The Motion Paradox | Joseph Mazur | Talks at Google

It's my pleasure to present Joe Mazur to Google today. He is the author of Euclid in the Rainforest, which was a finalist for the 2005 Penn Martha Albrand Award for First Nonfiction and chosen as one of Choice 2005 Outstanding Academic Titles of the Year. He received a Ph.D. in mathematics from MIT, was the recipient of the 2006 Gunheim Fellowship, and is currently professor of Mathematics at Marlboro College. He has authored many educational software programs, including Explorations in Calculus, the first interactive multimedia CD package of simulations for Kappa. So without further ado, please join me in welcoming Joe Mazuri.

Thank you, Jim. Thank you. There are a couple of corrections. One correction. I mean, Jimmy didn't consult with me on my background. This did not receive the pen. This was not a finalist for the PEN Award. It was a previous book that was a finalist for the PEN Award. So I just want to be sure that you understand that that's the case. What I first of all, thanks for coming and it's a great honor to be here, especially on this particular occasion when I know that Google went past 600. And what I plan to do is alternate between reading and telling something about the book. And then that would happen for about 30 minutes, 2530 minutes. And then I'll open it up for questions at that point. But if you have some burning questions that you really feel that you need to ask about, please do so at any time. So I am going to read the very first page of the book. Now, since I assume that few people have read the book, because I see the covers are all pristine and nobody's opened those books, they've all been sitting on those seats that nobody has read the book yet. And therefore I really need to get you to the point of understanding what the book is really about. Now, the book is about several different things and you will, I hope you will get a sense of what those things are in the next 25 minutes. So let me read a very short piece which is the preamble to the book. My father was the first person to tell me about paradoxes of time. He had never heard of Zeno's paradoxes, those peculiar arguments on motion that contradict common sense and that have been misunderstood these last two and a half millennia. But was a gentleman philosopher with instinctive wisdom about the world. My brother had just received a brand new Schwinn bicycle with chrome fenders, a speedometer and battery operated horn for his birthday. Boy, was that neat. The gentleman philosopher knew just what I was thinking to soothe My jealousy. He took me aside and told me that I was half my brother's age, but in eight years I would be 3/4 his age, and from then on, there would hardly be a difference. Of course, I had no idea what he meant by 3/4, let alone 3/4 of someone's age. When I asked how old I would have to be to catch up completely, he laughed and said that that would never happen, but that the difference would always be getting smaller. Years later, I thought I understood, but now, rapidly gaining on my brother as I pass 16, 17 of his age, I'm just beginning to more than 2,000 years before my father eased my bike envy with his thought experiment, Zeno had invented similar paradoxes. Zeno argued with flawless logic that contrary to what everyone experiences every day, nothing moves. Okay, so that's the setup for these paradoxes. The birthday convergence is. We will talk about that perhaps a bit later when I line these paradoxes up at the end and see the comparison between the paradoxes. But let me tell you a little bit about the paradoxes. I'm not sure how many of you are familiar with these paradoxes, but there are. Zeno had. Is reputed to have had 40 paradoxes. We only have four of them. The others are lost. I will tell you about three, because the fourth one is a little convoluted and a little bit difficult to understand without a real blackboard. And so I will just tell you about the three, and we can talk about the fourth at this time later. So the first is something you probably have heard before. It's called the dichotomy paradox. And it goes like this. If I want to leave this room, there are several doors in this room. But let's suppose I want to leave this room by that door. What I have to do first is get to a point that is halfway the distance between where I am and that door. But once I do that, I'm at a new position, and I have to continue by getting to a position that's halfway between where I will be and that door. And once I do that, I will have to repeat the process, and I will repeat the process indefinitely for an infinite amount of steps. So I will never get to the door because supposedly I can't complete an infinite number of things in a finite amount of time. Of course, you at Google probably could. But. Now the next is another paradox called Achilles. It's called Achilles the Achilles paradox, but by some 19th century modern names, it's called Achilles and the Tortoise paradox. And you probably have heard of that one as well. Achilles, swift footed Achilles, great warrior, is about to have a race with a slow moving tortoise. And the tortoise argues with Achilles and says, look, Achilles, you're so fast. It wouldn't be fair to have this race without giving me a head start. Give me a head start. Now, in most fables, it turns out that the animal is always the wise thing. In this case, it's Achilles. Achilles one thing is godlike, so he does have wisdom. Achilles says, no, if I give you a head start, it's impossible for me to win because I will have to get to the place where you started racing before. Well, I'd have to get to the place where you started racing. And at that point you will have moved further along and then I will have to get to the new position that you're at. But by that point you will be still further along and that will continue indefinitely. So I will never be able to catch up with you at all. Okay, so that's the Achilles and tortoise paradox. There's a third paradox, and I have a prop for that goes like this. You shoot an arrow across the room, and at some point somebody drops the arrow, drops another arrow. So there are two arrows. One's going