Mathematics is both a language and a form of artistic expression. Language because it has all its characteristics: it is symbolic, has a morphology and a syntax. It also represents something as an idiom does, in this case our natural world (or rather, our limited interpretation of it).
But Math is not tied to reality, unlike Physics for example. There is no need for experiment to validade its work. As an important mathematician one wrote:
The essence of mathematics lies in its freedom.
One is free to conjure any kind of creation to suit his needs. That is the great power of this field. It can be something that was motivated by a practical problem: How to keep track of a herd of sheep? How to measure the area of a irregular terrain? Will there be enough corn to feed the population? For practical reasons, this comprises most of it.
And it can be simply a thought experiment: What would a 9-th dimension cube be like? The problems in this pure Mathematics have a higher degree of abstraction, and are considered a larger set of the problems that arise from the real world. It’s possible for one of these problems to be found later having a counterpart in the physical.
The more you come close to this century Mathematics, the stepper the abstraction curve goes. That is the reason why most people will only have the practical knowledge of the Classical Era. And that is a bad thing in my humble people, but I guess Math just isn’t everyone’s cup of tea.
People could benefit from the gain in coping skills from studying it, though. It’s trying at times, but the satisfaction of grasping a new math tool is beyond words. And the variety of problems you can solve increase rapidly with more studying. I consider my Math skills the ultimate measure of knowledge, as everything I ever hope to learn has it in its foundation.
And it’s still under construction, even when it comes to the basics. Much is done of course, but one can easily find knots to tie or a new problem to tackle. A example of this is the prime numbers – is there a formula that can give any prime, given it’s position? This has haunted me for a very long time, it looks at first so simple. Solving it could render our cryptography useless, for those who want to see the world burn.
The mysterious pattern in prime numbers, known as Ulam’s Spiral. credits
There is much more that we can’t do than that we can. As an example, only a handful of integrals have a primitive in terms of defined functions. And the hardship of computing an integral increases dramatically if you start picking elaborate functions.
One might view this tendency as an evidence that the Math we developed is unsuitable to handle our intricate universe. We could be speaking Russian instead of plain English. Perhaps there is a different way of looking to things that is simpler and more effective. It’s hidden, waiting to be found.
From this freedom of creation comes the artistic aspect of Mathematics. In a carefully refined theory, one could leave a lasting work just as a painter or a composer does. I hope this has insipired some more courage when opening your next Calculus book.
[IMAGE: A computer generated fractal, Sierpinski. image source]