Infinities in physics

[Genady]

Does the fact that set of real numbers is uncountable, as opposed to rational numbers, for example, play any role anywhere in physics? More generally, do infinite cardinal numbers play any role anywhere in physics? 

For example, in an infinite dimensional Hilbert space, does it make a difference if the number of dimensions is countable or uncountable? 

[martillo]

8 minutes ago, Genady said:

Does the fact that set of real numbers is uncountable, as opposed to rational numbers, for example, play any role anywhere in physics? More generally, do infinite cardinal numbers play any role anywhere in physics? 

For example, in an infinite dimensional Hilbert space, does it make a difference if the number of dimensions is countable or uncountable? 

Could a difference be in if there exist a grid on the space or not? I mean, a difference would be if the movement of the particles would be in discrete steps or continuous.

 

[Genady]

9 minutes ago, martillo said:

Could a difference be in if there exist a grid on the space or not? I mean, a difference would be if the movement of the particles would be in discrete steps or continuous.

 

I don't think so, because in this case the number of steps in any finite time would be finite.

[martillo]

21 minutes ago, Genady said:

I don't think so, because in this case the number of steps in any finite time would be finite.

Yes, in the case of a discrete grid the number of the steps in any finite time would be finite, countable. If there's no grid, the movement is continuous and can be thought as in uncountable infinitely small steps. 

[Genady]

5 minutes ago, martillo said:

Yes, in the case of a discrete grid the number of the steps in any finite time would be finite, countable. If there's no grid, the movement is continuous and can be thought as in uncountable infinitely small steps. 

I am looking for a difference, in physics, between infinite countable and infinite uncountable.

[studiot]

55 minutes ago, Genady said:

Does the fact that set of real numbers is uncountable, as opposed to rational numbers, for example, play any role anywhere in physics? More generally, do infinite cardinal numbers play any role anywhere in physics? 

For example, in an infinite dimensional Hilbert space, does it make a difference if the number of dimensions is countable or uncountable? 

Don't see why you need to get all fancied up;  Hilbert space ?

What is wrong with good old fashioned Hippasus ' square root of 2 ?

Or Archimedes and his Pi ?

[Genady]

2 minutes ago, studiot said:

Don't see why you need to get all fancied up;  Hilbert space ?

What is wrong with good old fashioned Hippasus ' square root of 2 ?

Or Archimedes and his Pi ?

These are mathematical examples. I'm looking for an example in a theory in physics.

Pi, e, sqrt of 2 appear in physics many times. Of course, they are irrational. But would anything in physics change if they happened to be rational? Does the fact that they are irrational play any role in physics?

[studiot]

1 minute ago, Genady said:

These are mathematical examples. I'm looking for an example in a theory in physics.

Pi, e, sqrt of 2 appear in physics many times. Of course, they are irrational. But would anything in physics change if they happened to be rational? Does the fact that they are irrational play any role in physics?

Of course it would.

The diagonal of a unit square would be different for instance.

I'm not sure if the triangle inequality or Schwartz inequality which underlie quantum theory would also be different.

 

[Sensei]

Why do you go from theoretical physics?

Go from experimental physics.. to finding or not finding quantization..

 

For example.

1 cm^3 of water has mass 1 gram. 1 g/cm^3

1 g/cm^3 / 18.016 g/mol = ~ 0.0555 mol/cm^3

~ 0.0555 mol/cm^3 * 6.022141*10^23 mol^-1 = ~ 3.3427e+22

3.3427e+22 ^ (1/3) = 32.2 mln water molecules in any X,Y,Z axis in 3D world.

0.01m / 32.2 mln = 310 pm = 0.31 nm

Can you move a quantum object with a precision of more than 0.31 nm and be really very sure that this is the result of your action and not some other thing acting on the particle?

Can you move it? With what kind of precision?

Repeat the experiment with different particles until you find a boundary that cannot pass..

It would be interesting experiment, a single, literally single atom of some heavy element which can be detected by MRI or so, put in a piece of solid ice (or whatever else transparent to MRI or so), then record its path, when it goes from the top to the bottom sinking due to its weight, due to the gravity or whatever else..

