GMTA, Philosophy Edition: Patterico Vindicated on Zeno’s Paradox?
On July 3, 2016, I proposed what I thought was a completely original solution for Zeno’s paradox: a “minimum distance” in the real universe that cannot be subdivided:
You have probably heard of Zeno’s paradoxes, one of which is that to get anywhere, you first must travel half the distance, and then half the remaining distance, and then half the remaining distance, and so on. According to this theory, you never get there.
I have seen various solutions to this, but I am not sure anyone has proposed mine. Perhaps they have, but here it is anyway:
My proposed solution is that there is a minimum distance in the universe, which cannot be subdivided into smaller distances.
Imagine my amazement to learn that, at about the same time, a professional philosopher was bouncing the same theory off of a mathematics professor, who agreed with him!
Embedded for your listening pleasure is a discussion between philosopher Steve Patterson and Dr. Gary McGuire, the head of Mathematics and Statistics at the University College Dublin.
A little background: Patterson proposes a different description of Zeno’s paradox, in which you put together a pie by first creating half, then adding a fourth, then an eighth, and so on. Does the pie ever get completed? This is what they are referring to when they talk about the pie.
The most relevant part is at 40:27:
PATTERSON: If that’s true, then does that not mean that Zeno’s paradoxes are not solved by calculus? Because the claim is not that the runner will get ever so close to the final point, but that the runner will actually complete the race. That ultimately, the pie will ultimately be completed. Doesn’t that mean that Zeno, Zeno had a — was making a good point there?
DR. McGUIRE: Uh, yes. Yeah. Yeah. No, a really good point, I mean, I’m agreeing with you. I’m not disagreeing with you. So.
PATTERSON: So what do you think of this, this potential resolution: that the reason calculus does work in the real world is because reality is finite. It’s not infinitely divisible and therefore at some point the calculations terminate, and then, you know, you can complete the whole pie, and you would complete the race.
DR. McGUIRE: Well in the real world, as we were saying earlier, we don’t get into the infinite. So we have to approximate everything by a finite number. And so, in the real world, we would get, we wouldn’t be able to, if we were adding smaller and smaller and smaller pieces of the pie, we’d eventually have to stop somewhere. We can’t, we can’t get ever smaller and smaller and smaller pieces; we just can’t do that. So we have to stop at some smallest possible piece and then we add that in and then we finish the pie.
PATTERSON: But what about with something like distance? So could we say the same thing — that ultimately — this is what I think the resolution is to Zeno’s paradox, is that there is like a base, a base distance unit in the universe that you can’t actually divide in half. Because otherwise it seems like motion would be impossible. But if we’re — if there’s like a base unit of distance, then everything seems to resolve itself. Just like a base unit of pi.
DR. McGUIRE: I, yeah, I kind of agree — I agree with you. I think, in the real world, in practice, there is a base unit of distance, yeah.
I heard this in my car and almost involuntarily slammed on the brakes in surprise when I heard the bolded language — which, you’ll note, is the same as my own solution to Zeno’s paradox above: that “there is a minimum distance in the universe, which cannot be subdivided into smaller distances.”
Note that many commenters laughed at me in comments to the previous post — and yet this mathematics professor agrees with Patterson!
I am going to write Patterson about this rather bizarre confluence of thinking. I may have a hard time convincing him that I didn’t take my cue from him, since his podcast preceded my post by a little more than a month. But I never even heard of the guy before a couple of weeks ago.
Great minds think alike — and sometimes, so does mine!
P.S. If you listen to the whole interview, you’ll notice Patterson’s notion that numbers do not exist outside our conception, which I see as a kind of corollary to my hypothesis that — even abstractly and not purely as a “real-world” phenomenon — there is a smallest number (and a largest!) . . . but it’s beyond the limitations of humans to conceive of, or express. If that notion is right, it has real implications for the very concept of infinity.
everything counts in large amounts
it’s a competitive whirl
happyfeet (a037ad) — 9/23/2016 @ 7:27 amBy the way, I really enjoy Patterson’s podcast. If you like this sort of topic, you’ll enjoy him. He approaches life from a rationalist perspective that rejects academia and fufu nonsense, and marries that to a radically free-market economic outlook that embraces anarchocapitalism. Challenging and interesting ideas.
Patterico (bcf524) — 9/23/2016 @ 7:31 amMy knee jerk reaction is people, as is the philosopher’s wont, overthink these problems. Can the physical, dimensioned world represent noetic concepts, like transcendental numbers?
Hardly, mathematics is applied to the physical world, but much of it exists independently, whether we think it discovered or invented.
