Paradoxes

I was reading a book called Dream, Death, and the Self by J. J. Valberg. (Lately I’ve been enjoying reading rather random selections from random books that look interesting, always wishing I had time to read more.) At the beginning, the book is talking about philosophy, and mentions a list of paradoxes. I knew what some of the paradoxes were, but didn’t recognize all of them, so I done gone went and looked ’em up! So here they all are, for your mind to consider and my mind to remember:

Zeno’s paradox

This could really could refer to a set of similar motion paradoxes, but here’s one:

This ancient paradox reveals how we don’t understand the nature of motion, space, and time. If you walk a mile, you must first walk half a mile. If you walk half a mile, you must first walk a quarter mile. But first a 8th, a 16th, a 32nd, a 64th, on and on… you must pass an infinite number of halfway points! How is it possible to move at all?

(Along the same lines, we might just ask: is it really possible to divide something (time or distance) infinitely? What is an infinitely small distance or moment of time like? Could we give an infinitely small entity a numerical value to let us now where it goes in a sequence? Wouldn’t that be impossible since each infinitely small entity would be sandwiched between an infinite amount of other infinitely small entities on either side? Is it possible to take two thirds of those infinitely small entities, when two thirds and one thirds would both be made up of an infinite amount of entities and thus be equal? And then we could get into the nature of infinity… what is infinity minus infinity? Is infinity times infinity more than it already was? Why does saying “infinity” instantly win any argument? (Yes, it does, infinity.))

The Ravens paradox

You accept that “All Ravens are black.”

You accept that this means “All things that are not black are not ravens.” (And vice versa. These statements imply each other.)

But what if you apply specific examples?

You accept that “My pet Nevermore is a raven, and is black.”

Well, good, this seems to support our first premise that “All ravens are black.”

Finally you accept that “This green thing is an apple.”

Ah, good, and this seems to support our second premise that “All things that are not black are not ravens.” Which then implies our original premise that “All ravens are black.”

Woah, does it really? The sight of a green apple is evidence that all ravens are black? What if our premises said that all ravens were white? It seems the site of a green apple should support that all ravens are any color we want! Ahhhh! Paradox!

This paradox reveals the problem of induction. Does inductive reasoning really lead to knowledge? If all you ever see are white swans, you might conclude that “all swans are white.” But that’s a false assumption; you really can’t assume anything about the swans you have not yet observed. This also applies to many complex (or chaotic) systems, such as how well a movie does, how the stock market works, when terrorist attacks will happen, etc. (That is, you can not find things that hold true for every terrorist attack and then conclude “aha, a terrorist attack happens when such and such happen.” Even though this is what historians love to do in retrospect. Or you can not say “aha, this movie or book did so well because of these few factors… blah blah blah” even though that’s what news anchors and magazine articles like to do all the time.) Check out the book The Black Swan. It is all about this problem of induction. And most people live their whole lives unaware that they’re using this faulty logic daily. It’s natural, after all.

Surprise examination

Or the Unexpected hanging paradox…

Can you really expect the unexpected? If someone tells you that you will have a surprise exam at an unexpected time, or you will be executed at an unexpected time, or the end of the world will occur at the least expected time, won’t that lead you to constantly expect it? And if you constantly expect it, does that mean it will never happen? Or does expecting it lead you to not expect it?

What if there’s even a time limit? Someone says “you will have a surprise exam sometime this week, but on an unexpected day.” OK, if you haven’t had the exam by Friday, you know it has to be Friday, and thus Friday would be expected, so it can’t be Friday. And then can’t you just use the same logic to rule out all the days?

Sorites paradox

How wide can a chair get before it becomes a sofa? How many hairs does a man have to lose before being considered bald? How many grains of sand make a heap? (If you’re pro-choice, how many days of existence in the womb make a human?) The old Sorites paradox deals with the problem that some of our ideas have no defined boundaries; they’re vague. And then we wonder: how do we come to understand such vague ideas so well?

Preface paradox

This paradox basically says: aren’t there situations in which an individual would rationally hold two contradictory beliefs? For instance, in a book’s preface, an author might apologize for mistakes contained in the book, believing that there is likely at least one mistake just due to chance. At the same time, he might’ve fact-checked his book very carefully, and so know that there are no errors. So he believes that there both are and aren’t errors at the same time.

I don’t fully understand this paradox, as I don’t accept the second premise; an author could carefully fact-check his book and still believe it’s likely that he missed something.

Perhaps this paradox can be better illustrated with religious hypocrisy; a believer rationally believes it is right to donate to the poor for instance, but upon leaving his worshiping session he donates nothing. (Or he votes to raise taxes; the wonderful virtue of donating other people’s money!) But even in this case, someone would argue that he is being irrational in at least one belief.

Lottery paradox

This is the paradox you think of when you buy a lottery ticket. The chance of any one certain ticket winning is extremely small. Yet the chance of one ticket somewhere winning is extremely high. So you can safely conclude that your lottery ticket won’t win. Yet someone somewhere must have the winning ticket, and therefore must be wrong.

Russell’s paradox

From good old Bertrand Russell. Basically, we have two kinds of sets: normal and abnormal. Abnormal sets include themselves, normal sets do not. For example, the set of all squares would be a normal set; a set of all squares is not a square. However, a set of all non-squares would be abnormal; a set of non-squares is itself also not a square.

The paradox comes when we ask: is the set of all normal sets normal or abnormal? If the set includes itself, it’s abnormal. But if it doesn’t include itself, it’s normal, and should then include itself. Ahhh!! Paradox!

This is related to the more popular barber paradox: A barber only shaves all gentlemen who do not shave themselves. Does the barber shave himself? If he doesn’t, he should; if he does, he shouldn’t.

The Why paradox

I made this one up, though someone probably philosophized about it before me, and it probably has a more formal name (if you know, I’d love book recommendations on this paradox… Godel maybe?). Basically, if you can always ask “why?” to any statement and then any proceeding answer, where does logic end? If we can’t give reasons for everything, then does that mean we just have to assume certain things? Does that mean all logic is based on illogicalness?

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The main point of all these paradoxes is that they don’t really manifest themselves in the real world; we don’t directly observe anything tangible that makes no sense; we’re not observing magic. As Dream, Death, and the Self says, the generation, analysis, and solution of such a paradox are all purely philosophical. They don’t really create any problems in our everyday life, only problems in how we perceive and understand the world, only inner-conscious problems. (As opposed to, say, an optical illusion, or a limited understanding of some tangible science like physics. Gravity is a paradox, for instance, and not a purely philosophical one. As is the uncertainty principle.)

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In the film Inception, I’m not quite sure how an optical illusion involving a staircase is a paradox… ?? Makes a nice one-liner thought, I suppose.