The Unanswerable Question

[Aetius]

I have asked this question to many people, and no one can answer it. I believe it cannot be answered, but if you do answer it with skill I will gladly give you rep!

So lets say that an archer fires an arrow at a target.

Lets say it takes 1 second to reach the target.
Now it will take .5 seconds to reach halfway to the target.
It will take .25 seconds to a quarter distance to the target, ok?

Now this is the part that will blow your mind. We can keep dividing the second further and further and further. You come to realise that there are infinite moments in a second.

Now the final question is, if there are infinite moments in the second(which the arrow has to travel through all of them) how does the arrow ever reach the target? Because it obviously does.

Thats what I dont get. How can it travel through infinite moments in space/time and reach the target??

[Thanatos]

Oh, I've read of this problem before...

Just let me recollect as to what the answer was...

[boofhead]

I've heard it also but in this context: "A frog jumps half of the remaining distance to the edge of the table each jump. How many jumps until it reaches the end of the table?"

Maybe this is a little better than the archery analogy - as of course the arrow is faster and straighter when it leaves the bow than when it strikes the target (just nit-picking )

[Sétanta]

Perhaps I'm not understanding you, but of course the arrow reaches the target. You can keep dividing the amount of time that it takes (1 second to go all the way, 1/2 second to go half way, .25 second to go a quarter of the way and so on), but as the arrow travels, it slows down, thus making the amount of time it takes to go a certain distance a bit longer.

If the arrow is fired at 150 MPH, when it reaches the target it will not be travelling 150 MPH, unless the target is within 5 feet or so.

Now, I think I can answer your question, but you will be a better judge of that than I. If it takes 1 second to hit the target, the arrow doesn't travel through infinite moments within time. It travels exactly 1 second. The second can be divided up into infinite divisions, but that is irrelevant since the arrow is only concerned with taking the single second to reach it's target. Thus, not travelling though an infinite time.

[Niccolo Machiavelli]

It disregards the fact that an infinite series can have a finite sum.

[rathelios]

This is called Zeno's paradox, he invented it.

Calculus is one way of answering: “The sum of an infinite series need not be infinite.”
In calculus an infinitesimal is supposed to be an infinitely small quantity, smaller than any quantity conceivable and yet not zero. It is not zero because continuous lines are regarded as being comprised of infinitesimals. However, in the calculation of a derivative – such as the velocity of a moving object at a certain point in time – an infinitesimal is treated as 'no quantity'.
This process works in practice but is manifestly inconsistent.
So using calculus may be a technically correct answer but I think it misses the point Zeno was trying to make.

Zeno's problem strikes at the heart of mathematics. Is maths 'the establishment of exact truths concerning the abstract world of ideas' or rather 'a pragmatic series of useful techniques for solving problems'?
To me maths is a language used to symbolise the universe but it can do so only imperfectly because the universe is 'non obvious'.
No evidence can exist for continuity of reality as your knowledge of reality must be expressible in a finite amount of information – it is an assumption that reality is continuous.
Language is no less imperfect as a means of representing reality than maths. Words are symbols too. Infinite and finite are words that represent properties in the material world.
Zeno is not wrong in his central theme that the motion of the arrow is illusory. Maths, language, even conscious experience are imperfect means of representing reality.

[Aetius]

One thing is for sure, the TWC has the most intelligent of any forum I have ever visited (and most everyone I have ever met in my daily life laughter: )

This is called Zeno's paradox, he invented it.

Calculus is one way of answering: “The sum of an infinite series need not be infinite.”
In calculus an infinitesimal is supposed to be an infinitely small quantity, smaller than any quantity conceivable and yet not zero. It is not zero because continuous lines are regarded as being comprised of infinitesimals. However, in the calculation of a derivative – such as the velocity of a moving object at a certain point in time – an infinitesimal is treated as 'no quantity'.
This process works in practice but is manifestly inconsistent.
So using calculus may be a technically correct answer but I think it misses the point Zeno was trying to make.

