What Does It Mean to Know the Answer?

My sub-field isn’t big on philosophical debates. We don’t tend to get hung up on how to measure an infinite universe, or in arguing about how to interpret quantum mechanics. Instead, we develop new calculation techniques, which tends to nicely sidestep all of that.

If there’s anything we do get philosophical about, though, any question with a little bit of ambiguity, it’s this: What counts as an analytic result?

“Analytic” here is in contrast to “numerical”. If all we need is a number and we don’t care if it’s slightly off, we can use numerical methods. We have a computer use some estimation trick, repeating steps over and over again until we have approximately the right answer.

“Analytic”, then, refers to everything else. When you want an analytic result, you want something exact. Most of the time, you don’t just want a single number: you want a function, one that can give you numbers for whichever situation you’re interested in.

It might sound like there’s no ambiguity there. If it’s a function, with sines and cosines and the like, then it’s clearly analytic. If you can only get numbers out through some approximation, it’s numerical. But as the following example shows, things can get a bit more complicated.

Suppose you’re trying to calculate something, and you find the answer is some messy integral. Still, you’ve simplified the integral enough that you can do numerical integration and get some approximate numbers out. What’s more, you can express the integral as an infinite series, so that any finite number of terms will get close to the correct result. Maybe you even know a few special cases, situations where you plug specific numbers in and you do get an exact answer.

It might sound like you only know the answer numerically. As it turns out, though, this is roughly how your computer handles sines and cosines.

When your computer tries to calculate a sine or a cosine, it doesn’t have access to the exact solution all of the time. It does have some special cases, but the rest of the time it’s using an infinite series, or some other numerical trick. Type in a random sine into your calculator and it will be just as approximate as if you did a numerical integration.

So what’s the real difference?

Rather than how we get numbers out, think about what else we know. We know how to take derivatives of sines, and how to integrate them. We know how to take limits, and series expansions. And we know their relations to other functions, including how to express them in terms of other things.

If you can do that with your integral, then you’ve probably got an analytic result. If you can’t, then you don’t.

What if you have only some of the requirements, but not the others? What if you can take derivatives, but don’t know all of the identities between your functions? What if you can do series expansions, but only in some limits? What if you can do all the above, but can’t get numbers out without a supercomputer?

That’s where the ambiguity sets in.

In the end, whether or not we have the full analytic answer is a matter of degree. The closer we can get to functions that mathematicians have studied and understood, the better grasp we have of our answer and the more “analytic” it is. In practice, we end up with a very pragmatic approach to knowledge: whether we know the answer depends entirely on what we can do with it.

Where Do the Experts Go When They Need an Expert?

If your game crashes, or Windows keeps spitting out bizarre error messages, you google the problem. Chances are, you find someone on a help forum who had the same problem, and hopefully someone else posted the answer.

(If your preferred strategy is to ask a younger relative, then I’m sorry, but nine times out of ten they’re just doing that.)

What do scientists do, though? We’re at the cutting-edge of knowledge. When we have a problem, who do we turn to?

Typically, Stack Exchange.

The thing is, when we’re really confused about something, most of the time it’s not really a physics problem. We get mystified by the intricacies of Mathematica, or we need some quality trick from numerical methods. And while I haven’t done much with them yet, there are communities dedicated to answering actual physics questions, like Physics Overflow.

The idea I was working on last week? That came from a poster on the Mathematica Stack Exchange, who mentioned a handy little function called Association that I hadn’t heard of before. (It worked, by the way.)

Science is a collaborative process. Sometimes that means actual collaborators, but sometimes we need a little help from folks online, just like everyone else.

Numerics, or, Why can’t you just tell the computer to do it?

When most people think of math, they think of the math they did in school: repeated arithmetic until your brain goes numb, followed by basic algebra and trig. You weren’t allowed to use calculators on most tests for the simple reason that almost everything you did could be done by a calculator in a fraction of the time.

