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Accepting or rejecting the axiom of choice as "natural" is a red herring based on the idea that mathematics express external truths about the world. Mathematics
by murbard2 11y ago
Accepting or rejecting the axiom of choice as "natural" is a red herring based on the idea that mathematics express external truths about the world. Mathematics describe rules of reasoning which preserve truths about the world.
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.
- dietrichepp 11y agoThis "upcasting" and "downcasting" also work with mathematics itself. For example, we can consider adding infinitesimals to the real numbers, allowing us to do calculus without taking limits. Then we can downcast back to the real numbers.
- rndn 11y agoThe problem is that some concepts might not have an "isomorphism" to the real world (as Hofstadter calls it). In fact, finitists believe that the concept of infinity possibly introduces some discrepancy, making the "casting" a lossy transformation.
- platz 11y agoExactly,"upcasts" can be one-way
- murbard2 11y agoDo you have an example?
- Strilanc 11y agoPresumably the axiom of choice is a good example of something unlikely to be applicable in a real world context. Unless you're trying to anticipate when a rogue mathematician is about to attack, of course. (Also halting oracles.)
- deleted 11y ago[deleted]
- j2kun 11y agoAnother 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-consequences-of-the-axiom-of-determinacy http://mathoverflow.net/questions/129036/counterintuitive-co...
- reagency 11y agoThe 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.
- black_knight 11y agoIf 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.
- deleted 11y ago[deleted]
- snowwrestler 11y ago> 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.
- im3w1l 11y ago>Mathematics describe rules of reasoning which preserve truths about the world. 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.
- musername 11y agoZermalo-Frenkel Set theory with the Axiom of Choice is not proven to be consistent. One might argue that consistency is a natural requirement for a rule to be a rule. Especially for peano axioms, consistency is a given, so your comparison doesn't hold. Following the incomplefeness-theorem, the (dis-)proof might be external to any of ZFC's descriptions of the world, which believing in ZFC would be the world. disclaimer: I don't understand axiom of choice.