Please don’t mock and ridicule this question. OK, you can mock and ridicule it, but please answer it, too!
I asked this question on some “Ask the Expert” website, and the guy responded to me with, “I’m not here to help you with your homework.”
Hey, you smug %*(&^%, I’m college-edjumacated and I seriously don’t know the explanation for this!
Anyway …
Presumably fractions can be continually made smaller by making the denominator ever larger. Decimal places can always go further right by adding another zero.
But when it comes to distances, you do actually reach zero. What’s the smallest fraction or decimal place before that?
Scenario: I’m standing outside, an open beer can in my hand (we could have gone with a rubber ball, but why do that with beer available?).
I’m holding the beer about five feet above the ground. I drop the beer. The beer is now four feet from the ground … two feet … a foot … six inches … an inch … a half inch … a sixty-fourth of an inch … etc.
The beer hits the ground (goes everywhere, a tragedy to be sure).
It went from 4.99999999999999999999999999999999999999999 feet from the ground to 5 feet. It hit the ground.
The thing is, I could put 9s on there forever, but it hits the ground.
Doesn’t there have to be some last fraction of a number before reaching the complete distance? What would that number be?
Ultimately you get to quantum levels. Things really only move in discrete steps but those steps are so small that everything seems to move continuously in the everyday world.
So the beer never hit the ground?
Thanks for the Zeno’s Paradox, Achilles and the tortoise cite; it’s one of those things that unless you knew what to look for, you’d never find.
And yet beer splashed everywhere. It did reach “zero.” No matter what math and ever-expanding decimal points say.
I guess it points out that even something as black-and-white as math is still just an attempt to explain something that ultimately becomes elusive and ethereal.
The thing a lot of people ignore is: Yes, the remaining distance is getting smaller and smaller . . . but so is the remaining time. So the distance and time both approach zero simultaneously.
As pointed out above, you re-created Zeno’s Paradox - or at least one of them.
The resolution is easy, once you realise the flaw in the logic. There are an infinite number of rational numbers in any interval. (OK, for the pedants we won’t go into the results of Cantor’s life, so we will just call it infinite and have done.) Zeno’s paradoxes come from looking for some of that infinite number, whilst thinking in terms of a finite division of time to look for them. So each time you say “halve the distance” there is an implicit suggestion that this takes the same amount of time as the time taken on a previous step of halving the distance. But clearly it didn’t. If you ignore acceleration, and just think about constant velocity, the paradox should be “if I take a time interval half that of the the previous time interval, the beer bottle moves half as far as the distance it moved in the last time interval.” Which isn’t exacly earth shattering, or paradoxical at all. Indeed it is merely bottle shattering.
You are left with the issue that kept Cantor awake at night for most of his working life. That there are an infinite number of rational numbers in the interval. Then the rather wonderful result that they remain countable. Infinities come in lots of different forms, and they don’t work like other numbers.
0.99999999999… Is defined to equal 1. This is a result that is also slightly surprising. Note that 1/3 = 0.333333333333… and 2/3 = 0.66666666666666…
So clearly 1/3 + 2/3 = 1 = 0.999999999999…
There is an astoundingly deep amount of fun to be had starting from here.
Besides what Francis posted, I think you have misunderstood the post that you quoted. The poster stated that, once one gets to quantum levels, distance is actually discrete, so there is theoretically a distance that is one planck length, or smallest possible length, away from the destination.
I think the issue of whether the planck length and time actually exists, but you should ask someone such as Chronos about that.
I can take a tape-measure from where my hand was holding the beer to the ground where it hit, and it’s exactly 5 feet. So that measurement is finite, but within itself it’s infinitesimal.
That’s like a small side salad for lunch: Ultimately not satisfying.
You’re correct; I did misunderstand it. So then would it be correct to say that numbers fail us when it comes to this “one planck away from the destination?” In other words, could it be written down in a mathematical formula or a number?
Because again, I can go 0.999999999999, etc., from here to the sun and back.
Nope. It points out that while arithmetic is black and white, real math is much more interesting.
