CoffmanPhDIs space-time discrete or continuous?

[onemind]

Hi,

I am a total retard at math and have what is probably a naive philosophical question that either has an obvious answer you all know about or has no answer and is right up there with the meaning of life.

Anyway, as you probably know, Zeno said something like, "Movement is impossible because in order to get from a to b you need to travel half the distance and in order to travel half the distance you need to travel half that distance add infinitum".

Now, mathmematicins answer this with limits but are limits just an abstract concept or is the physical universe a set of infinite limits? I mean, is math in this case just a simplification in order to deal with this problem but doesn't represent the true physical reality of movement?

Sorry if that was confusing :P

Just trying to get my head around what is real and what is just human thought.

And please no pedantic arguments about how real is real :)

Thanks for any insight..

 

[neurocomp2003]

depends on if you define a fundamental stepsize(that is for each movement you make you must at least translate 1 stepsize unit)

If you do then at that fundamental scale Zeno's statement fails...however if you can take an infinitesimal stepsize then his statement holds.

and In pure mathematics the concept of infinitesimal is a powerful concept.

in physics, it is whether you believe that there exists a fundamental set of things.

 

[HallsofIvy]

Would asking you what you mean by "real" be pedantic? Certainly mathematics gives useful answers to such questions as "how fast" or "how far". If that doesn't mean that mathematics "represents true physical reality", then I don't understand your question. What more do you want?

 

[onemind]

Thanks Neurocomp, not sure what you mean by step size. In my mind, it doesn't matter what size the move is you would need to travel half that move making movement impossible.

The only way i can get around it is that life isn't real. :P

For example, when i am having a dream in deep sleep, i can see things moving and it seems real at the time. But there are no "things" and the whole thing is just an illusion. So maybe the physical univesre is also just an illusion and from the human perspective we get fooled into thinking there are discrete "things" when the whole things is one big dream :P

Too off topic for this forum I'm sure but for me, infinite series does not solve zenos paradox but only provides a handy tool for humans to use in their reality approximations.

 

[neurocomp2003]

stepsize- it has to move at least a certain amount of distance and cannot be any smaller..hence no infinitesimal but quantized length.

Or you could think of the velocity part...its got to make up this length in a certain amount of time.

 

[Werg22]

stepsize- it has to move at least a certain amount of distance and cannot be any smaller..hence no infinitesimal but quantized length.

Or you could think of the velocity part...its got to make up this length in a certain amount of time.

Isn't movement continuous?

 

[onemind]

Thanks again Neuro. I guess step size is another human invention to quantify models but i doubt it is reality.

And continuous makes no sense outside of human reasoning.

I guess i'll just die ignorant like everyone else :P

 

[Werg22]

And continuous makes no sense outside of human reasoning.

An utterly ridiculous assertion. First, sense is part of human reasoning, how can you even talk about sense "outside" human reasoning? Second, it is perfectly sensible in absolute terms: an object that moves from A to B on a straight path will pas through every singular point position between A and B. In other words, time (hence movement) can always be broken down in smaller quantities. Anything else would be admitting that teleportation exists.

 

[onemind]

an object that moves from A to B on a straight path will pas through every singular point position between A and B

An utterly ridiculous assertion. There are infinite singular point positions between point A and B therefore it would take an infinite amount of time to cross them all. It would go for eternity. Continuous only makes sense in the human mind, just like 3 dimensions when we find it hard if not impossible to visualise a 16 dimensional universe.

 

[Werg22]

There are infinite singular point positions between point A and B therefore it would take an infinite amount of time to cross them all.

That is fallacious logic. Look at things this way: for any point between A and B, we could find a certain moment at which the object was at that point. You either accept this, or try to convince me that objects teleport in space. And your comment about dimensions sounds straight out of a sci fi. Dimensions are merely frames of chosen parameters, not some kind of unimaginable alternate realities.

 

[Office_Shredder]

How long does it take to go across a point? Zero time/point.

You've just introduced zero time/point*infinite points to traverse gives you the time it takes to move from anywhere to anywhere else. Turns out that's really just not useful

Alternatively, we can accept that measuring the time it takes to moves across a point is kind of useless, and thus we're back to standard continuous movement as we know it today

 

[onemind]

I don't think anyone gets what i am saying.

I understand the concept of continuous and agree that thinking about the amount of time between time is infinite and the amount of space between to points is infinite and is not useful but i am not talking about being useful, i am talking about how insane the concept of infinite is when clearly day to day reality appears discrete which we in turn label "continuous" out of convenience.

