Say there is an object 100 feet in front of you. Now, you start walking towards the object. Each time you move closer, you are only moving 1/2 the remaining distance to the object. With this said, theoretically, if humans were small enough to move very, very small distances, would you ever reach the object and be able to physically touch it?
Depends on how you define “ever”…
If you mean “within a lifetime”, then the answer is “No”.
If you mean “within an infinite amount of time”, then the answer is “Yes, when an infinite amount of time has passed”. Keep in mind, however, that an infinite amount of time will never pass.
From a mathematical perspective, no you would never reach the object. If you take the progression from 100-50-25-12.5-6.25-3.125-etc. you can continually divide the number in half infinitely. So you would never actually reach zero, so technically you would never reach the object. Could you physically touch it. Yes, if you reached out your arm you could touch it from a couple of feet or more away. If you mean would your feet touch it, then theoretically no.
I do not want to get into the calculus or anything like that.
Historical note first – you’ve posed a question that first came up over 2000 years ago! It’s one of “Zeno’s Paradoxes” (Zeno was a Greek philosopher), and he wondered exactly the same thing (although he phrased it terms of an arrow reaching the target). So, you’re in good company. Fortunately, the question has been settled in the intervening couple of millenia, and I’m pleased to report that not only do you actually get where you are going, but you DO theoretically get where you are going as well.
Suppose you’re trying to walk a total of 2 feet in the manner you described. First, 1 foot, then 1/2, then 1/4 and so on.
1 + 1/2 + 1/4 +1/8 + …
You can see that you keep adding smaller and smaller amounts. What’s special is that this particular series adds those smaller amounts in just the right way (it’s called a geometric series) that the final SUM is a finite quantity; if you actually added all the inifinite bits, it would add up to exactly 2 (this is demonstrable using limits–trust me).
But, wait, you say, I can’t really add up all those infinite bits, it would take forever! Here’s the catch. If you were to STOP on each “marker” and call out “1”, I’d agree that theoretically you’d never get there, but’s that not how things travel. Suppose you walk at (conveniently) a rate of 1 foot per second. Then, it takes you 1 second to cover the first bit, then only 1/2 a second, then 1/4 and…hey, same pattern! Each successively shorter distance get covered in a successively shorter time. And the sum of all those terms is
1 + 1/2 + 1/4 + 1/8 + … = 2 seconds.
I think people get stuck on the “infinite” part because they think about how long it would take to write a sequence of terms. By doing so, they are implicitly assigned a fixed time to accomplish each chunk. Since the chunks get shorter, in the real world, they’re slowing down to a stop (and yes, if you do that, I promise you’ll never get there).
In general (through there are some bizzare branches of math that appear to have no real world application thus far), theoretical math does model the real world. It has to, if we’re going to use it as a tool to solve problems. So if you, or me, or anyone runs into a case of “well actually you do, but theoretically you shouldn’t”, it’s time to go check the theory!
I want to challange your statement on adding all the bits…
if you actually added all the inifinite bits, it would add up to exactly 2 (this is demonstrable using limits–trust me).
Ok, lets say that you take the first 10 segments of a two foot span. What you are saying is the first 10 portions of the short journey would add up to the total distance to be covered. Lets look at the first 10 steps and the distance traveled on each. Each increment is in feet…
Now, the sum of all these distances is…
This does not add up to the total distance to travel. Although close, it is still not two complete feet. Is it not?
So, with this said, you would never be able to reach the destination since the total distance traveled would never add to the total distance to travel.
I am not getting what you said correctly? Also, keep in mind this theory would only apply a flat, 2D person. If this was not the case, then what point of the human body would you use to measure the closet point to the object being traveled to? You could not say well the nose, because it sticks out further because another person could have a big toe that sticks out further than their nose. Also, what would the total thickness of the person in question need to be? Maybe that would not be the case since 2D would be flat anyway with a smooth surface to use as the closest point.
hmmm, I think you’re getting stuck on that time aspect. Let me see if I can clarify. You’re exactly right that if I add exactly 10 of the numbers, I get an answer that is not exactly 2, and of course, you wouldn’t expect it to be either, since we haven’t added up ALL the pieces, only 10 of them. Try adding the 11th piece, then the 12th, and none of them will be exactly 2, but each one will get a little closer (this is one of the best ways to get an intuitive understanding of a limit), and yes, you really would have to “add” an infinite number of pieces to get 2 exactly. So the question is still, does it take an infinite amount of time to add an infinite amount of pieces? I’ll repeat myself here for a moment, YES if you take the time to write (or say or even stop for just a moment to think the number to yourself), but when you’re in motion, you’re not pausing on each bit. Here’s a thought: draw two dots on a piece of paper an inch apart and move your pencil from one to the other. Congratulations! You just passed over an infinite number of points!! But you didn’t cover an infinite distance, right? Now, you know that (common sense), but it is also true that theoretical math has absolutely no problem with it either. The reason is that each point is infinitely small, and takes and infintely small amount of time to pass over. So…
take 10,000 points
pass over each single point in 1/10000 of a second
takes 1 second
take 10 million points
pass over each point in 1/10 millionth of a second
takes 1 second
take 10 gazillion points (yes, I know)
pass over each in, yep, 10 gazillionth of a second
takes 1 second
As you get closer and closer to an infinite number of points, your time gets correspondingly shorter to cross each point, and the whole thing balances out.
