The Barber paradox is attributed to the British philosopher Bertrand Russell. It highlights a fundamental problem in mathematics, exposing an inconsistency in the basic principles on which mathematics is founded.
The barber paradox asks us to consider the following situation:
<em>In a village, the barber shaves everyone who does not shave himself, but no one else.</em>
The question that prompts the paradox is this:
<em>Who shaves the barber?</em>
No matter how we try to answer this question, we get into trouble.
If we say that the barber shaves himself, then we get into trouble. The barber shaves only those who do not shave themselves, so if he shaves himself then he doesn’t shave himself, which is self-contradictory.
If we say that the barber does not shave himself, then problems also arise. The barber shaves everyone who does not shave himself, so if he doesn’t shave himself then he shaves himself, which is again absurd.
Both cases, then, are impossible; the barber can neither shave himself nor not shave himself. The question ‘Who shaves the barber?’ is unanswerable.
Bummer...
As stated, all that the "paradox" shows is that we can construct sentences that are nonsense or meaningless, even if they sound as if they should mean something. All that the analysis shows is that the situation of "a village in which the barber shaves everyone who does not shave himself, but no one else" is impossible.
(By the way, you didn't state it correctly. You have to say something about nobody in town having a beard... and probably something about men and women. As you stated it, a perfectly good "solution" is "nobody shaves the barber," either because the barber is a woman, or because the barber has a beard.)
A classic example is "what happens if an irresistible force meets an immovable object?" The resolution is simply that if there is such a thing as an irresistible force, then it is logically impossible for there to be an immovable object; and vice versa.
Another example is "If God is almighty, that means he can do anything. If he can do anything, can he make a rock that is so heavy that he can't lift it?" All this shows is that we need to be careful about how to define "almighty."
One that is strictly a word trick is:
1) Nothing is better than a good steak. 2) A hamburger is better than nothing. Ergo, 3) a hamburger is better than a good steak.
As a kid, I was bothered by the question that it seem inconceivable for the universe to be infinite, but it also seems inconceivable for it to have a boundary.
The real Russell's Paradox, https://en.wikipedia.org/wiki/Russell%27s_paradox is... well, it's too complicated for me to understand. It involves, not barbers, but the class of all classes that are not members of themselves.
There is by the way something that puzzles me. I don’t know if it’s a valid kind of puzzlement. Sometimes I get the impression from neuroscience that the brain creates our reality in the sense that it doesn’t show us the “real world”, but a kind of interpretation of the world or perhaps one could call it a model of the world. But if that is true, and I don’t know if it is, then the brain is a part of the model that it is creating, because the brain is part of the reality we perceive. Do you understand what I’m trying to express?
Alan, have you every read Flatland, by Edwin Abbott? Very sui generis, a fantasy about a two-dimensional world with (obviously) satirical references to Victorian society. It is however quite readable and a good way of understanding concepts of dimensionality. At any rate, the idea of the universe being a hypersphere or curved in four dimensions, thus "finite but unbounded," is not too difficult for me. I can sort of accept it. It is, after all, not too different from our experience on the surface of the earth.
If the universe were clear enough and you had a telescope powerful enough and were willing to wait about 100 billion years for light to travel around it, and could get far enough from the earth so that it was not in the way, you could point a telescope out into space and, wherever you looked, you'd see the back of your head.
Dan Smith and Mikkel made me think about "meaningful definitions"...when I try to understand quantum physics (which I don't), I'm baffled by statements that "the universe is curved in the fourth dimension"
This kind of statement sounds as meaningless to me as Noam Chomsky's famous nonsense sentence "Green ideas sleep furiously"
But that's just because I'm using a layperson's interpretation of "curved" and "dimension"...even when talking about Quantum Physics people discuss “String Theory" and idiots like me imagine a literal string...
Metaphors can be clarifying or very confusing, depending on how they're used!