Bah, forget it..

 

[Werg22]

There is nothing mystic about continuity, it's a mathematical concept. Spacetime is assumed to be continuous as the mathematical definition wishes.

 

[onemind]

it's a mathematical concept

Exactly, i am not talking about the concept but the actual reality. Humans make models because they are useful but it doesn't shed any light onto the actual, complete exact ultimate reality of existence and makes space and time look completely like creations of the mind. A 2 dimensional creature sees the universe as 2 dimensional, we see it as three because our brains evolved to create our little illusions via our senses but it is not how it is.

 

[Office_Shredder]

The universe doesn't look discrete at all! In fact, only in the past 100 years or so has the concept of the universe being discrete even been a scientifically studyable concept, before that we weren't able to actually see anything that wasn't a continuous process

 

[onemind]

Fair enough but by discrete i meant in terms of movement which is the current topic.

As in, you can stop time and see the object at point a, then stop time a second later and see the object at point b even though there was an infintite amount of time in that 1 second and an infinite amount of space between point a and b.

It looks discrete because there is movement but infinity is unexplainable. The continuos concept is purely mathematical and only a model, not the actual thing.

 

[Werg22]

But what do you mean the actual thing? Who are you to affirm that?

 

[Office_Shredder]

There isn't an infinite amount of time between 0 and 1, there's just one second. Just because there are infinite points doesn't mean you spend time passing through each one (I already addressed that point in fact)

 

[StatusX]

I never understood why this is a paradox. To go half the distance takes half the time, a quarter takes a quarter the time, and so on, so the total time taken is finite, since 1/2+1/4+... is certainly less than, say, 2.

Is the problem just in completing an infinite number of "tasks"? If the definition of tasks allows them to be these increasingly smaller movements, then I would say this argument is simply proof that you can in fact complete infinitely many tasks in a finite time, rather than any other conclusion you might reach from it.

Or is the objection something like "you can't start because what would be your first step?" This is just a restatement of the question "what is the smallest positive number?", which simply has no answer, just like the question "what is the largest positive number?"

 

[EnumaElish]

I never understood why this is a paradox. To go half the distance takes half the time, a quarter takes a quarter the time, and so on, so the total time taken is finite, since 1/2+1/4+... is certainly less than, say, 2.

Assuming constant velocity, you are right. But how should one define velocity as Δd ---> 0 and Δt ---> 0? Isn't v = 0/0 at "that point"?

Is the problem just in completing an infinite number of "tasks"? If the definition of tasks allows them to be these increasingly smaller movements, then I would say this argument is simply proof that you can in fact complete infinitely many tasks in a finite time, rather than any other conclusion you might reach from it.

I think this is a restatement of the OP.

Or is the objection something like "you can't start because what would be your first step?" This is just a restatement of the question "what is the smallest positive number?", which simply has no answer, just like the question "what is the largest positive number?"

I believe this is a related but separate paradox.

 

[neurocomp2003]

As for the paradox...to my understanding it implies that you will never reach the point by infinite actions...not that it takes infinite time. So you divide by say half each timestep so to go from Ai to B you would have to get to Ai+1 first which is half the distance...and thus never reaching B.

However with a discrete stepsize...you would have to claim that at some An...it will reach from A to B because of this stepsize...

as for whether "movement continuous?" I guess that would mean you could observe it at every instance of time? It could be. But if you couldn't observe it at every fraction of time would you claim it to be continuous?

 

[onemind]

But what do you mean the actual thing?

The actual reality, not just the version of it the human brain constructs via sensory input. For example, the electromagnetic spectrum is huge but the human eye can only see a slither of it. Obviously we now have instrumentation to pick up infrared and ultra violet ect but what about dark energy that we are intuitively aware of but can't build any instrumentaion to pick up? And what about things we haven't even thought of? What if it is a multiverse like some famous cosmologists theorise? We can only see this universe but will be unawarew of the one next to us. As Dawkins said at the TED conference, its queerer than we suppose.By actaul thing, i mean everything that exists beyond human perception or any other living things perception. And who are you to affirm that we know of all that exists?

Who are you to affirm that?

Why do i have do be anyone to affirm anything? Why does your ego butt into a geniun misunderstanding of mine?

There isn't an infinite amount of time between 0 and 1, there's just one second

1/2 second, 1/4 second, 1/8 second +infinity.

