What is mathematics?
The realm of mathematics Cruel humor
Loo-Keng Hua

# Mathematics

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 thought.

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 adequate mathematics.

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.

#### Number theory

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.

### Cruel Humor

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 (shudder) statistics.

"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 Loo-Keng Hua, (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 Will Hunting.