Physics and discrete mathematics

[1MileCrash]

Do you think that matter, energy, space, time, etc. are discrete, or continuous?

If they are discrete, is continuous mathematics limited to a very, very good approximation for modeling physical phenomena?

 

[Aero51]

Cellular Automatia assumes that the physical world is discreet and and yielded some very promising results. Personally, I don't know if anybody could determine if the universe is discreet or continuous; I believe they are both approximations to "reality" and are a reflection of humans' quantitative reasoning.

 

[1MileCrash]

But if they are approximations to reality, but a reflection of human perception, how could human perception perceive an inaccuracy?

Analogous to what I mean is, gold, the element, is "yellow" as humans percieve but that means nothing. It is just how or brains interpret a wave length, it doesn't objectively "look like" anything.. it only holds meaning in the realm of human perception, but that's the only realm humans can investigate, anyway..

 

[beercandyman]

I believe that the universe is discrete and that the problems we have now in coming to a more through understanding of the universe is caused by the use of the "real number system". There is no real world problem that needs the invention of continuous numbers to solve it. I also believe that transcendental and irrational numbers don't repersent anything in the real world. Imagine that (for example) the plank length is the discrete building block of length. Then the ratio of circumference to diameter (pi) has discrete values in a curved universe. This would mean that the local curvature of space can only have discrete values. This would also imply that gravitational force can only have discrete values.

I believe that the universe is compact and not continuous. There is no need for continuity and every experiment we perform shows us that the universe is quantified. Remember that the invention of the "real number system" and it's application to physics was before the discovery of quantum mechanics. Infinity does not exist and it's use, in any way, to describe reality will lead to confusion. Real numbers have an infinite number of points between any two non-identical points and this just does not happen in the real universe...

 

[1MileCrash]

I believe that the universe is discrete and that the problems we have now in coming to a more through understanding of the universe is caused by the use of the "real number system". There is no real world problem that needs the invention of continuous numbers to solve it. I also believe that transcendental and irrational numbers don't repersent anything in the real world. Imagine that (for example) the plank length is the discrete building block of length. Then the ratio of circumference to diameter (pi) has discrete values in a curved universe. This would mean that the local curvature of space can only have discrete values. This would also imply that gravitational force can only have discrete values.

I believe that the universe is compact and not continuous. There is no need for continuity and every experiment we perform shows us that the universe is quantified. Remember that the invention of the "real number system" and it's application to physics was before the discovery of quantum mechanics. Infinity does not exist and it's use, in any way, to describe reality will lead to confusion. Real numbers have an infinite number of points between any two non-identical points and this just does not happen in the real universe...

That is exactly what I feel. However, I am not far into my physics education, so I can't "authoritively" state anything.

But I can't count the number of times I've noticed the contradictions in my physics class. Two charges, when touched, will have the same charge, so (q1+q2)/2. Except what if one of those starting charges was an odd multiple of the elementary charge, and the other even? I end up with two non-integer elementary charge multiples.

The relativity work I've done regards finding a number that expresses how much length contracts/time dilates at a given relative speed but, what if the length is 10 Planck lengths? It shortens by a factor of some irrational number?

 

[mikeph]

Two charges, when touched, will have the same charge, so (q1+q2)/2. Except what if one of those starting charges was an odd multiple of the elementary charge, and the other even? I end up with two non-integer elementary charge multiples.

A charge is not a particle, it is a property of a particle or object. To be more precise you should say that two conductors, when touched, will share their charges. When you make electrical contact between two conductors they become a single new conductor, whose charge is the sum of the charges of the original conductors.

Electrons don't get split in half, and there's no physical justification for dividing your new composite conductor into two separate parts to the extent that you get fractions of the electron charge on each.

The relativity work I've done regards finding a number that expresses how much length contracts/time dilates at a given relative speed but, what if the length is 10 Planck lengths? It shortens by a factor of some irrational number?

It shortens by the Lorentz factor, which could be irrational regardless of the object whose length contraction it is describing.

 

[1MileCrash]

A charge is not a particle, it is a property of a particle or object. To be more precise you should say that two conductors, when touched, will share their charges. When you make electrical contact between two conductors they become a single new conductor, whose charge is the sum of the charges of the original conductors.

Electrons don't get split in half, and there's no physical justification for dividing your new composite conductor into two separate parts to the extent that you get fractions of the electron charge on each.

That's my point. By the way - I mean touched and then "untouched," looking at their charges separately.

It shortens by the Lorentz factor, which could be irrational regardless of the object whose length contraction it is describing.

Which implies that one could observe a non-integer amount of Planck lengths?

 

[mikeph]

That's my point. By the way - I mean touched and then "untouched," looking at their charges separately.

Then the odd electron will either be attached to one or the other. In an ideal world it may even hover in space between the two as they are separated, since the electric field will be symmetric, there should be no force on it towards either sphere. But it certainly doesn't split.

Which implies that one could observe a non-integer amount of Planck lengths?

I don't see why not, at least under the framework of special relativity.