this way and the other's going this way. And you pick a point in time where you photograph the two arrows, a very quick photograph. And you look at the photograph and you say which arrow is going in which direction? And if the photograph is good enough. And what do we mean by that? Of course, if it's good enough, you really can't tell. Two arrows just look like that. Well, I have to assume then that the arrows are stationary. The arrows are in a stationary position at that point in time when I've taken the photograph. But I can take that photograph at any time, almost any time. Maybe one arrow will be here and the other one will be here. But I will get the photograph so that I will have the two arrows at positions where you can't tell which one is going in which direction. Okay, so Zeno argues from that, that if every time you look at the arrow, it's stationary. How does it get from one end of the room to the other? And we can do this mathematically, surely. I mean, we just fix time. Time t equals 2, 2 seconds. Where's the arrow when I'm at 2 seconds? Well, it's someplace. And I look at it and the arrow is stationary. Well, that's the arrow paradox. Okay, so those are the three paradox. Three paradoxes that we'll talk about. Let me read what Zeno will be trying to get at in one of these chapters. Athena was the gray eyed goddess of war, fertility, art and wisdom. Her birthday was one of those rare days when women and freed slaves were permitted to appear leisurely in public places. Imagine being inside of the majestic Acropolis, near the northwest corner of the great Athens market and gathering place, looking southeast along the Panathenic way, the dusty path partly shaded by poplars and wild hardy carob trees. You would see preparations for the great Panathenia festival. You would see athletes rubbed with olive oil competing for prizes in foot races, boxing, long jump, javelin throwing, chariot racing, musicians competing with voice kithra and flute, and blind bards reciting Homer's epics. On this day in 450 BCE, it was four years since the last great festival, just one year after the signing of a five year truce between Athens and the other regional power of the city state of Sparver. Northwest past the marketplace, through the sacred gate of the city wall and to the right left, lay the Ceramicus, a public square and war cemetery in the Potter's district. This is really a good picture of what Athens was actually like in the 5th century BC. This is all quite accurate by the way. Pentalic marble stones were being stored for the anticipated construction of the the temple to honor Hephaestus, the skilled fire God of the anvil, with huge bulk, thin legs, sturdy neck and hairy chest. It was quieter there, away from the loud hawking butchers, bakers, apiarists, olive pressers, wine merchants and ironmongers lining the crushed limestone avenue leading to the festival. High on the hill, wild thyme grew through the limestone cracks near fruit vendors selling pears and figs. As Homer noted in the Odyssey, there such fruit comes at all seasons of the year, and there is never a time when the west wind's breath is not assisting here the bud and here the ripening fruit, so that pear after pear, apple after apple, cluster on cluster of grapes and fig upon fig are always coming to perfection. According to Plato, Antiphon the sophist, heard the story of Zeno's visit to Athens from his friend Pythodorus so many times that he could repeat it by heart. Parmenides, founder of the celebrated Iliadic school of philosophy, was sitting on a stone, a distinguished man in his 60s with Bone White hair. Sitting next to him was Pythodorus, a younger bearded philosopher, looking particularly alert. Next to him was Aristoteles, a sun bronzed man in his 30s, lost in contemplation. And young Socrates, not yet 20, a nearby donkey, was obstinately complaining about a load of barley on its back. Zeno Avilia, a tall and attractive intellectual revolutionary, was reading from his famous book on philosophy. He had come to Athens from Cortona in southern Italy with his teacher and lover Parmenides, to visit Pythodorus in the Ceramicus, just outside the city wall, and to attend the great festival. His lines of reasoning were terribly confusing. They seemed to rely on language tricks aimed toward the mystifying suggestion that there is only one single thing in this world, the thing he called being, and that all else is mere appearance. He argued that if a thing can be divided, its divided parts can also be divided, and such divisions can continue indefinitely. From this he concluded that change, and hence motion, is not possible. He finished reading, but his audience was confused. Even Socrates was confused. We have heard it all before. And God made the firmament and divided the waters which were under the firmament, from the waters which were above the firmament. And it was so. And God called the firmament heaven in the book of Genesis. From the waters came two distinct heaven and earth. Creation is the vision to mark opposites, light and darkness, day and night, summer and winter, land and sea, fish and fowl, even and odd, good and evil. What Zeno said makes sense. If two things exist, a third must exist to separate them. Otherwise there would not be two things, only one. If three things exist, the fourth and fifth must exist to separate the three. To distinguish between A and B, there must be a separator, C, and to distinguish between A and C, there must be another separator, D, and so on, thus proving that there must either be only one thing in this world or an infinite collection of things. Zeno went on to argue that nothing changes, because change would require a beginning and an end to being. Therefore, Parmenides said, the one which is not, not possessing being in any sense neither ceases to be nor comes to be. He and Zeno were thinking of that something in the act of change must perform that act in time. So change is equivalent to motion. Like the arrow that can never leave the bow, change is impossible. Zeno's arguments for motion may be applied to the ripening of a pear. The neurologist Oliver Sacks once wrote, I would come down to the garden in