Treating it as a function is the fiction (and the reason is probably that the average college freshman doesn't know what a distribution is). But it's "enough" like a function that unless you look too closely, no real problems arise from glossing over the function/distribution distinction.
You're right, but what I should have emphasized is that if you accept that cos/sin are based in physical reality because they model a circle, then you have to have just as much faith in the dirac delta.
Cos/sin are infinitely precise, and can't be reduced to algebra (as opposed to calculus/analysis). This in itself is an idealisation, a fiction.Sin/cos seem natural because they model the ideal unit circle, while the dirac delta is natural because it models an ideal impulse. The latter seems more abstract than the former because we all have been exposed to unit circles, but usually we are not exposed to unit impulses.
You can apply a step function to a mass-spring-damper system, then observer the response. Thereafter, you can make the step function narrower but taller and observer the response. As you continue this process, the response approaches something simple. Maybe this way of thinking about the dirac delta in obvious to everyone, but to me it was a major breakthrough because I couldn't imagine how something infinitely tall and infinitely narrow could model anything in the real world.
So my ultimate takeaway message is that cos/sin are to an n-gon what the dirac delta is to a step function, and that they are just as "real" as one another. Alternately, the dirac delta doesn't work for "symbolic" or algebraic reasons. It's an idealisation of reality, not an abstraction _away_ from reality for the sake of convenience (e.g. how we sometimes artificiallyl define 0^0 (zero to the zero) to be 1 in some cases or 0 in other cases)).
I was under the impression that we were talking about mathematics and not physical reality.
In mathematics, cos/sin are just functions cos:R -> R and sin:R -> R. As far as functions go, Dirac delta is kind of a fiction because we don't have a function δ: R -> R. That's why I think that your initial characterization was correct.
Of course if we define it with distribution or measure then we don't have this issue.
> Maybe this way of thinking about the dirac delta in obvious to everyone, but to me it was a major breakthrough because I couldn't imagine how something infinitely tall and infinitely narrow could model anything in the real world.
By the way, if you're looking for intuition on the Dirac delta I think a good way to think of it is just the identity element under convolution. Equivalently, you can think of it as a thing whose Fourier transform is 1.