Just because there are infinite points doesn't mean you spend time passing through each one (I already addressed that point in fact)

You addressed it with the concept of continuity which is just a concept that is apparent but it still doesn't explain the actual thing. You seem to just stop at what works and how it appears from the human perspective.

I realize now that this is an unaswerable question. We can partially understand it with mathematical concepts and models to make things work but is is only our version of it.

 

[onemind]

as for whether "movement continuous?" I guess that would mean you could observe it at every instance of time? It could be. But if you couldn't observe it at every fraction of time would you claim it to be continuous?

Thanks Neuro, i think you are the only one that kind of grasps where i am coming from.

 

[Hurkyl]

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. ~Albert Einstein, Sidelights on Relativity​

Since the discussion is not about mathematics, but how it pertains to describing reality, I figured it would be more appropriate in the philosophy section.



I could write a long reply, but I'll keep it short and simple.

onemind: you keep asserting that "real" "movement" is not "continuous". Therefore, you must prove your assertion. Make an argument supporting your case. Zeno's argument (at least how it's usually stated) is very poor -- there is nothing in his argument from which one can conclude that movement is impossible.


In all seriousness, there should not be a discussion -- neither Zeno nor you has proven that if space is a continuum then motion is impossible. You are not entitled to an "answer"; if you wish to assert that motion is not "continuous", then it's your job to prove it. It is not your job to sit and feel superior until someone else happens to provide a counterargument you happen to accept.

The point of giving you an "answer" is to try and help you work through your misconceptions. For example, as Office Shredder said,

There isn't an infinite amount of time between 0 and 1, there's just one second.​

You replied

1/2 second, 1/4 second, 1/8 second +infinity.​

Where the heck did that come from? What bearing does that have on anything?

 

[onemind]

Where the heck did that come from? What bearing does that have on anything?

High speed cameras can break the second down into 40000 frames. Better technology could break the second down further and potentially keep going forever. Whats hard to understand about infinite fractions?

. It is not your job to sit and feel superior

More like where did that come from? Who's feeling superior? Whats with all this ego bs?

I am not trying to give an answer, i was asking a question. So your saying i should just accept any answer that comes to me? I understand the concept of continuity as a mathematical concept and its useful for creating other models involving change but no matter what the model is, my brain can't get past infinite fractions.

 

[Hurkyl]

High speed cameras can break the second down into 40000 frames. Better technology could break the second down further and potentially keep going forever. Whats hard to understand about infinite fractions?

Well, the first problem is that every fraction is finite...

More like where did that come from? Who's feeling superior? Whats with all this ego bs?

I am not trying to give an answer, i was asking a question.

Shall we review?

for me, infinite series does not solve zenos paradox but only provides a handy tool for humans to use in their reality approximations.

I guess step size is another human invention to quantify models but i doubt it is reality.

And continuous makes no sense outside of human reasoning.

An utterly ridiculous assertion. There are infinite singular point positions between point A and B therefore it would take an infinite amount of time to cross them all. It would go for eternity. Continuous only makes sense in the human mind, just like 3 dimensions when we find it hard if not impossible to visualise a 16 dimensional universe.

i am talking about how insane the concept of infinite is when clearly day to day reality appears discrete which we in turn label "continuous" out of convenience.

there was an infintite amount of time in that 1 second and an infinite amount of space between point a and b.

It looks discrete because there is movement but infinity is unexplainable. The continuos concept is purely mathematical and only a model, not the actual thing.

I realize now that this is an unaswerable question. We can partially understand it with mathematical concepts and models to make things work but is is only our version of it.

These don't look like quotes from someone who is not trying to give answers.

 

[onemind]

I am not giving the ultimate answer, just not accepting a mathematical model as reality.

I can't prove it either way but for me personally, the model is not the thing and is not a suitable explanation in the philosophical sense.

I think you had it right when you quoted einstein

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. ~Albert Einstein, Sidelights on Relativity

Why you would take it a step further and tell me to accept the continuity model as reality after quoting einstein beats me.

 

[Hurkyl]

Why you would take it a step further and tell me to accept the continuity model as reality after quoting einstein beats me.

Huh? I said no such thing.


The point of what I think you mean by "continuity model" is to thoroughly defeat the notion that "motion is impossible" is a necessary consequence of "there are infinitely many points between here and there".

Of course, physical theories model space-time as a continuum because it works, whereas discrete models do not work.


The main thing I am advising you to do is to cease your assertion spree. It's really not helping.



I do have a comment on the content of your posts, not just on the form:

As in, you can stop time and see the object at point a, then stop time a second later and see the object at point b even though there was an infintite amount of time in that 1 second and an infinite amount of space between point a and b.

