Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?

OK, maybe the title is exaggerated, but is it true that the rest of math is just "good enough", or a good approximation of absolute truth - like Newtonian physics compared to general relativity? How do we know that our "approximation" is the right one? Another analogy: Fundamental physics is also not well-fundamented (where is the Higgs boson?) but most of the rest of the physics is on top of it, and it does its job well (it's a good-enough approximation).

Summary: According to the responses, math is indeed an imperfect domain, but can be seen as perfect for all practical purposes. In this case I wonder if math is indeed pure and identical across all possible universes. Maybe another universe comes up with a different set of axioms, more or less consistent than what we have now.

Solution 1:

The first incompleteness theorem says that, under appropriate conditions, and for appropriate definitions of "true" and "provable," there exist statements that are true but not provable. This is interesting, but not a big deal. It is more or less a consequence of the existence of nonstandard models in first-order logic, which is an interesting feature of first-order logic, but mathematics is more than first-order logic.

The second incompleteness theorem says that, again under appropriate conditions, a sufficiently strong formal system can't prove its own consistency. Okay, this is kind of a big deal, but in practice it doesn't matter as much as it sounds like it does. Mathematicians don't actually do everything in a fixed formal system. There is a whole subject called reverse mathematics dedicated to finding the weakest formal systems that are capable of proving various things, which vary widely.

There is a possibility that ZFC, the formal system that sounds like it's what mathematicians work in (but isn't really), could be inconsistent. So what? If that ever happened we would just find another set of axioms to use. ZFC is a ridiculously strong formal system and in practice we never use its full strength, so it wouldn't really matter if it were inconsistent.

Mathematics is not the study of what formal statements can be proven in ZFC (although there are mathematicians who study this kind of thing). Proof: mathematics is thousands of years old, and ZFC is not.

Solution 2:

Your writing demonstrates some pop-math-esque misconceptions surrounding Gödel's Incompleteness Theorems. The analogy between math and physics is inaccurate, for one, because physics is specific to the contingent governing rules of our universe, whereas math is transcendent in that it looks toward truths that must hold in every possible world, i.e. are logically necessary. If I draw a curve on a piece of paper and ask you to model it, just because you only find yourself capable of approximating the curve doesn't change the reality that there are true facts about curves in general that can be investigated and discovered. Just because you don't know everything about the user "anon" on Math.SE doesn't mean you're incapable of knowing things about human beings in general. Just because we can't yet for certain nail down the exact form of the universe doesn't mean we can't figure out the logic of space and time and combination at all.

The key to understanding this is: Gödel did not demonstrate any of mathematics was incorrect or inaccurate in any way. I'm not sure how you even came to that misinterpretation. The theorems show, in a nutshell, that any formal mathematical theory with a given set of axioms (starting assumptions) and a given set of inference rules (ways of deducing things), which is capable of expressing basic arithmetic, is only self-consistent if it is incomplete (there exists some true proposition that the theory is capable of expressing but not proving) and if it is incapable of proving its own consistency.

The consistency of mathematics isn't really a problem; we can be confident of all of our theorems exactly as much as we can be confident in all of our axioms taken as a whole. The only real contentious one I'm aware of is the Axiom of Choice, but it's instructive to know that we have yet to have ever generated a contradiction or falsehood of any sort from our standard axioms. So why not do mathematics, as a society (I leave out personal reasons for doing math, as that is essentially another discussion entirely), if it has a pristine, 100% perfect track record of getting everything right? If absolute certainty is the standard for the worth of human endeavor as you tacitly posit, then that would make mathematics literally the most respectable endeavor humans have ever achieved.

The incompleteness of mathematics is likewise not really a problem; all of the ideas we've discovered so far that are undecidable, are either so extremely far-removed from reality and our lives that they are effectively insignificant or meaningless to us, or they are still far-removed but capable of being proven within a stronger system. Incompleteness means we will never fully have all of truth, but in theory it also allows for the possibility that every truth has the potential to be found by us in ever stronger systems of math. (I say in theory because, technically, the human brain is finite so there is an automatic physical limit to what we can know.) In a way, instead of being unsettling, incompleteness should almost be reassuring and reinvigorating to mathematicians, because it means the adventure is never-ending.

Solution 3:

Okay. They used a differential equation in establishing the engineering specifications used for building a certain bridge near here. Now that the troublemaker Gödel has had his say, should I conclude that differential equations are no good, and therefore avoid using that bridge?

Seriously: we can still do mathematics. It is just "Hilbert's Program" that should be (and was) abandoned.

Solution 4:

I think this thread could also benefit from my answer to another Godel question. But more importantly, it could benefit from looking at this paper written last year by Scott Aaronson.

The OP question is absolutely a legitimate one. I hate to see people saying that "you don't understand Godel's theorems" if someone asks this... because by that logic then, if you are a reductionist, you must also believe that Roger Penrose doesn't understand Godel's theorems.