The word no one’s used so far is limit. What’s tripping up you and good old Zeno is the assumption that if you add an infinite number of sums you must get an infinite answer. But that turns out not to be true. Certain infinite sums reach an answer, called the limit. The limit of .9 + .09 + .009 + … = 1. You can always add another zero in there, even if you filled the universe with them a zillion times over. But the limit is always the same. The limit for 1/2 + 1/4 + 1/8 + 1/16 … also happens to be one. But 1/2 + 1/3 + 1/4 + 1/5 + 1/6 … really does expand and go to infinity. Infinite sums are hugely important in many branches of math. You can express pi and e and phi and lots of other concepts best as infinite sums, for example.
Whether space is discrete or not is completely irrelevant to this discussion. You’re asking a math questions about limits and infinite sums, not a physics question about reality.
A turtle is racing Achilles, and is given a head start. By the time Achilles catches up to where the turtle started, the turtle has moved on ahead. When Achilles gets to that point, the turtle has still moved on ahead. Sure, the turtle’s lead is less and less, but how does Achilles pass the turtle and win the race?
The answer is that we are also slicing time into smaller and smaller segments until it is stopped. Let’s say it takes Achilles six seconds to reach the turtle’s initial starting position. By the time Achilles reaches the point where the turtle previously had been, it is now only 1/2 second later. The next point we consider is a mere 1/50th of a second later. And, the next point is 1/1000 of a second later. By the time Achilles is about to pass the turtle, we have frozen time.
In theory, you could always divide the distance in 1/2 again and again until infinity. But, as my Physics teacher was found of saying “It’s close enough for government work”. What he meant is that the distance will sooner or later get so small, we no longer care about it. Can you even measure whether something was .00000000000000001 inches above the ground? If you can’t measure it, can you even care about it?
Also in our quantum based universe, you would also get to the point where you couldn’t accurately tell whether the beer can ends and the ground begins once you get down to small enough distances. In other words, you might not be able to even tell the exact moment when the can hit the ground. The best you could do is have a small time range when the can did land on the ground. That is, of course if time itself isn’t quantum based. (That is, time isn’t smooth but itself is made of discreet time units of time).
I’d also like to reiterate Exapno Mapcase’s point: whether actual, physical space is discrete or continuous is irrelevant; the “paradox” is only a confusion about an abstract mathematical idea and the implications therein. Appealing to contingent physical laws is neither necessary nor actually on-target in clearing away the puzzlement here, as this ultimately is not about what particular structure the physical universe happens to have, so much as about what is logically possible.
Here’s a simple answer: People are measuring wrong. The beer isn’t five feet above the Earth (well, OK; it is), but almost 4,000 miles from the middle of it. So the beer isn’t falling to the ground; it’s falling to the center of the Earth. The ground just gets in the way before the beer can fall more than a miniscule fraction of the distance. So the beer hits the ground, because the ground is closer than the point to which the beer is falling.
Dealing with this by saying “it gets so small we no longer care about it” isn’t very satisfying.
I think it’s more useful to point out that at the time the paradox was posed, it wasn’t understood that the sum of an infinite series of non-zero elements could be finite. Without that understanding, showing that that the time when Achilles overtakes the turtle can be represented as the sum of an infinte number of non-zero intervals (as it validly can be) made a persuasive argument that it could never happen.
We now understand that the sum of such a series can certainly be finite. This resolves the paradox without any need to wave hands or refuse to consider steps smaller than some size.
Simple and jocular. More seriously, it does bring up the point (for the OP): If you think there has to be a last position in space before the beer hits the ground, do you think there has to be a first position in space after the beer has left the hand? Is there necessarily a last point in time before the beer hits the ground and a first point in time after the beer has left the hand?
It is clearly conceptually possible for both time and space to lack such a structure; we can model this easily, as you yourself did.
Is there a lowest fraction above 0? Is there a greatest fraction below 5? (No! Of course not! You realized this in the very OP. So what’s the problem? That there are infinitely many fractions between 0 and 5? So what? The distance between them is still finite, clearly (since it’s |5 - 0| = 5, clearly). You’ve just discovered that you can have infinitely many numbers within an interval whose span is finite [in the sense of how far apart the endpoints are]. Perhaps this is surprising (though it oughtn’t be, or at least, oughtn’t remain so for long after realizing it), but there’s nothing contradictory about it.)
Yes, you can and should care about it in math. The math problem that we’re talking about is nothing at all like the physics problem you’re referring to. The notion of the limit is simply one of the most critical advances in all of human history. You can’t blow it off. There is nothing here worth talking about until you make the distinction between math and physics.
