# If space is not continuous, then is calculus wrong?

by jessjolt2
Tags: calculus, continuous, space
P: 1,718
 Quote by 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?
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.
PF Gold
P: 6,489
 Quote by 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.
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.
P: 120
 Quote by micromass 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.
Mentor
P: 15,201
 Quote by Studiot 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.

 Quote by SteveL27 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.
P: 1,718
Zeno did not have the math wrong. He was not talking about convergence of series at all.
PF Gold
P: 6,489
 Quote by lavinia 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.
P: 16
 Quote by Studiot good evening jess, Do you understand the difference between 'continuous' and infinitely divisible?
no?
doesnt infinitely divisible mean continuous?
P: 5,462
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.
P: 16
 Quote by 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.
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.
 Sci Advisor Thanks PF Gold P: 12,253 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.
Thanks
PF Gold
P: 12,253
 Quote by jessjolt2 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.
P: 800
 Quote by 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.
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 ...
 Sci Advisor Thanks PF Gold P: 12,253 Yes. Ultimate Reality is a naive goal because it needs, yet, to be defined.
P: 5,462
 That is asking too much, I think
Exactly

You have to study something as it is not as you want it to be.
 PF Gold P: 641 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.
Emeritus
PF Gold
P: 9,526
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").

 Quote by jessjolt2 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.
Emeritus
PF Gold
P: 9,772
 Quote by 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.
I disagree here. It is possible for mathematics to be inconsistent, in which case, I would consider it to be "wrong".
Mentor
P: 15,201
Quote by Hootenanny
 Quote by 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.
I disagree here. It is possible for mathematics to be inconsistent, in which case, I would consider it to be "wrong".
What's worse, we don't know if the continuum (the reals) are inconsistent or incomplete. Gödel's incompleteness theorems kinda get in the way.

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