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Comment by legulere | Hacker News Reader

Banach-Tarski is absurd because matter in reality is made out of particles. If you take this into account in your definitions you can't do absurd stuff like Banach-Tarski.

As such it offers an insight into what can happen if your mathematical model fails at representing the reality.

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Strictly speaking, math it self isn't intended to reflect reality. It's just logical system together with some initial axioms. But it does happen to model reality very well, given the right interpretations.

But uncountable sets do NOT model reality well.

Oh but they do. When taking measurements in a scientific setting, it usually gives the most accurate and simple model to assume that your measurement is an approximation of a certain real number, and understand more accurate measurements as better estimates of this real number. Using rational numbers or other countable sets in this position usually leads to undesirable biases and or circuity in the model. When you need a "continuum", the real numbers have the best properties for the job.

Let's say you've got a light-emitting flat surface and want to describe its power output. I recommend modeling it as a set of real coordinates with an intensity value at each point and integrating to compute the total power output. By all means I invite you to come up with a better model, without using uncountable sets, for which there isn't a straightforward analogue, just as good, that uses uncountable sets.

Edit: Might as well jump straight to explaining how wave functions work without using uncountable sets, and why that's a better description of reality.

Banach-Tarski is not absurd. Supposing the world should behave in a certain way because there is a mathematical theorem that seems to say it should, that is what is absurd.

Banach-Tarski is important and interesting, because it neatly illustrates why mathematics that seems to be sensibly related to reality sometimes does not describe reality very well. Relations between mathematics and reality are tricky.

BTW, the finiteness of particles is not sufficient to explain the problem. E.g. electrons and photons do not have a finite size. Yet you cannot something similar to a ball of photons or electrons either.

It's not just that the particles have to be zero size. There also have to be infinitely many. Otherwise, you could just count before and after and apply conservation of mass. (This also applies in mathematics: any finite set has its size preserved by euclidean motions, unless you put points on top of each other.) Needless to say, gathering an infinite number of electrons or protons will cause black hole-themed "reality breakdown" problems long before you get there.

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