the morning and find the hollyhocks a little higher, the roses more entwined around their trellis. But how? However patient I was, I could never catch them moving. We have all seen garden of flowers, but have we ever seen flowers growing like the hollyhocks? We can never catch a pear ripening. And though it may change in color, taste, texture and even shape, it remains a pear. How does the pear get from unripe to ripe if every instant we look at it, it is in a fixed state, somewhere between two extremes? Zeno's paradoxes are not only about locomotion, but also more generally about change in quality and quantity. So that's. And then it goes on from there. That's the idea here is this whole question of change. Can there really be change given these kind of logical arguments that are being professed by Zeno and his school of philosophy? Okay, so now I skip a little bit. I'm going to skip a little bit to the linear. This is a linear in time, this book. So this book starts in 450bc. It goes to the 20th century, and I'm going to skip to the 16th century. Galileo. Now, what Galileo is doing in the 16th century is clearly not the first person to observe things. I mean, we live in a world where we observe things, we experience things, and we know that the arrow goes from one side of the room to the other. So we have an intuition, we have a mathematical intuition, and we have a physical intuition because we've lived. I mean, we're humans and we experience things in time and space. If you look at some of the literature that seems to come out of the things before the 16th century and bit beyond the 2nd century, you find that it seems as if these philosophers are not really observing the world and everything seems to be done as if the only pure thought that tells you something about truth is about rational ideas that come from reasoning. I mean, that is reasoning. So mathematics can tell you something about the world, but if it doesn't tell you something about the world, that's fine. It just is. It can be pure reasoning. Now, Galileo, of course, seems to be historically the person who's really starting to think to observe these things. Say, let's see, truth might come from observation and not simply by reasoning. Well, clearly the Greeks were observing things. I mean, they're humans, so surely they saw arrows crossing a room or crossing the field and people leaving a room going out a door. And they experienced these things that were. There's no surprise that they would have experienced those things. And surely they talked about it. And surely their world was a world of experience. Now when Galileo comes along, his. His object is to experience the world from. You know, he's going to find the truth out about the world from experience. From looking at the world. So I have a I'm going to read a little bit about that, and part of that has something to do with Google. Surprisingly enough, on a Sunday morning in 1583, young Galileo Galilei would have walked along the cobblestone cay of Santa Maria della Spina to attend mass at Pisa Cathedral. Other parishioners would have filled the streets as bells from small churches accompanied the loud gongs coming from the leaning tower beside the cathedral. Walking along the banks of the Arno crossing Brunelleschi's magnificent Pante Amare, Galileo would have seen the mass of fishing boats slowly swaying against the backdrop of stone houses like metronomes to the regular beat of rippling river and ringing bells. That's a wordville. He was living in the house of a relative near the port of Fiorentina. It was a short walk from there to the cathedral, but long enough for the talented student to spot synchronization of sound and scene Flags atop mast waving, waving and flapping to the strict rhythm of rocking deck boats. He might have noticed that the time it took for a mast to complete its swing did not change, an observation that could have incubated this subconscious waiting for important connections before hatching a scientific discovery. The cathedral was modest, not very tall, and architecturally plain, with a facade mimicking that of a leaning bell tower. Inside the nave was a large, ornate bronze chandelier with 30 candles and three tiers balanced around the periphery. A drip tray hung by three short chains under each candle. To light the candles, the giant fixture would be lowered and raised by a chain through its center. If this were done just before Mass, as it likely was, the entire fixture would gently oscillate until it calmed down to faint undulations caused by vibrations of the organ. Most worshipers would hardly notice, but a genius like Galileo sitting through a dull sermon would have wandering thoughts. If a chandelier was swaying, he would watch it and marvel at its motion. One imagines him timing the oscillations against the timing of his own pulse and having revelations about how to measure time. History claims that that Galileo timed the chandelier while sitting through a humdrum mass at Pisa Cathedral. If the story were true, he would have noticed a magnificent phenomenon. Though the chandelier would slow down, the duration of each swing would not. Shorter swings would simply be slower. The time required for any complete to and fro swing of the chandelier would depend only on length, that is the length of the the arm or the chain. Surely this is a fictional Account of how Galileo made his first independent discovery about motion, because by other accounts, the cathedral chandelier was not installed before 1588, five years after Galileo claimed the discovery. Now, true or not, the story tells of how such a discovery does surface. Galileo could have come to the same revelation observing any swinging object. The mathematician Vincenzo Viviani, pupil and friend of Galileo, perpetuated the story, saying, having observed the unerring regularity of the oscillations of this lamp and other swinging bodies, the idea occurred to him that an instrument might be constructed on this principle, which should mark with accuracy the rate and variation of the pulse. Some say it was that incident of the church that influenced Galileo to study mathematics. Others say it was a mathematics lecture