DNF (ffe548) — 9/23/2016 @ 7:43 amI think, in the real world, in practice, there is a base unit of distance, yeah.
This would not “solve” Zeno’s paradox. The base unit of distance may be the Planck length, or some other length, or maybe there isn’t one.
Zeno’s paradox was a challenge to the understand of mathematics that existed at the time, and it has already been “solved” to the extent it needed to be.
Zeno’s paradox does not require a smallest unit of length or time for its resolution.
Patterico, I love you for trying, but you’re 300 years out of date on this and so you’re not going to add much to the discussion. It’s like if I walked into court tomorrow having just read Blackstone.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 7:45 amLook, math is best thought of as language with extremely strict rules. Zeno’s paradox is not really a statement about the physical world. It’s a statement about how we think about math, using the physical world to illustrate the problem.
His point was that there was some defect in our understanding that led to the paradoxical result, and that defect has been resolved.
It doesn’t matter if distances can “really” be subdivided indefinitely or not. That was never the issue.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 7:50 amIt worked for Lincoln.
Zeno’s paradox falls in the category of a praxeology wouldn’t you agree? The trick is to deal with that in the terms consistent with the original formulation. We don’t measure the circumference of a circle to determine pi, for example.
BobStewartatHome (b41b89) — 9/23/2016 @ 7:53 amAnother way to express the paradox is, “how can I add up an infinite number of something to get a finite result?”
You could restate it with money. I can transfer one dollar to you, in bill form. But the value of the dollar can be divided in half, or quarters, etc. So if I give you a dollar, I am asserting that the sum of that infinite number of divisions is a finite number.
It doesn’t matter that there is a such a thing as a penny. You can have things worth much less than a penny, for example one grain of rice. Monetary value is infinitely divisible and thus just as subject to Zeno’s paradox as distance is.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 7:57 amFurthermore, if there is a smallest meaningful physical difference, that’s a function of physical laws. A different physics gives you different laws, some physics would have infinitely divisible distances. Zeno’s paradox is unchanged by that, because it’s a statement about math.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 7:59 am@BobStewart@home:It worked for Lincoln.
Yes, 150 years ago it did. And in Blackstone’s day it did too. And it would be sorely inadequate preparation now.
Though perhaps all of today’s lawyers might benefit from studying Blackstone, it would hardly be sufficient preparation for a legal career today.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 8:00 amZeno’s paradox falls in the category of a praxeology wouldn’t you agree?
No. It’s not about procedures. It’s about pure math.
We don’t measure the circumference of a circle to determine pi, for example.
You don’t need to determine pi. It is. If you want to know a decimal approximation to it there are ways to do it. You can approximate it with fractions if you like. But pi itself simply is pi. It has the properties it has.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 8:04 amYou are speaking of penumbric emanations, I presume? I think we’d much better off if we just wrote off the last century of legal progress as a lost cause. After all is said and done, the key test according to one prominent jurist is whether she thinks a law is in agreement with “good social policy”. In fact, we should probably throw out everything since the 1880s.
BobStewartatHome (b41b89) — 9/23/2016 @ 8:12 amBut using the praxeology of geometry and trigonometry, one can determine formulas for pi that can determine the value of pi to any desired number of digits. We don’t need to measure it.
BobStewartatHome (b41b89) — 9/23/2016 @ 8:15 amAnd the Waals keep coming down, the Waals. Yes, there is a minimum distance between matter in the real world (not the imaginary one of mathematics) and it varies with temperature and pressure. It approaches zero but does not reach it.
(It also helps you understand why super glue works on some plastics and not on others.)
nk (dbc370) — 9/23/2016 @ 8:20 am@BobStweart@Home: pi is a number with properties. Some of these properties are seen in trig, and some not. Numbers are defined by their behavior. It doesn’t matter if you ever express pi digits or not, in fact mathematicians and physicists generally don’t, they just write it with a Greek letter.
@nk:Yes, there is a minimum distance between matter in the real world
Really? How close can two electrons get? And what about the fields around them, do those fields have undefined values if you are below the “minimum” distance?
I can tell you that the equations used to understand the interactions of matter use infinitely divisible distances.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 8:29 amI can tell you that the equations used to understand the interactions of matter use infinitely divisible distances.
And I can tell you a story that starts with “Once upon a time there was a beautiful Princess named Snow White”. Just because you can write fiction with numbers does not make it any less fiction.
