Newsworthy Math

By definition, the expected value of a one in five chance of winning $10 dollars is worth $2 ($10/5). But to human beings, with our finite life spans and our asymmetric financial needs, a one in a million chance of $1,000,000 is not necessarily worth $1, it might be worth more.

Lottery tickets cost far, far more than their expected value. The amount you would win times the probability of you winning is pretty close to zero. They are, as far as expected value goes, practically worthless.

But despite what Adam Smith says, buying lottery tickets is not necessarily a bad idea. They’re vastly overpriced, but unless I’m already rich the difference winning $1,000,000 would make to my life is worth far more than the extra $1 I’m getting charged for my ticket.

Given the choice between $2 and a one in a million chance of a million dollars, I know which choice is more exciting. The poorer I am the more enticing that is. (If I’m well off I have many more, and better, opportunities to make money.)

The injustice of lotteries is that they exploit the perfectly natural assumption that if buying one ticket is a good idea buying 20 is 20 times better. But Adam Smith is right again: buying all the tickets would guarantee that you won the lottery and lost a fortune.

Lotteries are not a tax on stupidity, they are a tax on poverty and lack of opportunity that exploits an almost universal miscalculation of the effects of scale on risk and value.

They are a lousy form of taxation. If you want more money for schools, roads or, cultural institutions, then raise taxes, or issue bonds. Don’t fund opera from lotteries, that way poor people pay for things they can’t afford to enjoy.

Simulation is a wonderful thing. It’s one of the foundations of mathematics.

Scale models are a good example. If you want to see what the new building will look like make a model. It isn’t going to tell you everything, the architectural models of housing projects usually look wonderful, but it’s a great example of using the external world to think for you, of externalizing your brain. Yes, you can read a description and imagine what something will look like, but making a model, reduces how much of your brain that uses, leaving more brain for other things, like deciding whether any child is really going to want to play under the gaze of a thousand anonymous windows.

The power of modeling is real. From the scale drawings my father used to make before decorating the kitchen, to the simulation that demonstrated that the proposed system for baggage handling at Denver Airport was guaranteed to fail horribly. (Unfortunately the simulation, costing a few thousand dollars, was done after the actual system was built at the cost of hundreds of millions, and then millions more in delays.) Dangerous, lengthy and expensive processes can be evaluated for a fraction of the cost of the actual experiment. Some people’s love this new power, others can’t forget what’s being lost.

What’s lost in any abstraction is the specific, the individual, everything that matters. Some people can’t get over that.

Simulations and models only work because they ignore details, knowing whether they are important details can be difficult. A scale model works fine for the forces on a house or a skyscraper, but is hopeless for a ship. Econometric models are notorious for their spurious certainty.

So the practical fitness of a model to it’s purpose is one question, but there are others. For some people it is impossible to imagine the classic absurdist math problem characters “a man”, or “a woman” without some humanity to hang on to. People differ on this.

I remember some fine junior high level course material on date arithmetic. A drawing of a set of gravestones: name, born, died. The question was: how old was each person when they died. One child faced with “James Brown: March 1823 to June 1839″ confounded his teacher by insisting on knowing why Jim died. I don’t share that need, but I rather like it. I’m glad it’s around. If we want to make more people more mathematically adept, this is the kind of fact we need to acknowledge and deal with.

They go on forever, of course, but one of the entertaining things about mathematics is its habit of defining and naming fabulously large numbers. This can also be useful. The Romans could barely write down the number of citizens in their empire. These days Roman numerals couldn’t write down the number of people on the planet. Not all number systems are created equal.

Millions

The Roman’s could count up to a few million, which turned out to be enough for them. (It is not a coincidence that people never “need” math that hasn’t been invented yet. That’s for essentially the same reason that programmers never need features the programming languages they know don’t have. It’s hard to think things you don’t have language for.)

Googols

Indian place notation made “orders of magnitude” meaningful – every new column increases the size of the largest number by a factor of 10, and that turns out to be plenty for the physical world. There are about 10^80 elementary particles in the universe. And even if we turn out to be missing 99.999999% of the universe, that still doesn’t takes us to 10^90.

Stupid Big

But while the number of “things” is relatively small, the number of relationships between things grows shockingly fast. If there are 20 people at your party there are 171 introductions to make. If you have four tables of five there is a truly ridiculous 11 billion ways of assigning guests to tables.

Does that matter? Sometimes it does. There are  53,644,737,765,488,792,839,237,440,000 different ways to deal a deck of cards to four players. This is a giant number – at one deal per millisecond that’s still about a hundred million times the age of the universe. And yet, if you play cards you care about this number. This is the basis of card odds in Bridge. Sometimes money is at stake.

Stupid Big, But Still Useful

If you think about it, it is extremely odd that such a practically uncountable number could ever have a practical use. This, I think, is one of the distinctive features of mathematics, and one that can either make it seem pointless – why should I care about numbers so big we can never count them? – or almost supernatural.

Here we have tools that let us calculate numbers so large we can’t possibly count to them, and yet we can reason about them, and come to practically important conclusions. We start in reality – 52 cards – we fly off into a fairy land of numbers so big we sometimes need new notations to write them down, and we come back with a discovery that lets us beat the other guy at poker.

Let’s hear it for fairy land.