# Zenos paradox - Is math real?

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 doesnt 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.

Thanks for any insight..

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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
Homework Helper
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?

Thanks Neurocomp, not sure what you mean by step size. In my mind, it doesnt 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 isnt 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.

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 gotta make up this length in a certain amount of time.

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 gotta make up this length in a certain amount of time.
Isn't movement continuous?

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

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.

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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.

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
Staff Emeritus
Gold Member
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

I dont 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..

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

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 doesnt 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.

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Office_Shredder
Staff Emeritus
Gold Member
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

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.

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

Office_Shredder
Staff Emeritus
Gold Member
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
Homework Helper
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?"

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EnumaElish
Homework Helper
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.

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?

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 cant build any instrumentaion to pick up? And what about things we havent 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 in to 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 doesnt explain the actual thing. You seem to just stop at what works and how it appears from the human perspective.

I realise 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.

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
Staff Emeritus
Gold Member
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 arguement 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?

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 cant get past infinite fractions.