As for how close electrons can get, I gave you a hint with the super glue. van der Waals also explains electrostatic forces.
nk (dbc370) — 9/23/2016 @ 8:49 amThe repulsive part of the van der Waals effect is interesting but really tiresome to calculate. But these are the steps:
You write the potential energy function for your two electrons. Usually you consider them each part of a hydrogen atom just so have something to write down, but this isn’t strictly necessary. There will be five terms in it: electron 1 wrt nucleus 1, electron 1 wrt nucleus 2, electron 2 wrt nucleus 1, electron 2 wrt nucleus 2, electron 1 wrt electron 2. (The nucleus-nucleus interaction is treated as a parameter, we assume the nuclei are momentarily “nailed down”, this is because nuclei moves thousands of times more slowly than electrons.)
Then you set up wave functions for your two electrons, but with a twist: if you switch the coordinates of electron 1 with electron 2, your wave function has to be -1 times your original. (This is for electrons with the same spin, if you use ones with opposite spin you won’t do this and you won’t get repulsion when you solve the equations.)
Then you solve it, and you find that the electrons do not allow the nuclei to be pushed very closely together. This is over and above the repulsion of the two positively charged nuclei.
At every step of this calculation, you assumed distance was infinitely divisible. And your nuclei can be pushed as closely together as you like, provided you have enough energy available.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 8:49 amJust because you can write fiction with numbers does not make it any less fiction.
The equations allow predictions that very closely model the real world, and have allowed the development of the machine you are doing your hand-waving on.
Fundamentally the reason neutral atoms repel each other is because electrons can’t tell themselves apart, and won’t do the same thing if their places are switched.
And you can dismiss it as Snow White fairy tales if you like, but you don’t know anything whatever about it, and enormous sectors of our economy dependent on the accuracy of these equations, and Snow White fairy tales don’t make transistors work while quantum mechanics does make transistors work.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 8:54 amI don’t speak Chinese, either. I guess a 100 million Chinese toddlers know more than I do. But how many of them speak Greek? Eh?
BTW, the machine I am doing my handwaving on was not formed in a perfect drop of sweat on the brow of the Boddhisatva Of Mathematics, He Of The Circle Without End. It was built on thousands of empirically proven physical experiments.
nk (dbc370) — 9/23/2016 @ 9:01 am@nk:It was built on thousands of empirically proven physical experiments.
Yes, I know. I performed some of those experiments. Using the math I described to determine what would happen.
Transistors and integrated circuits were improved and developed using that same math. You can’t just wave it away as a story, you have to explain why the alternative fairy-tales didn’t produce those results.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 9:05 amAnd quantum mechanics does not make transistors work. It explains, in its own language, why transistors work, and helps others who understand that language to make transistors that work.
nk (dbc370) — 9/23/2016 @ 9:06 amAs an engineer, this strikes me as a simple problem of how many decimals can you carry before you run out of the ability to divide one more time? Of course, one runs into an issue of significant figures long before then.
Advo (c20fd3) — 9/23/2016 @ 9:11 am@nk: ok, then why does it do that if it’s a fairy tale?
Gabriel Hanna (64d4e1) — 9/23/2016 @ 9:12 am@nk: ok, then why does it do that if it’s a fairy tale?
Sigh. I did not say that mathematics is only the language of fairy tales. I confess, freely and of my own will having been duly advised of my rights, that it is a language which can *also* describe observable physical reality, adequately if not perfectly, not to mention help tune a guitar.
I take exception to the conceit that it is a science in and of itself and its equations, whatever they may be, always represent physical reality; or, and let’s get to the the heart of matter because that was your dig at Patterico, that mathematics is the *only way* to describe physical reality.
nk (dbc370) — 9/23/2016 @ 9:38 amI don’t think he said what you thought he said. I took it to mean that in our world, there is a minimum distance perceivable…we may never be able to even contemplate true infinity… But the world our perceptions exist in is more limited than infinity. I suppose eventually you get to quantum distances that may extend into another universe, but we can’t get to it. Does that count?
SarahW (3164f0) — 9/23/2016 @ 10:04 amI mean, in mathematics, a point in space has only location, not size. If you could “arrive” at point perfectly, it doesn’t exist anymore, there is no end to subdivision mathematically except on a scale you can perceive. (Or conceive)
SarahW (3164f0) — 9/23/2016 @ 10:14 amIt’s a convenient model to estimate reality.
n.n (137e6e) — 9/23/2016 @ 11:01 amYou can’t get there from here.
mg (31009b) — 9/23/2016 @ 11:01 am. . . mathematics is applied to the physical world, but much of it exists independently, whether we think it discovered or invented.
It’s the old story of the physicist dismissively telling the mathematician, “Our discipline invented your discipline so that we could explain what we were observing.”
JVW (cf259a) — 9/23/2016 @ 11:43 amIt doesn’t add up.
