If space is not continuous, then is calculus wrong?

[jessjolt2]

Ok so mathematically you can divide any number by any other (nonzero) number and you can keep dividing that number however many times you want. Like dividing 1 by 2 and then by 2 again etc. And this is the basis of the famous paradox that mathematically, you can't really move from point a to b because first you need to get to the middle of a and b. and then to the middle of the middle. and then the middle of the middle of the middle, etc.

But what if space is not continuous, but quantized? Like what if there is a smallest possible length, and you cannot be in between that length, meaning you cannot physically divide that length by 2 to get to the middle (even though mathematically you could). Wouldnt that have some serious consequences on the physical application of calculus to the real world? (maybe not when working with large bodies, but definitely with small scales?) For example the intermediate value theorem wouldn't hold true...

Idk I am not calculus expert (only had calc I and II nd basic physics) but this thought occurred to me and has bothered me..

 

[micromass]

No, calculus has never had the ambition to give an exact description of space. There are many problems with calculus as a description of space: space and time being discrete, the existence of points which are infinitesimal small, lines which have a length but not width, etc.

Calculus should never be looked at as a complete description of our physical world, but merely as a very useful approximation. That is, when you throw a ball in the air, then its path isn't an exact parabola, but it can be approximated by parabolas. This is so with everything in physics: everything is an approximation of the real world. Exactness is never claimed.

But why are our approximations so good? We don't know. This is (in my opinion) the greatest mystery of the universe. Why is math so good in approximating the universal laws?

 

[Borek]

First of all - math doesn't deal with the real space. It doesn't have to care about whether the real space is continuous or not, it does have to care about properties of the idealized space. And we define this idealized space to be as we want it to be.

 

[sophiecentaur]

Space is 'out there'. Calculus is in your head, along with the other maths you use.

 

[jessjolt2]

First of all - math doesn't deal with the real space. It doesn't have to care about whether the real space is continuous or not, it does have to care about properties of the idealized space. And we define this idealized space to be as we want it to be.

But physics uses math and deals with real space...so it should care if space is continuous or not, at least when it is used to describe motion in space and watnot...

Does anyone know of any literature that discusses the relationship between the possible discreteness of space and mathematics?

 

[Studiot]

Good evening JessJolt.

You have touched on a deep and interesting question.

Shan Majid (professor of mathematics at London University) for instance offers exactly this quantisation as the reason for our difficulty in generating grand unified theories.

See his essay "Quantum spacetime and physical reality" in the book he edited

"On Space and Time"

(Cambridge University Press)

go well

 

[HallsofIvy]

However, those are questions about physics, not about Calculus or whatever other mathematics is used to model physics.

If space is, in fact, discreet, it might mean that, for some questions about space, Calculus would not be the appropriate mathematical tool (which is what you perhaps meant), but it would not mean the Calculus itself was wrong.

 

[phinds]

But physics uses math and deals with real space...so it should care if space is continuous or not, at least when it is used to describe motion in space and watnot...

Does anyone know of any literature that discusses the relationship between the possible discreteness of space and mathematics?

You seem to be missing the point made by the previous posters. The math does NOT deal directly with the real world, it is an approximation. It is a damned GOOD approximation and gives us excellent answers but the real world really doesn't care what the math says and the math really doesn't have to worry about the real world. WE have to worry about the correlation between the two, which you are rightly attempting to do, but this particular "problem" that you have brought up just ISN'T one and we'll keep using the math as long as it gives good answers.

 

[jessjolt2]

However, those are questions about physics, not about Calculus or whatever other mathematics is used to model physics.

If space is, in fact, discreet, it might mean that, for some questions about space, Calculus would not be the appropriate mathematical tool (which is what you perhaps meant), but it would not mean the Calculus itself was wrong.

Well from my experience it seems like calculus is the main mathematical tool in describing physics...

I read Majid's essay (thanks Studiot), and he describes some math that he and others developed to include the discrete nature of spacetime, although its too complex for me to understand lol, but it's interesting that in the future perhaps some more accurate mathematics will replace calculus as a main tool for describing physical reality..

 

[jessjolt2]

You seem to be missing the point made by the previous posters. The math does NOT deal directly with the real world, it is an approximation. It is a damned GOOD approximation and gives us excellent answers but the real world really doesn't care what the math says and the math really doesn't have to worry about the real world. WE have to worry about the correlation between the two, which you are rightly attempting to do, but this particular "problem" that you have brought up just ISN'T one and we'll keep using the math as long as it gives good answers.

