If space is not continuous, then is calculus wrong?

[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?

 

[SteveL27]

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.

Regrettably I'm out of my physics depth at this point. Is this something that's commonly understood to reconcile Zeno's paradox?

 

[Studiot]

Regrettably I'm out of my physics depth at this point. Is this something that's commonly understood to reconcile Zeno's paradox?

Atomic emission spectra was one of the founding physical phenomena which lead to the quantum theory.

Essentially light emissions from stimulated atoms does not form a continuous spectrum of frequencies.

Light appears as a series of spectral lines at specific frequencies, with darkness in between.

The frequency spacing between these lines forms a diminishing series, eventually culminating in a continuous spectrum of emitted light frequencies above a certain value.


For example
http://en.wikipedia.org/wiki/Balmer_series

Now the interesting thing is that the mathematical solution of the continuous quantum equations leads to the same specific frequencies and forbids the dark regions. They also predict the diminishing step size and the continuous region. Further these equations are differential equations.

So this takes us back to the OP and the link between quantisation and calculus.

 

[phinds]

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

Huh? I don't get that at all. If I want to go from point A to point B, I just do it. I don't see any paradox. What is it about that that you think doesn't make sense?

 

[phinds]

Your point of view seems to ... assert that experience is irrational.

Say WHAT? If I want to move from point A to point B, I just do it. What is it about that that you find irrational? I certainly don't find anything irrational about it.

 

[lavinia]

Huh? I don't get that at all. If I want to go from point A to point B, I just do it. I don't see any paradox. What is it about that that you think doesn't make sense?

While I understand your argument you are ignoring Zeno's whole point. You are saying if it happens it makes sense. Ok so a mirage then is real - it is not a mirage. Anything is real. Fine. But Zeno's point, and the point of many others is that there has to be rational consistency to the world. You say no - it is what it is. That is a different point and is irrelevant to solving the problem of motion.

 

[phinds]

While I understand your argument you are ignoring Zeno's whole point. You are saying if it happens it makes sense. Ok so a mirage then is real - it is not a mirage. Anything is real. Fine. But Zeno's point, and the point of many others is that there has to be rational consistency to the world. You say no - it is what it is. That is a different point and is irrelevant to solving the problem of motion.

No, I understand your point as well. What I was objecting to was your statement that I had asserted that experience is irrational when I had said no such thing.

I DO understand that there is worth to pursuing the kind of things behind Zeno's paradox, what I object to is the phrasing that says Zeno's paradox shows that motion is not real.

NO, Zeno's paradox clearly CANNOT show that motion is not real because motion IS real, so the phrasing should be more like "hey, we have this really nifty, clever way of looking at motion that seems to make it not possible and since it so clearly IS possible, we need to figure out what it is about our way of looking at it that leads to such an absurd conclusion". An it seems to me that exactly that has been DONE a couple of times already in this thread. Zeno had the math wrong. It's Zeno's mistake.

 

[Dr. Seafood]

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?

I've always thought that this was a result of a convenient choice of notation and measurement. Our units, although naturally chosen, are still human constructs. If we keep building on these constructs to develop things like calculus, then of course we will well-approximate physical phenomena -- these physical phenomena are "measured" by human constructed units anyway.

 

[D H]

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.

Exactly. It was not a mistake back then. It was a puzzle. We do have such a notion now. Continuing to harp on Zeno's paradoxes of motion as anything but a lack of understanding of regarding the nature of the reals and the nature of science on the part of those ancient Greeks is a modern mistake.

Another way to look at it: At this site we no longer accept threads that try to argue that $0.999\cdots\ne1$. Zeno's paradox is exactly the same thing, just in base 2: $0.111_2\cdots\equiv 1$.

Yet another way to look at it is a failing to understand how science works. In a perhaps too condensed a nutshell, mathematicians try to prove mathematical theorems while scientists try to disprove scientific theories. There are (at least) two ways to disprove a scientific theory. One way is to attack the logic that underlies the theory. Scientific theories must be logically sound, mathematically correct. A hypothesis that doesn't add up is invalid.

Another way is to attack a scientific theory is from an angle that does not necessarily apply to mathematics. Just because the underlying math of some scientific theory is absolutely beautiful and perfectly sound does not mean the theory is correct. Science has to describe the real world. A failure here (observing just one black swan, for example) means the theory is false or is of limited applicability. This connection with reality can never be proven to be true. Science depends on observation. While one observation can prove that a theory is incorrect, mountains of observation do not prove that a theory is correct. It is merely confirming evidence.

That one black swan rule does allow us to rule out a lot, including Zeno's paradoxes of motion. The seemingly naive answer, I just walked from A to B, does it in.

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.

This is exactly what I was talking about above. There is no need for a physical solution to Zeno's paradoxes of motion. I just walked from A to B. End of story. Zeno's dichotomy fails to comport with reality. It is a falsified scientific theory. Discussing it from a scientific point of view is pointless.

 

[lavinia]

No, I understand your point as well. What I was objecting to was your statement that I had asserted that experience is irrational when I had said no such thing.

I DO understand that there is worth to pursuing the kind of things behind Zeno's paradox, what I object to is the phrasing that says Zeno's paradox shows that motion is not real.