‘”Divide and Conquer” – Frank Capra, 1943.’
DCSCA (797bc0) — 9/23/2016 @ 11:54 ammy own solution to Zeno’s paradox above: that “there is a minimum distance in the universe, which cannot be subdivided into smaller distances.”
It’s an obvious idea – which you don;t hear thiugh, becasue it contradicts general relativity.
What I said at the tiemn was the Zeno in fact was right. It’s not that there is not a minimum distance or aminimumtime- there may be – but before you get to taht you run itno the uncertainty orinciple, and if you try to divide motion into smaller and smaller increments, you reach a point where Zeno’sparadox – or Xeno;as law – kicks in and
http://www.news.cornell.edu/stories/2015/10/zeno-effect-verified-atoms-wont-move-while-you-watch
You can never test Xeno’s paradox.
Sammy Finkelman (643dcd) — 9/23/2016 @ 12:19 pmSarahW is almost there.
I suppose, to use a law analogy, it’s like saying that you found some evidence that proves Miranda should have been found guilty that was overlooked at the time. Ok, yeah, but that is missing the point of Miranda v Arizona. And in fact Miranda WAS retried and WAS found guilty. But that’s not why we talk about that case and not why it’s important. It could have been some other guy and some other crime and the same issue would have been there.
Zeno was saying something about numbers using a real-world analogy. His paradox applies to money equally well.
So yes, you could count all the countable things in the universe and find that there is no number REPRESENTED BY OBJECTS which is larger than that one. But you haven’t legislated away infinity from mathematics, or solved any of the problems that involve infinity. You can determine the Planck length and say it’s the minimum physically meaningful distance in our universe, but you have not legislated infinitesimals from mathematics or overturned calculus.
Because mathematics deals in ideas and relationships, not in facts.
A simple example. Cop pulls you over, says you were traveling 72 miles per hour in a 60 mile per hour zone. And you say, that can’t be right. I’ve only been on the road 5 minutes, and I only traveled 6 miles. So there is no way I was going 70 miles per hour. The motorist is denying infinitesimals, and will always weasel out no matter what units the cop uses for the numerator and denominator.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 12:29 pm@Sammy Finkelman:You can never test Xeno’s paradox.
I did it right now. I walked across a room 30 feet across at 3 feet per second. I covered the first 15 feet in 5 seconds, the next 7.5 feet in 2.5 seconds, and so on.
And when I got to the end I had covered all those infinite segments in 10 seconds.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 12:31 pmWe could have had a discussion. Instead you chose the haughty and arrogant route. Not that I am surprised. So you want to play a stupid game of breast-beating and appeals to authority instead of discussion? Fine. Take it up with Prof. McGuire. I wager he knows more math than you, Gabriel. Why does he say my solution is right?
Patterico (fcc13e) — 9/23/2016 @ 12:43 pmPATTERSON: Because otherwise it seems like motion would be impossible.
It is impossible, if you really get down to the micro scale.
http://www.askamathematician.com/2012/03/q-is-the-quantum-zeno-effect-a-real-thing/
http://scitation.aip.org/content/aip/journal/jmp/18/4/10.1063/1.523304
This was done with atomic decay, but applies to motion, too.
https://www.inverse.com/article/11874-the-quantum-zeno-effect-explains-how-you-can-stop-time-using-physics
Sammy Finkelman (3915d0) — 9/23/2016 @ 12:52 pmGabriel Hanna (64d4e1) — 9/23/2016 @ 12:29 pm
And why not?
Because mathematics has nothing to do with reality? In that sense, maybe.
Not only is there no such thing as an irrational number, there is no such thing as a circle, or even a straight line. If you try to cvlasim that there is, I think quantum mechanics wll show you theer is not.
Sammy Finkelman (3915d0) — 9/23/2016 @ 12:56 pmthis explains this odd world we live in,
http://www.dailymail.co.uk/sciencetech/article-3800393/Are-black-holes-gateways-worlds-Bizarre-theory-says-NINE-dimensions-lead-alternate-universes.html
narciso (d1f714) — 9/23/2016 @ 1:05 pmIt truly pains me to say this, but Hanna is right. Xeno’s Paradox befuddled the Ancients, but it has not BEEN a paradox for quite some time now. Math has moved on from arithmetic.
Kevin M (25bbee) — 9/23/2016 @ 1:15 pmI always get my science from Daily Mail articles. Just like I get my government theory from The Wizard of Id.