Sorry, I didnt mean that calculus doesn't give good approximations nd that we should stop using it. But i still think its a problem that calculus can't describe nature at small scales if calculus assumes spacetime is continuous if it possibly isn't...my point is that there should be developed a math on the basis of discrete spacetime and to see if that math can work well with both small and large scales

 

[TylerH]

Well from my experience it seems like calculus is the main mathematical tool in describing physics...

I read Majid's essay (thanks Studiot), and he describes some math that he and others developed to include the discrete nature of spacetime, although its too complex for me to understand lol, but it's interesting that in the future perhaps some more accurate mathematics will replace calculus as a main tool for describing physical reality..

But math isn't a tool for the sake of being a tool. It's a study in its own right, independent of reality. It's the fact that calculus happens to approximate the real world that makes it a tool. It wasn't designed to be a tool.

Sorry, I didnt mean that calculus doesn't give good approximations nd that we should stop using it. But i still think its a problem that calculus can't describe nature at small scales if calculus assumes spacetime is continuous if it possibly isn't...my point is that there should be developed a math on the basis of discrete spacetime and to see if that math can work well with both small and large scales

Again, calculus isn't developed to describe reality. If one is worried about discrete intervals, one can just use difference quotients and Riemann sums; problem solved.

 

[pwsnafu]

my point is that there should be developed a math on the basis of discrete spacetime and to see if that math can work well with both small and large scales

http://en.wikipedia.org/wiki/Discrete_calculus
http://en.wikipedia.org/wiki/Quantum_calculus
http://en.wikipedia.org/wiki/Quantum_differential_calculus

The problem is that these are computationally hard.

 

[phinds]

Sorry, I didnt mean that calculus doesn't give good approximations nd that we should stop using it. But i still think its a problem that calculus can't describe nature at small scales if calculus assumes spacetime is continuous if it possibly isn't...my point is that there should be developed a math on the basis of discrete spacetime and to see if that math can work well with both small and large scales

Yes, but it is NOT a problem for CALCULUS. I guess that's what threw me about your reasoning. It is a problem for US in that we may need to find a better approximation tool if it comes to that, but as other posters have pointed out, this is NOT a flaw in calculus.

I do agree w/ you that it is unfortunate that it may be that calculus, which is one of our best tools, is not applicable in situations where we might wish it to be.

 

[dalcde]

You can't say Calculus is wrong unless it isn't consistent. Calculus is just a bunch of definition that turns out to be useful.

 

[Robert1986]

. It wasn't designed to be a tool.

I'm not sure this is correct. It was the need to describe natural phenomena that lead to the discovery/development of calculus. So, I think it was, at least initially, designed to be a tool.

 

[Hootenanny]

It was the need to describe natural phenomena that lead to the discovery/development of calculus.

That depends on whether you're a Leibniz or Newton man

 

[Robert1986]

That depends on whether you're a Leibniz or Newton man

It is my understanding that Newton developed calculus as a way to describe physical stuff, is this correct? I don't really know what Liebniz was doing, but was it purely mathematical? I always thought that Liebniz's mathematical interests were motivated by solving problems. Am I wrong in this?

Either way, I understand what you are saying :)

 

[Hootenanny]

It is my understanding that Newton developed calculus as a way to describe physical stuff, is this correct? I don't really know what Liebniz was doing, but was it purely mathematical? I always thought that Liebniz's mathematical interests were motivated by solving problems. Am I wrong in this?

Either way, I understand what you are saying :)

In a nutshell, as far as I know, Newton developed calculus in order to solve physical problems, Leibniz developed calculus to find the area under a curve.

 

[Robert1986]

In a nutshell, as far as I know, Newton developed calculus in order to solve physical problems, Leibniz developed calculus to find the area under a curve.

Ahhh, I see. So it looks like Liebniz was doing purely mathematical stuff. Interesting.

 

[Stephen Tashi]

An accurate title for the original post would be "If space is not a continuum then calculus is not applicable".

I wonder how Zeno's paradoxes get resolved if calculus is not applicable.

 

[SteveL27]

An accurate title for the original post would be "If space is not a continuum then calculus is not applicable".

I wonder how Zeno's paradoxes get resolved if calculus is not applicable.

At the risk of contradicting what everyone "knows" to be true, I don't think that Zeno's paradoxes have been resolved. Mathematically they have, because we can sum an infinite series. But in the physical universe, we'd have to accept that we complete infinitely many tasks every time we take a step. There's no physical evidence that this is true.

