Fourth of July, and the State of Mathematics

With everything that’s been going on lately, first Farrah Fawcett dying, then Michael Jackson died, then Billy Mays. Then Jeff Goldblum died, then Sarah Palin died. And Michael Jackson is still dead!

I know you must be thinking – Where did he go? Where has that Fox run off to? Where is that fantastic Mr. Fox?

In what is not a non-sequitur, Roald Dahl hosted a TV show in the early 1960’s called Way Out. Sci-Fi, Fantasy, Horror, it was very much in the spirit of The Twilight Zone. I remember it most clearly for the opening sequence, which involved a hand coming up out of a grave and bursting into flames.

Actually, I don’t remember that at all – but I do remember my father describing that to me. Or, at the very least, that’s how I remember him describing it to me. One of those many things he’s said, almost in passing – but it all adds up over time. Building this image of a time, a world. Important things, small things. James Burke. TV shows. People, places. The Wounded Lion statue in Switzerland. So many things.

Important things.

And, I meant to write something about that on Father’s Day, and didn’t. Sorry, DadFox. You are best.

Of course, I’ve been meaning to write so much lately.

Which in turn begs the question – just what is it I -have- been doing?

I’m still in Pittsburgh right now, finishing off some math for the summer. Most notably a course in Calculus of Variations – which is some very interesting and exciting stuff – and a research project/experiment in using Monte Carlo techniques (as well as the Cross-Entropy Method, new exciting stuff) to try and solve ‘hard’ graph theory problems. I will probably write about some of that, soon enough.

But, I would like to get back into the habit of writing. There are so many interesting things that need to be shared and explored.

For instance, consider the following game. Player A and Player B both have a die, and they are rolling them, each with a goal in mind. Player A wants to roll a 1, and then another 1, in order. Player B wants to roll a 1, and then a 2, in that order. They will continue to roll until they get their sequence, 11 and 12 respectively. The winner is whoever gets their sequence first. The question is – who is more likely to win? The obvious answer is that they are equally likely. Various arguments can be made, for example that the probability of rolling a 1 is equal to the probability of rolling a 2. However, since I’m asking the question, you should assume the obvious answer is wrong – and in fact, it is. On average, Player B needs 35 rolls to win, but Player A needs 42. Another reason I like this problem, aside from it being non-obvious, is that the answer is (in part, at least) 42.

One thing I’ve been thinking about a lot lately – and I think is a good starting point to get back into writing about maths – is this: Just what is math? I’ve been ‘doing math’ for an awfully long time, and I’m still not sure I know what it is. I know I’ve gone through phases of knowing, thinking I’ve known … but certainty always fades, leaving the question, “But what do you doooooo?”

At first, math is taught as a tool. Basic arithmetic. Algebra. Calculus. Tools to solve certain kinds of problems. To answer certain kinds of questions. How many sheep to I have? How many sheep do I need? What is the interest on this investment going to be? At this rate of increase, how long until our coastal cities are submerged in water? How much fuel do I need to put a man on the moon?

But, as you continue on – it’s the questions themselves that become important. Asked in slightly different ways, perhaps – what if we want to land on Mars, instead of the moon? What is the square root of negative sheep? Sometimes, the questions come, almost naturally, from the math itself – given a function, is there another function that has the first as a derivative? This leads, naturally, to the entirety of differential equations. Again, more tools to answer more questions. But, the questions continue – even in a branch of math as well studied as differential equations, it’s well known that most interesting differential equations cannot be solved analytically at all. Questions questions questions – always new questions. Bring your differential equation tools to bear on this little gem, which I may discuss sometime. I dare you.

The questions come faster than our ability to answer them, for the most part. Higher level math seems to be, from what I can tell, developing new ways of thinking about these questions – new tools, and frameworks to phrase them in, to try to get at an answer. But these new tools create more questions still.

After a while, it seems as though what math is isn’t the tools, it isn’t the arithmetic or the formulae, it’s the questions themselves. But, I feel as though it’s a symptom of something larger. The mere act of asking a question is a product of thinking about something in a certain way – expressing it in a certain framework. Processing an object, concept, or idea, in a way that allows the question to be asked.

And that, I think, is the core of mathematics. It’s a way of thinking about the world and processing it. And in my mind, it’s a fundamentally scientific way of thinking. Testing, exploring, and questioning.

A drunken physicist once told me that I wasn’t a scientist – I was a philosopher! Gauss, however, declared math to be the Queen of Sciences. I imagine it’s clear who I agree with.

But, at this point, I have veered so far off track it should be clear – I don’t know what math is. There is something I am very excited about. It’s … it’s thinking, it’s exploring. It’s fundamentally creative, inquisitive, and incredibly human. I identify it with math, but, as I think we can all agree, anyone who declares math to be anyone single thing is probably wrong.

I do not know what math is – but I will continue to do whatever it is I do do, because I have an excellent time doing it.

Next up! When I get a chance, several interesting calculations.

Happy Fourth of July!