"...my interest in most things lies in the nominal rather than the phenomenal aspect. Some fine day I intend to try to get to the bottom of WHAT'S GOING ON HERE -- the real world here, rather than the world of seeming. Are we all liars and humbugs and if so, why not?" -- Myles na gCopaleen
If the average person exists, then the average person in the United States has a smudge where the meaning of “paradox” should be. Of the persons who know the word without its -ical handmaiden (“It's paradoxical that so many people know the adjective and so few the noun”), the Liar's Paradox and Zeno's Paradox must be the only occupants of the set. Oh, “set of all sets” and other strangers wave from beyond the margins at certain professions, but Zeno's paradox of motion far outweighs the argument he made that there is no pain, and Cretans lie (aren't Cretins the only ones fooled by the Boolean trap, or do we hold out for prepubertal viewers of "Star Trek" like me, too?).
For me, Zeno's paradox has been something of a haunting presence. I have never been satisfied with the answers that I have gotten to it, except the answer of my senses (those flying turtles hurt). Furthermore, I am haunted by the fact that we have somehow managed to go about our business, playing Math and Physics for centuries without getting a good answer to the paradox. Not only that, but I am molested, every time I consider this, with just how pleasant the people are who believe that they have solved the 'problem.'
My reader is an above average reader in every way, and so my reader already knows all there is to know about the paradox itself.
[Above Average Reader Of Note: Yeah, but we want to hear you say it, so we can make fun of how much you get wrong, later.]
Well, Zeno was slaloming in between the pillars and posts of his porch, wondering when the Romans would come make it all intramural, and I'm told he was ranting about how much Parminedes bothered him. Parminedes's paradox of time and motion, he said, was stubborn. “The way I heard it,” he told his students, “Usain Bolt and Rafael, the teenaged mutant ninja turtle, were going to have a race. Because it was unfair on the face of things for Usain Bolt to race a fictional character, he decided to give the turtle a fifty yard head start. However, when the race began, Usain Bolt discovered that he could not win, because he was unable to pass the turtle.” “This is because the turtle is a poorly conceived cartoon character, isn't it,” one of Zeno's students asked. “No,” Zeno replied, “but because Bolt, to pass the turtle, had to cross half the distance between himself and it. He then had to make up half the remaining distance. After that, he had to make up the very slight half distance remaining, but only to discover that there was as much distance to go. He crossed half of that, but then there was half again remaining. Of this half, he crossed half, and then saw that he could only cross half of the remainder. This would go on, as it turned out, forever, or for as long as there were divisions and numbers, and Bolt simply could not pass the turtle.”
[Aaron: That's not how we heard it.]
I know. That's why I told it differently. The point is that “time and motion are both illusions,” like Parminedes said, except, of course, that they're not, no matter what you've heard to the contrary.
Aristotle said that the problem with Zeno's paradox is that he presupposes and infinite amount of space to cross, but not an infinite amount of time to do it in. If we segmented time just as infinitely as movement, then the turtle would be in the soup. Something or other to do with the definition of time or the definition of motion or something. I usually agree with Aristotle – which is to say that I usually understand Aristotle – but in this case I'm still left with that “infinite division” thing. Worse than him, though, mathematicians have told me that Zeno's paradox is not only not a paradox, it isn't even a problem. While all of us were sliding off our seats drunk at a wedding reception after-party, a mathematician told me that I was just confused because I hadn't realized that some infinities are larger than others. The infinity of fractions is larger than the infinity of whole numbers, sez him, but they are contained within the whole numbers, and so we were just playing a game with sets... or something... or one infinity overtook or ate the other. I thought it was drink talking until a sober mathematician (speaking of poorly conceived fictional characters!) attempted to tell me something very similar.
Ever since I received those explanations, I have used them as a test of soul, as a Procrustean bed. If Aristotle has it right, then you, Madam, are clearly meant for a physicist. If, on the other hand, new groom Bill's cousin the mathematics professor had it, then, Sir, you may be fit for any number of jobs, but you are no humanist. Both results are, of course, to your credit and health. You will, I have no doubt, sleep better at night, and your horror scope indicates that you will be able to accomplish deeds surprising and proficient.
[Aaron: Yes, very amusing, in a 19th century Gentleman's Magazine sort of way, but not a humanist? Exaggeration is one thing, but insult is another!]
I mean it, every word. Well, I mean the words I mean, but not the words that are meant in jest or gesture (watch carefully). However, the people who agree with Aristurtle or the math folk are not humanity centered, not placing the human reality at its proper position -- which is the center of all possible perception. This is not egoism, nor species egotism, but rather knowing one's place and recognizing that it's a great deal better than it might be otherwise. Let me explain.
[Aaron: And now he's asking permission? Where's my pouch of Red Man?]
The new groom whose wedding we were dunking till drunk went on to get a Ph.D. in electrical engineering. I would like to say that he's one of my life's friends. However, his doctorate was in analog signals, and this was in the middle of the 1990's. “I thought the point is that everything is, uh, digital,” I said to him one day, “so why analog signals?” “Because,” he answered, “at the high end, digital signals begin to act like analog.” I thought about this for a few moments, made no progress, and then went on to worry about chain marks in bibliography class or some other important problem. I still needed, after all, to distinguish between late 17th century usages of “lampoon,” “burlesque,” “satire,” and “parody.”
I was so ignorant then!
Sometimes a fact will wedge in the soil of the mind, and layers of information, passion, misdirection, reconnection, and agitation will eventually force it back up. The process is not vegetative. People have been speaking of ideas “growing” in the mind since at least Charlemagne, and probably ideas do grow, send out rhizomes, cross-pollinate, and need pruning, but facts do not. Facts are like colored glass beads to the magpie mind. They go into the nest. In the course of the seasons of straw, hatchlings, and matings, the bead will turn up again in a new context where it will – by itself still only a bead – become part of a new mosaic. So it was with the fact that digital becomes analog analogy for me.