he just attended by chance. In late 16th century Italy, professors were still entrenched in Aristotelian doctrine in the belief that the Greeks had already expressed all worthwhile knowledge, and in a mistrust of new ideas coming from creative minds. A century of unmatched explorations had just passed. More than doubled, doubling the size of the known world. America was discovered. Vasco da Gama had sailed around the Cape of Good Hope to reach India. Magellan had sailed clear around the entire world. The vast Pacific was discovered. Europeans had stood on the continent of Antarctica and had not fallen off. Stones still fell to the ground even on the other side of the planet. And the world was suddenly believed to be spherical. And more than just Europe, Asia, Africa and the Holy Land. For a thousand years, thinkers had been sitting in their dimly lit studies, university libraries and secluded monasteries, rationalizing and arguing about the shape of the planet, the makeup of the celestial sphere, or the laws of nature. But it took the courage of the Portuguese, Spanish and Italian sailors venturing into the dangers of vast, unknown seas to determine perceptible truth. Nobody would ever again deny that a great sphere of constellations completely surrounds our planet. The discovery of the Americas offered a new perception of the world. America was not discussed in the Bible. Ptolemy never mentioned it. Nor did Aristotle in his De Calo, nor Pliny in his Natural History. So when Spanish conquestadors returned to Europe laden with myriad wonders that had never been seen or written about, some folks at home began to experience exceptional curiosity, questioning why their handy classics never said a word about the existence of such exotic lands of improbable flora and fauna. The conquestors found delicious fruits, vegetables and nuts never seen in the Old World. Tomatoes, corn, avocados, pineapples, cranberries, blueberries, sunflower seeds, cashews and cochineal, giving the most intense concentrated Brilliant red dye the world had ever seen. A dyestuff so precious that it later set off wars and encouraged piracy in the Atlantic from the hidden coves of the Caribbean to the port of Cadiz. Hesitant to defy their Bible, they challenged the wisdom of established classical intellectual teachings and began to investigate nature by direct observation. And of course the movable type of Johann Gutenberg, his printing press in fort with replaceable wooden or metal letters and the invention of plant fiber paper were responsible for the publication of more books in the 16th century than had been produced in the 3500 year period since the first Babylonian author produced the first cuneiform tablet. Now, okay, I'm going to stop there for a second because I mean, I suppose you all, since you all work for Google, Google, you all understand what has happened to the world because of Google. I think a thousand years from now, I'm serious about this, I think, well, maybe not a thousand years. Let's go 500 years. 500 years from now, somebody is going to say exactly what is being said about Johann Gutenberg and his printing press and how that changed the information in the world and how it changed the entire world. Somebody's going to say that about Google. I mean, you are part of it and I really do believe that because you know, it's just, it's certainly changed the way I view the world. It's certainly changed the way I do my work and I think it changes the way almost everybody does their work these days. So you're part of it. I won't be here 500 years from now, but it'd be fun to think about what people will say about Google 500 years from now. I mean, we're talking about Johann Gutenberg 500 years since Gutenberg. Okay, I'm going to read one more little piece because that's the concluding piece that will probably bring this together and open it up to questions because I think this will, what I'm going to read here will certainly generate some questions. This is a piece that actually got me to write this book. So I have to tell you that I wrote this book backwards. Started with a little essay here and I filled it. This is chapter 15. I filled in the other 14 chapters just to lead up to this. This is the way it actually happened. I'm starting in the middle of the chapter though. Take sight. The Zoetrope, a 19th century parlor room toy that gives the illusion of motion. It is nothing more than a spinning cylindrical drum containing slits and no more than a dozen still images of a each Image is very much like the next, except for the slight difference of anatomical position. When the succession of discrete images is viewed through the slits of the turning drum, the viewer sees the images fused into a dynamically moving picture, say a person in motion. You know what I mean? The zoetrope is kind of like a drum and turns and you see inside the drum, you just see through these slits and you see there are maybe a dozen little pictures inside this drum. And you spin it, you see a motion picture. Still images in rapid succession are interpreted as real continuous motion. How does this happen? The celebrated 19th century physicist Hermann von Helmholtz thought that the eye held one image just long enough for the next to take over. Something of the sort actually does happen in the retina. Look at a black spot on a white background for a few seconds and then turn away. The black spot will linger for a few seconds more. This is even more pronounced when we see a spot of bright light in a dark room. Long after the spot of light is extinguished, the spot is temporarily burned onto the photosensitive retina. But we now know that the coordination of discrete visual images, real sight, takes place in the visual cortex, not in the eyes. So the question remains, how is it that the rapid succession of still images is construed as a moving picture, seamlessly flowing in time? Is there some biological necessity to see continuous movement? A frog is uniquely capable of catching bugs because it sees only the movement of its tiny prey, Undistracted by inconsequential surroundings. Humans do not need to catch flying insects with their tongues, but need far more than