Zeno's paradoxes are intimately related to the question of whether space is a continuum -- a question whose answer is currently unknown and may well be unknowable.

 

[phinds]

Zeno's paradoxes are intimately related to the question of whether space is a continuum

I don't think so. Zeno's paradox is just an example of how you can incorrectly apply a mathematical concept in a way that makes no sense. And it HAS been solved. We KNOW that its conclusion is silly by everyday practical evidence so the "paradox" is just a word game, based as I said, on the inappropriate application of a math concept it a nonsensical way.

 

[jessjolt2]

I don't think so. Zeno's paradox is just an example of how you can incorrectly apply a mathematical concept in a way that makes no sense. And it HAS been solved. We KNOW that its conclusion is silly by everyday practical evidence so the "paradox" is just a word game, based as I said, on the inappropriate application of a math concept it a nonsensical way.

Can't Zeno's paradox be taken as proof that space isn't continuous?

 

[Studiot]

good evening jess,

Do you understand the difference between 'continuous' and infinitely divisible?

 

[phinds]

Can't Zeno's paradox be taken as proof that space isn't continuous?

No, Zeno's paradox can be taken as proof that math can be applied inappropriately

As Korzybski famously said "the map is not the territory". Math is a model, not the real world.

 

[Studiot]

Math is a model, not the real world.

Don't you think that's a bit harsh?

Maths is a discipline in its own right with a very real world existence, even though it is abstract.

It is true that we can and do use mathematical constructs to obtain indications of the way other systems will behave because we can observe the same or very similar structures in both.

Doing so does not make either any less real or distinct any more than using balls on sticks to 'model' molecules makes invalidates either of these.

Maths, of course, is not the only abstract system in existence. Colour is another.

 

[phinds]

Don't you think that's a bit harsh?

could be ... doesn't seem that way to me

Maths, of course, is not the only abstract system in existence. Colour is another.

Color doesn't seem to be in the same league as math. Color is very subjective and even culturally variable (I don't mean just the names of the colors, I mean the perceptive training that comes with it, such as having dozens of names for hues of red and thus learning to see differences that you and I might not see at all). This doesn't happen with math.

 

[Studiot]

Colour was just the first and simplest example that came to mind.

 

[SteveL27]

No, Zeno's paradox can be taken as proof that math can be applied inappropriately

As Korzybski famously said "the map is not the territory". Math is a model, not the real world.

So would you say that Zeno's paradox has NOT been refuted or explained in physics? And if not, then would you say that it provides a proof that space can't be infinitely divisible? Where is math being applied inappropriately?

 

[phinds]

So would you say that Zeno's paradox has NOT been refuted or explained in physics? And if not, then would you say that it provides a proof that space can't be infinitely divisible? Where is math being applied inappropriately?

I'm an engineer. Zeno's paradox says you can't get from here to there. My experience says you CAN get from here to there. As an engineer, that pretty much ends it for me.

 

[SteveL27]

I'm an engineer. Zeno's paradox says you can't get from here to there. My experience says you CAN get from here to there. As an engineer, that pretty much ends it for me.

Then how do you refute/explain the paradox?

 

[D H]

Then how do you refute/explain the paradox?

Simple: It's nonsense.

The non-technical part of me sees that philosophers worry about the silliest things. Next.

The engineer in me sees this simple answer as obviously true. I get from A to B all the time. Next.

The physicst in me sees that until we know better, space and time are continuous. Motion is not a bunch of discrete tasks. Next.

The mathematician in me sees that under the right conditions, a countably infinite number of finite numbers can sum to form a finite number. This is one of those conditions. Next.

 

[phinds]

Simple: It's nonsense.

The non-technical part of me sees that philosophers worry about the silliest things. Next.

The engineer in me sees this simple answer as obviously true. I get from A to B all the time. Next.

The physicst in me sees that until we know better, space and time are continuous. Motion is not a bunch of discrete tasks. Next.

The mathematician in me sees that under the right conditions, a countably infinite number of finite numbers can sum to form a finite number. This is one of those conditions. Next.

Yep. Personally, I'm particularly attached to the engineer point of view but the others are just as good.

 

[SteveL27]

Simple: It's nonsense.

The non-technical part of me sees that philosophers worry about the silliest things. Next.

The engineer in me sees this simple answer as obviously true. I get from A to B all the time. Next.

The physicst in me sees that until we know better, space and time are continuous. Motion is not a bunch of discrete tasks. Next.

