Let’s continue our whirlwind tour of Cracked’s roundup of 20 Paradoxes Most Human Minds Can’t Wrap Themselves Around. If you haven’t already read them, you might want to check out Part 1 and Part 2.
Is this a paradox? As presented, yes. Remember: paradoxes don’t have to lead to true conclusions. They merely have to present a contradiction. Let’s have some background.
Zeno of Elea was a philosopher who lived from about 490 BCE to about 430 BCE (the 500 BCE estimate is therefore a bit too early). Zeno actually created several paradoxes of motion. This one in particular is known as the arrow paradox, or sometimes the fletcher’s paradox.
As originally formulated, the arrow paradox did not mention an observer “stopping time”. According to Zeno, in any instant – which Zeno treats as a frame of time with zero length – the arrow isn’t moving. And if the arrow isn’t moving during one particular instant, then it isn’t moving at all; in fact, it cannot move, because the flow of time is nothing but a progression of motionless instants.
The paradox here is obvious: In one corner, Zeno purports to show that motion is logically impossible. In the other corner, reality.
Obviously motion does occur. It occurs ceaselessly. Our entire existence is based on motion. And yet Zeno does present a case that is at least superficially convincing.
Is there a resolution? There must be. Zeno has argued that an arrow’s instantaneous velocity must always be zero, and therefore that it’s overall velocity is zero. (By extension, nothing else in the Universe must be capable of moving either.)
Actually, using Zeno’s description of an arrow at an instant in time, and applying the velocity formula – displacement divided by time – an arrow’s instantaneous velocity must be indeterminate. The duration of an instant is zero; likewise, the arrow has zero displacement during an instant. Ergo, its velocity is zero divided by zero – indeterminate.
But that’s not very helpful, is it? An indeterminate velocity gets us no closer to resolving Zeno’s paradox of the arrow. For a more fitting solution, we must turn to calculus.
Wait, where are you going? Come back!
If you took a calculus class in school, you probably learned about limits. I’ll not get into a detailed discussion about limits here, but suffice it to say that we can apply the concept to answer Zeno’s paradox.
You cannot measure an arrow’s velocity from a single photograph (which we’ll treat as an instant in time), but suppose you had two photographs of the arrow, taken at different, precisely-recorded times. From two photographs you can work out the difference in the arrow’s position – its displacement – and since you know the time interval between the photos, it’s a cinch to calculate the arrow’s velocity.
Only you don’t get the arrow’s instantaneous velocity; you get its average velocity. But now imagine you have two more photographs, this time taken even closer together in time. You will get a better approximation of the arrow’s instantaneous velocity between the two photos. The closer together the photographs are taken, the more you zero in on the arrow’s velocity at a particular instant. In math terms, then, the arrow’s instantaneous velocity is the limit of the arrow’s displacement divided by the time interval, as the time interval approaches (but is not necessarily equal to) zero.
Confusing? Maybe, but it eliminates the need to think of the arrow’s flight as a disconnected series of instants. Now you could get into a lengthy philosophical discussion about whether instants – zero-length moments of time – exist in any meaningful way, but regardless, motion is clearly possible, and any realistic interpretation of the world must reflect that.
Is this a paradox? Not really.
Titus Lucretius Carus is better known as a poet than as a philosopher, although his most enduring work, De rerum natura (On the Nature of Things), is a poetic effort to explain the philosophy of Epicurus.
Lucretius, like Epicurus, was strongly influenced by the atomism of Democritus and his mentor Leucippus, both of whom lived several centuries earlier. Democritus, you may recall, postulated a world made only of minuscule particles called atoms (from the Greek for indivisible) and empty space. Democritus believed that the motions and interactions of atoms could be described by succinct physical laws. By extension, each atom’s position and motion had a direct physical cause. In Democritus’s Universe, there was no room for chance.
This philosophy, which Democritus believed sprang naturally from atomism, is called determinism. While Lucretius subscribed to atomism, he wished to avoid a purely deterministic Universe because he perceived determinism to be incompatible with free will. If a person’s decisions are made in his brain, and if the brain is made of atoms, and if atoms obey unyielding physical laws, then it seems a person doesn’t really have the freedom of choice. He may have the illusion of choice, but each of the choices he thinks he is making were actually decided by the states of the numerous atoms in his brain, which were in turn affected by the states that existed just prior, and just prior to that, and so on.
Lucretius believed that free will was essential to personal accountability. For example, in a purely deterministic world, one might argue that a murderer is not truly accountable for his crime, since his actions were merely the result of atomic states in his mind and body over which he had no control.
Lucretius tried to dodge this sticky problem by suggesting that atoms occasionally have the ability to swerve, or to randomly move in a completely unpredictable manner. In that way, the Universe isn’t completely deterministic. There is still room for chance, and also for choice.
This meme doesn’t represent a paradox because Lucretius never meant to simultaneously champion a deterministic Universe and free will. Lucretius’s question is meant to draw the reader toward the conclusion that the Universe cannot be fully deterministic.
For what it’s worth, there is a school of philosophy that subscribes to compatibilism, in which the Universe is deterministic and yet humans are still free. Perhaps the meme’s author, in search of paradoxes, should have focused on a statement made by that school.