Zeno's problem strikes at the heart of mathematics. Is maths 'the establishment of exact truths concerning the abstract world of ideas' or rather 'a pragmatic series of useful techniques for solving problems'?
To me maths is a language used to symbolise the universe but it can do so only imperfectly because the universe is 'non obvious'.
No evidence can exist for continuity of reality as your knowledge of reality must be expressible in a finite amount of information – it is an assumption that reality is continuous.
Language is no less imperfect as a means of representing reality than maths. Words are symbols too. Infinite and finite are words that represent properties in the material world.
Zeno is not wrong in his central theme that the motion of the arrow is illusory. Maths, language, even conscious experience are imperfect means of representing reality.

This is Zeno's "Paradox of the Arrow". He created it to support Parmenides doctrine of "All is One" - i.e. that all of creation is one indivisible whole and that therefore plurality and motion are illusions.

He attempts to prove by Reductio Ad Absurdum that the arrow is stationary at each of the infinite set of positions between archer and target and therefore is never actually "in motion".

Mathematically this can be refuted with calculus (i.e. that the sum of an infinite set can be finite) as has been pointed out by previous posters.

Physically, there is reason to believe that the real universe is discrete rather than continuous. If time and space are truly quantised, there may be no distance shorter than the Planck length (1.6 x 10-35 metres) or time shorter than Planck time (approx 5.4 x 10-44 seconds).

The archer should get himself a quantum tunnelling upgrade for his weapon. He can then have the arrow travel instantly to the target without traversing the intervening distance.

These two solutions best answer the question to me conceptually. I never knew there was such a space/time measurment called a Plank before either (you learn something new everday ) Although it all comes down to whether infinite series can add up to a finite answer.

I was in a hurry to get to work, therefore my rather short comment.

Another way to disprove it would be the consideration that at each point in time t, the arrow is in a location s(t). At the next timepoint t'>t, the arrow is in s(t'). The velocity v = (s(t') - s(t)) / (t' - t) is equal for all t' In an inertial reference frame. Therefore at each location s(t') at a given timepoint t', also the constant velocity v is present.

But as others have pointed out, the problem at hand lies within his lack of a concept of the limit of a function, either because Zenon wasn't aware of it or he successfully exploited this limitation. Although his intentions weren't passed down, I would agree with those given above to be most likely to be correct.

This was the best mathematical answer. Although I admit, that took awhile for me to fully understand you.

Rep to all three of you!

[Juvenal]

Now the final question is, if there are infinite moments in the second (which the arrow has to travel through all of them) how does the arrow ever reach the target? Because it obviously does.
That’s what I don’t get. How can it travel through infinite moments in space/time and reach the target??

This is Zeno's "Paradox of the Arrow". He created it to support Parmenides doctrine of "All is One" - i.e. that all of creation is one indivisible whole and that therefore plurality and motion are illusions.

He attempts to prove by Reductio Ad Absurdum that the arrow is stationary at each of the infinite set of positions between archer and target and therefore is never actually "in motion".

Mathematically this can be refuted with calculus (i.e. that the sum of an infinite set can be finite) as has been pointed out by previous posters.

Physically, there is reason to believe that the real universe is discrete rather than continuous. If time and space are truly quantised, there may be no distance shorter than the Planck length (1.6 x 10-35 metres) or time shorter than Planck time (approx 5.4 x 10-44 seconds).

The archer should get himself a quantum tunnelling upgrade for his weapon. He can then have the arrow travel instantly to the target without traversing the intervening distance.

[Banned]

I have asked this question to many people, and no one can answer it. I believe it cannot be answered, but if you do answer it with skill I will gladly give you rep!

So lets say that an archer fires an arrow at a target.

Lets say it takes 1 second to reach the target.
Now it will take .5 seconds to reach halfway to the target.
It will take .25 seconds to a quarter distance to the target, ok?

Now this is the part that will blow your mind. We can keep dividing the second further and further and further. You come to realise that there are infinite moments in a second.

