[1]: http://mathoverflow.net/questions/129036/counterintuitive-co...
The solution is either to distinguish between countable and enumerable and decidable, or to use intuitionistic logic.
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.
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.
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.
(Also halting oracles.)
I'd disagree even with this, and state something even weaker. Mathematics describes rules of reasoning. A sound mathemathical theory of the something in the real world is a (partial) correspondence between statements about the real world and mathematical statements, such that the mathematical manipulation of statements preserves correspondence to truths about the world, or produces statements that don't correspond to the world.
I think this distinction is important because it emphasizes that it is the scientists' responsibility to pick an appropriate mathemathical model.
disclaimer: I don't understand axiom of choice.
For instance, if I am trying to count my goats, the concept of natural integer allows me to count two groups, sum the results, and still have an accurate understanding of the number of goats I have.
As it turns out, the theory of natural integers allows one to define arbitrarily large numbers, even though I cannot possibly have more than, say, 1000 goats. Yet, a theory that deals only with numbers less than 1000 might be more cumbersome to work with. Try to define addition there and see for yourself.
The theory of natural numbers is useful because it provides a simple formal way to accurately count the number of goats (and possibly other things). It doesn't matter that it allows us to conceive of an absurd number of goats, like 1,000,000. It does the job, and as long you use it in the real world on real data you won't get absurd results.
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The same goes for the axiom of choice. Whether or not you accept it isn't going to make a practical difference once it comes to measuring the volume of helium needed to fill your balloon.
Think of it as upcasting and downcasting. You're dealing with a finite problem which is hard to solve. So you upcast it to a more general theory, solve it, and then downcast it back to the original setting.
There are as many ways to upcast as there are ways to pick independent axioms, so pick whichever make the problem easiest to solve, and don't worry about the "truth" of these propositions.