just visual receptors to sense movement. Humans were once capable of hunting and fishing, as well as watching out for ambushing saber toothed tigers or stealthy reptiles. But why the need for continuous movement? Wouldn't staccato motion be enough to protect us from wild beasts and enable us to farm and live as we always have? Perhaps the real world that we live in is truly disjointed thing to think about here, where every movement behaves as though it is a flickering Mack, Senate, Keystone Comedy. If it were, would we know it? Or would we simply wrap what we see into a blanket of personal sensations relative to what we expect to see. If the brain is truly the thing doing the seeing, then it hardly matters what we see, as long as we give enough information to the brain to interpret reality. Helmholtz experimented with prism glasses, which turned his field of vision upside down. It didn't take long for his brain to compensate with the help of his physical existence. In the real world, and it put the world right side up. So shouldn't our magnificent brains be capable of adjusting the flickering world of keystone comedy so it appears smoothly, continuous, as long as we give it time to do so? And what about the other way around? What if the real world behaved smoothly and we could only see it through stroboscopic light? Would we feel discomfort at seeing a disjointed succession of images? Would we know the difference? Or do we simply adjust, as we did to foods that had bothered us in infancy? Our perception of Zeno's continually moving arrow may have nothing to do with its true movement. And yet the mathematics we install to model his arrow's flight must also model our sense of that flight. So the paradox stands between the crossroads of reality and our perceptions of it. The same holds for the dichotomy, or the Achilles and tortoise paradox, in which Zeno claims that an infinite number of events must take place before anything is accomplished. David Hilbert said, and the verdict is that nowhere in reality does there exist a homogeneous continuum in which unlimited divisibility is possible, in which the infinitely small can be realized. Hilbert is suggesting a friendly quarrel between the concept of infinity, the principal ingredient of continuity, and the perpetual, the perceptible world as mathematics brushes close for a clear look. And the friendly quarrel it is because, like very young siblings, they still need and nurture each other. We know the physical world by how we perceive it, and we perceive it by how we measure it. The moment we attempt to measure by ruler, scale, gauge, compass or thermometer, we are sanctifying number. In that moment, we are assuming continuity through a precise correspondence between what we see and what is really there to be seen. But we don't measure with infinitesimal instruments and so must be content with relatively rough estimates of reality. For each paradox in Zeno's quiver, we have an answer. Continuity is merely a conscious impression, a fabrication of the mind, elevating illusion to reality. Though mathematicians may try to explain the paradoxes by logical models of motion, phenomena such as algebra or infinite series, they miss the target and give a to give a phenomena phenomenological explanation of the unavoidable sense of harmony between the fantasy of time and the continuously flowing universe. Yes, they can tell us precisely where the arrow is, when Achilles will overtake the torus, or when we will come to the other side of a room. But they cannot tell us why without bending our perception of space to fit our inflexible intuition of time's continuous nature. Ask why Achilles overtakes the tortoise. And the response will inevitably be. Because we see that the algebra tells us that it happens when. Ask again and the response will point to a mathematical model. We know that the world, the one constructed to give the answer, is based on the continuous nature of the real number line, which cannot precisely imitate a phenomenological nature of real matter composed of atoms with their excited electrons permitted to change orbit only by discrete jumps and their energies changing by discontinuous quanta packets. Ask why the Earth orbits the sun according to Kepler's laws, and the answer will be universal gravitation. Hurrah. Universal gravitation tells us that two bodies attract with a force inversely proportional to the square distance between them. But why? The answer is clear. Force is proportional to acceleration. That's the old F equals MA formula relating force to mass and acceleration. But these are merely physics terms, suggesting from experiment, of course, that there is a pull that can be felt and measured and an increase in speed that can also be measured. All we need to know is that the pull increases as the acceleration does. Then F equals ma. But why? Why does the ball continue on its trajectory when it leaves the hand that throws it? The answer invokes a combination of laws. An object will continue to move in uniform motion unless acted upon. And F equals ma. Whenever we ask why, we are cornered into a mathematical formula that was initially constructed by relating physical phenomena. Why do electric and magnetic fields have such strong symmetry in the sense that a changing electric field induces a changing magnetic field and vice versa? Why are they both just two forms of radiation? Why is mass just another form of energy? The list goes on. Ask and you will be given some mathematical model as the answer. Mathematics has done a pretty good job of formulating the laws of physics. Does that give it the right to be at the end of a long line of Y's? Why does Achilles overtake the tortoise? Because a geometric series with ratio less than one converges. Satisfied? No. We are more satisfied after watching Achilles beat the tortoise. Why is mass just another form of energy? Because equals MC squared. Because equals MC squared, says the physicist. Satisfied? No. The horrific bombings of Hiroshima and Nagasaki were more convincing. So I'll stop there because of time and open it up for questions. I think that that last piece probably will generate a bunch of questions. Yes.