Kevin M (25bbee) — 9/23/2016 @ 1:18 pmit comes from cern, last seen enabling the antimatter particles in angels and demons, so something 1/10 -35 power should be the smallest distance,
narciso (d1f714) — 9/23/2016 @ 1:21 pmXeno’s Paradox befuddled the Ancients,
It may have befuddled Sophists who accepted Zeno’s red herring that the way to calculate the distance in question was by division (instead of subtraction) and to whom the Arabs had not yet brought the concept of zero. It would not have befuddled a plowman who, having plowed two hundred furrows back and forth in a field, stopped for lunch and started plowing again when he reached into his bag for one more fig and found it empty.
nk (dbc370) — 9/23/2016 @ 2:03 pmJust as space may be a particle, so to might time. In physics, something peculiar is theorized to occur at: a Planck Lenth and a Planck Second.
The reason is that our current understanding of the rules of relativity force the fabric of space time to become a singlarity, at this size.
Distance (and time) cease to exist as we know it, in the ordinary sense.
The author makes the analogy that a mathematician can simply double the speed of light, but a physicist cannot, as such a construct would have zero physical meaning.
Pon Ansinorum (e44a65) — 9/23/2016 @ 4:21 pm…
Similarly, a mathematician can simply divide a Planck Length, but a physicist cannot, as such a construct would have zero physical meaning.
Pon Ansinorum (e44a65) — 9/23/2016 @ 4:27 pmyou could double c, but only tachyons could travel at that rate of speed, excepting alcubierre drive of course,
narciso (d1f714) — 9/23/2016 @ 4:51 pmwhat I was referring to:
http://www.andersoninstitute.com/alcubierre-warp-drive.html
narciso (d1f714) — 9/23/2016 @ 4:54 pm@Patterico:Why does he say my solution is right?
He doesn’t say you solved Zeno’s Paradox by assuming a smallest indivisible distance. He acknowledges that real objects cannot be subdivided infinitely, and we already knew that.
So you want to play a stupid game of breast-beating and appeals to authority instead of discussion?
That’s your characterization, but it’s false. I explained a lot of things, provided examples, and responded to things people said, I didn’t say shut up because Newton said so and that’s that.
We could have had a discussion.
We certainly still could. There is a lot of background material on this one that you are missing, and that’s not a shame or a crime. It’s just that you’re not an expert, why would you be? Why be so mad about it?
Gabriel Hanna (64d4e1) — 9/23/2016 @ 5:01 pm@Kevin M: Arithmetic does solve this problem. You just have to carry it further than the Greeks did. Like non-Euclidean geometry, if the Greeks had thought harder about their assumptions they’d have had the answer themselves.
Remember that they considered negative numbers to be meaningless, and they actually killed one of the people who discovered irrational numbers.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 5:05 pmTake it up with the professor.
Patterico (fcc13e) — 9/23/2016 @ 5:21 pmBecause instead of offering to talk about it you get haughty and arrogant. “Patterico, I love you for trying, but you’re 300 years out of date on this and so you’re not going to add much to the discussion.” That translates as “fuck you,” in case your people skills are so poor that you don’t realize how you are coming across. So: fuck you too.
Patterico (fcc13e) — 9/23/2016 @ 5:25 pmthey actually killed one of the people who discovered irrational numbers.
Hippasus? That’s mythical and there are several versions. Personally, I like a commingling of a couple of them: He was drowned by the gods at sea for sacrilege against Pythagoras — he discovered that the square root of 2 was irrational by testing the Pythagorean Theorem with a triangle with equal a and b sides. ☺
nk (dbc370) — 9/23/2016 @ 5:46 pm@Patterico: That translates as..
…a jocular way of saying that someone who doesn’t have expert knowledge on this is probably not going to say anything that wasn’t said already, likely centuries ago.
When I was young I thought I figured out how to trisect an angle. I didn’t see why it was supposed to be impossible, in fact I thought it was fairly obvious how to go about it. It’s because I lacked the background to understand the real problem.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 5:55 pm@nk:Hippasus? That’s mythical and there are several versions.
Yep. Even in more modern times, there’s a lot of romantic(?) folklore around Galois that turned out not to be really true.
When I was young I loved to read Isaac Asimov’s short non-fiction. It wasn’t until a few years ago that I realized how poorly sourced some of it was and I’m beginning to wonder how good any of it was.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 5:58 pmreally they seem to think it’s possible,
http://www-history.mcs.st-and.ac.uk/HistTopics/Trisecting_an_angle.html
narciso (d1f714) — 9/23/2016 @ 6:00 pm@narcisco:really they seem to think it’s possible,
The problem is to devise a method that exactly trisects any possible angle using only compass and straightedge. The reason the conditions are so restrictive, is because it makes the problem correspond to solving an algebraic equation, and it has been proved that this equation does not have real solutions for every possible angle, and that’s because of the kinds of numbers that are involved (transcendental).