 

[1MileCrash]

Then the odd electron will either be attached to one or the other. In an ideal world it may even hover in space between the two as they are separated, since the electric field will be symmetric, there should be no force on it towards either sphere. But it certainly doesn't split.

...

Maybe I'm not making myself clear.

I know it doesn't split. Everyone knows it doesn't split. The physical reality of what is going on is not hard to deduce.

Yet our mathematics says it splits. Which doesn't make sense. We can't just take the average if the number is discrete. This is just one example. I can name formulas all day long that will almost never give actual integer multiples of quantized charge/etc.

 

[mikeph]

Yet our mathematics says it splits. Which doesn't make sense. We can't just take the average if the number is discrete. This is just one example. I can name formulas all day long that will almost never give actual integer multiples of quantized charge/etc.

You asserted that that the charge on each sphere is (c1+c2)/2. That is not necessarily correct.

"Mathematics" doesn't say the electron splits, it only leads you to a nonsensical answer from an incorrect premise.

 

[1MileCrash]

That's not my assertion, that's what my textbook claims, and any other reference I could find. I agree it is an incorrect assertion for an odd (q1+q2). That's the whole point, again.

Can you show me a way to express the charges on each sphere, correctly, without a piecewise function, using the real number line, then?

 

[Studiot]

Two charges, when touched, will have the same charge, so (q1+q2)/2.

Please post the exact words from you textbook.

 

[1MileCrash]

I don't have it. If you are saying that the formula is incorrect (as I do as well) when considering that elementary charges can't be split, please show me how to express the charge on each conductor after they have been connected, and then disconnected, using the real number line. I don't see a way without a piecewise function.

 

[mikeph]

Can you show me a way to express the charges on each sphere, correctly, without a piecewise function, using the real number line, then?

I don't really understand the question and I think it's a bit distracting. I'm just saying that I can't see any logical link between this conceptual scenario and the conclusion that an electron must split in half. The electron will probably end up on one sphere or another, based on a whole range of things. It might leak if the conductor is too negatively charged.

If you really want a way, formulate a function describing the energy of the system as a function of N charges positioned around the edges of the conductor, it will be based on the energy of Coulomb repulsion. Whatever configuration minimises that energy will be the one that the electrons adopt. If you separate the conductors, you are changing the boundary conditions and must re-formulate the problem, bearing in mind that the electrons do not necessarily have to stay on the surface of the conductor.

 

[1MileCrash]

I am not concluding that the electron must split in half. I am concluding that numbers that can be split in half and then again and then represented as a series thus creating an irrational andsoforth are "overkill" for modeling situations like this.

I personally think the electron would be caught in the conductor used to bridge the two "spheres" or what have you. That may be wrong but it makes sense to me. In that case the expression (q1+q2)/2 as an integer valued expression does explain it rather well. If one charge is 2e and the other is 3e we get (5e)/2. Integer value for a result, 2 goes into 5 two times, so each sphere has a resulting charge of 2e.

Conservation wouldn't be violated because I made the assumption that the lone electron would be stuck in the wire. That could be wrong, I'm not sure on that, but in my head it seems as though the electron would move to the more positively charged 2e but then at the middle of its journey would have a net electric field of zero acting on it.

(Note that by 5e, I mean 5 electrons, or -5 elementary charges.)

 

[Studiot]

I don't have it.

Then this is wasting everone's time until the exact passage from your textbook is known.

What is the textbook?
Someone else may have it.

 

[1MileCrash]

Then this is wasting everone's time until the exact passage from your textbook is known.

What is the textbook?
Someone else may have it.

This discussion is not about the textbook, or even about that specific example. It's not a derivation mystery, its just the average of the two charges. But since charge isn't continuous, the continuous real numbers aren't a good model for charge at small scales. That's all I am saying, this doesn't hinge on some single thing from one textbook, this isn't even about charge specifically, it was just analogous to what I am talking about.

If you say the total charge must be conserved, and both charges will be equal afterwards, that's the average of the two charges. Which is how any undergrad would answer that question. Its probably in any textbook.

 

[Studiot]

If you say the total charge must be conserved,

I am not saying anything without knowing the facts, but I would like to help.
I am not the first in this thread to wonder what your direction is.

However you seem to have an oddly antagonistic attitude towards your physics classes.

But I can't count the number of times I've noticed the contradictions in my physics class. Two charges, when touched, will have the same charge, so (q1+q2)/2. Except what if one of those starting charges was an odd multiple of the elementary charge, and the other even? I end up with two non-integer elementary charge multiples.

The relativity work I've done regards finding a number that expresses how much length contracts/time dilates at a given relative speed but, what if the length is 10 Planck lengths? It shortens by a factor of some irrational number?

If you are genuinely studying physics at a level to encounter relativity then this book may be of interest

On Space and Time edited by Majid - it explore exactly the granularity v continuity of space and time and contains essays from several famous physicists and mathematicians.

PS I still don't understand what you mean by (q₁ + q₂)/2 - nor does Google.