If space is not continuous, then is calculus wrong?

Studiot

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

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

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

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

No, I understand your point as well. What I was objecting to was your statment 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

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

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 [itex]0.999\cdots\ne1[/itex]. Zeno's paradox is exactly the same thing, just in base 2: [itex]0.111_2\cdots\equiv 1[/itex].

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.

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

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

phinds

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

jessjolt2

no?
doesnt infinitely divisible mean continuous?

Studiot

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.

jessjolt2

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

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

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

Exactly

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