What is Mathematics
Mathematics is an artificial language intended to be imbedded in a natural language for the expression and manipulation of sense data. And it was my first love, and always will be. First love is a bit like virginity. It only happens once.
The word mathematics takes its meaning from the Greek μανθάνω which means I seek, I learn, I understand. So it means, to write down what you know. It is the art and science of doing that precisely.
Where does Mathematics come from?
A related discussion is, how is new mathematics created?
Some mathematicians think they are creating something new. Others think they are describing something that already exists. Still others think it's a bit of both. Leopold_Kronecker is supposed to have said God made the integers, all else is the work of man (or at least that's the standard translation of his comment which was in German).
I think it's both at once. I even think this is the only approach worthy of any Mathematician. We have two notations for the same thing, and that's good. We work with whichever works best. They are both tools. This fits well with my belief in what I call the medium-strong-and-weak principle of linguistic_relativity. Language is one of the things your mind is made of. The more languages you speak, the richer your mind becomes. And the same with mathematics. See my own essay on linguistic relativity.
Pure Mathematics, Applied Mathematics, Hobby Mathematics
Morris Kline famously called for abandoning Pure Mathematics completely.
With respect, he didn't mean Pure Mathematics. He meant what Pure Mathematicians spent most of the 20th century doing. And he was right in that. But his mistake and theirs was in thinking this was Pure Mathematics. It wasn't Mathematics at all. So Kline was right in another way too. They were wasting the Mathematics department budget in doing it.
Hobby Mathematics
Years after I left Macquarie University I returned to speak with one of my former lecturers who graciously gave me some of his time to explore some of my ideas. He said
I'm sure if something I create turns out to be useful, we're all delighted. But that isn't the point. What I am creating is just a beautiful structure.
If my definition above is correct, then that isn't Mathematics, because it is not motivated by the desire to explore and understand the real world. Mathematics is, to use the subtitle of Eric Temple Bell's 1951 book, Queen and Servant of Science. So exploring these beautiful structures for their own sake doesn't belong in the Department of Mathematics at all, and much less in the Macquarie University School of Mathematics and Physics.
It does belong in the University, but in the Department of Philosophy. Because that's what it is. Similarly, Lord Rutherford famously said The great beauty of my work is that it will never be any use to anybody. He was describing his discovery of the atomic nucleus. But more important, he was describing what was then called Natural Philosophy rather than Physics.
Applied Mathematics
When I first attended Macquarie University as a first year undergraduate enrolled in the School of Mathematics and Physics, I attended the introductory lecture given by Professor Ward, FRS, then head of the School. He told us that we would be educated in Mathematics in the same way that students of musical composition learned the rules of harmony, so that we would be able to follow and break them in an educated fashion.
It's a good approach. And he was very good at it. And it was reflected in the Macquarie University program of allowing the best HSC students to skip first year Mathematics subjects and go straight to second year subjects (which was probably a good idea for many others too... first year maths at Macquarie was far less rigorous and comprehensive than Level 1 mathematics at High School).
But it's not the only approach.
Pure Mathematics
Pure Mathematics is what Newton and Leibnitz and Gibbs and Hamilton and Levi-Cevita and Einstein did. They didn't break the rules. To an applied mathematician that may appear to be what Hamilton for example did in abandoning the Commutative Law in his definition of Quaternions, but it was far more profound than that. They recognised that the rules didn't describe the Universe as well as they wanted to, and came up with new ones that were a better fit.
And that is exactly what is needed to make sense of the enormous amount of confusing data that we now have.
To an Applied Mathematician this may seem like laziness. And there's an element of truth to this! One of my age-mates as an undergraduate was advised to do physical science rather than Mathematics as his honours year (he was one of those exceptional students who had done so many final year subjects and received such good grades that he had the choice) because he "lacked the essential laziness" that motivated the very best mathematicians. We both knew what they meant by this advice, and it came from some excellent mathematicians, and he took their advice.
There was a lot of Mathematics done in the 20th century, and much published in what was called Pure Mathematics. And some of it was real Pure Mathematics. Banach Spaces were for example used to solve whole families of previously insoluble differential equations. Professor Ward himself solved some important integral equations that had not been believed to be capable of analytic solution.
But there are some real questions as to whether our calculus, based as it is on so-called Real Numbers, is adequate to measure the universe on the scale of quarks or of galaxies. And these questions have not been resolved, or even as far as I can see attempted. (And I am very keen to learn of any such attempts. Whether by professional mathematicians or by amateurs such as Descartes and Einstein.)
See also
logical equivalence
More to follow!
Comments (0)
You don't have permission to comment on this page.