Most good Mathematics is done somewhere on the middle ground, and what is Pure Mathematics one day (Complex Numbers, Complex Functions, Matrices) is mysteriously Applied Mathematics a short time later. And Chaos theory is taught at UWA as Applied Mathematics, although nobody has made a buck out of it so far, unless you count academic paper production. One day, a great many bucks could be made. Our descendants will know in a century or so.
All Mathematics lives in the tension between two kinds of mental processing, the intuitive and geometric on the one side and the formal, logical and computational on the other. Without the intuitive, Mathematics is sterile, without the formal it is wishy-washy. This is as true of Pure as it is of Applied Mathematics.
It is hard to say what the difference between Pure and Applied is: it seems to be a matter of style. It isn't that Pure Mathematics is more formal, and Applied more intuitive, since Pure Mathematicians complain about (bad) Applied Mathematicians pushing recipes and formulae down students throats. It isn't that Pure Mathematics is more intuitive and less formal, or Appied Mathematicians wouldn't whinge about (bad) Pure Mathematicians leaving out the motivation.
It isn't that Pure Mathematics is useless; look at any area of theoretical Physics and you will find Pure Mathematical ideas from Algebra and Geometry. Some of the Pure Mathematicians here at the University of Western Australia are working on making sense of objects in images, a subject otherwise known as computer vision. This is so robots with vision may become a reality one day. Other work which has been done by Pure Mathematicians at this University involves CAD (Computer Aided Design), Control Theory, theoretical Computer Science, and scattering theory, as well as the more obviously `pure' topics of finite groups, finite geometries, linear analysis and singularities of smooth maps. All things considered, Pure Mathematicians are a versatile lot.
If there is a difference, it is perhaps similar to the difference between a physicist and an engineer. The former wants to understand how things work, the latter can be pretty shaky about the understanding so long as he knows what happens when he does certain things. Mind you, modern Engineering isn't just a matter of knowing a few tricks either. Pure Mathematicians like understanding things; they like seeing the patterns that make sense of a subject. For that reason Applied Mathematics often consists of Mathematics which was once Pure. On the other hand, much good Pure Mathematics has come about from making sense of ideas put forward in a scruffy (but useful) form by engineers or physicists. For example, Functional Analysis is a subject which owes its origins to Heaviside, an engineer, and the theory of distributions was devised to make sense of nonsensical (but useful) ideas such as the Dirac Delta function, invented by P.A.M. Dirac, a theoretical physicist. And the Exterior Calculus of Differential Forms, which has many uses in Physics and Engineering, was fed back recently to the physicists by pure mathematicians who sorted out the subject in the twenties and thirties after it was invented in a fairly scruffy way in the late nineteenth century by Lord Kelvin, a physicist (with some help from Stokes and Maxwell).
In the old days, before television, mathematicians usually did a lot of calculations. These days the computer does the sums. This means that the emphasis has to be on figuring out the rules the computer has to follow in order to get the right answers, that is, to devise the algorithms. It is this which is the sharp end of Mathematics these days; but devising algorithms has to be done in the context of a theory. Building theories is needed to devise algorithms. And theories are based on data, which again these days may be provided by a computer. Mathematics has been turned into an experimental science. Well, it always was, but now we can do a LOT of sums when we do our explorations. Well, we and the computer.
Pure Mathematicians go for clarity and insight. But it can be a long road to get to clarity, and from a thoroughly muddled starting place.
What the Applied Mathematicians go for we are not quite sure; perhaps it is better not to ask.
Anyway, as you can see, like any family members, we bicker a certain amount. The thing to do is to talk to people in both camps and make up your own mind. And don't buy any cheap bridges.
The answer is that former Pure graduates are found all over the place. One of our former algebraists is now with the Bureau of Statistics, another algebraist from ANU is also a senior staff member there. Some wind up as academics. Some wind up in the public service, others are lawyers. IBM tends to prefer Mathematics graduates to Computer Science graduates, all other things being equal: maybe snobbery has something to do with it. Pure Mathematicians are generally thought to be people who have an ability to think creatively and critically, and to learn some abstract ideas. Whatever it is, there are less unemployed Pure Mathematics graduates than there are any other sort. We also suspect that those Pure Mathematicians who ARE unemployed want to be.