If you want to take an angle and divide by three, and close is good enough, there are lots of ways to do it. Some special angles (like right angles) can be trisected exactly. But these variations on the problem don’t reveal anything about math in general.
It’s like focusing on whether or not Miranda was guilty and not talking about why people need to be informed of their rights when they are arrested.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 6:10 pmdepending on the angle, you could end up with an i value, no?
narciso (d1f714) — 9/23/2016 @ 6:13 pmwell some took issue with that widespread view,
In dissent, Justice John Marshall Harlan II wrote that “nothing in the letter or the spirit of the Constitution or in the precedents squares with the heavy-handed and one-sided action that is so precipitously taken by the Court in the name of fulfilling its constitutional responsibilities”. Harlan closed his remarks by quoting former Justice Robert H. Jackson: “This Court is forever adding new stories to the temples of constitutional law, and the temples have a way of collapsing when one story too many is added.”
narciso (d1f714) — 9/23/2016 @ 6:16 pm@narcisco:depending on the angle, you could end up with an i value, no?
There is that, but there is also another element to it. The problem shows that there are numbers which no polynomial can produce as a solution. That’s actually pretty staggering. Here you are on the line of all possible numbers that you can get out of an equation, and there’s these weird random holes in it.
But then complex numbers are staggering too. You may be able to trisect an angle with imaginary lines on paper, but we won’t know how to draw them. Nonetheless they are perfectly cromulent numbers.
well some took issue with that widespread view,
Exactly! It was much bigger than did Miranda do it. He was eventually convicted.
Gabriel Hanna (64d4e1) — 9/23/2016 @ 6:20 pmeven if the underlying argument was correct, as with brown, the warren court exceeded their authority or used shortcuts, like social science in lieu of statutory record,
narciso (d1f714) — 9/23/2016 @ 6:41 pmA man’s got to know his limits..
Ken in Camarillo (17aa36) — 9/24/2016 @ 9:37 amHave you never heard of the Planck Length? https://en.wikipedia.org/wiki/Planck_length?
“[T]he Planck length is, in principle, within a factor of 10, the shortest measurable length…”
WarEagle82 (5bf75f) — 9/24/2016 @ 9:56 amAs usual an expert in one field of human endeavor often thinks that confers on himself expertise in all other areas in which he has taken slightly more than casual interest. Zeno as a paradox dies when you see what is happening. However, it is a handy tool for explaining how calculus works to the mathematically challenged. If you charge that fool wall at a constant speed no matter how much you believe you will never reach it you will get bloodied. And momentum will tend to assure that more than your leading edge gets bloodied.
The key is speed. Note that I said “constant speed” above. No matter how you divide up the distance the time it takes to cover that distance divided into that distance is a constant. The time slices keep getting smaller as the distance slices get smaller. The concept is that the ratio remains constant. That’s the first step into the world of mathematical descriptions of reality. Conceptually speaking no matter how small you make the distance and the time some wag can come along and divide by two. But, that wag always gets the same number. Later on this is transferred into the realm of tangents to a smooth curve and you’ve developed the root of calculus.
One other feature of that successive halving of the distance to the wall and the time it will take to reach the wall is that the sum of all those wee little pieces will add up to the starting distance and the time taken will be the distance divided by the speed. Times beyond that time lead to the runner’s body trying to occupy the same space as the wall. So no matter how much you believe you can never reach that wall and damage yourself, you will reach the wall and physically interact with it. Hence, no paradox exists. All you have is a trip into infinities and an intro to a concept behind calculus.
Attempts to break Zeno’s Paradox by invoking the Planck length and Planck time MAY be hazardous to your mental health and preconceptions. If everything breaks down at that point in your divide by two (or any other number less than one by however small an amount you wish) it suggests something interesting. It asks the obvious question, “Are we real or are we simply calculations taking place in some uber-Cosmic scale computer?” In the latter case one must wonder, “Whose?” The next interesting question might become, “Is that computer programmer really God or does he have his own God?” And soon after you become thoroughly derailed one way or another while the engineers in the world simply ignore all this and build new cities, computing tools, traveling tools, and everything else we need for our comfortable lives.
I wish I didn’t have such a jaundiced view of lawyers becoming fascinated by Zeno and half understanding it. The implications I see coming from THAT plus my native cynicism leave me really scared for the future of humanity. (You know who I mean, those people I’ve spent a lifetime as an engineer trying to add my little increment to making lives better for everybody possible.)