The mathematician in me sees that under the right conditions, a countably infinite number of finite numbers can sum to form a finite number. This is one of those conditions. Next.

Just for my own understanding ... could you please point out the specific logical flaw in Zeno's paradox? Saying "it's nonsense" has a certain rhetorical finality; but it's lacking in the intellectual satisfaction department.

 

[Hootenanny]

Just for my own understanding ... could you please point out the specific logical flaw in Zeno's paradox? Saying "it's nonsense" has a certain rhetorical finality; but it's lacking in the intellectual satisfaction department.

See all the other points in D_H's post. For me, his final point is the most persuasive. Zeno maintained that an infinite number of finite steps could not be finite. This was his mistake.

 

[Studiot]

Hello, Steve,

There is actually slightly more to Zeno than can just be dismissed with a wave of a paw.

Not all infinite series have a finite total.

1 + 2 + 3 + 4 + 5 + 6... \to \infty

However take "the arrow can never reach its target because before it can travel the whole distance it must travel half the distance. Before it can travel the remaining half it must travel half of that and so on."

Here the series does sum to a finite total

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}}... \to 1

So we have to take care with infinite series.

 

[lavinia]

Ok so mathematically you can divide any number by any other (nonzero) number and you can keep dividing that number however many times you want. Like dividing 1 by 2 and then by 2 again etc. And this is the basis of the famous paradox that mathematically, you can't really move from point a to b because first you need to get to the middle of a and b. and then to the middle of the middle. and then the middle of the middle of the middle, etc.

But what if space is not continuous, but quantized? Like what if there is a smallest possible length, and you cannot be in between that length, meaning you cannot physically divide that length by 2 to get to the middle (even though mathematically you could). Wouldnt that have some serious consequences on the physical application of calculus to the real world? (maybe not when working with large bodies, but definitely with small scales?) For example the intermediate value theorem wouldn't hold true...

Idk I am not calculus expert (only had calc I and II nd basic physics) but this thought occurred to me and has bothered me..

Not necessarily. One might still use a continuous model to accurately predict quantized measurements.

 

[SteveL27]

Hello, Steve,

There is actually slightly more to Zeno than can just be dismissed with a wave of a paw.

Not all infinite series have a finite total.

1 + 2 + 3 + 4 + 5 + 6... \to \infty

However take "the arrow can never reach its target because before it can travel the whole distance it must travel half the distance. Before it can travel the remaining half it must travel half of that and so on."

Here the series does sum to a finite total

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}}... \to 1

So we have to take care with infinite series.

This is exactly my point. That is a mathematical solution, which of course I've known about for years. The question on the table today is, what is the physical solution?

We have no scientific evidence or even a good theory to support the idea that a convergent infinite series can be summed in the physical universe. Nobody has ever shown the physical existence of real numbers, infinite sets, arbitrarily small intervals, and all the rest of the set theoretic mechanism needed to develop the mathematical theory of convergent infinite series.

In fact my understanding is that a physical solution to Zeno's paradox does not yet exist. Of course one can always wave one's hands and say, "Well ... it's nonsense!" but that type of argument carries no weight on a physics forum.

 

[Hootenanny]

This is exactly my point. That is a mathematical solution, which of course I've known about for years. The question on the table today is, what is the physical solution?

We have no scientific evidence or even a good theory to support the idea that a convergent infinite series can be summed in the physical universe

In fact my understanding is that a physical solution to Zeno's paradox has not been proposed. Of course one can always wave one's hands and say, "Well ... it's nonsense!" but of course that type of argument carries no weight in science.

Zeno's argument, and his "paradox", was based on his (incorrect) belief that a infinite series cannot be summed to a finite number. Therefore, refutation of this fact is a refutation of Zeno's entire paradox, with all its physical implications. In fact, there is no paradox at all, rather it should be called "Zeno's mistake".

However, if you need a physical demonstration take a pad and a pencil. Draw two points a finite distance apart. Draw a line connecting the two points.

 

[Studiot]

Hello again Steve,

How about spectral series?

By this I do not mean the mathematical spectral decomposition theorem.
I mean emission/absorption spectral series.
Here there is a diminishing step size which eventually leads a quantized system to a continuum.

@Lavinia

What would happen if your exact continuous model predicted a state between two permitted quantized ones?

This subject is beginning to be studied further by the great and the ?good? so is certainly worth further discussion at PF.

