How to tell a paradox from a "paradox"?

Question

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?

Answer 1

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:

1. 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!").

2. 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.

Answer 2

Both the Russell paradox and the Banach-Tarski "paradox" show that certain ideas are contradictory. It seems to me that the key difference between the two is that, in Russell's case, the ideas in question had been proposed (by Frege) as axioms for a foundation of mathematics, and they seemed sufficiently basic to be accepted, until the paradox appeared. In the Banach-Tarski case, one of the ideas involved in the contradiction is the idea that one can meaningfully talk about the "volume" of arbitrary sets in $\mathbb R^3$. (Here "meaningfully" is intended to include additivity and invariance under Euclidean motions.) Although that is a very appealing idea intuitively, I'm not aware of anyone's proposing it as an axiom (or even as a conjecture). The development of Lebesgue's measure theory had already shown that the intuition is not reliable and the measurability of general sets is a delicate issue.

Answer 3

Although I am not so good with philosophical subtleties, I have always found useful to make a distinction between an antinomy and a paradox. The first leads to a formal contradiction, i.e., a logical inconsistency in your theory (you can prove both a formula and its negation).

The second `merely' defies human intuition, without being a (known) antinomy. Much less worrying (ask Frege :)).

Many just use `paradox' for both things, but I find this highly confusing.

Answer 4

I don't see how these two a so fundamentally different. Russell's paradox tells us that we have to think more carefully about what a set actually is, and Banach-Tarski tells us that we have to think more carefully about what the measure of a set is and which sets are measurable.

When arriving at a counterintuitive statement, there are two possible conclusions: First, the previous intuition was wrong, and in this case the result is genuinely counterintuitive, and the second possibility is that the definitions or the logic were wrong and need to be changed. Banach-Tarski falls into the second category, because one would not conclude that matter can be created by clever cuts and rearrangements, but rather that one needs a thorough definition of measure.

Answer 5

What you're describing as a "true paradox" is sometimes called an "antimony" and it means an actual logical inconsistency in the underlying theory. The Burali-Forti paradox is another example and it means there can't be a set of all ordinals (the ordinals are a proper class). By contrast, the Banach-Tarski theorem is consistent; it's just counterintuitive. The reason we don't hear much about "true paradoxes" (antimonies) these days is that logicians in the 1920's got the earlier inconsistencies under control, and (in all likelihood) we're not dealing with any actual inconsistent systems today, at least in everyday mathematics.