 

 

 

[Genady]

7 minutes ago, Sensei said:

Why do you go from theoretical physics?

Go from experimental physics.. to finding or not finding quantization..

 

For example.

1 cm^3 of water has mass 1 gram. 1 g/cm^3

1 g/cm^3 / 18.016 g/mol = ~ 0.0555 mol/cm^3

~ 0.0555 mol/cm^3 * 6.022141*10^23 mol^-1 = ~ 3.3427e+22

3.3427e+22 ^ (1/3) = 32.2 mln water molecules in any X,Y,Z axis in 3D world.

0.01m / 32.2 mln = 310 pm = 0.31 nm

Can you move a quantum object with a precision of more than 0.31 nm and be really very sure that this is the result of your action and not some other thing acting on the particle?

Can you move it? With what kind of precision?

Repeat the experiment with different particles until you find a boundary that cannot pass..

It would be interesting experiment, a single, literally single atom of some heavy element which can be detected by MRI or so, put in a piece of solid ice (or whatever else transparent to MRI or so), then record its path, when it goes from the top to the bottom sinking due to its weight, due to the gravity or whatever else..

 

 

 

I don't see any role of uncountability is this experiment. In fact, I don't know of any experiment where an experimental result of measurement is an irrational number.

[studiot]

3 minutes ago, Genady said:

In fact, I don't know of any experiment where an experimental result of measurement is an irrational number.

Hippasus already gave you one.

Remember the ancient greeks were primarily geometers, not algebraicists.

But not only geometers but constructive geometers.

Note that most of Euclid is couched in terms like  Proposition XXX  To construct ........

So imagine their consternation when they discovered that they could not measure the length of that diagonal, although they could undoubtedly construct it.

[Sensei]

1 minute ago, Genady said:

I don't see any role of uncountability is this experiment. In fact, I don't know of any experiment where an experimental result of measurement is an irrational number.

A simpler to visualize example. In a computer, you have an array of bits. 0 and 1. The bit index is an integer. But there are millions/billions/trillions of them in this array.

You can store PI or e, in binary in this array, but a human won't look at it deeply enough.. 6-7 decimal digits from an ordinary person is far too much..

 

An irrational number is irrational only in theory, when you use it in everyday mathematics, you don't use an irrational number, but something that is very close to it, with enough precision that is needed at the time of calculation.

 

[Genady]

Perhaps I need to make my question more specific:

Are irrationality of a number or uncountability of a set used in physics to derive a result, to prove a theorem, to formulate a model, etc.?

[Sensei]

26 minutes ago, Genady said:

In fact, I don't know of any experiment where an experimental result of measurement is an irrational number.

Is d=2 x PI x r a rational or irrational number?

The experimenter has it all turned around:

PI = d/(2 x r)

 

Edited March 16, 2023 by Sensei

[Genady]

2 minutes ago, Sensei said:

Is d=2 x PI x r a rational or irrational number?

 

It depends on r, of course.

But even when it is irrational, any physical measurement or computer representation can only give a rational approximation of it.

[Sensei]

10 minutes ago, Genady said:

15 minutes ago, Sensei said:

Is d=2 x PI x r a rational or irrational number?

It depends on r, of course.

Interesting point. 2^(1/2) * 2^(1/2) = 2. From two irrational numbers we get, a rational number.

So what rational number can be obtained from 2 x PI^2 (or whatever you meant) ?

Is PI^2 a rational number?

Is it possible to find such an r that d=2 x PI x r is rational.. ?

 

Is it possible to find such an r=PI^n that d=2 x PI x PI^n is rational.. ?

 

[Genady]

8 minutes ago, Sensei said:

Is it possible to find such an r that d=2 x PI x r is rational.. ?

Yes.

r = 1/Pi

for example.

[Sensei]

4 minutes ago, Genady said:

Yes.

r = 1/Pi

for example.

