revised June 2020
What is mathematics?
Let’s begin by comparing mathematics to science. Is mathematics a
science? There are similarities. In both, experiment and imagination
lead to hypotheses, which of course you try to prove. But there are
also differences between sciences that study the natural world and
mathematics, which studies abstract idealized objects such as
numbers, algebra, and geometric objects that exist only in
Therefore I do not claim that mathematics is a science, but
opinions differ. Stanford awards a Bachelor of Science degree for
graduating in mathematics; UCLA, a Bachelor of Arts. Gauss himself
a scientist by any standard (also the deepest and the strongest
mathematical genius) said that mathematics is "the queen of the
sciences." Go figure.
The realm of Mathematics
The realm of mathematics is pure reason. Only in mathematics, can one
with absolute certainty distinguish right from wrong. Conversely,
with no particular brilliance, no particular facility at computation,
no particular talent, except the rather mechanical ability to
distinguish rigorous thinking from bluster, there still can be
I asked Gordon, my advisor, "How far can you get in mathematics
without being smart?"
"Quite far," he said.
Pure and Applied Mathematics
Mathematicians distinguish pure and applied
mathematics. Actually neither of them applies directly to the real
world. But applied is the name given to those parts of
mathematics that are motivated by the expectation that eventually
they will be applied to the world. Mathematics that will be useful in
engineering, for example, or in statistics.
Obviously the really interesting mathematical problems are therefore on the
pure side. For here you can do anything. The field of Number Theory (defined as
studying problems about the integers, like
Fermat’s Last Theorem) is attractive for two reasons: One, it is absolutely pure, and
two, at least the problems, if not the solutions, are often comprehensible to a lay person. A
child, for example.
Gauss felt that number theory is the queen of mathematics.
I studied number theory, which is far removed from what I’m doing
now. Suffice it to say that I had to modify my prejudices and
motivations to come to this point. I also had to pick up areas
like statistics on the street.
The guest speakerstory! at the weekly mathematics colloquium was
scheduled to speak on "Sums of squares." Sounds very number theoretic. But then I had an awful
thought, so I asked Professor
Straus whether the talk was indeed on number theory, or whether it could be instead on
"Number theory of course," said Straus, "Look who he’s talking to." I noticed the stranger to
whom Straus referred, our visiting speaker, engaged in conversation with a wispily bearded
algebraist fellow from the department. Algebra is somewhat allied to number theory, and is
certainly a "pure" discipline.
"Do you mean," I asked, "that pure mathematicians never talk to statisticians?"
"Well," Straus assured me, "certainly algebraists don’t."
story: The speaker was in fact
the emminent number theorist
(1910-1985) who was the most famous mathematician in China and whose
remarkable history may be the inspiration for part of the story in the movie Good
Home page of Charles Brenner