Zeno and the Extent of Space and Time

Among the many ancient Greeks noted for enjoying mind games was a certain Zeno of Elea. He liked to conjure up apparently self contradictory ideas called paradoxes, and one of his most famous runs something along the lines of this: Suppose you’d like walk to the door of a room. In your first step you travel half way to the door. In your next step you travel half way again (which is one fourth of the original distance). You keep this up, moving only half way with each step… and so, even if you keep moving forever, eeking your way forward by ever tinier bits, you can never actually make it to the door. Zeno believed this because he believed space was a continuum which could be forever divided. The paradoxes associated with this conundrum has been tackled by great philosophers from Aristotle to Bertrand Russell and by mathematicians using such techniques as calculus and “convergent infinite series.” But don’t worry, I’m not going to invoke mathematics here. What interests me about this paradox is not if I will ever get to the door, but rather – just how small is it possible to go?! At first it is easy to consider one half of the distance. But what about after a trillion trillion years of going half way every single second? Just how far would such an infinitesimal distance be? Can you even imagine such a thing going on forever? It would be a voyage to inner space.

But now try going in the opposite direction, “out” this time, instead of “in”, first to the door, then twice as far as the door, and twice as far again with each step. I’m guessing you may have considered this before, maybe when you were a kid and didn’t have a mortgage or rent to think about. In no time you’d be as far as the moon, then across the solar system, leaping out of our Milky Way galaxy on your way to the edge of the known Universe…until…? What happens when you get to the edge of the Universe? Can the Universe even have an edge? What would be on the other side? There must always be farther to go! Right?! Space must just go on forever and ever and ever.

And for that matter, the same goes for time too.  When was ten minutes before the creation of the Universe in the Big Bang 14.5 billion years ago? And then just keep going back infinitely. How about longer than a trillion trillion years after tomorrow? How can time possibly be bounded? It’s unfathomable. And somewhat terrifying. Honestly, the only thing that seems real to me is right now.

Magnitudes and the Scale of the Universe

As we try to fathom the scale of the Universe from the depths of inner space to the mind boggling extent of outer space, it’s helpful if we take bigger leaps than Zeno’s one half step. A common device used by mathematicians and scientists in cases where they have to conceptualize, and measure, some vast quantity is to divide and multiply by tens rather than two. This gets us moving more quickly and it is also intelligible due to our familiarity with the base ten numeral system that we use. So, modifying Zeno’s instructions, our first step will bring us 1/10 the original distance from the door, the next will take us 1/100 the distance, followed by 1/1000, etc… Proceeding outward, we would first be 10 times the distance, followed by 100 times and so on. Mathmematically this would look something like this: […10^-3, 10^-2, 10^1, 1, 10^1, 10^2, 10^3…]. To the scientist, steps that move by tens are called “magnitudes” and measuring this way helps in more situations than distance. When measuring the strength of an earthquake, for example, geophysicists use the Richter Scale which measures the quake’s force in magnitudes, so that a six on the Richter Scale has actually ten times the force released in a quake that measures five. Astronomers use a similar scale in measure the brightness of stars so that a star of magnitude two is ten times dimmer thant a star of magnitude one (the direction may be opposite from what you’d expect).

Since we determined above, using Zeno’s example, that the smallest and the largest scales of space are unfathomable, we shouldn’t start measuring at the extremes, but rather on the scale of, say, our own bodies. The meter is the most common measure of space in the world today, a distance most people have a grasp of, and so we can start there. Two magnitudes below the meter is the centimeter, 1/100 of a meter, or about the width of your fingernail. Three magnitudes higher than the meter is the kilometer, about how far a fit person can walk in ten mintues. Luckily for us, some enterprising and imaginative people at IBM have created a video to help us envision what it like to travel by leaps of ten from the width fo a proton (1/10^-16 meters) atom to about “the limit of our vision” (10^24 meters), a distance of forty magnitudes. This worthwhile nine minute video can be accessed on vimeo by pressing here. And if you’d like a bit more control, you can play with this brilliant widget which takes you all the way down to the Planck length (see below) at negative forty magnitudes out to the edge of the know universe at positive 27 magnitudes; if you do this fast you really get the sense of what it means to travel to inner and outer space.