Real math isn’t like that. Mathematicians handle proofs and abstract concepts, definitions and constructions and functions and generally not a single actual number in sight. That much, at least, shouldn’t be surprising.

What might be surprising is that even tasks which seem very much like things computers could do easily take a fair bit of human ingenuity.

In physics, I do a lot of integrals. For those of you unfamiliar with calculus, integrals can be thought of as the area between a curve and the x-axis.

Areas seem like the sort of thing it would be easy for a computer to find. Chop the space into little rectangles, add up all the rectangles under the curve, and if your rectangles are small enough you should get the right answer. Broadly, this is the method of numerical integration. Since computers can do billions of calculations per second, you can chop things up into billions of rectangles and get as close as you’d like, right?

Heck, ten is a lot. Can we just do ten?

For some curves, this works fine. For others, though…

Ten might not be enough for this one.

See how the left side of that plot goes off the chart? That curve goes to infinity. No matter how many rectangles you put on that side, you still won’t have any that are infinitely tall, so you’ll still miss that part of the curve.

Surprisingly enough, the area under this curve isn’t infinite. Do the integral correctly, and you get a result of 2. Set a computer to calculate this integral via the sort of naïve numerical integration discussed above though, and you’ll never find that answer. You need smarter methods: smart humans doing the math, or smart humans programming the computer.

Another way this can come up is if you’re adding up two parts of something that go to infinity in opposite directions. Try to integrate each part by itself and you’ll be stuck.

But add them together, and you get something quite a bit more tractable.

Yeah, definitely a ten-rectangle job.

Numerical integration, and computers in general, are a very important tool in a scientist’s arsenal. But in order to use them, we have to be smart, and know what we’re doing. Knowing how to use our tools right can take almost as much expertise and care as working without tools.

So no, I can’t just tell the computer to do it.

“Super” Computers: Using a Cluster

When I join a new department or institute, the first thing I ask is “do we have a cluster?”

Most of what I do, I do on a computer. Gone are the days when theorists would always do all their work on notepads and chalkboards (though many still do!). Instead, we use specialized computer programs like Mathematica and Maple. Using a program helps keep us from forgetting pesky minus signs, and it allows working with equations far too long to fit on a sheet of paper.

Now if computers help, more computer should help more. Since physicists like to add “super” to things, what about a supercomputer?

The Jaguars of the computing world.

Supercomputers are great, but they’re also expensive. The people who use supercomputers are the ones who model large, complicated systems, like the weather, or supernovae. For most theorists, you still want power, but you don’t need quite that much. That’s where computer clusters come in.

A computer cluster is pretty much what it sounds like: several computers wired together. Different clusters contain different numbers of computers. For example, my department has a ten-node cluster. Sure, that doesn’t stack up to a supercomputer, but it’s still ten times as fast as an ordinary computer, right?

The power of ten computers!

Well, not exactly. As several of my friends have been surprised to learn, the computers on our cluster are actually slower than most of our laptops.

The power of ten old computers!

Still, ten older computers is still faster than one new one, yes?

Even then, it depends how you use it.

Run a normal task on a cluster, and it’s just going to run on one of the computers, which, as I’ve said, are slower than a modern laptop. You need to get smarter.

There are two big advantages of clusters: time, and parallelization.

Sometimes, you want to do a calculation that will take a long time. Your computer is going to be busy for a day or two, and that’s inconvenient when you want to do…well, pretty much anything else. A cluster is a space to run those long calculations. You put the calculation on one of the nodes, you go back to doing your work, and you check back in a day or two to see if it’s finished.

Clusters are at their most powerful when you can parallelize. If you need to do ten versions of the same calculation, each slightly different, then rather than doing them one at a time a cluster lets you do them all at once. At that point, it really is making you ten times faster.

If you ever program, I’d encourage you to look into the resources you have available. A cluster is a very handy thing to have access to, no matter what you’re doing!