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Comment by mrob | Hacker News Reader

Infinities aren't real, so you shouldn't be surprised if unrealistic things happen when you invoke infinities. That you can duplicate a sphere by cutting it into a finite number of pieces and reassembling it is a "fact" in the same sense as "Luke Skywalker destroyed the Death Star". It might be interesting and culturally important, but it's talking about fictional entities. Both spheres and arbitrarily detailed pieces of spheres only exist in the imaginations of mathematicians. The Axiom of Choice is obviously true, and the fact that it lets you invent weird sounding stories from weird components is no evidence against it. Don't confuse mathematical tools with reality.

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Mathematics is formalised is to avoid this sort of philosophizing.

I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack". But then in an engineering control systems class, we did an experiment where we used a step function to approximate a dirac delta function. I could see the results both on the computer screen and in physical reality through a mass-spring-damper system. From that moment on I saw the dirac delta function in the same way that I see cosine/sine: The reason it works in math is because it has a basis in physical reality.

The lesson to be learned here, is that you don't know in advance whether something is obvious or not. To me, it doesn't make sense to decide whether the axiom of choice can be justified by looking at the axiom itself. You have to look at where it's used and required, and whether the proofs convey something that matches your physical intuition.

The Dirac delta function is a mathematical fiction. It doesn't look weird in your case because you didn't mix it with any other weird mathematical fictions. But maybe somebody could invent a system that violated conservation of energy by using Dirac deltas. That wouldn't mean the Dirac delta is a bad fiction, because if you actually tried to build it using your step function approximation it wouldn't work.

> I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack".

Can you expand on this? The Dirac delta function is not a function from R to R for example.

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.

Right. That was kind of my point. It is a "mathematical trick" in the sense that it's not a function, but a distribution.

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.

Maybe I'm missing your point, but for δ I think the hack is using the word "function"; as a measure or distribution it's a pedestrian object.

You need to draw a line between mathematical entities that have "a basis in physical reality" or not. For most people, performing an infinite computation is what puts the axiom of choice firmly on the fiction side, as an impossible procedure, regardless of the application and the outcome.

There are a small number of people who think it's obviously false! (I'm not one of them). And a larger number who think it's not obviously true nor obviously false.

There's the old joke that the axiom of choice is obviously true, the well-ordering theorem is obviously false, and Zorn's lemma is too complicated to say.

"The GEODE" in GPS navigation is at least a culturally significant fiction. Of course, it's obviously not very infinite. But it's something very much like a sphere that also happens to have significant practical import.

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