Right now, I have iTunes playing in the background. It is playing a song recorded by The Ventures in monoaural in the 1960's that was converted to stereo, pressed onto an album, converted to a digital CD, purchased by me (I buy CD's, and not because I'm old and bewildered, either), and then converted to the super-magic iTunes format, and now it is playing. I am enjoying it, too, in case you were wondering. However, the way those lovely surf guitar sounds were made and recorded was analog.
In analog recording...
[Aaron: Sheesh! Now you're going to explain analog? Plug it in and go.]
...voltage went to a microphone beneath the strings and one in front of the amplifier, and an electromagnet recorded increasing and decreasing voltage continuously that was then recorded, continuously up and down on magnetic film. [I know! 'To learn more about analog recording, consult your local library.'] The vinyl record was a cut groove that continuously reflected the pushing of a stylus in response to voltage increases. The digital, on the other hand, is based on sampling. A computer is told to capture sound levels and take an exact picture of all voltage levels on all microphones and to write this data down. It will do this a number of times a second. It might do it sixteen times a second (low quality) or thirty-two times a second (standard) or one hundred and twenty-eight times a second. Regardless of how many times a second the computer is taking a 'snapshot,' it is still taking a “sample,” which is an audio record of all values, at discrete moments.
You can see now what I did not see, I am sure. The super high end samples will begin to act as if they were continuous. Bright man, my friend Bill.
[...And yet you want to connect this to Zeno's paradox by …]
Quiet you! I do not. I may not be as clever as Bill, but I'm not as clumsy as that.
You see, the digital music that I am listening to right now, which, incidentally, is Pink Floyd, is tricking me. [Pink Floyd? I thought the air smelled a little sweet!] It is playing back thirty-two separate clips of music per second, and I am too sluggish of thought to hear the silences between them. [Yep. That's what happens.] This is because I am a man – sufficient reason to be miserable, as the Greek said. However, if it were slower in its clips, would I still perceive the music? Think about watching television with your dog. Your poor dog never liked your old analog television set, because it emitted a high pitched whine and because he couldn't see a picture. That is because there wasn't a picture. Only a third of the screen was lit up at a time,and the third rolled from bottom to top, and not that quickly, either. If you took a picture of a television screen, all that you saw, in the old days, as a third of the screen. Well, that's what your poor dog saw when you tapped the glass and said, “Look, Rufus! Another doggie!”
A movie is twenty-four frames per second of pictures showing in sequence. A cartoon is a sequence of drawings flipped past the eye. There is nothing continuous in any of these three things. The old television, the movie, the cartoon: all of them were discrete objects that we insisted we experienced as continuous, as analog. Digital music is therefore no surprise.
They call the human blur/blend illusion the Phi phenomenon. Phenomenologically speaking, one can see where this is going – which is around and around, like a fly in a Coca-Cola bottle on Wittgenstein's lectern.
[I really must object! That's neither funny nor necessary.]
They laughed at Heidelberg.
[Yeah, but not at Heisenberg. No one could tell where the punchline was.]
There is always the possibility that Zeno's was 'right,' as it were, or that Parminedes was, and that we don't move, that motion is a phenomenon, a Phi phenomenon. Perhaps analog is digital, and, if we were to look at that “continuous” electrical charge going up and down in a microphone line, it would be made up of greater and lesser infinities of incremental measurements. It's just that there sure seems to be a difference. It sure feels different. Actual movement, with persistent objects, really seems to have some distinction against representations presented sequentially, no matter how rapidly.
If, though, you have enough of a humanist in you to think, as I do, that there is something about the world, about living, that is different, and even superior, to super-rapid sequence, then we're right back at Zeno's question: Where is it? If the digital acts like analog when chopped finely enough and the analog can be chopped into digital, where is the whole number?
You see, the answer to Zeno's paradox that does not require phantasms of infinity or time is to declare that there is an indivisible unit of space. If you deny infinity itself, then Zeno has to stop cutting the space in half. If you say that “there is no more distance,” then Usain Bolt finally gets to the finish line and the autograph seekers. The alternative is to say that a mystic Something Happens whereby the infinities surpass one another in a glibly invisible and imperceptible leapfrog as we move in whole numbers, unmindful of the limitless fractions we toss by. In either case, the skeptic (or Eleatic) has the right to ask you, “Ok, Buck, so tell me where it is.”
It is to avoid that question that everyone else seeks purely logical, non-mystical answers, no matter how much they hurt the brain.
I don't have a bead on the answer. As I said, Zeno's paradox keeps giving me the bad touch when I least want it. I was wondering, though, the other day, that we are so happy to ask no questions about another impossibility. We mumble and mutter along every day with circles and other oddities and their attendant irrational numbers. Pi, like movement, lodges like a pretzel in our windpipes. It just won't go away, even though it is generally a simple expression. What's more, like running around in a circle, it is infuriatingly rooted in the real! We know that the circumference of a circle is pi multiplied by the squared radius. That means that we get pi always from every danged circle. If circles are actual (real, existing), then so is pi. Well, that's simply intolerable.
Is pi proof that there has to be an irrational or a-rational or suprarational answer to the question of divisibility of space, time, and motion? I'm not that juvenile. Pi floats out there, infinitely, and waves at us from beyond our capacity to limit and define, just as motion does, just as the difference between the infinitely divided and the continuous does. All of these weave from the margins of our minds, producing a garland of phenomena and rational limit. They humble us. They insist that we are products of reality and consequently can never judge it entirely.