Arithmetic as a Formal System

[King Kong]

Recently, I read the short story Division by Zero by Ted Chiang, which is about a mathematician who proves that arithmetic as a formal system is inconsistent (i.e. 1 = 2) and is driven insane/suicidal by this realization. The title refers to the popular math trick that shows 1 = 2 through a hidden division by zero (In the story, the protagonist is able to do the same thing without any illegal operations).

Since I have zero experience in higher math, I was hoping that some of the more knowledgeable board members could answer a few questions about this subject for me:

What is the current status of the proof of the consistency of arithmetic in the math world? Why is this an especially difficult problem?

What would a proof showing its inconsistency (or consistency) mean for mathematics or philosophy? Would this mean that math is empirical?

Am I just incredibly gullible?

[Darth Wong]

Math is defined as a set of formal rules, so it is impossible for it to violate those rules. You have to remember that math is an entirely man-made system, and it doesn't exist in any form other than as a set of rules. A particular math theorem could be proven wrong, but that would only mean there must have been something wrong with the original proof used to justify it in the first place.

[Surlethe]

In higher math, arithmetic is not really a mysterious way of putting 2 and 2 together and getting four. Instead, the notion has been generalized to that of a binary operation. Given a set A, a binary operation on A is a function *: AxA → A. This means a binary operation is simply a rule that, given a pair of elements of A, spits out an element from A. For example, under the usual operation of addition, *(1,1) = 2. Given such a set and a function, there are whole bodies of research based around certain assumptions you can make about the function.

There are ways of constructing the natural numbers and their associated operations from the Zermelo-Fraenkel axioms of set theory; in the course of this construction, you should naturally (hah) show that the operations of addition and multiplication (and their inverses, through field extensions) are again consistent with the axioms. So, as far as I know, basic arithmetic is consistent with itself, although I haven't personally gotten my hands dirty with the proof. And it's only a difficult problem, I'd think, if you take a naive approach (that is, assuming the operations without precisely defining them). Most mathematicians, by the way, take a naive approach to the arithmetical operations and assume them to be true without ever really worrying about it.

A proof of inconsistency would be pretty devastating, in that it would reveal a contradiction in the set theory axioms themselves. Of course, there's no way of showing that the set theory axioms don't contradict themselves (cf. Goedel's incompleteness proof), since you can build the natural numbers from them, but I wouldn't expect any counterexample to set theory's consistency to come from the direction of arithmetic.

[Surlethe]

It's interesting to note that addition, and hence the other three primary operations, is defined in a very natural way on the natural numbers. N is a well-ordered set, so define on it the 'usual' order relation. In fact, you can define the order relation based on the cardinality of its elements (N = {0,1,2,3,...}, where n = {0,1,...,n-1} and 0 = ⌀, so 0 < a < b iff there exists an injection f:a→b, and 0 < n for all n). N is, of course, a well-ordered set; so for each element n, define the "immediate successor" S(n) = min{k:n<k}. Then addition is merely repeated application of the successor function, and the other functions are defined naturally in terms of addition.

Looking at it merely in terms of a binary operation, there is not really any reason why nature should favor the usual addition in its counting over any of the other functions from NxN to N.

[Durandal]

What would a proof showing its inconsistency (or consistency) mean for mathematics or philosophy? Would this mean that math is empirical?

Math is most certainly not empirical. Numbers don't exist anymore than "true" or "false" exist. Math is a system for determining truth or falsehood. It's a branch of philosophy. It exists in its own little world. And in that world, it is possible to discern perfect truth.

Put simply, we know that 1 + 1 = 2 because that's how we've defined the number line. 1 and 2 are not numbers. They are symbols that represent numbers. 1 + 1 will always equal 2. We can change the symbols used to represent the numbers (say, 11 + 11 = 1567854), but the result will be the same number as long as we remain consistent in our nomenclature.

And in case you're curious, yes, you can have two different symbols representing the same number. For example, 1 and 0.999... represent the same number.

[Glass Pearl Player]

What is the current status of the proof of the consistency of arithmetic in the math world? Why is this an especially difficult problem?

To sum the other responses and my own meager knowledge up:
If set theory is consistent, then arithmetic is consistent too - it can be modeled by set theory.
Is set theory consistent? I dare say yes, it is. But now comes the catch: We can't prove set theory to be consistent if we restrict ourselves to set theory. This is basically Gödel's theorem: To prove the consistency of a sufficiently "useful" (read: can be used to do arithmetics) theory, one needs a meta-theory; one has to add new axioms.
Do we have a meta-theory for set theory? I don't think there is any. Topos theory seems to be a generalization, but it seems to be formulated entirely within set theory. And set theory is what's under the hood of today's mathematics.