I agree with Sime, and I also gave a solution in that thread that is similar to his. — Relativist

What is your solution to the paradox? Could you explain @sime's solution to me?

Suppose spacetime is continuous. We still can't distinguish spatial measurements that differ less than a planck length, nor time measurements less than a planck time. — Relativist

I think if spacetime is discrete and our capacity to measure spacetime interval is much higher than Planck length and time then we can treat spacetime continuously, hence we can use the continuous physical models that describe reality well. We however still have to deal with Zeno's and infinite staircases paradoxes.

This suggests to me that we will make no errors by treating space and time as discrete, even if it is continuous. What's your thoughts? — Relativist

If spacetime is continuous then we are dealing with an error in treating space and time as discrete. There are numerical methods that allow us to minimize the error but at the end of the day, we cannot avoid the error at all. In most cases, we are safe if we discretize space and use good numerical methods. In the case of time, however, the error accumulates over time so we can find significant errors in our calculation in the long term. This error in the predicted variables can be catastrophic over time if the system is chaotic, such as the weather processes.

I am not a mathematician.
Proof:

Let Mx <-> x is a mathematician
Let Rx <-> x produces results in mathematics
Let t = TonesInDeepFreeze

1. Ax(Mx -> Rx)
2. Mt -> Rt {1}
3. ~Rt
4. ~Mt {1 3}
QED CIA FBI DHS MLB NBA NFL NBC CBS ABC JFK LBJ FDR ETC

I can express my solution in everyday language.

The scenario entails reaching the bottom of a staircase through a process consisting of stepping, from one step to the next. Reaching that bottom entails taking a final step.

The infinite series entails an unending series of steps. So a final step is logically impossible.

If spacetime is continuous then we are dealing with an error in treating space and time as discrete. ...

I think if spacetime is discrete and our capacity to measure spacetime interval is much higher than Planck length and time then we can treat spacetime continuously, hence we can use the continuous physical models that describe reality well. We however still have to deal with Zeno's and infinite staircases paradoxes. — MoK

OK, so we risk introducing error if we treat spacetime as discrete, but if it IS discrete, we introduce no errors by treating it (mathematically) as continuous. So treat it as continuous and use the math. Problem solved, right?

Why do we have to deal with Zeno's paradox? Is there some problem in physics where it makes a difference, or are you like the rest of us navel-gazers around here - and just curious the logical implications?

The infinite series entails an unending series of steps. So a final step is logically impossible. — Relativist

That is not a solution but the point of Zeno. If the final step is logically impossible then you cannot complete an infinite series of finite steps therefore you cannot finish the task.

OK, so we risk introducing error if we treat spacetime as discrete, — Relativist

Yes, if the spacetime is continuous and we treat it as discrete then we are introducing error.

but if it IS discrete, we introduce no errors by treating it (mathematically) as continuous. — Relativist

If spacetime is discrete we introduce error by treating it as continuous. We however might not be able to observe the error if our measurement devices are not precise enough.

So treat it as continuous and use the math. Problem solved, right? — Relativist

Yes, we can use a continuous model as far as our measurement devices are not precise enough. Otherwise, we have to use a discrete model.

Why do we have to deal with Zeno's paradox? Is there some problem in physics where it makes a difference, or are you like the rest of us navel-gazers around here - and just curious the logical implications? — Relativist

Zeno paradox is a metaphysical problem rather than a physical one. It tells us something about reality without a need for any measurements.

I suppose it can be useful to consider that if spacetime is continuous then this, and if discrete then that. But I haven't seen where it is made clear just what discrete and continuous mean. My bad if I missed it; please point me to it. Or can either of you in a sentence or two make those concepts clear?

That is not a solution but the point of Zeno. If the final step is logically impossible then you cannot complete an infinite series of finite steps therefore you cannot finish the task. — MoK

How is that not a solution? It can be framed as reductio ad absurdum:


1. Arriving at the bottom entails taking a final step
2. The defined infinite process of descent has no end
3. Therefore the infinite process of descent has no final step
4. (1)&(3) are contradictory

Zeno paradox is a metaphysical problem rather than a physical one. It tells us something about reality without a need for any measurements. — MoK

If the question can't be answered via measurement, or any other physical means, then it's unknowable. Quantum mechanics demonstrates that intuition isn't a reliable means of deciding physical* truths, so it shouldn't be too surprising.

*Although it's a metaphysical question, it pertains to the physical world.

Consider a set of points. We say that the set is continuous if there is a point between any arbitrary pair of points. We say that the set is discrete if there is a minimal distance between a pair of points. In other words, there is no point between such a pair. The example of a continuum is the real numbers and the example of a discrete is the natural numbers (the minimal distance between a pair of points is 1).

There are an infinite number of steps but we cannot complete them. The fact that we cannot complete the steps does not mean that they do not exist.

Although it's a metaphysical question, it pertains to the physical world. — Relativist

Yes, it has an implication. I think it means that spacetime is discrete.

