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.