Statistics are hard to come by, but most people in industry take the view that the average graduate knows almost nothing of value to his company, but the graduates who have tackled the hard subjects have shown an ability to learn some difficult ideas. So they must be smart. This is a good thing, up to a point. And past that point, graduates become post graduates and do Ph.D's anyway.
There is more information about Job prospects if you want it.
`Aha!' said the Statistician. `All Scottish Sheep are black.'
`Oh no', said the Physicist. `SOME Scottish sheep are black. Isn't that so?',
turning to the Mathematician.
`I fear, gentlemen', said the Mathematician, `that you both go far beyond the data. All that can be said, and that with grave reservations, is that there exists in Scotland a field, in which there exists a sheep, AT LEAST ONE SIDE OF WHICH IS BLACK'.
This will make more sense to you after you have studied the definition of a continuous map.
Mathematics is difficult. It isn't supposed to be easy. On the other hand most mathematicians are quite lazy. They like to find the quick and easy way of doing something. Faced with a boring calculation which could be done in thirty minutes, the true Mathematician will spend a week looking for the quick way. This sounds daft, but isn't. If he finds a way of cutting it down to twenty minutes, and then he tells a few people who tell others, and so on, the total profit to the world is enormous. And the Mathematician had a lot more fun figuring out what is going on instead of doing something boring.
Some people are brash and assume that if they can't do something quickly then it must be hard. Others lack confidence and assume that they must be stupid, or just lacking in talent. Also most people have had bad Mathematics teachers at school, because the number of people who are both good at Mathematics and like teaching is rather small. If you had such a teacher, send him or her a box of chocolates now. Most people have to make do with teachers who are very hazy about why those formulae work. This is the most interesting bit, actually plugging numbers in is not something normal people find exciting. Whereas almost everybody enjoys the little light flashing on over their head when it all clicks into place.
THAT is why Pure Mathematicians do Mathematics. They get a kick out of seeing the pattern, seeing why it makes sense. It may not have happened to you yet, when it does it feels like scoring the winning goal only a million times better.
Getting to be good at Mathematics is almost all a matter of practice. Like learning to ride a bike, you fall off a lot at first and feel stupid. After a while you master some of it and feel grand. You want self-esteem? The best way to get it is to tackle hard things and survive. The difference between riding a bike and learning Mathematics is that after a while you take your bike-riding skill for granted; that's IT. You have to move up to cars or horses. But there's no end to Mathematics. It just keeps on getting more mind-boggling the more you do.
There was a little girl who went to the Circus and turned to her mother at some point and asked `Mummy, are we having fun?' Mummy assured her that she was. Many people assume that they must be enjoying themselves if they are doing what everybody tells them is fun. Now the idea that Mathematics is fun is too bizarre a concept for most people to credit. So if you need to take an opinion poll to decide if you are enjoying yourself, you probably aren't. Doing Mathematics can't possibly be enjoyable can it? Isn't it just some sort of nerd thing, done only to impress, and failing miserably? Well, try it. You might get a pleasant surprise.
It undoubtedly helps if you have a sense of humour, are inclined to ask questions and have a good imagination. It helps if you are intelligent, whatever that means. But most people have some sense of humour, and most people like understanding things. As long as you are prepared to do some slogging, as long as you are persistent and know when you are muddled and don't like it, you can do Mathematics.
It is also useful for Physicists, Engineers, Economists, Psychologists, Biologists, Miners, Physiologists, Computer Scientists, and quite a lot of other ers and ists. In any real Science, or in Engineering, it isn't just useful, it's essential. The people who don't need Mathematics are people who don't mind living in caves.
So Mathematics is awfully useful stuff in terms of keeping society running. But this is why people pay us to do it, it isn't why we do it.
Still, it can give you some sort of idea about what the point of it all is.
Follow the links to learn more about
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If you have questions, comments or suggestions, email mike@maths.uwa.edu.au