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Almost 2,500 years ago, the philosopher Zeno of Elea set out to challenge the way we understand the physical world through a set of brain teasers that have stuck with us for millennia. The most powerful of Zeno's paradoxes grapple with the concept of infinity while pitting observable reality against the scientific language we use to describe that reality, suggesting that elements of the everyday, like motion and speed, are actually illusory.
Example paradoxes are:
The millet paradox, which states that one falling grain of millet makes no sound but a ton of falling millet makes a big one, is more of a stoner observation than a profound question about the physical world. His paradoxes of motion and space, on the other hand, are legendary. Four of the more than 40 thought experiments he is said to have devised are most often employed as vivid introductions to the intersection of math and philosophy, where something readily apparent is a challenge to definitively prove.
Dichotomy paradox: If you want to walk across the room, you have to first walk half that distance, then half the remaining distance, ad infinitum, so how do you ever get there?
Achilles paradox: If a turtle gets a head start in a race against Achilles, Achilles has to cover half the distance between himself and the turtle in order to catch up. Then half that. And half again. And again. In an upset, the turtle wins!
Arrow paradox: At any given instant, an arrow in flight occupies a certain space, no more and no less. At the next instant, it occupies a different space. If you assume an instant is indivisible, the arrow is not in motion. So how does it move? "It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever," as Bertrand Russell put it.
Stadium paradox: Imagine three sets of three bodies in stadium rows: three As, three Bs, three Cs. The As are stationary; the Bs are moving right; the Cs are moving left at the same speed. In the same timeframe, the Cs will pass just one of the As, but two of the Bs. Crazy, right? (It doesn't seem like it, but if you think of space and time atomistically, they pass without passing.)
It took more than 2,000 years to break the dichotomy and Achilles paradoxes, and the people to do it were the French mathematical prodigy Augustin-Louis Cauchy and the German Karl Weierstrass. The mathematical answer can be summed up by the intuitive answer: Eventually, you get there.
In mathematical terms, one way of putting it is "the limit of an infinite sequence of ever-improving approximations is the precise value" (pdf). By going from one side of the room to another, you go 100% of the way across. You can chop that 100% up into infinite pieces, but those pieces converge on a limit of 100, and the sum of those pieces is the value—the infinite number of increasingly small pieces adds up to a finite number. ½ + ½ = 1, of course. ½ + ¼ + ¼ also equals 1. And so forth: the numbers you add up to get to 1 can expand to infinity, but it's not changing the end result. Not all infinite geometric series converge to a limit, but some do (pdf), predictably: "All those (and only those) in which the ratio of consecutive terms is greater than –1 and less than +1, so that the absolute values of the terms get progressively smaller."
(Score: 0) by Anonymous Coward on Sunday February 23 2020, @08:37AM
These concepts sound a lot like limits in calculus, such as the idea of a Riemann sum to approximate the area under a curve or the distance between two points on a curve to estimate its slope. As the width of the rectangles or trapezoids used to approximate the area goes to zero or the distance between the two points approaches zero, there are mathematical issues if these quantities go to zero. What is the sum of the area of infinitely many trapezoids with zero width? What is the slope of two points along a curve with zero distance between them? The concept of a limit avoids the issues involved with transitioning from something that is quantized to a continuous function. In short, calculus has the answer for this mathematically.
However, there are alternative physical ideas that may avoid this, such as the theory that distances in spacetime are quantized so that there are minimum very small increments in both space and time dimensions. The concept of a fraction of a minimum step in space and time would be meaningless. By no means is any of this accepted physics, but the idea has been proposed. The steps are so small as to make space and time appear continuous on the macroscopic scale.
(Score: 1, Informative) by aristarchus on Sunday February 23 2020, @08:57AM (3 children)
Hmm, the editors have bulloxed the entire point of the submission again. Of course, if you just assume that space is not infinitely divisible, because of some supposed limit or "dark matter", . . . did I just Gaaark?
(Score: 5, Informative) by janrinok on Sunday February 23 2020, @09:16AM (2 children)
Your original submission contained such useful titbits as the following:
but did not include any of his Zeno's paradoxes - the choice was re-write it or reject it. Your submissions tend to be more of your attempts at witty comments than fact or information, you're lucky that this one got through. I'm surprised that you didn't accuse Zeno of being alt-right too.