Now the final question is, if there are infinite moments in the second(which the arrow has to travel through all of them) how does the arrow ever reach the target? Because it obviously does.

Thats what I dont get. How can it travel through infinite moments in space/time and reach the target??

Infinite refers to something that's beyond our level of ascertainment, but it necessarily doesn't imply perpetuity. It simply means a quantity that's incalculable. Concisely, we cannot calculate the limits of infinity and therefore, we conclude it as interminable. That's where you get the answer to this question.

[Niccolo Machiavelli]

I was in a hurry to get to work, therefore my rather short comment.

Another way to disprove it would be the consideration that at each point in time t, the arrow is in a location s(t). At the next timepoint t'>t, the arrow is in s(t'). The velocity v = (s(t') - s(t)) / (t' - t) is equal for all t' In an inertial reference frame. Therefore at each location s(t') at a given timepoint t', also the constant velocity v is present.

But as others have pointed out, the problem at hand lies within his lack of a concept of the limit of a function, either because Zenon wasn't aware of it or he successfully exploited this limitation. Although his intentions weren't passed down, I would agree with those given above to be most likely to be correct.

[mightyfenrir]

I've heard it also but in this context: "A frog jumps half of the remaining distance to the edge of the table each jump. How many jumps until it reaches the end of the table?"

Maybe this is a little better than the archery analogy - as of course the arrow is faster and straighter when it leaves the bow than when it strikes the target (just nit-picking )

Philip Dick wrote an excellent short story concerning the frog idea. It has two professors creating a system to settle their arguments where a frog is decreased in size with every successive jump within the system. You really have to read it.

I was in a hurry to get to work, therefore my rather short comment.

Another way to disprove it would be the consideration that at each point in time t, the arrow is in a location s(t). At the next timepoint t'>t, the arrow is in s(t'). The velocity v = (s(t') - s(t)) / (t' - t) is equal for all t' In an inertial reference frame. Therefore at each location s(t') at a given timepoint t', also the constant velocity v is present.

But as others have pointed out, the problem at hand lies within his lack of a concept of the limit of a function, either because Zenon wasn't aware of it or he successfully exploited this limitation. Although his intentions weren't passed down, I would agree with those given above to be most likely to be correct.

Obviously there's no point in furthering the mathematic answer to the question

[PointOfViewGun]

First of all if an arrow reaches it's target in 1 second, it doesn't mean that it reaches the halfway in 0.5 seconds. Second, you've already given your answer. The time is not infinite in your example. It has a limit. The example of boofhead is much more logical as a paradox.

[Aetius]

First of all if an arrow reaches it's target in 1 second, it doesn't mean that it reaches the halfway in 0.5 seconds. Second, you've already given your answer. The time is not infinite in your example. It has a limit. The example of boofhead is much more logical as a paradox.

OK, I dont know why people are getting wound up about the damn acceleration of the arrow. Just assume the arrow loses none of its velocity.

Maybe the arrow was a bad example(Even though Zeno himself used it), but it is sure better than frogs that jump on tables halfway to the end. I mean when does that ever happen in reality? Why would a frog even want to do that?

[Niccolo Machiavelli]

The one with the frog on the table is a disambiguation of Achilles and the turtle (by Zenon) btw...

[Simetrical]

Zeno's problem strikes at the heart of mathematics. Is maths 'the establishment of exact truths concerning the abstract world of ideas' or rather 'a pragmatic series of useful techniques for solving problems'?

Both. They aren't contradictory. Math is a set of statements about totally abstract and non-real quantities that may, in some cases (very few for modern innovations), have practical applications.

To me maths is a language used to symbolise the universe but it can do so only imperfectly because the universe is 'non obvious'.

Theorems on, e.g., generalizations of the derivative to arbitrary topological spaces can't really be called symbolic of anything in the universe, but they're still mathematics.