 

[lavinia]

Zeno's argument, and his "paradox", was based on his (incorrect) belief that a infinite series cannot be summed to a finite number. Therefore, refutation of this fact is a refutation of Zeno's entire paradox, with all its physical implications. In fact, there is no paradox at all, rather it should be called "Zeno's mistake".

However, if you need a physical demonstration take a pad and a pencil. Draw two points a finite distance apart. Draw a line connecting the two points.

Zeno's point may have been that infinity doesn't exist in reality and therefore that motion doesn't exist. It may have had had nothing to do with summability of series.

 

[SteveL27]

Zeno's argument, and his "paradox", was based on his (incorrect) belief that a infinite series cannot be summed to a finite number. Therefore, refutation of this fact is a refutation of Zeno's entire paradox, with all its physical implications. In fact, there is no paradox at all, rather it should be called "Zeno's mistake".

However, if you need a physical demonstration take a pad and a pencil. Draw two points a finite distance apart. Draw a line connecting the two points.

A line? Perhaps you mean, "sprinkle particles of graphite here and there on the fibers of a piece of paper. If you were to magnify your "line" you would find it full of gaping holes, and quite irregular.

This thread really brings home the distinction between a mathematician's view of the difference between math and physics; and a physicist's or engineer's view of that difference.

You can't draw anything with pencil and paper that deserves being called a line. I might call your attention to Weirstrass's everywhere-continuous but nowhere-differentiable function. Since a continuous function is one whose graph you can draw "without lifting your pencil from the paper," I'd like to see someone draw it!

Now I do take your point ... I can travel from point A to point B, hence motion must be possible. But that's why they call it Zeno's paradox. I can't travel from point A to point B because first I have to travel half the distance, etc. ... so I can't even get started.

So what is the physical resolution of this mystery?

 

[phinds]

So what is the physical resolution of this mystery?

Physically, there IS no mystery. Just walk from point A to point B. The mystery is why mathemeticians don't see it as Zeno's mistake as Hootenanny suggests.

 

[lavinia]

Physically, there IS no mystery. Just walk from point A to point B. The mystery is why mathemeticians don't see it as Zeno's mistake as Hootenanny suggests.

To me Zeno's point was that physically there was a mystery - that is that there is apparent motion but that it can not actually occur. He therefore argued that it must be an illusion - I think.

 

[Studiot]

So what is the physical resolution of this mystery?

Surely that's obvious by now?

The time taken to go from A to B is finite.

So that time is composed of an infinite number of steps. But each step is finite and diminishing.
It is purely because each step is diminishing sufficiently fast that the total remains finite.

We can show this mathematically with series theory, knowing that there are diminishing divergent series, eg the harmonic series

http://en.wikipedia.org/wiki/Divergent_series

But you were seeking a physical demonstration, what about my example?

 

[phinds]

To me Zeno's point was that physically there was a mystery - that is that there is apparent motion but that it can not actually occur. He therefore argued that it must be an illusion - I think.

So do you think that the argument that motion is an illusion is an argument worth spending time trying to resolve? You HAVE moved from point A to point B haven't you?

 

[lavinia]

I'm an engineer. Zeno's paradox says you can't get from here to there. My experience says you CAN get from here to there. As an engineer, that pretty much ends it for me.

The way I was told it Zeno argued that motion was an illusion because it was conceptually contradictory. Your point of view seems to deny the possibility of illusions and seems to assert that experience is irrational. I do not think Zeno would have found this point of view very relevant or correct.

 

[lavinia]

So do you think that the argument that motion is an illusion is an argument worth spending time trying to resolve? You HAVE moved from point A to point B haven't you?

If you believe that the world must make sense it seems that you must deal with this paradox.

 

[Studiot]

The mystery is why mathemeticians don't see it as Zeno's mistake as Hootenanny suggests.

Isn't 20 -20 hindsight wonderful?

I don't think the ancient Greeks had a theory of convergence for infinite series.
Zeno did the best he could at the time and pointed out an inconsistency in the then available theory and knowledge.

This is actually taking us away from the OP but I am labouring the point since the OP has done exactly the same with modern knowledge

Further, Professor Majid has offered this as a possible cause /reason for the current irreconciliability of relativity and quantum theory. A worthy prize indeed if such can be achieved.

So I repeat this is a subject worth serious adult consideration, rather than flippant dismissal.

 

[lavinia]

Just to show why I think convergent series was not at all on Zenos' mind but rather the proof that motion was illusory consider his second argument. An archer shoots an arrow and we watch it sail through the sky and hit its target. But at each point in time it is just where it is. So how can it be moving?