NO, Zeno's paradox clearly CANNOT show that motion is not real because motion IS real, so the phrasing should be more like "hey, we have this really nifty, clever way of looking at motion that seems to make it not possible and since it so clearly IS possible, we need to figure out what it is about our way of looking at it that leads to such an absurd conclusion". An it seems to me that exactly that has been DONE a couple of times already in this thread. Zeno had the math wrong. It's Zeno's mistake.

Zeno did not have the math wrong. He was not talking about convergence of series at all.

 

[phinds]

Zeno did not have the math wrong. He was not talking about convergence of series at all.

If you believe that Zeno was right, then good luck getting from point A to point B.

 

[jessjolt2]

good evening jess,

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

no?
doesnt infinitely divisible mean continuous?

 

[Studiot]

Good evening jesse

no?
doesnt infinitely divisible mean continuous?

Consider the following rather strange function which consists of all the numbers between 0 and 1, none of the numbers between 1 and 2, all the numbers between 2 and 3, none of the numbers between 3 and 4 ... and so on.

Is is infinite? Yes

Is it continuous? No

Is it infinitely divisible? Yes

This function is, of course, all the tops or bottoms of a perfect square wave.

 

[jessjolt2]

Good evening jesse
Consider the following rather strange function which consists of all the numbers between 0 and 1, none of the numbers between 1 and 2, all the numbers between 2 and 3, none of the numbers between 3 and 4 ... and so on.

Is is infinite? Yes

Is it continuous? No

Is it infinitely divisible? Yes

This function is, of course, all the tops or bottoms of a perfect square wave.

i kind of see your point, but i do not see how this relates to my question?

and based on current mathematics, this function is infinitely divisible on the interval (0,1), (2,3), etc, but it is not divisible at all on (1,2), etc...

my question is, what if space is not infinitely divisible on any interval?...and in this case we would need to use mathematics which takes this discreteness of space/time into account. I notice many people here are saying how good calculus is as a model of reality. but that is not my point. i do not want to know what MODELS reality, i want to know what IS reality...basically i want to know what math mirrors and PERFECTLY describes reality.

 

[sophiecentaur]

What seems to be missing from Zeno's paradox is the fact that the successively smaller and smaller distances require successively shorter and shorter times to travel over. If the speed is constant then the time taken is the same whether you divide the total distance by the speed or do it the hard way by summing thsee smaller and smaller times. So it isn't the maths that disagrees with reality. What's wrong is the way that people interpret what the maths is telling them.

 

[sophiecentaur]

i do not want to know what MODELS reality, i want to know what IS reality...basically i want to know what math mirrors and PERFECTLY describes reality.

That is asking too much, I think. All we can expect is to produce models that are closer and closer to 'reality'. By closer to reality, I mean to be able to predict things with better and better accuracy.
I could be wrong. One day I could wake up, having recently died, and find some geezer in a long white beard telling me the exact answer to everything - but I won't hold my breath.

There are other views about the purpose and meaning of Science, of course but they haven't yet been proven, any more than my view.

 

[SteveL27]

That is asking too much, I think. All we can expect is to produce models that are closer and closer to 'reality'. By closer to reality, I mean to be able to predict things with better and better accuracy.
I could be wrong. One day I could wake up, having recently died, and find some geezer in a long white beard telling me the exact answer to everything - but I won't hold my breath.

There are other views about the purpose and meaning of Science, of course but they haven't yet been proven, any more than my view.

If science is measurement; and if all measurement is approximate; then science must always be approximate.

The question of whether there even is anything that counts as "ultimate reality" is an unknowable mystery.

I often think that if we discovered an equation that would fit on a t-shirt that explains everything there is to know about the working of the universe ... that would tell us more about ourselves than it does about the universe.

The universe is not an equation.

Ok enough philosophy for one day. I'm off to San Francisco. But first I have to go halfway there ...

 

[sophiecentaur]

Yes. Ultimate Reality is a naive goal because it needs, yet, to be defined.

 

[Studiot]

That is asking too much, I think

Exactly

You have to study something as it is not as you want it to be.

 

[Dembadon]

Without getting too philosophical, I don't accept the premise that mathematics has the ability to be "wrong." It can be used/applied incorrectly, but to suggest that it can be wrong is analogous to assigning blame to a tool for being used improperly, rather than the person who used the tool.

 

[Fredrik]

The axioms that define the real numbers were inspired by human intuition about positions along a straight line. However, in modern mathematics, an "axiom" isn't "something that's so obvious that it doesn't need to be proved". (This is how my high school teacher defined the word "axiom", but it's completely incorrect). It's just a statement that's a part of a definition. A definition simply associates an English word or a phrase with a set that does satisfy the axioms. So once we have defined the real numbers, it's impossible for theorems about real numbers to be objectively wrong. The theorems will hold for what the definition calls "real numbers". (This would be the members of a set that satisfies the axioms that define "the set of real numbers").

i do not want to know what MODELS reality, i want to know what IS reality...basically i want to know what math mirrors and PERFECTLY describes reality.

We all do, but we will never know this. There's no method we can use to obtain that information, and even if we already had it, it would be impossible to prove that what we have is a perfect description of reality.