No evidence can exist for continuity of reality as your knowledge of reality must be expressible in a finite amount of information – it is an assumption that reality is continuous.

All a continuous universe means is that you can't know the position (or whatever) of an arbitrary entity to infinite precision. It doesn't mean you can't gather arbitrarily fine data, as fine or coarse as you like until your information storage runs out. There's no contradiction here.

[Richard]

Simetrical, just what the hell is it you do for a living? You certainly know your maths =/

Perhaps I'm not understanding you, but of course the arrow reaches the target. You can keep dividing the amount of time that it takes (1 second to go all the way, 1/2 second to go half way, .25 second to go a quarter of the way and so on), but as the arrow travels, it slows down, thus making the amount of time it takes to go a certain distance a bit longer.

If the arrow is fired at 150 MPH, when it reaches the target it will not be travelling 150 MPH, unless the target is within 5 feet or so.

Now, I think I can answer your question, but you will be a better judge of that than I. If it takes 1 second to hit the target, the arrow doesn't travel through infinite moments within time. It travels exactly 1 second. The second can be divided up into infinite divisions, but that is irrelevant since the arrow is only concerned with taking the single second to reach it's target. Thus, not travelling though an infinite time.

Summed up very well.

[Feliks]

It's almost, but not really, like the hotel one.

There's a hotel with an infinite of rooms, all filled with an infinite amount of guests. How could hotel accommodate for an infinite amount of additional guests?

The standard answer, of course, is:

The hotel moves each current guest to the room with the room number exactly double their current room number. Thus freeing up an infinite amount of rooms.


You see? When you're dealing with infinite numbers, things don't always turn out like you'd think.

[Chaigidel]

actually the only way that an infinite calculation or movement could take place and still produce relevant and physical results -- is because the entirety of the universe occurs in something that is essentially one thing.

since the infinite amount of things occurs within something that is necessarily everywhere and everything at the same time. the distances and calculations become meaningless.

it doesnt seem meaningless because of relative time; yet all time is occuring at the same time--

[Simetrical]

Simetrical, just what the hell is it you do for a living? You certainly know your maths =/

I'm an undergraduate (third-year, starting a week from Monday) in, yes, pure mathematics. With a minor in physics, and I do programming/system administration as a hobby.

There's a hotel with an infinite of rooms, all filled with an infinite amount of guests. How could hotel accommodate for an infinite amount of additional guests?

The standard answer, of course, is:

The hotel moves each current guest to the room with the room number exactly double their current room number. Thus freeing up an infinite amount of rooms.

Of course it would take an infinite amount of time for all those rooms to be freed, since the occupant of the Nth room would have to walk N rooms down, with N unbounded. Then again, if you had a countably infinite number of new guests to accommodate, it would take an infinite amount of time for all of them to be able to walk to their rooms anyway, so I guess that's fine.

actually the only way that an infinite calculation or movement could take place and still produce relevant and physical results

A hotel with an infinite number of people in it is physically impossible. Any results will be, perhaps, imaginable, but not physically relevant.

-- is because the entirety of the universe occurs in something that is essentially one thing.

since the infinite amount of things occurs within something that is necessarily everywhere and everything at the same time. the distances and calculations become meaningless.

it doesnt seem meaningless because of relative time; yet all time is occuring at the same time--

None of that makes any physical or mathematical sense, I can tell you that.

[Silver Guard]

actually the only way that an infinite calculation or movement could take place and still produce relevant and physical results -- is because the entirety of the universe occurs in something that is essentially one thing.

According to a twisted sense of the universe I have come across from you, I have yet to find any evidence of galaxies in my shoe.

since the infinite amount of things occurs within something that is necessarily everywhere and everything at the same time. the distances and calculations become meaningless.

If the big bang theory is correct, the universe is not infinite, it is finite, just so large it is incomprehensible, and still expanding rapidly.

it doesnt seem meaningless because of relative time; yet all time is occuring at the same time

This comment makes no sense. The only thing which passes in one second is one second, not all eternity.