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Comment by j2kun | Hacker News Reader

Another perspective is that mathematics is not intended to describe the real world, and so what makes a mathematical assumption "natural" is how elegant or inelegant its consequences are. There is a long list of strange things that would be true if you assume the axiom of choice is false [1], and I'm pretty sure nobody who has ever considered the pros and cons of AoC has cared about the amount of helium in their balloon. Esp. considering that it's not possible to construct nonmeasurable sets in the physical world (since you would need to do operations at a scale infinitely smaller than the Planck constant, for one!).

[1]: http://mathoverflow.net/questions/129036/counterintuitive-co...

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The reason those things are strange is that uncountable sets have absolutely no bearing on computable objects (== enduring in math and science) but everyone mistakenly applies their countable intuition to uncountable set theory.

If one uses classical logic there are even many countable, even finite sets which are beyond computable distinction. The set of computable functions on say natural numbers is a countable set, but even distinguishing when two such are equal cannot be effectively done.

The solution is either to distinguish between countable and enumerable and decidable, or to use intuitionistic logic.

> Another perspective is that mathematics is not intended to describe the real world

Isn't that a fairly recent perspective, though? My memory is that the concept of math for its own sake (purposefully divorced from "real world" applications) is only a couple hundred years old at least. Whereas for most of its history, from the Egyptians up to Newton, mathematics was developed as a tool to understand the physical world.

So while some people today might consider that mathematics is not intended to describe the real world, they're working on top of a deep system that was.

> what makes a mathematical assumption "natural" is how elegant or inelegant its consequences are

So now we start to argue about concepts that are not well defined in the context, introducing new terms, even?

Art for art's sake is just one aspect of science and craft. You might argue that mathematical knowledge as it is incorporated in the material world, e.g. in brains, books and soundwaves, has become a real matter which entails all the complexity that makes it so difficult to grasp.

> introducing new terms, even?

We do, but you have to remember that mathematics is an activity invented by humans largely for their own amusement. This is especially true when it gets down to the foundations of mathematics.

So let me put it this way. Your description of mathematics is not very amusing to me.

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