The Digital versus Analog Universe

By the end of the nineteenth century most phenomenon which could be observed in the world around us appeared to be explained by what we now call classical physics. This included Newton’s laws of motion, Maxwell’s laws of electricity and magnitism and Kelvin’s laws of thermodyamics. It was common at the time to believe that we had in fact reached the end of science. There were a few “minor” questions which still needed to be answered if one thought very big, like at the speed of light, or very small like at the cause of radiation or the structure of the atom, but surely, the extremes would just be an extention of the natural laws we already knew.

Then, in the twentieth century, came the scientific revolutions of quantum mechanics, the physics of the most wee, and relativity, the physics of the most immense, which revolutionized our understanding of space and time. Thank you Mr. Einstein.

Let’s examine just how our understanding of space and time have changed. The reason that Zeno’s Paradox is appealing is because it is natural to assume his premise that space is a continuum and not constructed from discrete bits at any level. In modern lingo we call this the difference between an analog (continuum) world and a digital (discrete) world. In case you aren’t familiar with this difference, it’s germane to our modern world of gadgetry. Let’s, consider the clock. The only clocks our great grandmothers knew where the kind where the hands moved gradually across the face. This was an analog device that required a small bit of judgement as to when the time changed from three minutes after twelve to four minutes after twelve.  This is how the world appears to us and how it appeared to Zeno; between any point from here to there was a point in between. We gracefully move across the stage of life. Today, however, we have created digital devices (often with the aid of quantum mechanics) that deal with the world in discrete lumps. (Scientists call these lumps “quanta.”) When you go to bed at night your modern digital clock may say that it is either 12:03am or 12:04am but not something in between. And well, it turns out that much to our surprise, the entire universe, is digital. Everything comes in lumps. Time, space, energy, mass, everything.

With this in mind, let’s turn our attention back toward Zeno and his paradox concerning space. It was Max Planck, one of the

Max Planck, discoverer of the smallest possible length.

Max Planck, discoverer of the smallest possible length.

founders of quantum mechanics, who discovered, using tricky mathematics and such things as the speed of light and the strength of gravity, that there actually is a kind of “smallest amount of space”. Today this is called the “Planck length” in his honor and it is indeed unimaginably small:

.0000000000000000000000000000000000016 meters   or  1.6 x 10^(-35) meters

If it means anything more to you, perhaps after clicking on the widget link above, that would be about one ten millionth of a trilltionth of a trillionth of the width of a hydrogen atom. It’s WAY down there! But I said “kind of the smallest amount of space” above because it isn’t even very clear what space means at that point. We certainly can’t do any experiments at that scale. What we can say is something like this. Imagine a ruler the size of the Planck length. Call one end A and the other end B. It is possible to be at A or B at any one time but not somewhere in between. And if we were able to look at something “being” at A or B it is just as likely to be at one as the other. To answer Zeno, once you have finally arrived at two Planck lengths from the door to the room, you can go one more Planck length…but no more! You have arrived. Zeno has been answered.

The Scale our Brains Live On; The Human Sense of Reality

The smooth analog way that the world looks to us and the classical physics we use to explain this world is something that is comfortable to us. Our brains evolved to understand the world at this scale we live on, and no other, for this is the only kind of understanding that would be useful to our ancient troglodyte ancestors. Allow me to expand on this idea. At the shortest end of our experience of time is something that we might consider to be “now.” It might include something like how long it takes the neurons in our brain to assimilate the image of a tiger on our retina, form the concept of “tiger,” recognize the danger, and then send signals down to our muscles to “run”! At the most optimistic, let’s say this takes 1/100 of a second. And while that seemingly instantaneous moment is long enough for light to travel almost 1,863 miles, it is the shortest span of time that we can sense. Shorter than “now” is indistinct. Our sense of time also begins to blur on the long end of the time scale. We might comfortably think we can experience the time between the planting of a tree and its harvest years later or how our pollution might effect the lives of our grandchildren, but millions of years of time are really only understood academically, not experiencially.

