Mathematics and Design: Yes, But Will it Fly?
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Visiting Scholar UC Berkeley
Professor Emeritus, New York University
Associate Professor, Mathematics and Statistics
University of Missouri-Rolla
MO 65409 USA
Does anyone believe that the difference between the Lebesgue
and Riemann integrals can have physical significance, and that
whether, say, an airplane would or would not fly could depend
on this difference? If such were claimed, I should not care to
fly in that plane. --Richard W. Hamming
MATT INSALL WROTE:
there any historical record of anyone ever having scientifically
tested whether the use of one mathematical concept, as opposed
to another, hampered the design of something? In my experience,
mathematical applications are not treated this way. Designers
typically know little or no abstract mathematics, and so they
typically do a series of tests. Frequently, they refer to them
as statistical tests, but even the statisticians will at times
shudder at what is being referred to as "statistical testing".
Thus, for instance, I would say that this quote of Hamming's
is a "straw man" attack on abstraction.
Personally, I think it is likely to make a difference whether
Riemann or Lebesgue integration is used, but that the difference
will only be noticeable provided that the designers rigorously
adhere to the requirements that that particular integration theory
be used for all the integrals. As soon as they revert to numerical
approximations, instead of using calculus or analysis, they have
muddied the waters so that no direct comparison between one integration
theory and another can be made. Another way to test this might
be to allow numerical approximations of all but one integral
used in the design. For that integral, design the aircraft once
using Riemann integration, and design it again using Lebesgue
integration, but make sure that each of the other integrals approximated
using numerical techniques uses the same techniques and has the
same approximate value in the two different designs. Then try
each aircraft out. An obvious problem with this "experimental
design" is that scientists will be unwilling to part with
the cost of an aircraft - let alone two aircraft - to test the
usefulness of Lebesgue integration in this way. Another obvious
problem is more technical: When the one integral being tested
for its effect on the design of the aircraft is computed, using
Riemann integration, the value so obtained may be used as an
input to some other portion of the design. This would skew the
results of other
numerically approximated integrals in the design. Similarly,
when computing that particular integral using Lebesgue integration,
if the value obtained is different from that obtained using Riemann
integration, then the values elsewhere in the design will be
skewed. Thus, the test may be inconclusive on the basis that
the effects of the change in this single item is not isolated
from the other. Finally, one may do statistical testing of the
use of Riemann integration versus Lebesgue integration. This
would be possible if there are some designers who typically use
Lebesgue integration when the choice of which integral to use
comes up, and when exact calculation of an integral is possible.
(Of course, I expect that most designers of aircraft who use
integration in the design actually use Riemann integration when
they actually compute the integral in question. Part of the problem
here is that the Lebesgue and Riemann integral agree with each
other in most cases in which a designer would find that they
are capable of computing the integral anyway, so the murkiness
remains.) This type of statistical testing is likely, I believe,
to not demonstrate much about the technical differences between
the integration theories of Riemann and Lebesgue, but instead,
it would point out whether it seems, at this time in history,
to be more useful to a company to hire graduates who have learned
the Lebesgue theory of integration, as opposed to hiring only
those who only know the Riemann theory of integration.
Unfortunately, there are many complications involved in this
issue: On the one hand, graduates who have studied Lebesgue integration
are more likely to have taken some graduate mathematics or science
or engineering courses, whereas many designers of various things
-- such as aircraft --
typically, at least in the United States, have an undergraduate
degree in some engineering discipline. Thus, in order to study
the relevant statistical correlations in this instance, one needs
correlations such as these, which involve, usually, a host of
confounding variables: postgraduates are typically older, and
so may be more mature, leading to better overall workmanship
and greater levels of production overall, but also, being older,
they may typically have less energy than their counterparts with
no postgraduate training, leading to an attenuation of any productivity
advantage they may have had over the others. On the other hand,
those who are aware of the theoretical limitations of Riemann
integration may have less practice with Riemann integration,
having spent less time doing these integrals quickly than they
have spent learning the theory of Lebesgue integration over the
last few years of their training. This will in some cases attenuate
their production levels even more than those for graduates who
are oblivious to the distinctions between the two main theories
of integration. In a sense, they are "overqualified",
meaning they know more, and know it better, but may not be practiced
at doing things that the corporations need done quickly, in order
their stockholders' need for dividend infusions.
Needless to say, I am not enamoured by such anti-mathematical
quotations as this one attributed to Hamming.
MARTIN DAVIS REPLIED:
06:39 AM 8/26/2002, Matt Insall wrote:
<< Personally, I think it is likely to make a difference
whether Riemann or Lebesgue integration is used, but that the
difference will only be noticeable provided that the designers
rigorously adhere to the requirements that that particular integration
theory be used for all the integrals. As soon as they revert
to numerical approximations, instead of using calculus or analysis,
they have muddied the waters so that no direct comparison between
one integration theory and another can be made. >>
This is with reference to Hamming's quip about this difference
not being significant for engineering design.
