Is space-time discrete?

A set S in a topological space is discrete if every element x in S has a neighborhood U such that S ∩ U = {x}. In the case of the real line, using the standard topology, this would mean each real number is an element of an open interval that contains no other real number. That's quite false, of course. No open interval containing 0, for instance, fails to contain other real numbers.

As a so-called "totally ordered set," however, the real line qualifies as a "continuous set."

Originally posted by @wolfgang59
No.
A line is continuous.
No matter how you look at it.

A line can be used to represent discrete data (for example population growth)
but that is not the same as saying it is discrete, that doesn't make sense.

If you look at the points within the line, the individual points are discrete.

Originally posted by @eladar
If you look at the points within the line, the individual points are discrete.

but the individual points are not the continuous line so, although they are discrete, that doesn't mean the line is. And a continuous line isn't discrete no matter how you look at it. So wolfgang's assertion was correct and yours was incorrect.

Originally posted by @humy
but the individual points are not the continuous line so, although they are discrete, that doesn't mean the line is. And a continuous line isn't discrete no matter how you look at it. So wolfgang's assertion was correct and yours was incorrect.

As I said, it depends on how you choose to look at it. If you choose to look at the entire line it would be continuous. If you look at individual points it is discrete.

Originally posted by @eladar
As I said, it depends on how you choose to look at it. If you choose to look at the entire line it would be continuous. If you look at individual points it is discrete.

You're equivocating. Continuity is a property of a set. If you take an individual element of the set and then point out that it doesn't have the property, in this case continuity, you cannot then go on to say that this must be a property of the containing set. Aside from which, if your statement were correct you'd have a contradiction and have disproven the existence of lines which might be a problem for geometry.

Originally posted by @deepthought
You're equivocating. Continuity is a property of a set. If you take an individual element of the set and then point out that it doesn't have the property, in this case continuity, you cannot then go on to say that this must be a property of the containing set.

Exactly!
We often have;

property of a set ≠ property of individual elements of the set

The set of all identical twins doesn't itself have an identical twin.

The set of all things that are not a set isn't itself not a set.

Eladar

You should study what is known as the logical error of equivocation.
Wiki doesn't in my opinion do a good job of explaining it but, this is nevertheless a start;
https://en.wikipedia.org/wiki/Equivocation
BUT, unlike what that above link suggests, equivocation is more generic in its scope than merely 'calling two different things by the same name' because it may not involve 'names' of things.

Originally posted by @deepthought
You're equivocating. Continuity is a property of a set. If you take an individual element of the set and then point out that it doesn't have the property, in this case continuity, you cannot then go on to say that this must be a property of the containing set. Aside from which, if your statement were correct you'd have a contradiction and have disproven the existence of lines which might be a problem for geometry.

I was just pointing out that it all depends on how you wish to look at it. Much of life is how you wish to look at it.

Originally posted by @eladar
it all depends on how you wish to look at it.

no, it clearly isn't. We have now very clearly explained to you how it isn't.

Originally posted by @humy
no, it clearly isn't. We have now very clearly explained to you how it isn't.

You are like the puppy dog in cartoons yapping after the bigger dogs do something.

Originally posted by @soothfast
Lee Smolin is good at raising thought-provoking questions, but often falls short of furnishing convincing answers. And it is during those times when he is the least convincing that he frequently sounds the most sure of himself. Maybe it's just his writing style, but some of his chapters have the feel of going to church or listening to a sermon on the mo ...[text shortened]... a proponent of something called quantum loop gravity, which is an alternative to string theory.

The Trouble with Physics was also the first book by Smolin that I read. Like you I also thought it was much more engaging. Three roads to quantum gravity is though provoking and even interesting at times, but I thought Smolin omitted the case against his own biases. Although I have nothing against having biases as we all do, I think he should have made a better effort to inform the reader of the cons as well as the pros for his opinions.

Smolin seems to think magnetic fields indicate that space-time is discrete because of the field lines observed with iron particles. I'm not sure why he finds that so convincing, because the Casimir Effect seems more compelling to me personally and I don't recall him mentioning it at all.

https://www.scientificamerican.com/article/what-is-the-casimir-effec/

Smolin thinks that physicists in both the string theory and loop quantum gravity camps are like religions that are divided. I think he does lean toward loop quantum gravity though. He said as much in his postscript in 2002.