So, replace 1 by any rational, and Pis will cancel, and you have a rational result..

ps. How about making it harder: working with both 2D (obeying above) and 3D (V=3/4 x PI x r^3) ?

 

Then, there are more physical equations taking irrational as parameters, find whether they have rational answers like above..

 

[Genady]

12 minutes ago, Sensei said:

So, replace 1 by any rational, and Pis will cancel, and you have a rational result..

ps. How about making it harder: working with both 2D (obeying above) and 3D (V=3/4 x PI x r^3) ?

 

Then, there are more physical equations taking irrational as parameters, find whether they have rational answers like above..

 

I don't see what this game has to do with the OP.

I have specified my question here:

 

53 minutes ago, Genady said:

Are irrationality of a number or uncountability of a set used in physics to derive a result, to prove a theorem, to formulate a model, etc.?

 

[TheVat]

1 hour ago, Genady said:

It depends on r, of course.

But even when it is irrational, any physical measurement or computer representation can only give a rational approximation of it.

Measurements cannot even provide many rational numbers.  No one can measure 1/3. Maybe you will get 0.33332. With some margin of error.  That’s not 1/3 and it doesn't seem you can improve the accuracy of the experiment to make it so.  1/3 will always lie between error margins and beyond our observation. 

Is there any physical distinction that matters between a rational number and an irrational number if the two are closer to each other than the experimental error?

Edited March 17, 2023 by TheVat
njrfojg

[Genady]

4 minutes ago, TheVat said:

No one can measure 1/3.

I can have a line segment and, using a compas and a ruler, make a 1/3 mark on this segment. Doesn't it measure 1/3? 

[TheVat]

I think it doesn't.  I think any measurement will fall short of that precision.  And at the Planck level, it may not be possible even in principle?  You may get 0.3333333329 with some error margin.  Rational numbers may only lie in some Platonic realm. 

(similar idea is that all surfaces in the real world are not precise 2D, but some fractal expression, like around 2.11112 D)

Same thing for irrational numbers.  

And plus 1, cool thread.

Edited March 17, 2023 by TheVat
add

[Genady]

7 minutes ago, TheVat said:

I think it doesn't.  I think any measurement will fall short of that precision.  And at the Planck level, it may not be possible even in principle?  You may get 0.3333333329 with some error margin.  Rational numbers may only lie in some Platonic realm. 

(similar idea is that all surfaces in the real world are not precise 2D, but some fractal expression, like around 2.11112 D)

Same thing for irrational numbers.  

And plus 1, cool thread.

Thank you.

Yes, I agree. The numbers we get experimentally are at best rational, but we don't get all rational numbers.

But in theory? (Which is the question in OP.) Is it important anywhere in physics theory if a number is rational or irrational, or a set is countable or uncountable? 

[Genady]

@TheVat,

I think we can measure 1/3 (or any other rational number) in cases when we can measure by counting. I mean, if we have counted 1000 of something and we know that there are 3000 of them in total, we have measured 1/3, right?

Edited March 17, 2023 by Genady

[Sensei]

6 hours ago, Genady said:

I have specified my question here:

Is that the only thing that matters to you? When you can become "Planck v2.0"?

You didn't give enough thought and attention to what I said in my previous posts..

 

 

5 hours ago, TheVat said:

And plus 1, cool thread.

Indeed, interesting.  That is why I participated..

 

5 hours ago, TheVat said:

I think it doesn't.  I think any measurement will fall short of that precision.  And at the Planck level, it may not be possible even in principle?  You may get 0.3333333329 with some error margin.  Rational numbers may only lie in some Platonic realm. 

Molecules/atoms/molecules are not constant.. they are in constant motion depending on the temperature.... (or vice versa: temperature tells how much they move and/or rotate etc.)..

If you make a line of atoms/particles/molecules that will form 1/3 in what time and length should maintain this state.. ?

In which frame of reference should Y=1, and X=3, or whatever?

 

 

 

In which FoR you measure inverse-square-law.. ?

 

Edited March 17, 2023 by Sensei