So now this is the really interesting thing. It turns out that the way the world works on really small scales and really large scales don’t just look foreign to our human minds, but impossible. Things happen on these scales that we cannot begin to wrap our minds around.

On the scale that we live on, concepts such as “cause and effect” and the three dimensional extent of space are easily grasped by our minds. If I want to shoot an arrow at some target I sense how I need to shoot it with this specific force and in this specific direction. Maybe I’ll catch some tasty deer for dinner. Sending a man to the moon is complicated, but it is also not unintelligible, because it required mostly classical physics. On very small scales, however, say the size of an atom and smaller, classical physics is of no use at all. In the model of the atom that most of us may have learned in school, electrons “orbit” about the nucleus. This model and vocabulary conjures up notions that the atom is something like the earth and the moon or any solar system with orbiting planets. But the electrons and nucleus of an atom are nothing like that. The position of the moon around the earth can be predicted thousands of years in advance by just knowing the postion and speed of the moon around the earth right now. But if you want to find the position of an electron around an atomic nucleus you can either determine its position or its speed, but not both. This makes it impossible to determine where an electron will be for certain in the future. No, it’s more than that. An electron has no deterministic position in the future. The best it has are odds that it could be found here or there and odds that it is moving at such and such a speed. They exist in a zone of probability. This is pretty disturbing to our concept of cause and effect! We rely on cause and effect to escape the jaws of a tiger. Only a world of cause and effect makes sense to our human minds.

Large scales also mess with our sense of time and space. I have a sense that time marches on like a metronome, tick-tock, passing along out there in the world, outside my mind. Distance too seems immutable. Speed, however, seems like it is not a constant thing and that nothing could stop you from going faster and faster. But it’s just the opposite! It turns out that time and space can be stretched or compressed depending on relative speed or the presence of strong gravitational field, while the Universe has a speed limit which is precisely the speed of light. If you were to take off on a space ship at close to the speed of light and then return, you will have aged less than me; perhaps significantly depending on how far you went and how close to the speed of light you approached. This is readily calculated using a mathematical formula called the Lorentz transform and has been verified perfectly using atomic clocks aboard the space station (not even remotely approaching the speed of light of course). The same is also true in the presence of a strong gravitational field; time literally slows and space contracts. Furthermore, our sense of simultaneous events is also affected by speed and distance. Can you imagine what a loved one is doing right now on the other side of planet or in their home across the city? (both are nearby) How about on a planet a billion light years away and receding from us rapidly? To someone on that planet, “now” would be sometime in our past and we are living in their future. On large scales our concept of “simultaneous” fails as well.

So, which is it? Does quantum mechanics and relativity describe the way our Universe works or does the classical physics of Newton? The answer is that neither of them actually contradict classical physics, they simply add to the spectrum of things we can now measure. What happens is that the probability of events on the quantum scale collapses to deterministic events according the law of large numbers. If you flip a coin just once, the odds that you’ll get one heads is exactly one half. But it you flip it one billion times the odds of getting just head heads are exceedingly rare. It could happen, theoretically, but then it could also happen that you walk right through the wall if given enough time, like trillions of years. Barring these exceedingly rare events, for us up here on our relatively large world, Newton works just fine.

What truly excites “Craig’s sense of wonder” is the discovery that my mind is limited. imagesI simply don’t have the mental hardware to understand the world of the tiny or large. Stuff that looks miraculous to me is really happening and I don’t get it. I can’t fathom it, can’t really picture it, and no human can, not even Einstein could. It’s not possible to grasp something that we have no mental apparatus for, just like your dog can’t tell you if he’d like to be buried or cremated and you can’t smell you lover’s lover’s apartment on their coat like a dog. Imagine a being that evolved on the quantum scale that found it perfect normal that it was never quite certain what would happen, or a vastly huge being across a galaxy that could feel how much slower time passes near the black hole in the center of its body. Alright, alright! Perhaps it’s time to settle back to my reality before I venture too far into science fiction. How about ending with a visit to the writing of that master observer, Henry David Thoreau? From Zeno to Thoreau. That I can do.