I might add that my jaundiced view comes from seeing one too many instances similar to Hillary Clinton celebrating getting a rapist off in a court trial when she knew he was guilty as hell. I have this vision of lawyers getting their kicks out of the game rather than honest facts and judgements. (Miranda does NOT help this view. It punishes society for the misbehavior of some of its members rather than holding those members PERSONALLY to a suitable standard.)
{o.o} Joanne
JDow (255762) — 9/24/2016 @ 2:04 pmgood video of planck’s solution to black body radiation problem which is strangely similar to Zeno’s paradox . . .
https://www.youtube.com/watch?v=tQSbms5MDvY
marc (4a86fa) — 9/24/2016 @ 2:49 pmI approximated plank length to induction, assuming the dimension of the current universe and working backwards,
narciso (d1f714) — 9/24/2016 @ 2:52 pmGabriel Hanna (64d4e1) — 9/23/2016 @ 6:10 pm
My position is that transcendental numbers, not only are not real, they don’t exist at all, and neither do irrational numnbers.
All you can do is say if there was a solution to certain equations (that usually involve division) it would be somewhere close to a certain number, and since you can carry a number to many decimal places, you can approach it as closely as you want – much closer than reality in fact.
If the number 2 had a square root – but it doesn’t – it would be approximately 1.4142135… If there was such a ratio as Pi, and there was such a thing as an exact circle, it would be about 3.1415926335.. If there was a number like e, it would be 2.7181281828… (it doesn’t repeat after that) If there was a golden ratio, it would be 1.618034…
ALl the proofs that irrational and transcendental numbers are something taht exists in the real world were devised before Twentieth Century physics: If you try to give any example from the real world, I believe it can be shown that all these proofs fall apart.
It is high time that mathematics was revised, and then we might gain some insights. They should stop teaching these things in schools.
Sammy Finkelman (3915d0) — 9/25/2016 @ 6:37 am60. JDow (255762) — 9/24/2016 @ 2:04 pm
Actually you can’t. You run into the uncertainty principle at some point. which says that you annot measure position and time at the same time, which measn you cannot measure speed.
Calculus is inaccurate at the subatomic level, and maybe even at a larger scale.
Sammy Finkelman (3915d0) — 9/25/2016 @ 6:48 am@Sammy Finkelman:My position is that transcendental numbers, not only are not real, they don’t exist at all, and neither do irrational numnbers.
What is “real”? What is “exist”? Can you hold transcendental numbers in your hand? No. In that sense “freedom” and “justice” are not “real” either.
Numbers do not correspond to things. They are relations among ideas.
Once you accept that numbers can be abstracted from what they are representing: once you step away from “this apple and that apple make two apples” into “1 + 1 = 2”, at that point you have accepted all that logically follows from it, including irrational and transcendental numbers.
Because “this apple” and “that apple” are not interchangeable. No two apples are identical, so this apple and that apple do not make “two apples” but instead are still “this apple” and “that apple”.
So if we’re going to argue the “real world” then even integer arithmetic is suspect.
It’s a good idea to pick up the background before waded into waters where you cannot see the bottom. Saves a lot of time and confusion.
Gabriel Hanna (10e685) — 9/25/2016 @ 11:47 am@Sammy finkelman:Actually you can’t. You run into the uncertainty principle at some point. which says that you annot measure position and time at the same time, which measn you cannot measure speed.
No, it does not say this. Again, there is a al ot of background you are missing here.
The uncertainty principle links conjugate pairs. Position and momentum are conjugate pairs. So are energy and time. There’s no problem with measuring position and time as accurately as you can, their uncertainties don’t affect one another. And it’s not a postulate, it’s a consequence of describing matter with waves.
Gabriel Hanna (10e685) — 9/25/2016 @ 11:50 am@Sammy Finkelman:Calculus is inaccurate at the subatomic level, and maybe even at a larger scale.
That’s a pretty strange statement, since calculus is used to describe virtually everything we know about the subatomic level.
Gabriel Hanna (10e685) — 9/25/2016 @ 11:53 am@Sammy: Calculus is like a language. There are things easy to say using it, and things hard to say. You would never say that English is an inaccurate language simply because we might use more words to describe a concept than another language that has a single word. And you would never say that English is “inaccurate” simply because it can be used to describe things that are not real or say things that are false.
It’s just a system for describing particular kinds of relationships. That’s mostly what math is about. There’s no necessity that everything sayable using math has to be real anymore than it is for English.
Gabriel Hanna (10e685) — 9/25/2016 @ 11:56 amYES. Yes, it does.