Is this a paradox? Many sources present this as a paradox, and I will defer to their judgement, although I will go on record as saying that this is a weak paradox.
For one, the paradox has not aged well. This story comes from ancient Greece, where legal training and court proceedings were remarkably different than they are today. Modern lawyers are not taught by a single over-priced instructor, but by a network of instructors at an over-priced law school. You could replace John with, say, Harvard Law School, but then the story doesn’t work, because I’m pretty sure Harvard would never make a deal like the one John made with Bill (or Protagoras made with Euathlus in the original version.)
Also, John was foolish not to include a stipulation that Bill must enter the legal profession and accept cases, lest he breach the contract. Had he done so, then John would have a clear case against Bill. Alas, what’s done is done.
The contradiction is obvious enough: Regardless of the outcome of the case, Bill does and does not have to pay John. What can the court do to bypass this legal conundrum?
Is there a resolution? There are many resolutions. In my (admittedly non-expert) opinion, the difficulty posed by this situation is trivial, hence my initial assertion that this paradox is weak.
The court could simply dismiss the case without hearing it. Arguably, then, Bill would not have won or lost; in fact, he would not have argued a case at all, which means that he still is not obligated to pay John.
Bill could hire someone else to represent him in court. In this way, Bill would not be personally arguing a case. The judge could rule in Bill’s favor without violating the terms of the contract (and simultaneously send a message to John to exercise caution when deciding on the precise wording of future contracts.)
The court could rule in favor of John, thereby overriding the contract because Bill would be forced to pay despite not having won any cases. As I understand it, a court order is the law of the land and supersedes any other agreements that were previously established.
For the same reason, the court could rule in favor of Bill, freeing him from his contractual obligation to pay. The court could also uphold the contract, but decline to force Bill to pay since he had not yet won any cases. (His first win doesn’t happen until the judgement is delivered.) Of course, this would provide an opportunity for John to launch another suit against Bill – one he would be much more likely to win.
Is this a paradox? I’ve tossed this back and forth in my mind countless times, and I have come to the conclusion that it is not a paradox. However, I’m so ambivalent about this one that I am willing to be corrected by any philosophy majors who happen upon this blog.
In my reasoning, this cannot serve as a paradox because a paradox is built from two or more premises which are by themselves logically sound, but which nevertheless lead to a contradiction. Archytas (who, by the way, believed the Universe to be infinite) used the second part of his question to show why the first part was not logically sound. If you can extend your staff further into space anywhere you go, then there is no place that can truly be called the edge of the Universe. These are not two propositions which stand against each other on equal footing. This is a case where one proposition destroys the other.
Here’s a similar example: If I count to the largest possible number, can I then count beyond that? The first part of my statement assumes that there is a largest possible number, which we know to be nonsense. The second part explains why it is nonsense. Of course you can add one to any number, no matter how huge. Ergo, the number line is infinite.
Although I agree with Archytas that the Universe is probably infinite (We’ll talk more about it tomorrow), that need not be the case in order for the Universe to be edgeless. Some believe that the Universe wraps back around on itself in every direction, so that if you were to start off in a straight line, and if there were no speed restrictions, you would eventually return to your starting point. In any case, no modern cosmologist believes that there is some “wall” in outer space that marks the boundary of the Universe, beyond which you cannot pass.
Before we leave this one behind, let’s take a minute to address the second part of the meme: What is the Universe expanding into? It’s a common question asked by those who are naïve to the Big Bang theory and modern cosmology. Unfortunately, the answer is hard to fathom. You see, the Universe isn’t expanding into anything…it’s just expanding. This is true regardless of whether the Universe is finite or infinite. Scientists believe there is no edge to the Universe – no boundary between Universe and not-Universe. It is a fallacy to think of the Universe as being held inside a glass ball that is somehow expanding into a region beyond the ball. There is no beyond, because there is no ball. Even if the Universe is finite in volume – a doubtful proposition in itself – it may still be edge-free if space wraps around on itself in every direction.
This is not a concept we accept lightly, because in our limited human experience, we are used to finite things that have distinct boundaries. Space defies every one of our expectations, and we just have to learn to live with that.
Is this a paradox? Unequivocally, no.
This is a thought experiment, similar to the one we covered in Part 1, which asked whether an erstwhile blind person is able to visually identify three-dimensional shapes based on previous tactile familiarity.
The sensation of seeing color – indeed, any sensation – is known to philosophers as a quale (plural: qualia). Qualia are entirely subjective, which means each person experiences his or her own set of qualia that cannot be adequately described to anybody else. When you see the greenness of grass, or the redness of a sunset, that experience is yours alone. There doesn’t appear to be any way you can describe your qualia to me so that we can compare. Did you ever wonder if your red is the same as my red when we both look at a red object? If so, you were wondering about qualia. Unfortunately, your question is unanswerable. Sorry.
So the solution seems to be yes. If Mary, previously color-blind, suddenly gains the ability to perceive color after a lifetime of studying color vision, she will indeed gain some indescribable knowledge about color. Until then, it doesn’t matter how thorough her research is concerning color vision – until she has seen color, color will be alien to her.
Three days down: one more. Today’s tally of actual paradoxes: 2 (maybe 3) / 5. Running tally: 7.5 (maybe 8.5) / 15. Fifty percent…not great.