Exploring the Concept of Infinity: Movement from A to B Explained

[Deepak K Kapur]

Hi

Any line joining points A and B has infinite points.

So, when I move from A to B, do I cross infinity?

 

[jtbell]

What exactly do you mean by "cross infinity"?

 

[Nugatory]

Hi
So, when I move from A to B, do I cross infinity?

No. You cross the finite distance between A and B.

 

[rumborak]

You cross an infinite number of infinitely small distances. They cancel out.

 

[Deepak K Kapur]

Thanks for responses..

I mean if I 'move point by point from A' i.e one point at a time, can I reach B?

 

[Deepak K Kapur]

You cross an infinite number of infinitely small distances. They cancel out.

What do you mean by 'they cancel out'?

Thanks.

 

[rumborak]

I highly suggest you read the Wikipedia article on Zeno's paradox, that is "Aristotle and the tortoise". That's what you are asking here essentially.

 

[Deepak K Kapur]

What exactly do you mean by "cross infinity"?

I mean do I travel an infinity of points when I move from A to B?

 

[Nugatory]

I mean do I travel an infinity of points when I move from A to B?

You do pass through an infinite number of points, but you also pass through a finite distance in a finite amount of time.

@rumborak's advice about reading up on Zeno's paradox is good - don't post again until you've followed it.

 

[jtbell]

Zeno's paradox, that is "Aristotle and the tortoise".

I think it's more commonly known as "Achilles and the tortoise." IIRC Aristotle wasn't much of an athlete.

 

[Deepak K Kapur]

I highly suggest you read the Wikipedia article on Zeno's paradox, that is "Aristotle and the tortoise". That's what you are asking here essentially.

I have read this article with things like dichotomy, arrow paradox, heap paradox, grain of millet etc.etc.

But... Still my question is

Space is continuous and therefore there are infinite points between A and B. If I move from A to B point by point, how can I ever reach B.

Thanks.

 

[Drakkith]

Space is continuous and therefore there are infinite points between A and B. If I move from A to B point by point, how can I ever reach B.

Don't let the math confuse you. Points are mathematical objects that help us describe the universe. The fact that there are an infinite amount of points between A and B simply doesn't stop you from moving between A and B.

 

[Deepak K Kapur]

Don't let the math confuse you. Points are mathematical objects that help us describe the universe. The fact that there are an infinite amount of points between A and B simply doesn't stop you from moving between A and B.

Your post has raised a few questions in my mind...

1. It means math does not describe nature fully/accurately. If so what about all the equations of Physics that contain an awesome amount of math?

2. What is it that stops any physical distance from being infinitely divided. If you say it's the plank's length, what is the reason for not being able to go smaller than the plank's length?

Please don't mind but your post seems to be faith oriented and not logic oriented...

Thanks.

 

[Drakkith]

1. It means math does not describe nature fully/accurately. If so what about all the equations of Physics that contain an awesome amount of math?

On the contrary. The math works just fine. There is nothing else that describes the universe more accurately that the correct application of math. I think the issue here is that you're searching for an 'intuitive' answer when there really isn't one.

To elaborate a bit on my previous post, this is actually a well known 'paradox' and there really isn't a single solution to it outside of math. Obviously we can move from point A to point B. Since we model the universe using math which uses 'points', it follows that even though there are an infinite number of points in between A and B, we also travel through all of these points as we move from A to B. The resolution is simply to accept this as a fact, much like how we accept certain things as axioms in math. Math deals with infinities and infinitesimals just fine and there's nothing paradoxical about moving between two points from the standpoint of math.

This 'resolution' may not be the one you wanted or even be a resolution at all, but I feel it's the only real answer you can get. We have to start somewhere after all. So when I say that you shouldn't let the math confuse you, I mean that this is a confusing situation with no clear answer outside of math and you shouldn't worry too much about it. Not everything is going to have an easy answer and sometimes there simply isn't an answer at all. Which is perfectly okay.

2. What is it that stops any physical distance from being infinitely divided. If you say it's the plank's length, what is the reason for not being able to go beyond the plank's length?

Nothing stops an arbitrary distance from being divided into smaller distances. But realize that we aren't taking some physical object and cutting it into pieces. We're talking about math. As long as we model space as being a continuum, there we can divide any distance up however we want.

Please don't mind but your post seems to be faith oriented and not logic oriented...

By this what you really mean is that the explanation doesn't fit your logic. And I don't mean to negatively criticize or to insult, but to bring it to your attention that our own personal logic is very rarely the same logic that math and science uses.