I grasp this is difficult for non-mathematicians to follow, but Zeno’s paradox isn’t. It derives from the fact that the Greeks had done nothing on the study of “Limits” nd, more specifically, infinite series. The idea that an infinite series converged to a finite value was foreign to the Greeks. But by the 17th Century, the notion had been demonstrated to everyone’s satisfaction.
This is important, because much of what **Newton** did with mathematics deals with infinite series…
Calculus, in particular, both Integral and Differential, depend on having fixed, non-infinite values for an infinite series or a sum of values as they approach a fixed limit by infinitesimal amounts.
Specifically, the sum of “1/2x” as “x goes to infinity” IS ONE… Mathematicians have known this == been able to prove it == for about 4 centuries now.
“Zeno’s Paradox” just flat out isn’t.
IGotBupkis, "Si tacuisses, philosophus mansisses." (225d0d) — 9/25/2016 @ 7:47 pmLincoln was dealing with Legal precepts 180 years ago. Did you want to go to a doctor practicing medicine as it was practiced 180 years ago? No? Me neither. The Law has advanced — probably negatively in some regards, but Lincoln’s law could not deal with modern telecommunications law without considerable enhancement.
No, as a mathematician, Zeno’s Paradox falls into the category of “long-ago solved problems”.
IGotBupkis, "Si tacuisses, philosophus mansisses." (225d0d) — 9/25/2016 @ 7:52 pm😉
how have the legal principles changed,
https://www.quora.com/Is-it-true-that-nothing-is-smaller-than-the-Planck-length
narciso (d1f714) — 9/25/2016 @ 7:54 pmOK, nk, not to insult you, but you — seriously — are now being an example of the Dunning-Kruger Effect. Your understanding of things is so poor you don’t grasp how poor it is.
Why does the above “story” have more validity than a fairy tale? Because it’s used all throughout engineering and physics. Because all that stuff interacts, and if one part of it didn’t work then the whole thing would start to fall apart, and someone would have to come up with a variant on the theory which explained the shifted things — much as Einstein explained minor perturbations in Mercury’s orbit which Newton’s theory could not explain…and our understanding of motion got THAT much better, it got extended into realms which Newton had not the technology to follow.
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The real test of our knowledge of physics and engineering — the stuff we cannot begin to see or touch or taste or feel? — is that the fucking magic WORKS!!.
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The assemblies of plastic and wires and “carefully dirtied SAND” actually glows and makes pictures of something happening a thousand miles away.
The little boxes we carry around in our pockets actually allow us to talk halfway around the world while standing in the middle of a park.
It’s all freakin’ MAGIC, yes, but it stands there and says that our understanding of The Magic is VALID. The stories are MOSTLY RIGHT.
So, please — realize you’re dipping into areas where your understanding is utterly inadequate for you to even have a STAB at a valid opinion.
IGotBupkis, "Si tacuisses, philosophus mansisses." (225d0d) — 9/25/2016 @ 8:05 pmWell, there’s also Smith Drive
:^D
IGotBupkis, "Si tacuisses, philosophus mansisses." (225d0d) — 9/25/2016 @ 8:10 pmCould we possibly be overthinking this? Achilles got shot in the heel with a poisoned arrow, remember? Hydra venom, the deadliest known. It may very well be that a tortoise could have outrun him? 🙂
nk (dbc370) — 9/25/2016 @ 8:26 pmSome thoughts re: Zeno solution
If a runner halves a distance with each step, then that runner must converge on one-mile, mathematically:
1/2 + 1/4 + 1/8…
= Σ 1/2^n –> where n starts at 1, ends at ∞
= Σ (1/2)^n
= ( (1/2)^1 ) / (1 – 1/2) –> by geometric series test, which is a Convergence Theorem.
= 1
In short, there are different sizes of infinity (the infinity of numbers between 0 and 1 is different than the infinity of numbers between 0 and ∞).
An infinite number (of steps, for example) can exist within a finite unit (like distance, for example.)
This specific Zeno paradox assumes that an infinite number of finite steps within a finite distance, cannot be finite. This is false mathematically by convergence theorem and false physically by the Planck Length limitation (@41 & @59.)
Pons Asinorum (8d8929) — 9/25/2016 @ 9:02 pmone-mile = on that distance
sorry for the typo
Pons Asinorum (8d8929) — 9/25/2016 @ 9:06 pmPons Asinorum (8d8929) — 9/25/2016 @ 9:02 pm
The Plancck length means you can’t have an infinite number of steps. And the uncertainty principle means you can’t watch anything move between the two (or actually much higher) Planck lengths. If you try, motion stops.
Sammy Finkelman (643dcd) — 9/26/2016 @ 10:26 am