I want to make sure I follow correctly. So you're not suggesting that these paradoxes are unanswered. You're saying they are answered satisfactorily. But the answer. But the Mode of the answer isn't satisfying to our intuition.

That's pretty close to what I'm saying. Yes, that's pretty close. Yeah. There's this whole question of how we perceive the world and what we expect as an answer to those long, that long list of whys. I mean, we normally say we answer this by giving some mathematical formula or a model or some mathematical model. Is that really, you know, there are things in the world that we want to understand that are not generally expressed or answered by mathematical models. What is love, for example? How much mathematics can you put into the question of what is love? There are things that we try to answer and there are answers for some of those things, reasonable answers. But when we try to talk about experiences, physical experiences in the world, physical phenomena in the world, we tend to say, well, this is because of the mathematics predicts that this will happen. So there's. Now the question is, what was Zeno getting at when he was formulating these paradoxes? Could it have been that he was getting at precisely this question and he wasn't simply a troublemaker making paradoxes, he was actually perhaps asking the question, what does it mean to answer the question, why that's a possibility. I mean, I don't think anybody knows what was on his mind, but that is a possibility. Does that answer your question?

Well, it feels like you're hinting at something that we're supposed to take away in order to see things differently or do things differently. And I feel like I'm missing it. I can sense it's there, but.