The business about the 1/2 and the 1/4 and so on is really a distraction; we could phrase it as 1/3 of the distance, then 1/9, but it comes out the same. It just makes the question look so much more convincing!
Like I said, I don’t think we’re discussing whether walking works, but whether math actually models the way walking works. The theory (that limit thing) formalizes the sense we get looking at patterns, and there isn’t any conflict.
SlimFlem writes, “So, with this said, you would never be able to reach the destination since the total distance traveled would never add to the total distance to travel.”
Actually, what Lynne posted was right on the money. You keep adding smaller and smaller amounts, but (this is the important bit), you add an infinite number of them.
When you wrote, “1.998046875 feet.”, you just stopped early! You didn’t yet add the rest of the infinite series up There are actually ways to do this without needing to spend an infinite amount of time with a calculator.
You can model it as a sum from 0 to infinity of 1/2^n. I.e, you go half as far with each successive step (but of course it only takes half as long). It all works out both in theory and in practice
If you’re curious about the topic, a good place to start is a freshman calculus text - it will tell you lots of stuff about these kinds of things, and you can just ignore all the really nitty gritty stuff and focus on the basic concepts.
Well put, Lynne.
From another perspective: draw the two dots, and we want to connect them with a line, as in Lynne’s example. We have defined mathematically that there are an infinite number of points between the two dots. If you try to draw in each one of the points with a very, very sharp pencil, you will never succeed. But you don’t have to draw each dot separately, you can draw a line that encompases the theoretic infinite number of dots, just by pulling the pencil along the paper.
The mathematical model works very well as a model of the real world. But there is a slight disconnect between the real world and the mathematical model: in the real world, no “point” can be any thinner than (say) an electron. So you would cram a line full of electrons, and you would have a finite number (a very large number, indeed, but still a finite number) of points. Or, if you would prefer, a point can’t be any smaller than the smallest pin-prick that you make with that very, ery sharp pencil. Again, that pinprick has a certain diameter, no matter how tiny, and a finite number of pencil-dots will eventually fill in the line.
In the mathematical model, however, a point has no diameter at all, and the line is infinitely divisible into smaller and smaller units.
Yet another perspective (following Lynne’s example of walking a foot a second): The paradox arises because you are thinking about walking that next half of each remaining distance in a discrete time unit, pausing, then taking the next half distance. That is, your mind first is thinking that takes me 1/2 second to walk the first 1/2 foot, then 1/2 second to walk the next 1/4, then 1/2 seconds to walk the next 1/8, then… and 1/2 second to walk the next 1/1024… and … At that rate, it is true, I will never get to the end of the line. And the paradox is resolved when you understand that you’re THINKING of time in just that way.
If you instead recognize that at a constant speed of 1 foot per second, it takes 1/2 seconds to walk the first 1/2, then 1/4 seconds to walk the next 1/4, then 1/8 seconds to walk the next 1/8, then… 1/1024 seconds to walk the next 1/1024… etc. Since time is infinitely divisible as well as space, you can cover the fixed space in the fixed time.
You would never reach the object, as you would first be run over by a turtle and then by Achilles.
This is similar to the following riddle:
If you can’t grasp the concept of infinity, both of these problems probably give you a headache.
If the first person to reply to SlimFlem’s post provides half the answer, and the next person provides half of the remaining explanation, and so forth, will this thread ever end?
Designated Optional Signature at Bottom of Post
Xeno’s paradox originates from the faulty way of describing motion. Motion is distance per time. Xeno kept chopping up the distance into smaller and then even smaller and then even smaller bits. The recurring divisions leads to a limit of zero. At zero distance over zero time, of course there’s no motion. Duh. No wonder why he never got where he was going. If we start with that kind of infinitesmal reduction, we don’t even begin the journey at all! [After all, how can you get to the first half-way point if you have to first travel half way to get there…]
Between any two points, there are an infinite number of infinitesmal points. To calculate motion by counting each individual infinitesmal point (of which there are an infinite quantity) is way too time consuming (to put it ironically).
[Actually, one can count the infinitesmal points if one integrates an algebraic formula that acurately describes that distance. The half and half again series that Xeno describes just so happens to be that kind of formula which calculus shows does reach the full length at infinity – this has been adequately described here in this thread.]
IOW, if you don’t chop up my time into a standstill frame, I’ll get where I’m going.