Welcome To A Crazy World

This sentence is false.

That innocuous looking statement is the simplest but nonetheless revealing introduction to the world of paradoxes. That particular one, called Russell's paradox (actually, Russell's paradox is quite a bit more and involves set theory and self-swallowing sets), can send one into spirals of self-referential logic. To elaborate, consider the sentence carefully. If it is true, then it cannot be true and must be false as it itself says. And if it is false, then it is clearly true, contrary to what it says.

And if it is true, then it is false, and so on ad infinitum.

Paradoxes are interesting and surprisingly common in mathematics, logic and language. And some of them can be difficult to even think about, even after they have been laid out for you. An example is the Barber's paradox, another non-mathematical corollary to Russell's paradox which goes like this: Consider a town in which there is one barber and he shaves all those men who do not shave themselves, and no one else. Can such a town/barber exist?

I invite you to think about this and it is my solemn guarantee that your head will hurt after a while.

Spoiler: Such a town/barber cannot exist because if the barber does not shave himself, then he is included in the definition of "all men who do not shave themselves" and thus must shave himself according to the rule. Conversely, if he does shave himself, then he cannot be the barber who is being spoken of, because by definition, he only shaves those who do not shave themselves.

Most paradoxes exhibit self-reference. If a sentence or a subject refers to itself, there is a chance that it'll lead to a contradiction. The first sentence of this post refers to itself and this leads to circular logic. The Barber paradox has self-reference because it refers to "shaving himself".

Another example of self-reference:

Last Saturday, Shona and I were driving to Chicago. I like these drives, it gives us a chance to talk of many things. We got to talking about how different people were good at different things. Some people, for example, Shona said, were just so good at music. And some others couldn't make music to save their sorry behinds if you held a gun to their heads.

This is true. I mean, you take two different people and present them with the same notes and one of them takes 'em and creates a symphony while the other creates baboon noises. The reason of course, is that their brains are constructed differently. But what is the anatomy of that difference? We don't know. Because we don't yet fully understand how the human mind works.

When will we ever understand the human mind, Shona asked with a drawn out sigh, and then promptly lost interest.

I don't know the answer to that, honey, and maybe, just maybe we can never understand. Because there may just be a degree of self-reference there. Because we are proposing that the human mind try to understand itself. Can the human mind ever understand itself? Or are we setting ourselves up for a infinite loop in the attempt? Maybe all those crazies in our institutions are not really crazy. Maybe they are among the handful that have really understood how the mind works. And driven insane in the process.

Yet another example. Can we have a list of all lists that do not contain themselves? This is a paradox again, because if such a list listed itself, then it would be incorrect (it would contain at least one list that happens to contain itself, namely itself). On the other hand, if it omitted to list itself, it would satisfy the definition of "lists that do not contain themselves" and thus must list itself.

Self-reference is a general, all-around bitch, but it can actually be quite useful at times. Indeed, many algorithms in computer science depend on self-reference--or recursion, in the parlance--to come up with elegant solutions that would otherwise be ponderous to solve through iterative methods. But such solutions, if designed incorrectly, can run away from you in a rapidly growing infinite loop. We recognize this and always build in a trap-door clause, a clause that will ensure that the algorithm will always terminate for a particular set of input conditions.