Simplicity – the word was a recent daily prompt. When I read it I was reminded of something I read recently, and something I have been aware of for some years – simplicity in mathematics, of all things.
I’m no mathematician. I chose a balance of arts and sciences in high school, and although I’m glad I did, I do wish I could have somehow done advanced maths as well (they used to call it pure and applied maths – not sure what they call it now, or what they call it in your country). So I don’t understand what simplicity in mathematics means in practice, but I love the idea.
Mathematics is anything but simple. And the most complex mathematics in the world is tied in with science, physics in particular. I can’t begin to understand any of it, and I believe that when Einstein came up with his theories of relativity, only a handful of mathematicians in the world were capable of understanding it. But what interests me most about solving physics problems with mathematics, is that scientists look for the simplest, or most elegant, solution.
Elegant – what a lovely word. I remember watching “A Beautiful Mind” a film about the life of a famous mathematician, John Nash, who at one point in the movie, discussing a student’s submission to a maths problem, said it was an elegant solution but was unfortunately incorrect. The fact is, scientists love elegant solutions – mathematical equations that solve as many problems as possible, as easily as possible. They are particularly in search of the holy grail – a mathematical solution which combines all the different disciplines of physics (relativity theories, quantum theories and goodness knows what else). It’s called (wait for it) – the Theory of Everything.
I wish I could provide examples of elegant mathematical solutions but they are way beyond me. I just love the idea. I rather expect that, by elegant, they mean something that doesn’t have to have lots of “fixes” attached to it. Let me try, by way of contrast, to give a human example of an inelegant solution.
We don’t have to look any further than the laws of our land. It doesn’t matter which country you are in, you will have exhaustive and mind numbingly detailed laws that try to apply justice to every conceivable situation. Laws that say “this is the rule” and then go on to list all the exceptions to that specific rule. This is necessary, though unwieldy, because all laws are blunt instruments, and have unintended consequences. In order to minimise those unintended consequences, a whole lot of exceptions have to be included. A bad law is one that has lots of unintended consequences, and either needs masses of exceptions to try and curb the damage, or it just creates a whole lot of damage and that’s it. I thought about providing an example of the unintended consequences of an otherwise good law, but it was so depressing I decided against it. This blog, after all, is about “simplicity”, not the ugly side effects of our legal system.
It’s a bit scary to think that laws are like that, but they are designed by humans and that’s probably why they are so unwieldy, and certainly inelegant. Physics, however, seeks to find out the fundamental building blocks of the universe, and I find it very attractive and wondrous that the equations that work best are those that are elegant, and don’t have any “work arounds” or “fix it clauses”.
Not that ‘fixes’ are never used in science – apparently they are. But the best laws don’t need any fixes. Einstein’s theory of relativity predicted an expanding universe. At the time, it was held that the universe was constant, and did not expand. He therefore applied a ‘fix’ to his theory to accommodate this, and lo and behold, only a few years later it was proved that the universe was indeed expanding! Einstein felt very sheepish, but this was simply more proof that his original theory was indeed an elegant solution.
It amazes me that Isaac Newton came up with the laws of gravity – that two bodies attract each other (think about that – and we’re not talking about magnetism here), and then found a whole set of equations to predict that behaviour. And although they were elegant and accurate in almost every situation, along came Einstein with his theory of relativity (which he worked out all on his own, an amazing feat) which provided an even more complete description of how gravity works, even though his starting point, his premise, was completely different to Newton’s.
That astounds me in itself. Newton’s theories worked in all but the most exacting of situations, but a better solution answered the questions of gravity even more elegantly and completely. As I said I’m no mathematician, but I stand in awe of the minds that can see the problems and find such amazing solutions.
Here’s to simplicity and elegance in the very framework of our universe, even though it takes geniuses to find them.