Archive for 2008

Simulation and Modeling

Wednesday, July 30th, 2008

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.

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Physical Data

Wednesday, July 16th, 2008

RDF stores in hardware.

Whatever else it turns out to be, the semantic web has already given us RDF. It’s here, it’s now, it rocks.

RDF says that every data structure can be modeled using three part statements consisting of Subject, Predicate and Object. Each of these being a simple piece of data that can be represented as a string.

This is a surprisingly substantive claim. It means two part statements (Subject/Property, or Name/Value) are not enough, and four part statements unnecessary. It means that in structured data RDF triples play the role that triangles play in graphics. If you can draw triangles you can draw everything. If you optimize your triangle drawing you optimize all your drawing. In networking TCP/IP packets play a similar role. Routers, switches and network cards don’t care what new applications are built on top of packets. Make passing them efficient, and everybody benefits, the cost of switching packets tends to zero.

The completeness property of triangles in graphics makes graphic cards rewarding – custom silicon that solves a specific problem insanely efficiently. RDF has similar potential. RDF can scale in hardware. The memory storing the statement can be smart enough to parse and query it. The structure is so simple, basic filtering so straightforward, that it can be done on a per triple basis in hardware, implemented directly in transistors alongside the non-volatile memory, built scaleable to plug into an RDF bus. As you add data you add processing power. Since one of the sweetest properties of RDF is the fact that it can be munged together indiscriminately – you can always add more triples without compromising what’s already there – this means we can have physical data. Physical data scales really well.

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Six Suggestions That Can Make You a Better Maker

Friday, June 6th, 2008

A designer friend recently sent me a link to an excellent post by Eric Karjaluto on design.

The original post is design specific – and well worth reading – but it also struck me as completely relevant to programming, which I do professionally, and to a great deal else that I’ve done and tried to do over the years.

Here’s my version of the six suggestions:

1: Original ideas come as solutions to problems, not when you go looking for original ideas. Collect good problems.

2: The goal is not to impress people with the complexity of your ideas, but to stun them with their simplicity. There’s nothing better than an “obvious” solution that no one else has thought of.

3: Ideas are there to be shared. If you can take someone else’s idea to a new place, or someone can take something you’ve done and use it for a new purpose, then everyone wins. Ideas are not used up by being applied. Intellectual Property is Theft©

4: Always be growing your vocabulary and your capabilities – expanding the collection of things you’re confident you know how to do. This opens up new landscapes. If it isn’t in (or close to…) your repertoire then it won’t be in your imagination. You can’t want to do things you don’t know about.

5: Lose the fear of doing the wrong thing by minimizing the cost of doing the wrong thing. Errors are not the problem; expensive errors are the problem. Reduce the cost and increase the rewards of your mistakes. Then make plenty of them. If you don’t fail some (or most) of the time then you’re not trying hard enough.

6: Do something. Actually take the first step and see what the landscape looks like from there.

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What’s the Biggest Number in the World?

Sunday, May 18th, 2008

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.

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Who Needs Algebra?

Thursday, May 15th, 2008

One of the biggest questions about mathematical education has to be: Why bother? Anyone teaching math gets lots of practice answering that. It’s a good question.

There are two types of justification, the aesthetic and the practical. The aesthetic case is actually the easy one. Math is one of the least obvious, most complex, profound and uniquely creative things our species has done, right up there with music, painting and dancing. As a liberal art math needs no justification: if you want to understand the glory of what it is to be human then you need to be understand something of these spectacular creations.

But if that was all there was then we certainly wouldn’t be spending all this money teaching math. We had no problem dropping music from the curriculum. (Doubly unfortunate given that music lessons help math scores.) Math is taught for economic reasons: we think the return justifies the expense.

It probably does, but there are many unexamined assumptions here. Consider some justifications for teaching math.

  • it’s useful
  • it’s good for the brain
  • it separates the wheat from the chaff
  • engineers need it, and we need engineers
Usefulness

It’s only useful if you use it, or if being able to use it unlocks some new ability. (Being able to swim enables me to go on the boat, even if I don’t fall in and never have to swim.) But if you never use it, never consider it, are never empowered by it, can never do something only because you have the knowledge, then it is not useful. A lot of the math people are taught is like that.

Good Brains

Any kind of study is good for the brain. Latin and Greek are excellent. Memorizing decks of cards, wine regions of France, the plays of Shakespeare, all good. The brain is a muscle. Use it and make it strong.

That doesn’t explain why mathematics instead of Greek, but there might actually be an answer to this one. I believe that there are useful, brain reshaping ideas in math that none of us is likely to come up with in a single lifetime. Things like the reductio ad absurdum, like proof by induction. The kind of idea some human being comes up with every few hundred years. Things worth sharing. I believe that there are brain rewiring mathematical ideas that most people should be exposed to. But that needs to be proved. And which specific ideas should be prioritized and taught should be determined by something other than habit and inertia.

Math as Gatekeeper

If you need to narrow down the field then math works about as well as height. If you need to eliminate 50% you can choose one type of math and one cut-off score. If you need to eliminate 90% then, hey, let’s require calculus… If the math skills are needed then fair enough, otherwise this is as unfair as discriminating on any other irrelevant criterion. And yes, I’m talking about requiring nurses to know algebra.(this caused a stink) I want nurses to be bright. I’m not sure that requiring algebra measures the kind of smarts that really matter. I might be wrong, but it isn’t obvious. Using math as a proxy for an IQ test is just dishonest, and lazy.

We Need the Eggs

Suppose traditional math education ruined almost everyone’s chances of understanding or enjoying math. And suppose that the only way to train engineers, physicists, quants, etc, was to teach them that way from an early age, before they could even be identified as prospective quants. If that were true then it might well be worth teaching everyone that way. We do need the engineers. (Under the right circumstances even I would swap understanding how the defibrillator works for having a defibrillator.) But this is implausible. Colleges and universities need to be fed numerate students, and the burden imposed on them by the need to supply remedial math classes is considerable, but if the best answer we can come up with is to teach in ways that are known not to work for most people then we need to keep looking.

What To Do?

What is useful and necessary? It depends on personal accident, but there are probably lots of overlaps. This is the kind of thing a rational; research policy would research: what math gets used? What is unknown but of most value? We already spend an unimaginable fortune on math education (This is the good news. Can you imagine how jealous art teachers are right now?), we could spend a few more bucks on measuring needs and results. Every scrap of curriculum needs to earn its place. Historical importance is no more relevant in math than any other historical importance. The speeches of Cicero are of enormous historical importance, but we gave up teaching Latin. Quadratic equations are the Cicero of high school mathematics.

What actually is good for the brain? (This is a special case of ‘usefulness’.) And it’s an empirical question. We ought to be able to answer it and to agree upon the answer.

What do engineers need to learn, and when? In most things we learn what we need to know close to having to use it. That helps with motivation, understanding, and retention. On the other hand musicians, sportsmen and dancers, are generally considered to need an early start. That might also be true for mathematicians. This is an interesting and difficult question, but it’s the right question.

There are real questions here, and issues that should generate research instead of debate. Rather than argue about the facts we should find out what they are.

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