Originally posted by @metal-brain
The Trouble with Physics was also the first book by Smolin that I read. Like you I also thought it was much more engaging. Three roads to quantum gravity is though provoking and even interesting at times, but I thought Smolin omitted the case against his own biases. Although I have nothing against having biases as we all do, I think he should have made ...[text shortened]... hink he does lean toward loop quantum gravity though. He said as much in his postscript in 2002.

Why do you equate Casimir effect with discrete space? The effect, as you know, is a shielding of a volume of space where the amount of spontaneous generation and destruction of virtual particles and that leads to a force causing the shielding plates to try to push the plates together. It seems to me more tied to simply proving the idea of virtual particles as being real and not a math construct.

Originally posted by @sonhouse
Why do you equate Casimir effect with discrete space? The effect, as you know, is a shielding of a volume of space where the amount of spontaneous generation and destruction of virtual particles and that leads to a force causing the shielding plates to try to push the plates together. It seems to me more tied to simply proving the idea of virtual particles as being real and not a math construct.

I don't think you can use the Casimir effect to prove the existence of virtual particles. They don't enter into the standard derivation of the effect. One simply notes that each mode of a field in a box is an harmonic oscillator, and the ground state of a harmonic oscillator has energy hf/2, where f is the frequency of the harmonic and h is Planck's constant.

Virtual particles enter physics in quantum field theory as the intermediate particles that carry forces, so you may as well say that the observed force between two magnets prove the existence of virtual particles.

The problem is that you are trying to use the existence of an effect that quantum field theory was invented to describe to prove the existence of an artifact of the theory, which seems to me to be begging the question.

Originally posted by @sonhouse
Why do you equate Casimir effect with discrete space? The effect, as you know, is a shielding of a volume of space where the amount of spontaneous generation and destruction of virtual particles and that leads to a force causing the shielding plates to try to push the plates together. It seems to me more tied to simply proving the idea of virtual particles as being real and not a math construct.

The spontaneous generation and destruction of virtual particles is what I meant. I'm not sure destruction is the best word to use though.
It was just easier to say the Casimir Effect rather than search for a more specific description. I thought you would know what I meant.

Explaining the appearance and disappearance of virtual particles in empty space seems lacking to me, but if we consider space-time is discrete there is an atomic like structure of space-time where these particles can come from. This seems like a more acceptable explanation than saying they came from nothing and science cannot explain it at all.

Quantum Entanglement seems unlikely to me unless space-time is discrete as well. When the two particles are separated it is as if there is a trail in space-time connecting them from a distance. If space-time is NOT discrete how can this spooky action at a distance take place? It seems more mysterious than not to explain it under those conditions.

I'm just saying that there is a certain convenience associated with the theory of discrete space-time. I'm not confident of it like Smolin is, but the theory would have some advantages in some ways.

I was hoping someone would make a case against space-time being discrete because I would get both sides of it then. I still have no idea either way, but I am starting to see why Smolin is compelled by the idea.

Originally posted by @deepthought
I don't think you can use the Casimir effect to prove the existence of virtual particles. They don't enter into the standard derivation of the effect. One simply notes that each mode of a field in a box is an harmonic oscillator, and the ground state of a harmonic oscillator has energy hf/2, where f is the frequency of the harmonic and h is Planck's c ...[text shortened]... prove the existence of an artifact of the theory, which seems to me to be begging the question.

But the Casimir effect is real and now measurable, a small but measureable force tending to bring two plates closer together so what else would explain that effect other than virtual particles? It has nothing to do with electric. magnetic or gravitational fields, or strong force or weak force, none of that has anything to do with Casimir, directly anyway. It's like as if there was a small vacuum between two plates and some air pressure on the other sides which would push the plates together. So it sounds an awful lot like physical pressure of virtual particles on the outside V a smaller physical pressure of virtual particles in the space between the two plates.

I wonder if this effect could be measured by say a box with balloon material around the box and of course vacuum inside the box and outside and a conductive layer on the balloon faces and conductors leading to electronice that could measure the change in capacitance CASIMIR would make. Assuming the two balloon faces are very close together to begin with.
I guess since the force would be dependent on the distance between the faces, there would be a perhaps measureable change in capacitance as you would have the two faces some distance apart, say 100 mm or so where the CASIMIR effect would most likely to be not measureable then bringing the faces closer and closer together, that force would be just like reading a capacitance manometer. We use probaby a dozen of these divices to measure actual vacuum on our machines. It would seem to me there would be a tiny change in capacitance as the faces get very close together.