton:
Certainly Lebesgue. He developed all the tools to characterize,
construct the integral, and to investigate the class of absolutely
integrable functions [i.e., both f and f are integrable].
Me: And ...
Newton:
Well that leaves only Denjoy who gave an elaborate treatment for the
constructibility of the integral of the nonabsolutely integrable
functions.
Me: What about the Riemann integral?
Newton:
A minor observation. Also misleading. It pushed integration theory in
the wrong direction for too long. His socalled "integral" only works
for bounded functions and derivatives need not be bounded. Also it
doesn't handle even bounded derivatives. As soon as that was realized
then Lebesgue took it up as a problem: How to integrate constructively
all bounded derivatives. He succeeded, of course.
Me: About the HenstockKurzweil integral, do you ....
Newton:
That's just another descriptive integral. It looks kind of
constructive, only because it is so similar to the Riemann integral. Its
equivalent to my integral. In fact you can show it is included in my
integral in a few short steps. Some people think you could start
teaching integration theory with it, but I much prefer my own integral
as a starting point. As I said, integration on the real line is
antidifferentiation. Start there ...
end of transcript
[Sorry I don't recall any more, except that he began railing against Leibnitz, Locke, Pepys and some others.]
Posted by BST at 9:18 AM, Friday, February 8, 2008
Top ten reasons for dumping the Riemann integral: #10, #9, #8, #7, #6, #5, #4, #3, #2, #1