Well, that's the trick. I think that's the trick is that the paradoxes, I feel, are real. By the way, you know that we call them paradoxes. Somebody somewhere along the line called them paradoxes. Way, way back. Are they really paradoxes? Not in the true sense of the word paradox. They're not statements that are self contradictory in any way. They are statements that deal with the question of how mathematics fits the real world or how logic fits the real world. So you have, you're taking the real world and it doesn't seem to fit what we're expecting from the logic. So in that sense we're calling it a paradox. It's not a true paradox, but that's what people mean when they talk about the Zeno paradox.

I mean, it sounds like you're saying what I sometimes feel, which is when somebody gives you an equation in response to a question, it's a way of begging the question. Yeah, it's a way of formulating it in Terms that are different than the way you ask, but it doesn't give you any information.

That's right.

So has anybody in your experience answered it appropriately about what a good answer is?

No, I'm not sure there is a good answer. I mean, that's what's so puzzling about this whole. You know, every once in a while, over quite a few centuries now, the paradoxes seem to surface and die out. People start to bring them up for one good reason or another. Surely when calculus was invented, the xenoparadoxes were alive because of this whole question of. Now we can answer the xenoparadoxes because we know what it means to take a limit down to that photograph of the two arrows, for example. We know that that photograph has some fuzz in it. One of the arrows is going to have fuzz horizontally, the other vertically, so we can tell the difference. That's a calculus kind of argument for telling which arrow is going in which direction. So what happens is that every once in a while the xenoparadoxes surface and people think, well, you know, we have a real paradox here. Let's try to answer it. And they answer it by some mathematical or by models, and then it dies off and then it surfaces again. The last time it surfaced from what I know is, oh, about in the 1930s or so, there was a lot of literature on it in the 30s, for some strange reason, then it dies off. And now I think I'm sort of reviving it a little bit. I'm trying to revive it, because there is something there. I really do think there's something there to be talked about.

What do you think is there?

Well, it's this phenomenological question that we're really not answering the phenomenologically, a phenomenological question of why things happen. And all we are doing is saying, yeah, they happen, because I can prove it mathematically. It may be the best we have. I mean, I'm not saying that that's not good, but it's just maybe the best we have. Yes.

Is this a problem in the sense that, like, for example, we perceive the world as flat, but then we got some really great evidence that it's not flat? And that evidence, you can see it in pictures, and now it makes intuitive sense to us. But the questions you're asking, I think what you're saying is that because the way our brains are wired, we're never going to be satisfied with mathematical models of the way we perceive the. So is the problem the disconnect between Our perception of motion and the way motion actually is.

Well, what Zeno is saying way back then is that it is an illusion. That's a possibility. I mean, you know, when he says. When he says there's one thing in the world being, what does he mean by being? What do you think he might mean by being. One thing in the world? When you break it down, there's one thing in the world. I mean, look, you take the chair and you start chopping it up. You keep chopping it up. Eventually, you know, we know where it. What happens. You know, it's pieces of plastic and the plastic breaks down. And then you keep doing that, and eventually you don't get plastic anymore, you get molecules. And then you break the molecules up and you don't have molecules anymore, you have atoms. Then you break the atoms down if you can, and you keep doing that. And eventually you get some energy or you get something. You get energy out of it. Well, maybe it's all just energy. Maybe it's all illusion. Maybe it's, you know, it's. It's a. You remember that he's talking about. He's Talking in the 5th century BC the concept of monotheistic deity is rare. I mean, there's Judaism, of course, but there's no Christianity yet. And, you know, so there's a hint of theology behind this being that he's talking about, I think.

But the concept of Maya, the concept of everything is actually just an illusion that whenever we actually experience this, just on the surface.

Yes.

It seems like eerily summer.

Yeah, very similar. Very similar. And it could be. What if it were all taking place in your head? Everything, this lecture. It's a bizarre thought, of course, for us, but not for everybody. There are people who might believe that kind of thing. Maybe it's just one thing. It's really there. Yes. Would you say that the paradoxes depend on the assumption that matter is infinitely divisible? Well, that's what they're getting at. Very good question, because that's exactly what they're getting at. Two of the paradoxes really suggest that they are infinitely divisible. The other two suggest that. No, in fact, you get to discrete things. I didn't mention the fourth paradox, which really clinches it, which tells you. Which suggests that, look, the world cannot possibly be discrete. It can't possibly be discrete. It has to be a continuum. Because if it were discrete, nothing could move. There would be no movement. And that's a good. It's a good paradox, but it's a little too Complicated to informally, if anybody wants to talk to me about it, that would be. I could try to explain it. Not always so easy to explain that fourth paradox. I tried several times, and people walk away. I'm very confused by that. Anyone else? Yeah.

Maybe not directly related, but isn't this idea that, you know, either everything's one or everything's infinitely, or they're infinite things?