Henry David Thoreau and the Battle of the Ants

In 1845, Henry David Thoreau went to live in the woods near Walden pond in Massachusetts. One of his aims was to expand his consciousness and become a better observer of nature. One fall day he happened to look at what was happening about two magnitudes below his usual world, which you now know is about the size of your fingernail. What he found was remarkable, for just below his peaceable world he witnessed a very vicious war.


The Battle of the Ants

Henry David Thoreau

One day when I went out to my wood-pile, or rather my pile of stumps, I observed two large ants, the one red, the other much larger, nearly half an inch long, and black, fiercely contending with one another. Having once got hold they never let go, but struggled and wrestled and rolled on the chips incessantly. Looking farther, I was surprised to find that the chips were covered with such combatants, that it was not a duellum, but a vellum, a war between two races of ants, the red always pitted

Henry David Thoreau's shack on Walden Pond. Two magnitudes beneath this idyllic scene a fierce battle raged.

Henry David Thoreau’s shack on Walden Pond. Two magnitudes beneath this idyllic scene a fierce battle raged.

against the black, and frequently two red ones to one black. The legions of these Myrmidons covered all the hills and vales in my woodyard, and the ground was already strewn with the dead and dying, both red and black. It was the only battle which I have ever witnessed, the only battle-field I ever trod while the battle was raging; internecine war; the red republicans on the one hand, the black imperialists on the other. On every side they were engaged in deadly combat, yet without any noise that I could hear, and human soldiers never fought so resolutely. I watched a couple that were fast locked in each other’s embraces, in a little sunny valley amid the chips, now at noonday prepared th fight till the sun went down, or life went out. The smaller red champion had fastened himself like a vise to his adversary’s front, and through all the tumblings on that field never for an instant ceased to gnaw at one of his feelers near the root, having already caused the other to go by the board; while the stronger black one dashed him from side to side, and, as I saw on looking nearer, had already divested him of several of his members. They fought with more pertinacity than bulldogs. Neither manifested the least disposition to retreat. It was evident that their battle cry was “Conquer or die.” In the meanwhile there came along a single red ant on the hillside of this valley, evidently full of excitement, who either had dispatched his foe, or had not yet taken part in the battle; probably the latter, for he had lost none of his limbs; whose mother had charged him to return with his shield or upon it. Or perchance he was some Achilles, who had nourished his wrath apart, and had now come to avenge or rescue his Patroclus. He saw this unequal combat from afar,–for the blacks were nearly twice the size of the red,–he drew near with rapid pace till he stood on his guard within half an inch of the combatants; when, watching his opportunity, he sprang upon the black warrior, and commenced his operations near the root of his right fore leg, leaving the foe to select among his own members; and so there were three united for life, as if a new kind of attraction had been invented which put all other locks and cements to shame. I should not have wondered by this time to find that they had their respective musical bands stationed on some eminent chip, and playing their national airs the while, to excite the slow and cheer the dying combatant. I was myself excited somewhat even as if they had been men. The more you think of it, the less the difference. And certainly there is not the fight recorded in Concord history, at least, if in the history of America, that will bear a moment’s comparison with this, whether for the numbers engaged in it, or for the patriotism and heroism displayed. For numbers and for carnage it was an Austerlitz or Dresden. Concord fight! Two killed on the patriot’s side, and Luther Blanchard wounded! Why here every ant was a Buttrick,–“Fire, for God’s sake fire!”–and thousands shared the fate of Davis and Hosmer. There was not one hireling there. I have no doubt that it was a principle they fought for, as much as our ancestors, and not to avoid a three-penny tax on their tea; and the results of this battle will be as important and memorable to those whom it concerns as those of the battle of Bunker Hill, at least.