(Score: -1, Troll) by aristarchus on Sunday February 23 2020, @11:15AM (1 child)
Ah, janrinok, you think the biographical details are irrelevant? What do you think Parmenides was trying to do? And what was Zeno doing with the paradoxes? Mathematicians and modern physicists just take them as problems to be solved, and they just do that by positing totally unsubstantiated metaphysical assumptions. But back to the point? Philosophers have always opposed tryants, far from being alt-right, Zeno was anti-fascist, as are all right thinking people, except, evidently, the editores of SoylentNews. I await your continued censorship, old friend!
(Score: 5, Touché) by janrinok on Sunday February 23 2020, @12:08PM
You chose the title of the submission - if you had wanted to write about the life and times of Zeno and his association with various philosophers, then perhaps you should have titled it more appropriately.
(Score: 4, Interesting) by Coward, Anonymous on Sunday February 23 2020, @09:21AM (2 children)
Don't forget the Quantum Zeno effect [wikipedia.org], named in his honor. It can be summarised as "a watched pot never boils" which is true for some quantum systems where observing them changes the behavior drastically.
(Score: 0) by Anonymous Coward on Sunday February 23 2020, @09:58PM (1 child)
hahaha, lol. if you dont observe how can you beforehand know that it changed drastically?
nevermind this bollocks. all this confusion comes from education not being free and popular science books trying to interest "new age" crowd (in buying the book).
ofc, the world IS mostly boring and imagining stuff is fun and no hard work (then again you dont really gain any superpowers if you dont actually handle the world. not implying that life is only worth living w/superpowers).
(Score: 2) by Coward, Anonymous on Monday February 24 2020, @04:29AM
1. Prepare in a known state by e.g. optical pumping.
2. Apply oscillating field to drive Rabi-flopping (cyclic transitions between initial and target state).
3. Measure more or less often and watch the evolution rate change.
It is well understood and has been observed many times in experiments.
(Score: 3, Funny) by Nuke on Sunday February 23 2020, @11:16AM (11 children)
2500 year old news in fact. Is SN getting desparate? Next up, the Universe is created.
(Score: 2) by Gaaark on Sunday February 23 2020, @11:23AM (2 children)
It was pushed through, after a MAJOR rewrite, to make aristarchus happy.
--- Please remind me if I haven't been civil to you: I'm channeling MDC. ---Gaaark 2.0 ---
(Score: 4, Funny) by Anonymous Coward on Sunday February 23 2020, @12:27PM (1 child)
Isn't any attempt to make aristarchus happy a paradox in and of itself?
(Score: 2, Insightful) by aristarchus on Sunday February 23 2020, @08:33PM
aristarchus is not happy. He is giddy, ecstatic, consummated with bliss. Now if only janrinok understood what a slam to STEM the paradoxes are.
(Score: 2) by MostCynical on Sunday February 23 2020, @11:50AM
Only if the git request went to the correct workflow.
Likely, we're alpha/pre-release, and the updates are still waiting in a request queue..
"I guess once you start doubting, there's no end to it." -Batou, Ghost in the Shell: Stand Alone Complex
(Score: 2) by janrinok on Sunday February 23 2020, @12:12PM (6 children)
STEM includes mathematics. Many of our community like brain teasers such as this, and they will have tried to think the paradoxes through for themselves before reading how and when they were solved mathematically.
They are as relevant today as they were 2500 years ago, IMO.
(Score: 2) by Azuma Hazuki on Sunday February 23 2020, @05:57PM (5 children)
I don't really get the mathematics behind most of this, or anything honestly, but in the specific case of the arrow and the tortoise, isn't it just a converging series? 1/1 + 1/10 + 1/100 + 1/1000...etc, should sum to a number much less than infinity.
Is this a classical proof that spacetime is quantized, actually? That's what I took from this, that at some point there has to be a leas unit of spacetime...
I am "that girl" your mother warned you about...
(Score: 0) by Anonymous Coward on Sunday February 23 2020, @09:03PM
The dichotomy paradox is a series of halfs, not how they gave it. They gave the limit version, which is not the original one given. Basically, the repeated is that you have to halfway to get somewhere, which repeats for that half, etc. Which means that your first step is to cross the smallest half. This is what is paradoxical, obviously we move but then how can one do an infinite number of actions in a finite period of time? Even a discrete universe seems to suffer from that problem, as to do otherwise would seem to make geometry and other mathematics contradictory at such a scale.
(Score: 0) by Anonymous Coward on Sunday February 23 2020, @09:20PM (3 children)
Just realized you were talking about the other one, but the underlying idea is the same. You'd have to do an infinite number of fractions in a finite period of time.