 

[Deepak K Kapur]

@Drakkith
Great many thanks for such a considerate answer.

I think I will have to accept this as a mystery even more deeper and confusion than the 'cause of big bang'

 

[CWatters]

If there are an infinite number of points then each is infinitely small. So if you move point-to-point you move infinitely slowly. So I'd say no you won't reach B. At least not in finite time.

 

[jbriggs444]

If there are an infinite number of points then each is infinitely small. So if you move point-to-point you move infinitely slowly. So I'd say no you won't reach B. At least not in finite time.

There is no "sequence" of point to point moves that both touches all points in an interval and respects the natural ordering of the reals.

[A sequence as is commonly understood -- where each step is indexed by a countable step number]

 

[Nidum]

Think of it this way . You are walking from A to B . The path between A and B has graduation lines like a tape measure . Your rate of progress from A to B is just set by your natural walking speed . In moving from A to B you pass over all the graduation lines - no matter how many there are - but the graduation lines on the path have no effect at all on your walking speed ..

 

[rumborak]

The Wikipedia article on Zeno's paradox has several counterarguments, but I particularly liked the one that pointed out that these "infinite points" arguments implicitly consider the object at rest at each exact point, i.e. that the object kinda jump from one point to the next. That already is erroneous; the motion is an irreducible aspect of the object. That is, you can place as many markers on the smooth motion of an object, even infinity, but that doesn't mean you're reducing motion into a collection of point jumps.

 

[russ_watters]

@Drakkith
Great many thanks for such a considerate answer.

I think I will have to accept this as a mystery even more deeper and confusion than the 'cause of big bang'

I would prefer if you learned it instead of thinking it was illogical and that you have to accept it on faith. This issue is not difficult. It ain't quantum mechanics!

Try this: if you need to measure the length of a 1m object and have a choice of meter sticks with tick marks in meters, centimeters, millimeters or micrometers, does your choice of meter stick change the length you are measuring?

 

[russ_watters]

If there are an infinite number of points then each is infinitely small. So if you move point-to-point you move infinitely slowly. So I'd say no you won't reach B. At least not in finite time.

Only if you don't divide the time intervals by the same factor as you divide the distance intervals.

 

[rumborak]

This is probably partially the usual "does physics/math describe reality, or only reality's phenomena?" discussion. Points, limits (in the mathematical sense) etc are excellent tools to describe all of our experiments, but that in turn does not necessarily mean they are physically "real" (nor does physics make a claim that they are). In that sense, one must be careful using those mathematical tools for thought experiments like this.

 

[sophiecentaur]

This is probably partially the usual "does physics/math describe reality, or only reality's phenomena?" discussion. Points, limits (in the mathematical sense) etc are excellent tools to describe all of our experiments, but that in turn does not necessarily mean they are physically "real" (nor does physics make a claim that they are). In that sense, one must be careful using those mathematical tools for thought experiments like this.

Perhaps that could be put another way. The initial description of the phenomenon is where the problem starts. The situation cannot be described in terms of finite steps because it is a continuum of states. Initially describing it in the wrong way is what introduces the 'paradox'. Maths should not beat itself up about this.
Maths is no more an artificial description of the 'real world' than any 'verbal / hand waving description. It is better in (i suggest) every case because it is more rigorous.

 

[Deepak K Kapur]

I would prefer if you learned it instead of thinking it was illogical and that you have to accept it on faith. This issue is not difficult. It ain't quantum mechanics!

Try this: if you need to measure the length of a 1m object and have a choice of meter sticks with tick marks in meters, centimeters, millimeters or micrometers, does your choice of meter stick change the length you are measuring?

You are right. The length will not change.

But suppose..

I use infinitometers (infinitely small unit of length), would I be able to measure the length of the said object?

Thanks.

 

[rumborak]

Deepak,

it is tempting to think about infinities, but without the proper math, these types of discussions often descend into contradictions and nonsensical answers.
If you *actually* want to know, you will not get around than to pick up a textbook and try to learn the math necessary.

 

[russ_watters]

You are right. The length will not change.

But suppose..

I use infinitometers (infinitely small unit of length), would I be able to measure the length of the said object?

The point is that it doesn't matter how many points there are (how small the divisions are), the length does not change: it is still just 1 meter. Even if there are an infinite number of points -- which there are.

 

[Deepak K Kapur]

The point is that it doesn't matter how many points there are (how small the divisions are), the length does not change: it is still just 1 meter. Even if there are an infinite number of points -- which there are.

I think the question is not of the sameness of
length but that of 'traversing' the said length...

 

[russ_watters]

I think the question is not of the sameness of
length but that of 'traversing' the said length...

