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I was reading some of these "mathematical paradoxes", and trying to understand why the list presents only counterintuitive mathematical results.

Is there room in mathematics for logical paradoxes?

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up vote 17 down vote accepted

I'm reading between the lines a bit, but you seem to be confusing "paradox" with "contradiction".

A paradox is a true result that is surprising to our human sensibilities. These are the kinds of things on the list you provide. There is nothing wrong with paradoxes of this sort. Indeed, having our intuition turned on its head is (in my opinion) one of the great things about mathematics.

A contradiction would arise in a logical system that purports two opposite statements to be true (for example, a model of arithmetic in which you can derive both $1 + 1 = 2$ and $1 + 1 \neq 2$). Logical contradictions are not permissible in mathematics, since one can derive any statement (true or false) from a contradiction.

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I say that paradoxes must be true statements, but this is not always how the word is used. Russell's Paradox, for example, would be more rightly called Russell's Contradiction in my terminology. Perhaps it is a historical accident that it is called a paradox (or else I am being to strict in my use of the word). – Austin Mohr Jan 1 '13 at 22:25
I think your understanding of "paradox" is too limited and is at odds with the way the word has been used historically. W.V.O. Quine identifies several sorts of paradoxes. One he calls a "veridical" paradox, which is an absurd-seeming claim that is nevertheless true; the Banach-Tarski paradox is of this type. A "falsidical" paradox is a statement arrived at by seemingly correct logic that is nevertheless clearly false; Zeno's paradoxes are of this sort. – MJD Jan 1 '13 at 23:41
@MJD I am inclined to agree that my use of the word here is too limited. Still, I think OP's confusion arose from the overloading of the word to refer both to counterintuitive results and logical contradictions. – Austin Mohr Jan 2 '13 at 0:58

There are two meanings in which one can interpret "paradox" when talking about mathematics:

  1. Contradiction which is naturally derived from assumptions, e.g. Russell's paradox, or Cantor's paradox. These are sentences which exhibit the inconsistency of a definition. For example Burali-Forti shows that there is no set which contains "all the ordinals". Therefore any system of definitions in which every collection is a set will be inconsistent (granted we can define an ordinal in this system, which is something we expect to be able to do in a reasonable set theory).

  2. Counterintuitive results which show how strange mathematics can be. The best known example is the Banach-Tarski paradox which states that assuming the axiom of choice (and in fact a very weak variant of it) we can prove the the unit ball in $\mathbb R^3$ can be partitioned into five pieces and recomposed as two balls whose volume is twice the original.

    Much less known is the paradox arises from assuming all sets are Lebesgue measurable (in ZF+DC). In such model we can partition the real numbers into more parts than sets. Yes, we can cut a set into a strictly greater number of parts than we have elements to partition. Sounds strange? Well, this is just one of the things that can go wrong when not assuming the axiom of choice!

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Does Cantor's paradox (presumably "there is no greatest cardinal number" or in less precise terms "there are more infinities than any infinity") belong in 1 or 2? – Henry Mar 7 at 15:16

In classical logic logical paradoxes are not allowed but in paraconsistent logic they are allowed.

There are several types of paradox. There is one that we may call a phenomenological paradox, one where the mathematical results contradict basic truths about what the mathematics is supposed to model. For example, The Banach-Tarski paradox can be considered such a paradox. Then there are logical paradoxes, that is a statement that is provably both true and false.

In classical logic (where every statement is either true or false, but not both) it can easily be shown that a logical contradiction entails every other statement and thus if a logical contradiction exists in a logical system then that system is quite useless. Therefore, paradoxes must be banished. Phenomenological paradoxes are banished by either fine-tuning the axioms so that the paradoxical result no longer follows or by accepting the result as true and announce that our intuition has been refined. Logical paradoxes are usually resolved by carefully fine-tuning the axioms or by somehow disallowing the paradoxical creatures away from the discussion.

In paraconsistent logic, where a statement may, without causing any logical maladies, be both true and false (see for a nice introduction) logical paradoxes are quite different and can, sometimes, be harmlessly accepted.

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No, there isn't. Within sound mathematics, one cannot obtain inconsiatnt results. Then again, we simply cannot prove consistency of mathematics (and it has been proven that we cannot, at least not if it is true). Thus - paradoxically - yes, there is room in mathematics for logical paradoxes, though it is just some tiny little corner and we know there is none, but we cannot prove that.

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+1 I would like to find more information about "consistency of mathematics", and the lack of a proof – rraallvv Jan 1 '13 at 22:04
@rraallvv You'll find Godel's incompleteness theorems interesting. – Austin Mohr Jan 1 '13 at 22:06
@AustinMohr If I've got your point, mathematics cannot be used to prove consistency of mathematics... interesting! – rraallvv Jan 1 '13 at 22:14

You can quibble about what precisely is meant by the term "paradox," but if I understand your meaning, the so-called liar paradoxes can be seen as logical paradoxes. The original liar paradox, for example, is based on a claim by the Cretan poet, Epimenides (circa 600 BC), that Cretans are "always liars." Some will argue, using what I think is an overly strict interpretation, that this is not a "true paradox" since it can be resolved by simply pointing out that Epimenides' claim must itself have been a lie, and that, therefore, at least one Cretan must once have told the truth. (One logically consistent scenario that satisfies this requirement would be that all Cretans but Epimenides himself always told the truth!)

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I think this question is a much more significant than might seem at first. And a mathematician's perspective on it would depend a lot on what kind of math they're involved in.

First of all, there is no mathematical definition of the word "paradox." Notice how the definitions you'll find in dictionaries have something to do with how a statement is perceived. I've never seen a mathematical definition that depends on how something is subjectively perceived.

Most mathematicians are interested primarily in distinguishing the true from the false, and do not concern themselves with things that have no truth value. Of course a lot of things are hard to distinguish; you could call those paradoxes, or you could just call them difficult or confusing, and it would just be semantics.

In set theory though, paradoxes are very important. They still do not have an actual rigorous definition there, but that's part of why they are interesting. Set theorists study what statements dwell within a theory given certain axioms, and what ones do not. A paradoxical statement is just the kind of tricky elusive thing that makes that kind of work interesting.

@MJD commented on Quine on here somewhere. I'd recommend looking at him too, especially the paper The Way of Paradoxes (I found it here at least for how:

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Paradox is the antithesis of classic deductive logic. By the principle of explosion from any one contradiction, you can both prove and disprove everything.

Thus, when paradoxes arise in mathematics, they get taken extremely seriously; the foundations of mathematics have gone through revisions to eliminate the known ones. I'm thinking specifically of modern set theory (e.g. Zermelo set theory and everything sense) which requires a more sophisticated notion of 'construction' to avoid things like Russell's paradox, and the various modern forms of logic which have to go out of their way to avoid the more problematic variants of the liar's paradox.

This is the reason why the list you mention contains only counter-intuitive results (i.e. pseudo-paradoxes): mathematics has been fixed so that there are simply no known hard paradoxes.

As the other answer points out, some people use the term "paradox" to include both paradoxes and pseudo-paradoxes.

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Many paradoxes were only taken seriously from the 19th century: several parts of calculus were previously justified on a scientific "it seems to work" basis before limits were made rigorous, and some early statements about divergent series now look startling. – Henry Mar 7 at 15:26

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