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.
My take on definitional issues is that the correct definition depends upon what you intend to use the answers reached with that definition to do.
In the case of determining whether an answer is analytic (incidentally, a term that has a pretty simple meaning that isn’t defined until far too late in the curriculum, making it seem more mystical and complicated than it is), for practice purposes, a question like whether the sine of an angle is an analytic answer depends a lot on whether the numerically evaluated trig function is in the final result (in which case it is probably fine), or in an intermediate step in the analysis, in which case it may very well not be sufficient (especially if your function is chaotic in the sense of being highly sensitive to slight variations in initial conditions). Of course, it is often possible to decompose trig functions into analytic functions that define them, so you could, for example, use trig functions a bit like a Fourier transform to get something easier to work with mathematically and then transform it back based on the trig function definition rather than using a numerical evaluation of it, at the end, to keep everything exact.
“What if you can do series expansions, but only in some limits?”
Coming from physicists, who routinely say, “all other terms are of order alpah^3 or higher and can be neglected”, need one even ask? Of course, compact exact representations of infinite series when they are known are like crown jewels of mathematics that are precious in the extreme, and when you can dissect an infinite series into several other infinite series that can be reduced that way you’ve hit a gold mine of genius. I wonder how many breakthroughs we could make with just a couple more of those.
“What if you can do all the above, but can’t get numbers out without a supercomputer?”
Then you have the state of QCD for the last four decades. Honestly, it is really impressive how universally QCD is accepted as a correct theory in the absence of the ability to compute and test the predictions against experiment using non-approximations of the true QCD equations directly.