If the length is always the same, then traversing it is always the same. Take a large step. Did you just tavel 1m, 100cm or 1000mm? All of them: they are all the same.

 

[Deepak K Kapur]

If the length is always the same, then traversing it is always the same. Take a large step. Did you just tavel 1m, 100cm or 1000mm? All of them: they are all the same.

I may be wrong but by 'traversing' I mean point by point 'traversing'.

If point are infinite, point by point 'traversing' cannot take me from A to B. Infact, I think I even will not be able to move from A in this way.

 

[russ_watters]

I may be wrong but by 'traversing' I mean point by point 'traversing'.

If point are infinite, point by point 'traversing' cannot take me from A to B. Infact, I think I even will not be able to move from A in this way.

How does this relate to your original question? Aren't we talking about reality here? If you take 1 step, you take one step. Not 100, not 1000, not a million, not infinity, one. That's reality. It is almost like you are trying to disprove reality by making up a scenario that isn't reality and disproving that!

[edit] From your second post (which it doesn't appear anyone really answered directly):

I mean if I 'move point by point from A' i.e one point at a time, can I reach B?

No. But so what? You don't do that. You can't do that -- the way you actually move is different. So what use does this question have?

 

[Deepak K Kapur]

How does this relate to your original question? Aren't we talking about reality here? If you take 1 step, you take one step. Not 100, not 1000, not a million, not infinity, one. That's reality. It is almost like you are trying to disprove reality by making up a scenario that isn't reality and disproving that!

[edit] From your second post (which it doesn't appear anyone really answered directly):

No. But so what? You don't do that. You can't do that -- the way you actually move is different. So what use does this question have?

My original question also meant point by point 'traversing'. May be I was not able to make it clear..

I am not trying to disprove anything,I just want to clear my doubts...So, please tell me...

When I take a step how many points do I traverse?

If it's infinite points, how can infinity be ever traversed?

 

[russ_watters]

My original question also meant point by point 'traversing'. May be I was not able to make it clear..

When I take a step how many points do I traverse?

If it's infinite points, how can infinity be ever traversed?

These questions are not identical to each other. When you take a step you take one step. You don't do "point by point traversing" (stopping at every point), but you do cross an infinite number of points.

So again: "point by point traversing" is not a thing. It doesn't exist. It isn't how reality works.

 

[Deepak K Kapur]

You don't do "point by point traversing" (stopping at every point), but you do cross an infinite number of points.

I think the two ideas in the above statement are contradictory.

If there is no stopping, the concept of a point ceases to exist,IMO.

Then, a 'point' is just a fictitious tool. Points don't correspond to reality.

Perhaps, a 'continuum' has no proper definition.

 

[ahiddenvariable]

The way I resolve this issue both mathematically and intuitively is to recall that a sum of an infinite series of fractions can be finite. For a convergent series, this it true. For a divergent series, this is not true. Take the convergent series 1/2 + 1/4 + 1/8 + ... The sum of this series is 1. One statement of Zeno's paradox is, in order to cover a given distance, you first have to go 1/2 the distance, then a little later, 1/4 the distance, later still, 1/8 the distance, and since there are an infinite number of such steps, with each step taking a finite time, how can you cross the entire distance in finite time? For constant velocity of transit, time and difference are transferable, and the sum of the infinite series of time steps is thus finite.

 

[russ_watters]

I think the two ideas in the above statement are contradictory.

Sorry, they are not.

If there is no stopping, the concept of a point ceases to exist,IMO.

Why? That makes no sense. The universe is filled with points that people don't travel to and stop. Whether you stop at a point has nothing to do with whether they exist (as real entities or non-physical, mathematical tools).

Then, a 'point' is just a fictitious tool. Points don't correspond to reality.

Fictitious tool or not is not a critical issue, but they most certainly do correspond to reality; The math works.

Perhaps, a 'continuum' has no proper definition.

That's just not true. We have a definition that works: you invented your own definition, that doesn't work, and are trying to use it to say reality is wrong. It makes no sense. Why do that? Why not just use what works?

You kind of jumped the shark with that post!

We've reached a point where I think we both realize that what you are saying does not match reality, but instead of accepting that it means what you are saying is wrong, you are clinging to the idea that reality is wrong. Reality can't be right or wrong, it just is. So you have to choose for yourself whether you want to describe it accurately or not. But don't say there is no logic to it: I've given you a method whereby you can physically demonstrate to yourself that what you are saying doesn't match reality and a logic/definitions that do. What works, works! What does not work does not work! Choose to accept what works!