Functions that are Riemann integrable are automatically Lebesgue
integrable and the values of the integrals will be the same.
There are two ways in which a function can be Lebesgue integrable
but not Riemann integrable. Any bounded measurable function is
L-integrable, but not all of them are R-integrable. Some unbounded
measurable functions are also L-integrable, but not Riemann integrable,
since all R-integrable functions are bounded. An example: the
function 1/sqrt(x) has the L-integral 2/3 on the interval [0,1].
Being unbounded in that interval, it has no R-integral there.
Of course, this function does have an "improper integral"
(sometimes called a Riemann-Cauchy integral) in that interval
with the very same value. Only functions having absolutely convergent
improper integrals have Lebesgue integrals. The classical example
is sinx/x on the interval [0,\infnty] which has the value \pi/2
but no Lebesgue integral. This fact is sometimes confusingly
stated as a case where something is R-integrable but not L-integrable.
But one can perfectly well define "improper"
L-integrals, so that this comparison is inappropriate. It's just
that the definition of L-integration automatically includes cases
that in the Riemann case require this extra limiting step for
which the inappropriate term "improper" is used.
Hamming of course hadn't meant to be taken so literally. His
aphorism was intended to say that the fine points of mathematical
analysis are not relevant to engineering considerations. And,
he was perfectly right.
In the long-ago days when I had occasion to be on committees
administering an oral qualifying exam for the doctorate, I would
often ask the hapless student why analysts prefer the L-integral.
This was a trap question: students who fell for the trap would
tell me that the function on [0,1] that is 1 on the irrationals
and 0 on the rationals is L-integrable, but not R-integrable.
The right answer is that the L-integral has useful convergence
properties not enjoyed by the R-integral.
Now whatever did Mat Insall have in mind? sin x/x on [0,infty]?
Surely not in any engineering analysis.
MATT INSALL REPLIED:
course, I have no specific examples in mind where one needs to
integrate the sinc function on an interval including zero, or
I would have written them in my previous off-the-cuff reply to
the quote from Hamming. However, this function, and others like
it, which are improperly R-integrable over some interval, but
which are not (improperly) Lebesgue integrable on the same interval,
are used reasonably often in various ways in engineering, and
as we continue to need to make smaller objects with the properties
expressible by the use of such functions, which fit in some significantly
larger global structure, the tendency of certain engineers may
well be to estimate certain values by computing integrals over
intervals such as (0,\infty). (This is especially the case if
they have no introduction at all to higher analysis, and so do
not know that the sinc function behaves so badly.) [The smallness
of such an object would typically lead to a need to perform computations
relating to nondegenerate intervals containing 0, whereas the
remote location of the boundary of a region under consideration
leads to models having lazy eight as the right endpoint of an
interval over which one would like to
Examples of articles I dug up (but have not read) by engineers,
in which the sinc function and its relatives are used are below
[in the bibliography]. I shall keep
an eye open for other places where such functions are used. In
the meantime, remember that the original question had to do with
dependency of the design on the use of Riemann versus Lebesgue
integration. Thus although I focused on the version of this discussion
in which a problem might involve a specific (improper, of course)
integral for which the two theories diverge, my main point was
that whether or not the choice of an integration theory is significant,
in any specific way, to the design of an aircraft, it is unlikely
that a reasonable answer to such a question will be found, and
that factors that come into play in making the answering of such
a question so unlikely include quite a few computational and
societal difficulties that are not faced as often, for instance,
in the fom community, where the question, for example, of whether
the axiom of choice is a necessary ingredient in one's set-theoretic
assumptions to obtain some specific proof in analysis from the
axioms of Zermelo-Fraenkel set theory. (I certainly do not contend
that one problem or the other is truly "easier". However,
they are definitely different problems, and in each case, there
are significant hurdles to overcome.)
Smit, T., Smith, M.R.
and Nichols, S.T., "Efficient sinc function interpolation
technique for centred padded data", IEEE, A.S.S.P., Vol.
38, No. 9, pp. 1512-1517, September, 1990.
"Enhanced Period-Peak Analysis of the
Electroencephalogram Using a Fast Sinc Function,"with M.
Ferdjallah, Medical and Biological Engineering and Computing,
ABOUT THE AUTHORS
The correct citation for
this article is:
Davis and Matt Insall, "Mathematics and Design: Yes, But
Will it Fly?", Nexus Network Journal, vol. 4, no.
4 (Autumn 2002), http://www.nexusjournal.com/DavIns-HM.html
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Copyright ©2002 Kim Williams
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