Here’s an interesting line from Brad Osgood:
Isn’t it a little embarrassing that multibillion dollar industries seem to depend on integrals that don’t converge?
In context, he’s not saying that huge companies are blithely using bad math. Some are, but that’s not what he’s getting at here. His discussion is an example of coming full circle, where experts and novices come to the same conclusion for different reasons.
The divergent integrals Osgood refers to are Fourier transforms of certain functions. A beginner might not notice that said integrals don’t converge. An expert knows that the calculations are justified by a more sophisticated theory. Someone in-between would have objections. Experts can be casual, not because they’re ignorant of technical difficulties but because they’ve mastered these difficulties. [1]
The expert in Fourier analysis has all the technicalities in the back of his or her mind. Often these don’t need to be explicitly exercised. You can blithely go about using formal calculations that aren’t justified by the classical theory.
But the expert doesn’t entirely come full circle, not in the sense of walking in circles in the woods. It’s more like winding around a parking garage, coming back to the same (x, y) location but one level up. Sometimes the expert needs to pull out the technical machinery to avoid an error the beginner could fall into. The theory of tempered distributions, for example, doesn’t justify every calculation a novice might try.
***
[1] In a nutshell, here’s the theory that justifies apparently sloppy calculations with Fourier transforms. The key is to view the function you want to transform not as a function on the real line but as a tempered distribution, a linear functional on the space of smooth, rapidly decaying test functions. A function acts on a test function by forming their product and integrating. Then use Parseval’s theorem from the classical theory as the definition in this new context, moving the transform operation from the original function to the test function. Simple, right?