Hi,
Enjoy reading the following article about maths research
that has been written by Anton Zettl. Thank you.
Cheers,
Wen Shih
I want to start by saying a few words about research in
Mathematics generally. Describing research in mathematics to an
audience not familiar with the language of the subject is a
formidable task, one that may well be beyond my capabilities.
Nevertheless I will make an attempt.
It is more difficult to explain, to a general audience,
research in mathematics
than it is to explain research in many other fields, including
the
Sciences. Not because mathematics is deeper than the sciences, but
because
scientists have the advantage of being able to build on common
human experiences
and to rely on intuitive meanings of commonly used terms.
Thus
a Physicist can talk about an atom and all of us prescribe some
meaning
to this term. She can then go on to tell us what it really is and
describe its
fascinating properties. Similarly a geologist can talk about
ancient volcanic
rocks in lake superior and a biologist can discuss the genetic and
molecular
analysis of certain viruses.
A mathematician does not enjoy such a luxury. People
outside the field
do not normally encounter the spectrum of a differential operator
and even
if they did they probably wouldn't be aware of it.
Everything in mathematics, even the numbers 1,2,3, ... has been
painstakingly
and rigorously defined in terms of basic set theory. So when a
mathematician
discusses the differential equation
y'' + q(t)y = 0 (1)
every other mathematician knows precisely what he is talking
about. There
is no ambiguity, no room for different "interpretations".
Research in mathematics consists of proving theorems. A
theorem has
two parts: (i) a statement of a fact and (ii) its proof. The
statement must
be clear and precise, the proof completely rigorous and based
entirely on
logic. All terms used in the statement must be painstakingly and
accurately
defined, the proof cannot leave any doubt about the validity of
the
statement. Here mathematics differs from the sciences. Trying to
prove a
theorem merely by confirming its validity in 27,650 special cases
is totally
unacceptable in mathematics; no graduate student survives a basic
course
with such an attempt. Some 5,000,000 special cases out of
infinitely many
possibilities is an insignificant number.
Once a theorem has been proven it becomes part of the
heritage of the
human race forever. The Pythagorean Theorem established by the
Greeks
more than 2,000 years ago is as valid (and as useful) today as it
was then;
the Chinese Remainder Theorem is more than 3,000 years old. A
research
Mathematician simply adds to the collection of theorems. The number
of
known theorems increases every day. The total accumulation of
theorems
is so large that no human knows even one percent of them. Only God,
and
possibly someone somewhere in the Andromeda galaxy, knows them all
and
also those yet to be discovered.
How does one discover a theorem? Often part (i) expresses
a pattern
observed in numerous examples. For part (ii) the challenge then is
to prove
that, given the conditions of (i), this pattern always prevails.
There are two
extremes: On the one hand you can take a known conjecture - a
pattern
observed by someone else - and prove it. A prime example of this is
Andrew
Wiles' proof of "Fermat's Last Theorem" (a misnomer - it should
have
been called Fermat's Last Conjecture). A French lawyer named
Fermat,
whose hobby was mathematics, claimed, in the margin of a book he
was
reading at the time, to have proved a theorem. But only part (i)
survived
him because he died before he was able to publish part (ii). This
occured
more than 350 years ago. Fermat's part (i) became so famous that
many
people devoted their life to finding part (ii) but only Wiles
succeeded. If his
proof holds up - it is still being checked by experts - Wiles will
join Euclid,
Archimedes, Newton, Leibnitz, Gauss, Fermat, Euler, etc. as one of
the
immortals of Mathematics.
At the other extreme are people who find a new pattern
for part (i)
and succeed in proving it. This way of discovering theorems can be
utterly
worthless, if no one else is interested, or it can be a major
breakthrough.
Often asking a good question is more important, and requires more
insight,
than finding the answer.