(Score: 2) by pgc on Monday February 24 2020, @11:57AM (2 children)
It is not a paradox, since the time needed for each action will eventually approach 0, therefore the total time taken will fit in a limited amount of time.
(Score: 0) by Anonymous Coward on Monday February 24 2020, @11:41PM (1 child)
But you don't catch up to the turtle until it does reach zero. The fact that it approaches zero doesn't change the fact that it will never be zero.
(Score: 2) by Azuma Hazuki on Tuesday February 25 2020, @03:36AM
This is what introduced me to the idea of quantized spacetime as a child, though I didn't have any words for it other than "well we KNOW we can get past any distance we want if we take long enough, so maybe that means somewhere there's a smallest piece of distance?" I was 9 or 10 at the time, IIRC.
I am "that girl" your mother warned you about...
(Score: 3, Funny) by Gaaark on Sunday February 23 2020, @11:20AM
Janrinok should get credit for this submission: aristarchus' submission was junk... He submitted half an article, then the other half, then cut out 50%, then half that again, then brought in a turtle to finish up for him: the turtle puked half its dinner on it, then pooped on the other half.
Janrinok eventually got it there.
Good rewrite!
--- Please remind me if I haven't been civil to you: I'm channeling MDC. ---Gaaark 2.0 ---
(Score: 2) by theluggage on Sunday February 23 2020, @12:47PM (3 children)
...but to walk even "half that distance" first requires that you walk 1/4 of the distance, then 1/8th... so if the assertion is valid, you can't even get half the way. So we have an assertion that assumes its own falsehood... so really it's just a rip off of the old faithful "this statement is false"...
Then there's the uncertainty thing - even bog-standard classical margin of error: no need to interrupt Heisenberg's cooking - you don't have to sub-divide time infinitely just fine enough to get you within a few atomic radii of your destination... Or just aim for a point a few angstroms in front of the turtle...
So that just about wraps it up for Zeno - I'm going to be very careful on zebra crossings today... :-)
(Score: 0) by Anonymous Coward on Sunday February 23 2020, @01:20PM (2 children)
Zeno's accomplishments weren't about the paradoxes themselves, but the questions they raised. It is easy for us to dismiss his questions or disprove his assertions, because we have since formulated the scientific tools to answer these questions. Try to disprove Zeno's paradox without using math's limit theory; both the solution to the paradox and your refutal rely on it. Or explain the arrow paradox from the summary without using the physics concept of momentum.
(Score: 1) by khallow on Sunday February 23 2020, @04:53PM
I would, of course, use Dedekind cuts [wikipedia.org] which are equivalent. But if you try to solve something without using the tools that are intrinsic to solving that something, then you're not going to get anywhere. Here, limits, Dedekind cuts, or an equivalent are necessary to solve the problem. You simply can't do it otherwise.
(Score: 2) by theluggage on Sunday February 23 2020, @06:20PM
Absolutely - but maybe they should be called "Zeno's really good questions" instead?
Yeah, but the tool in question isn't the sum of an infinite series - it is the scientific method, which in this case involves the simple act of testing the assertion by experiment and getting up and walking across the room.
(Score: 5, Interesting) by stormwyrm on Sunday February 23 2020, @03:44PM
Funny that I got into this sort of discussion very recently with another maths-head on a different forum. We were discussing the concept of infinitesimals, and while the concept was historically important in the development of the calculus and is superficially easy to understand, it is fraught with logical difficulties. The Jesuits basically banned the teaching of the concept in their schools in the 17th century, because the idea of an infinitesimal smacked of atomism, which caused philosophical issues with the Catholic doctrine of transubstantiation. Back in those days no one made any distinction between the idea of an abstract mathematical continuum space and a physical continuum (this distinction is really a very modern idea), and it was this same lack of distinction that made the concept of non-Euclidean geometry so difficult to accept later on. Mathematicians also recognised for more than two thousand years the logical paradoxes inherent to the idea of infinitesimals, and Zeno's Paradoxes are the oldest known examples. Infinitesimals thus caused a lot of difficulties in attempts to put the calculus on a logically rigorous foundation, with Bishop George Berkeley deriding them as "ghosts of departed quantities" and Georg Cantor blasted them as "the cholera bacillus of mathematics". And so in the nineteenth century, mathematicians such as Karl Weierstrass, Augustin-Louis Cauchy, Georg Cantor, Bernard Bolzano, and Richard Dedekind developed the epsilon–delta definition of a limit, and they built up the calculus from there on a firm logical basis free of any paradoxes such as Zeno's. So now every beginning student of elementary analysis starts from there. However, in the 1960s Abraham Robinson developed non-standard analysis that includes infinitesimals in a rigorous way that avoids the paradoxes.