We say that the set is continuous if there is a point between any arbitrary pair of points — MoK

Are the rationals continuous?

Between any two rationals there is another rational, right?

But the rationals are full of holes. For example the set of all rationals whose square is less than 2 has no least upper bound in the rationals. Can you see that?

Besides, a totally ordered set with the property that there is a point between any other two is called dense. Have I not previously drawn your attention to this fact? It's the definition.

https://en.wikipedia.org/wiki/Dense_set

I wasn't claiming it disproved the existence of infinitely many stairs, but it proves that an infinite number of steps cannot be completely traversed in a sequence of of steps of finite temporal duration.

This is in spite of the fact that the set of steps (the activity) maps 1:1 to the set of physical steps that comprise the stairway. The more important conclusion is that there's a logical disconnect between this logical mapping and the analogous temporal process; IOW, the mapping doesn't fully describe the temporal process; something is missing - and it would be worthwhile to develop a mathematics that accounted for this.

Many important metaphysical questions have implications for the physical world. Metaphysics tries to figure things out with conceptual analysis (which can include math and logic) and intuition. In this case, it appears the process can't reach a definitive conclusion.

But I wonder: is it really hopeless for physics? You said that treating spacetime as discrete would lead to errors if it's actually continuous. Couldn't this be tested?

Consider a set of points. — MoK

Ok. The surface of a table-top. Discrete or continuous? A sandy beach? Or the surface of a liquid? Certainly by your definition the number line continuous, but made up of discrete points - how can that be? It would seem that "discrete" and "continuous" are abstract convenient fictions their utility depending on usage in context. Thus when misused you might bet on the tortoise, but I'll bet on Achilles every time.

Correct. How about considering the point between two arbitrary points, namely a and b, to be mean, namely (a+b)/2? If not, could you please define the continuum for @tim wood in plain English?

I wasn't claiming it disproved the existence of infinitely many stairs, but it proves that an infinite number of steps cannot be completely traversed in a sequence of of steps of finite temporal duration. — Relativist

Correct. So we are on the same page.

This is in spite of the fact that the set of steps (the activity) maps 1:1 to the set of physical steps that comprise the stairway. — Relativist

Isn't the set of steps the set of physical steps? If yes why do you use a one-to-one map?

The more important conclusion is that there's a logical disconnect between this logical mapping and the analogous temporal process; IOW, the mapping doesn't fully describe the temporal process; something is missing - and it would be worthwhile to develop a mathematics that accounted for this. — Relativist

I cannot figure out what you are trying to say here. Do you mind elaborating?

Isn't the set of steps the set of physical steps? If yes why do you use a one-to-one map? — MoK

Step (the verb) = the act of setting ones foot onto the next step (the noun; a thing).

The set of actions maps to the set of things.
The stairway consists of the set of steps, which we're stipulating as being infinite. Unlike the staircase, the acts of stepping don't exist (they are actions).

please define the continuum for @tim wood in plain English? — MoK

He didn't ask for a definition of 'the continuum'. 'the continuum' is a noun. He asked for the distinction between 'continuous' and 'discrete'. 'continuous' and 'discrete' are adjectives.

'the continuum' has been defined at least three times already in this thread.

'continuous function' is the defined as usual in chapter 1 of any Calculus 1 textbook.

Other senses of 'continuous' depend on context. And definitions of 'discrete' depend on context.

How about considering the point between two arbitrary points, namely a and b, to be mean, namely (a+b)/2? — MoK

We've been considering it at least fifty times already in this thread. What about it do you want to say?

Many important metaphysical questions have implications for the physical world. Metaphysics tries to figure things out with conceptual analysis (which can include math and logic) and intuition. In this case, it appears the process can't reach a definitive conclusion. — Relativist

What about the conclusion that spacetime is discrete?

But I wonder: is it really hopeless for physics? You said that treating spacetime as discrete would lead to errors if it's actually continuous. Couldn't this be tested? — Relativist

Yes, weather forecast for example. Any chaotic system in general. Even nonchaotic systems show the error in the long term.

We say that the set is continuous if there is a point between any arbitrary pair of points. — MoK

Who is "we"? Other than you? What does "between" mean for sets in general? Or do you mean for real intervals? Usually, 'continuous' refers to functions. Perhaps there is an even more general notion of 'continuous sets', but we'd have to see it mathematically defined, in which case it's not going to be "there is a point between any two points".

Why don't you look up some mathematics rather than just making up claims about it?

Ok. The surface of a table-top. Discrete or continuous? A sandy beach? Or the surface of a liquid? — tim wood

You are talking about physical objects that have extensions in space so their location is not definable unless you talk about their center of mass. Do you know what the center of mass is? If not think of an ice cube. The center of an ice cube is its center of mass. The center of mass of the ice cube is definable though hence you can define the location of the center of mass of the ice cube. Now, you can move the ice cube along a line. This means that its center of mass moves from one point to another point along the line. So, by now you have a definition of a point, the center of mass of the ice cube, and a line, its motion along the line.