It looks discrete because there is movement but infinity is unexplainable. The continuos concept is purely mathematical and only a model, not the actual thing.

To restate what Office Shredder said, there is not an infinite amount of time in one second.

The amount of time in one second is exactly one second.
What is infinite is the number of instants that occur during one second. (at least, according to our leading physical theories)

In what way could you possibly mean "infinity is unexplainable"?

 

[onemind]

In what way could you possibly mean "infinity is unexplainable"?

1 second is finite but it is a measure of infinite points. Finite infinity.

 

[kant]

Hi,

I am a total retard at math and have what is probably a naive philosophical question that either has an obvious answer you all know about or has no answer and is right up there with the meaning of life.

Anyway, as you probably know, Zeno said something like, "Movement is impossible because in order to get from a to b you need to travel half the distance and in order to travel half the distance you need to travel half that distance add infinitum".

Now, mathmematicins answer this with limits but are limits just an abstract concept or is the physical universe a set of infinite limits? I mean, is math in this case just a simplification in order to deal with this problem but doesn't represent the true physical reality of movement?

Sorry if that was confusing :P

Just trying to get my head around what is real and what is just human thought.

And please no pedantic arguments about how real is real :)

Thanks for any insight..

Zenos thinks that the distence that is needed to complete a infinite number of steps is also infinite. This is wrong, since we know that the series below is convergent.

1+1/2+ 1/2^2 +1/2^3....

 

[onemind]

But it never actually gets to the limit.

 

[arunbg]

From the wikipedia article,

Issues with the proposed calculus-based solution

A suggested problem with using calculus to try to solve Zeno's paradoxes is that this only addresses the geometry of the situation, and not its dynamics. It has been argued that the core of Zeno's paradoxes is the idea that one cannot finish the act of sequentially going through an infinite sequence, and while calculus shows that the sum of an infinite number of terms can be finite, calculus does not explain how one is able to finish going through an infinite number of points, if one has to go through these points one by one. Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself, independent of how much time such an act would require.

Another way of putting this is as follows: If Zeno's paradox would say that "adding an infinite number of time intervals together would amount to an infinite amount of time", then the calculus-solution is perfectly correct in pointing out that adding an infinite number of intervals can add up to a finite amount of time. However, any descriptions of Zeno's paradox that talk about time make the paradox into a straw man: a weak (and indeed invalid) caricature of the much stronger and much simpler inherent paradox that does not at all consider any quantifications of time. Rather, this much simpler paradox simply states that: "for Achilles to capture the tortoise will require him to go beyond, and hence to finish, going through a series that has no finish, which is logically impossible". The calculus-based solution offers no insight into this much simpler, much more stinging, paradox.

A thought experiment used against the calculus-based solution is as follows. Imagine that Achilles notes the position occupied by the tortoise, and calls it first; after reaching that position, he once again notes the position the turtle has moved to, calling it second, and so on. If he catches up with the turtle in finite time, the counting process will be complete, and we could ask Achilles what the greatest number he counted to was. Here we encounter another paradox: while there is no "largest" number in the sequence, as for every finite number the turtle is still ahead of Achilles, there must be such a number because Achilles did stop counting.

 

[neurocomp2003]

To cleanup that quote of arunbg:
3 concepts - time, space,action (inregards to the paradox).
the time between [0,1] is 1 time-metric (doesn't matter which metric scale)
the space between [0,1] is 1 space-metric (doesn't matter which metric scale)
the action(dynamics) is the event of going from point Ai to point Ai+1

So to my understanding of onemind' grasp of the paradox.

the traveller going from 0-1 space in 0-1 time should never reach either because they must perform inf# of actions. So according to onemind(IMO), he's asking how do we justify using math in physics. Which i would answer they're ain't no other TOOL. As for the paradox itself...it would be resolved with a stepsize or fundamental speed. But if space was truly continuous, you would still have to account for velocity.

 

[Hurkyl]

But it never actually gets to the limit.

What does it matter if any of its partial sums are unequal to the limit?

 

[Hurkyl]

The calculus-based solution offers no insight into this much simpler, much more stinging, paradox.
...
there must be such a number because Achilles did stop counting.

It's not very stinging at all; this statement is flawed. If Achilles counted as stated in the article, then he finished counting; he went through every natural number.

Of course, if he was counting in ordinal numbers instead of natural numbers, then he passes the turtle as he says $\omega$.