Numquam ponenda est pluralitas sine necessitate.
(Score: 2) by HiThere on Sunday February 23 2020, @04:27PM (1 child)
Most of Zeno's paradoxes depend on continuity, the rest depend on not noticing really small effects, but being increasingly likely to notice them if the effects get larger.
N.B.: There *are* solutions which accept continuity, e.g. in the Achilles and the Tortoise you can can assume that covering shorter distances takes increasingly less time, but the real answer is that step size isn't a continuous variable, and that distance is more finely divisible. (Also that the step size for a human runner isn't the same as the step size for a Tortoise. Etc.)
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 1, Insightful) by Anonymous Coward on Sunday February 23 2020, @05:04PM
(Score: 2) by opinionated_science on Sunday February 23 2020, @04:39PM
A long but very interesting read. Won a prize apparently.
If you want a 2nd book, Donald Knuth's AOCP.
Not related to the article, but since I'm referring to dead trees....;-)
(Score: 2, Disagree) by edinlinux on Sunday February 23 2020, @06:07PM (1 child)
The solutions..
>Dichotomy paradox: If you want to walk across the room, you have to first walk half that distance, then half the remaining distance, ad infinitum, so how do you ever get there?
Because space time is quantized (i.e., its 'pixels'), eventually, there is a minimum unit you can move, and you cannot split it in half..(choice is move the minimum unit, or do not move at all)
>Achilles paradox: If a turtle gets a head start in a race against Achilles, Achilles has to cover half the distance between himself and the turtle in order to catch up. Then half that. And half again. And again. In an upset, the turtle wins!
Same for this one.. space time is quantized, eventually, there is a minimum unit you can move, and you cannot split it in half..
>Arrow paradox: At any given instant, an arrow in flight occupies a certain space, no more and no less. At the next instant, it occupies a different space. If you assume an instant is indivisible, the arrow is not in motion. So how does it move? "It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever," as Bertrand Russell put it.
When in motion, the arrow is heavier, and is slightly compressed in the direction it is moving (relativity theory). This defines is speed and direction from frame to frame.
>Stadium paradox: Imagine three sets of three bodies in stadium rows: three As, three Bs, three Cs. The As are stationary; the Bs are moving right; the Cs are moving left at the same speed. In the same timeframe, the Cs will pass just one of the As, but two of the Bs. Crazy, right? (It doesn't seem like it, but if you think of space and time atomistically, they pass without passing.)
I don't see a problem with this one, it behaves as expected..
(Score: 0) by Anonymous Coward on Sunday February 23 2020, @06:20PM
(Score: 4, Insightful) by sjames on Sunday February 23 2020, @08:14PM (1 child)
There are a number of posts here that claim a simple solution to Zeno and question the value of his questions. Invariably those simple solutions use mathematical tools of concepts in physics that did not exist in Zeno's time. His very good questions questions lead to those tools.
(Score: 2, Informative) by Anonymous Coward on Sunday February 23 2020, @09:28PM
Also note that many of them don't actually solve them without bringing their own problems. For example, a quantized universe causes mathematics to break down at said small scales or entail contradictions. Or that the solutions that require a B theory of time either don't explain across reference frames or assume total ordering.
(Score: 1, Funny) by Anonymous Coward on Sunday February 23 2020, @10:12PM
well, after archilles and turtle heard about zenons objection they decided (after an infinite about of time past inbetween the decision process) to first declare they where racing to pluto even tho they secretly already agreed on a length of 5 turtles and 2 archilles so that if zenon showed up to the race and would start deviding halfs they would both already be past the finish line. then secondly they both agreed to race ... sideways.
(Score: 1, Informative) by Anonymous Coward on Monday February 24 2020, @04:06AM (1 child)
The paradoxes only occur because people are bad at dealing with "infinity." For one thing, what does infinity even MEAN? You will notice no rigorous definition here. The only folk-definition Zeno uses is that something "continues" without end... but he always stops the "sum" and says, "See, there is some distance left! The sum doesn't include this last bit of the distance yet to be travelled!" You see, Zeno ISN'T DOING an infinite sum, but a partial sum. Zeno is adding in a complication he doesn't understand by breaking things down into an infinite sum.
(Score: 0) by Anonymous Coward on Monday February 24 2020, @12:56PM