Certainly by your definition the number line continuous, but made up of discrete points - how can that be? — tim wood

Mathematicians work on abstract objects like points and lines all the time. They define a line as a set of dimensionless points and show that things are consistent. Whether these objects are real or not is subject to discussion.

It would seem that "discrete" and "continuous" are abstract convenient fictions their utility depending on usage in context. Thus when misused you might bet on the tortoise, but I'll bet on Achilles every time. — tim wood

Well, the Zeno paradox certainly threatens mathematics, especially the continuum concept. I also bet on Achilles since my common sense tells me he will win.

Many important metaphysical questions have implications for the physical world. Metaphysics tries to figure things out with conceptual analysis (which can include math and logic) and intuition. In this case, it appears the process can't reach a definitive conclusion. — Relativist
What about the conclusion that spacetime is discrete? — MoK

I haven't seen a conceptual analysis that concludes it is discrete, but my impression is that it's typically assumed to be continuous.

But I wonder: is it really hopeless for physics? You said that treating spacetime as discrete would lead to errors if it's actually continuous. Couldn't this be tested? — Relativist

Yes, weather forecast for example. Any chaotic system in general. Even nonchaotic systems show the error in the long term. — MoK

Is it your opinion, as a physicist, that chaotic systems are not (in principle) reducible to deterministic laws of physics? My impression is that the math related to chaotic systems is pertains to identifying functional patterns to make predictions. That, at least, seems to be the nature of weather forecasts - it's not that the movement of air molecules is fundamentally indeterminstic, rather it's that it's that the quantity of data that would be needed to identify the locations and trajectory of each molecule is orders of magnitude too large to be practical to compute.

Well, the Zeno paradox certainly threatens mathematics, especially the continuum concept. — MoK

Oh please :roll: While the word "continuum" is everywhere on this thread, what is really the heart of the subject is "connectedness" of sets. And the Zeno paradox does not threaten mathematics.

Correct. — MoK

Correct meaning you understand that the rationals are dense but not continuous?

How about considering the point between two arbitrary points, namely a and b, to be mean, namely (a+b)/2? — MoK

Haven't we been doing that all along? Not sure what you mean. The rationals are dense and so are the reals. But the rationals are not Cauchy-complete. They lack the least upper bound property. So you can't use denseness to characterize the continuum, since the rationals are dense but not complete. The rationals are full of holes.

If not, could you please define the continuum for tim wood in plain English? — MoK

The set of standard real numbers, as you yourself have defined it since the first post in this thread, when claiming it doesn't exist. I believe you've now come around to accepting that it does exist. So that's the mathematical continuum. The real numbers.

ps -- Technically, what I've described is a linear continuum.

Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound. — Wikipedia

The official definition of a continuum is too general for our purposes here.

The thing about Zeno's paradoxes is that there is no finite time involved. The time would be divided infinitely just as much as the space or distance. Kant's second antinomy is Zeno's paradox. Maybe Banach-Tarski is too, idn. Everything becomes like infinite balloons stretching without end as they stay in space. Discrete space sounds like the answer for whomever don't like that acid nondual or whatever approach. It's hard to say what a discrete thing would be if it couldn't be divided. The back and forth would end. "Now space does not consist of simple parts, but of spaces. Thus every part of the composite must occupy a space. But the absolutely primary parts of the composite are simple. Thus the simple occupies space. Now since everything real that occupies a space contains within itself a manifold of elements external to one another..." Kant (second antinomy, antithesis).

I still think the mathematics used in physics can already address this question. What about Conformal Cyclic Cosmology (CCC). Penrose explains how the universe goes from the big bang to infinity, how we can used compactification to bring the infinite into the finite, and have a finite beginning after the infinite "forgets" it's infinite (his idea). Relations between that which ends and that which doesn's is the essence of this debate

the Zeno paradox certainly threatens mathematics — MoK

The speed of Achilles is 10meters/1second. The speed of Tortoise is 1meter/1000seconds.

I applied mathematics to determine that, in the 100 meter race, Achilles will cross the finish line in 10 seconds and that Tortoise will cross the finish line in 100000 seconds.

Then I called my bookie Zeus "The Moose" to place my 1000 euro bet on Achilles (favored 1000000000 to 1) and turned on the TV to watch the race on MSPN (the Mythic Sports Programming Network). After Achilles won, I called The Moose to collect. He started to say that my payoff is infinitesimal, so there's no way he can pay me; but I corrected him by telling him that there are no infinitesimals in the reals and that though the payoff is small, it is not infinitesimal. So he said he'd that he'd apply the fraction of a cent to my account. Meanwhile, there were a lot of Parmenideans who lost their togas betting on Tortoise.

Seems math did a pretty good job. Maybe math threatens Zeno's paradox.