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Paradoxes arise in mathematics; a famous one being Russells; usually its taken as a sign that the theoretical ediface needs to change. The difficult question is how; for example Russell developed his theory of types to dissolve the threatening nature of his.

But can there be actual paradoxes? For example, surely we can take it for granted that the individual & collective intelligence of humans is limited; thus there is a limit to ingenuity in resolving difficult questions; taking these questions to the 'limit' provides a sense of questions whose nature is beyond the ability of humans, alone or enhanced (ie by computers) to understand; they will then be to us refractory and paradoxical; possibly in every possible way.

If they are not actual paradoxes; then what are they? One might say that it is sufficient to posit that such a question is paradoxical; one must prove that it cannot be made 'unparadoxical'; but can this be possible when we have placed it beyond any human endeavour to decipher? Under such circumstances the question of proof itself, appears to becomes irrelevant.

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    Logic of Sense and its reading of Caroll's paradoxes might not be the worst place to go here – Joseph Weissman Aug 23 '14 at 21:56
  • @weissman:interesting suggestion - i'll look it up. thanks. – Mozibur Ullah Aug 24 '14 at 1:42
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    I always considered the liar or the sorites to be among the 'actual' paradoxes. – sequitur Aug 24 '14 at 13:16
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I will address the issue of whether unprovable mathematical results constitute a paradox: they don't.

Many people make an error when discussing this topic of assuming it is a matter of pure mathematics. A mathematician might point out correctly that there is no largest prime number. It could be the case that there exists some prime p such that nobody will ever use a prime larger than p, but that wouldn't change the fact that there is some larger prime. So mathematics is not a branch of physics. However, the study of mathematical methods is a branch of epistemology. And one of the relevant sets of limitations on what we can know is provided by the laws of physics. The reason that Turing machines are useful as a model of computation is that the set of Turing computable functions happens to coincide with the set of functions that can be computed by machines allowed by the laws of physics. If the set of functions that could be calculated by some physical device was larger or smaller than that set then the set allowed by the laws of physics would be more relevant to what we can calculate than the set of Turing computable functions.

A formal mathematical proof is an argument to the effect that some suitably interpreted formal system that we have instantiated in some physical device (a human brain or computer) implies that some statements follow from some other statements. The fact that some results can't be proven means that the laws of physics don't allow an instantiation of a system suitable for providing such a proof. There is nothing particularly paradoxical about this.

Nor does it stop us from doing mathematics with undecidable results. We can just follow the consequences of a particular result being true and the consequences of it being false and look for criticisms of each option. All that is different is that a particular method of criticism, that of constructing a formal system and instantiating it in a computation, can't be used in such cases. There is a lot of knowledge for which that method can't be used, so I don't see that there is anything particularly terrible or puzzling or paradoxical about it.

For more see

http://www.daviddeutsch.org.uk/wp-content/PPQT.pdf

and "The Beginning of Infinity" by David Deutsch, Chapter 8.

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    I am not entirely sure, but I do not believe that a quantum computer would qualify as a Turing Machine. If not, is it correct to say that "the set of Turing computable functions happens to ..." – Nick R Aug 26 '14 at 17:18
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    A quantum computer has the same set of computable functions as a Turing machine: cs.berkeley.edu/~christos/classics/Deutsch_quantum_theory.pdf. What I said about the set of Turing computable functions happening to coincide with the set of functions that can be computed by any device obeying the laws of physics is true. There is no explanation for why that set is picked out as being the only set realised by the laws of physics beyond the fact that it happens to be a consequence of those laws. – alanf Aug 27 '14 at 8:46
  • That's interesting and surprising. I would have thought (naivley) that a quantum computer could, for example, generate a random number (in the true sense of the word) while a Turing Machine could not. Personally, I find it a bit disappointing that a Turing Machine is complete in this sense. – Nick R Aug 27 '14 at 17:21
  • Regarding my previous comment, I realise now that my understanding was ill-formed. A random number is not a computable number and any function using a random number in its definition would be neither a computable function nor a function. Further, the definition of a computable function strongly suggests that a Turing machine will suffice. – Nick R Aug 27 '14 at 18:05
  • Nick, from my experience, I know random numbers can be, and are, "computed" by a computer. In addition, the "output" of a random number generator is used as the input to other functions. Maybe I am not understanding correctly your comment? – Guill Aug 28 '14 at 9:02
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Let's expand @nir 's posted answer.

The so-called particle-wave duality of quantum physics does appear to be a real paradox. How can something be simultaneously a particle and a wave - simultaneously both discreet and diffuse.

One might try to argue that this apparently impossible state of affair would be resolved if we either extended quantum theory in some way, or if quantum theory was not a complete view of the situation. But it would appear that neither of these options would succeed, at least not in the mainstream, Copenhagen interpretation of quantum theory.

A result in quantum theory know as Bell's Theorem tell's us that, as far as quantum theory is concerned, there is no information missing from our picture of this state of affairs; i.e., there are no hidden variables. One might say that our quantum description is complete.

Similarly, completeness means (by definition) that no extension of quantum theory will resolve this apparent paradox.

All of this appears to leave us in a position of having to reject quantum theory as the only way of resolving this paradox. Given the observable quantized nature of energy, this would also appear not to be an option for science.

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Isn't the observable behavior of elementary particles a real paradox?

Here is how David Albert describes superposition in Quantum mechanics, in his book Quantum Mechanics and Experience:

So what we're faced with is this: Electrons passing through this apparatus, in so far as we are able to fathom the matter, do not take route h and do not take route s and do not take both of those routes and do not take neither of those routes; and the trouble is that those four possibilities are simply all of the logical possibilities that we have any notion whatever of how to entertain!

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Honestly im not sure what you mean by Paradoxes. As i konw it a paradox is a statement that contradicts itself. It doesnt have to be a question of uber human nature. ex:

This sentence is wrong.

The paradox in this example is that the sentence states that itself is wrong, but then the the statement becomes wrong which means that the sentence isnt wrong after all but then it becomes true .... and on and on and on

most examples i know are paradoxes in language:

The line below is wrong

The line above is right

Or the russell example

Does the set of all those sets, that do not contain themselves, contain itself?

So i would say that paradoxes arise in language because it is imperfect and not necessarily math

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