Russell's paradox showed that naive set theory leads to a contradiction. This was something that was taken seriously and caused a lot of work.
Now, Banach–Tarski paradox is arises from a result that a ball can be decomposed into finite amount of pieces and the pieces can be used to built two identical copies of the decomposed ball. Banach-Tarski paradox is often treated as a "paradox", basicly meaning that, yes, it is counter intuitive but yet there is no problem – mathematics just occasionally is counter intuitive.
To be honest, I have never understood why Banach-Tarski is not a "real" paradox but not being expert of measure theory I chose to accept the common view.
Is there some high level explanation on how to tell a paradox from a "paradox"? What is it that makes a counter intuitive result to a "real mathematical paradox" that we should start worrying about?
Best Answer
Many paradoxes are first expressed in a semi-formal way, for example "the least number not describable by fewer than eleven words". They are warning signs that lead us to further analysis and can be resolved in different ways:
We can just get used to a "paradox" and accept it as "truth", e.g., there are infinite sets of different sizes, or there is a real function which is continuous at irrational arguments and discontinuous at rational arguments. There are famous paradoxes in philosophy which would not be considered paradoxes today, such as Zeno's paradox ("How can an infinite sum of positive numbers be finite? No movemement is possible!") and various arguments from Prime Cause ("How could we have an infinite descending chain of causality? God must exist!").
We find the paradox unacceptable and so we need to change something. We might change rules of logic, definitions, or axioms, everything is up in the air.
A paradox which actually proves falsehood, or a statement as well as its negation, is more properly called an inconsistency. An inconsistency is something we can never get used to and so we have to change something. A milder form of paradox is one which does not prove falsehood but just something very counter-intuitive, in which case we have to decide whether to accept it, or admit that our attempt to bring something into the realm of mathematics worked in unexpected ways.
I think this question is about how to tell whether a given "paradox" is of the first or second kind. When should we just "get used" to a paradox and when should we "change things"? In the case of Russell paradox we had no choice but to change something. In the case of Banach-Tarski paradox there is a choice. The accepted view is that we should just get used to it, but there are interesting alterantives which force us to rethink the notion of space. Even though these alternative notions of space are far better suited for probability, measure and randomness than the classical approach, mathematicians are unlikely to adopt them widely out of sheer inertia and historical coincidence. But mathematicians do not like to admit that mathematics is a human activity, and as such subject to sociological and historical trends.
So I suppose my answer is this: when faced with an unacceptable counter-intuitive statement which offers several mathematical resolutions, the choice will be made through social interaction which has some mathematical content, but not as much as we would like to think. Other factors, such as arguments from authority and social intertia will play an important role.