I took up the chip on which the three I have particularly described were struggliing, carried it into my house, and placed it under a tumbler on my window-sill, in order to see the issue. holding a microscope to the first-mentioned red ant, I saw that, though he was assicuously gnawing at the near fore leg of his enemy, havng severed his remaining feeler, his own breast was all torn away, exposing what vitals he had there to the jaws of the black warrior, whose breastplate was apparently too thick for him to pierce; and the dark carbuncles of the sufferer’s eyes shone with ferocity such as war only could excite. They struggled half an hour longer under the tumbler, and when I looked again the black soldier had severed the heads of his foes from their bodies, and the still living heads were hanging on either side of him like ghastly trophies at his saddle-bow, still apparently as firmly fastened as ever, and he was endeavoring with feeble struggles, being without feelers and with only the remnant of a leg, and I know not how many other wounds, to divest himself of them; which at length, after half an hour more, he accomplished. I raised the glass, and he went off over the window-sill in that crippled state. Whether he finally survived that combat, and spent the remainder of his days in some Hotel des Invalides, I do now know; but I thought that his industry would not be worth much thereafter. I never learned which party was victorious, nor the cause of the war; but I felt for the rest of that day as if I had had my feelings excited and harrowed by witnessing the struggle, the ferocity and carnage of a human battle before my door.


What do you suppose aliens would think if they were to travel across vast distances of space and time only to observe us humans at war?



Posted on November 25, 2014, in Uncategorized and tagged , , , , , , , . Bookmark the permalink. 9 Comments.

  1. Holy. I didn’t expect a post on scale to cause me to cry. Well done. More in a private message.

  2. Hey 🙂
    Very interesting topic indeed! And you leave hooks to develop more thoughts later, I expect you’ll write more on related topics at some point! Great video illustration of what happens (or our humble understanding of…) when zooming in and out to unintuitive magnitudes.

    And this makes me wonder: have you seen “Interstellar”? It brings up a lot of theory that we sometimes have heard of but never really fathomed, in a scenario that to some extent is conceivable. In any case, the movie and your post are both good food for thought and conversation 😉

  3. Lovely work Craig. Zach was excited to see a math related post 😀 Elegantly written. We likey!

  4. Hi Craig…I’m wondering how you can write with such depth on this subject. When my mind has wandered in wonder regarding space it always comforted me to think the end of space was God…or my image (unknown) of God being their protecting us. Also…I’ve always felt that our thoughts could travel faster than anything…so they could be in Milky Way at the very moment they were experienced. Well, guess it doesn’t matter if there isn’t anyone there to receive them? Now I have to go and find out what Thoreau’s “internecine war” means. You never fail to educate and joggle the mind. Thanks for sharing your supreme being!

    Love, Beth

    • Hi Beth, It’s interesting that in your mind you think of going “out” to find God. For me, I would imagine finding God by going “in”. Going “in” is where we are likely to find other dimensions which seems like a good place for a God to inhabit! 😉 In the big bang it does appear that the Universe came from some place (a singularity) “in” and has been expanding out ever since.
      “Internecine wars” is strife between members of a group; family, nation, religion. E.g. red ants vs. the black ants or Sunnis vs. Shiite.

  5. Love the post! It is a good reminder to recognize that we are hard-wired with certain ‘scale-biases’. Imagine the time that the telescope was invented, or the microscope! Suddenly, whole new worlds open up that before was just not part of our understanding. (what? there are tiny things living all over me?!)

    Then there is the strange scale of humanity and society. Try to figure out how something like macro-economics work for example. It is hugely complex, and we obviously don’t have a good grasp of that one yet, but if we are ever to ‘solve’ it, then you have to start with recognizing that things work very differently on different scales.

    • I would love to have been in the room when Leeuwenhoek discovered microorganisms (he called them animalcules). Or when he first saw his spermatozoa…that would have been exciting! Truly a paradigm shifting mind expanding moment. As to your second point, solving macroeconomics is an exercise in conquering chaos. Chaos is an excellent topic for a future post too since I’m such a weather geek. That’s a good idea! Thanks for your thoughts!

  6. CRaig, I am saving this I haven’t had a chance to read it yet…..

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