Yeah.

Isn't that also, you know, I mean, if things are one, if there's only one thing, then this couldn't be happening in my head because there's movement, there's change. So.

And you can.

One thing if you start, because that kind of support.

Right, that's true. If there's only one thing, then there's only one thing. And it can't be any time because there's no change. I mean, you can't have change unless you have time and vice versa. Right. I mean, for example, if nothing moved in the entire world, entire universe, nothing moved for, let's say, what we would call 10 minutes, then what would the 10 minutes mean? There wouldn't be any meaning to that 10 minutes, that gap of 10 minutes, so that the movement and time are so intimately connected that you just can't have one without the other. Can't be gaps in time without movement. But does.

Does that mean time can be quantized or can't it be.

Well, you know, if you go to the Zeno paradoxes, it's what could be said for space can be said for time, too. I mean, is it a string of beads? You know, you go from one bead to the next. And that's part of the problem is because we're thinking of the world, you know, we're thinking of moving, you know, I move my hand from one side, from this position to this position. You assume that I've moved through every point in between, Right. Is everybody with me on that? You agree with that? I've moved it from here to here, so I've moved it through every point in between. But the mathematics of the continuum says that there's an infinite number of points, a large number of points, and if you pick any one of those points, there's no next point. There is no next point. And pick a point.

Why is there no.

Well, let's say here's this wall, this is zero, and this is, let's say, 10ft away. Pick up any point, like one foot away. One foot. Exactly one foot. What's the next point?

Because it's anything else, you can't specify, Right.

You can't specify it. So mathematically, there's no next point. If it's a continuum, so there's no next point. So if there's no next point, how does it get from one point to the other? What other? You have to ask that. It's gone. It's at one foot. Exactly. And it's got to get to the next point before it gets to two feet. But there's no next point. Mathematically, there's no next point. So how does it do it? Well, it can't do it mathematically and physically at the same time. So there is this jump, that leap that we have to make when we're talking about the mathematical model of physical reality. There has to be that leap. Unless you're an engineer. And then you make.

For all practical purposes, you've reached the point.

That's what you do. That's what you do. Right. That's what we do. We say, well, we fudge it. But I'm more interested in the phenomenological nature of how I get from that wall over this podium. Maybe I disappear every. You know, as you walk, you sort of appear and disappear as the Keystone Comedy. Right. As an old movie, flickering movie. Maybe you do. And you wouldn't know it just as you don't know it in a movie. You know this. If you. Are we going over time? We're okay. Two minutes. Okay, great. Let's look at it this way. You know what happens if you look at. Let's say you look at something. You look at that thing, whatever that is. Camera, I assume. It's a camera, right? You look at that. You stare at it. But you look at it at a completely fixed, no movement. It doesn't move. You don't move, Your eyes don't move. What happens? Is it on the camera? No. Whatever it is, this Google message here. Emergency dial. Okay, what happens if you look at it? Anybody know? The light hits your eye. Yeah, yeah, but what. Okay, I'll tell you what happens. Yes. Light bounces if you look at it for a short time, but very short time, but really stare at it without any movement of your eye. You don't see it anymore. It disappears. We only see movement. That's amazing. Now you say, well, wait a second. I can stare at that thing and I'm looking at it. I'm looking at it. I'm looking at it. Well, you can't really, because your eye is moving infinitesimally. It's averaging out. That's what's going on. You're right. It's it's got a slight movement in it and it's averaging all the information that's coming from that Google sign. And if you stop that movement, it doesn't know how to average because it only has one piece of information, not enough information, so it just doesn't know how to do it. Actually, if you could see this room and just keep your eye fixed without any movement and the room. No movement, which is an impossibility, of course, you wouldn't see the room.

Has anyone proposed an experiment to test whether there's.

I think that. I think there is. There are experiments. Yeah. You know, vision. People know this as a property.

Yeah, well, I meant more like, you know, the hand thing.

Experiment to see whether it disappears. But I don't know.

To test the validity of some of the mathematical constructs.

I don't know how you test it because, you know, at faster than a speed of a film, which we see perfectly nicely, we wouldn't know the difference. I mean.

Well, there are implications. If you have a good theory, you can design something where the implications, even though you can't observe the way it gets there.

I assume that somebody could. That's true. Well, that's. Yeah, I assume that that's a possibility. Anyhow, thank you all for coming. That's great, thanks.