In my view, that isn’t even true for nonnegative integers. What’s the physical representation of the relatively tiny (compared to ‘most integers’) Graham’s number (https://en.wikipedia.org/wiki/Graham's_number)?

Back to the reals: in your view, do reals that cannot be computed have good physical representations?

I think these questions mostly only matter when one tries to understand their own relation to these concepts, as GP asked.

What about if I have a rock and I pick up another rock that is slightly bigger. Do I now have 2 rocks or a bit more than 2 rocks? Which one of my rocks is 1? Maybe the second rock, so when I picked up the first rock I was actually wrong - I didn’t have one rock I had a little bit less than one rock. So now I have a little bit less than 2 rocks actually. How can I ever hope to do arithmetic in this physical representation?

The more I think through this physical representation thing the less sense it makes to me.

OK so say somehow I have 2 rocks in spite of all that. The room I am in also has 2 doors. What does the 2-ness of the rocks have in common with the 2-ness of the doors? You could say I can put a rock by each door (a one-to-one correspondence) and maybe that works with rocks and doors but if you take two pieces of chocolate cake and give one to each of two children you had better be sure that your pieces of chocolate cake are goddam indistinguishable or you will find that a one-to-one correspondence is not possible.

To me, numbers only make sense as a totally abstract concept.

Rational numbers I guess, but real numbers? Nothing physical requires numbers of which the decimal expansion is infinite and never repeating (the overwhelming majority of real numbers).

I'm not a physicist, but do we actually know if distance and time can vary continuously or is there a smallest unit of distance or time? A physics equation might tell you a particle moves Pi meters in sqrt(2) seconds but are those even possible physical quantities? I'm not sure if we even know for sure whether the universe's size is infinite or finite?

I searched what's the smallest time unit and its also planck's time constant

The smallest unit of time is called Planck time, which is approximately 5.39 × 10⁻⁴⁴ seconds. It is theorized to be the shortest meaningful time interval that can be measured. Wikipedia (Pasted from DDG AI)

From what I can tell there can be smaller time units from these but they would be impossible to measure.

I also don't know but from this I feel as if heisenberg's principle (where you can only accurately know either velocity or position but not both at the same time) might also be applicable here?

> A physics equation might tell you a particle moves Pi meters in sqrt(2) seconds but are those even possible physical quantities

To be honest, once again (I am not a physicist) but Pi is the circumference/diameter and sqrt(2) is the length of an isoceles triangle ,I feel as if a set of experiment could be generated where a particle does indeed move pi meters in sqrt(2) meters but the thing is that both of them would be approximations in the real world.

Pi in a real world sense made up of the planck's length/planck's time in my opinion can only be measured so much. So would the sqrt(2)

The thing is, it might take infinitely minute changes which would be unmeasurable.

So what I am trying to say is that suppose we have infinite number of an machine which can have such particle which moves pi meters in sqrt(2) seconds with only infinitely minute differences. There might be one which might be accurate within all the infinite

But we literally can't know which because we physically can't measure after a point.

I think that these definitions of pi / sqrt 2 also lie in a more abstract world with useful approximations in the real world which can also change given on how much error might be okay (I have seen some jokes about engineers approximating pi as 3)

They are useful constructs which actually help in practical/engineering purposes while they still lie in a line which we can comprehend (we can put pi between 3 and 4, we can comprehend it)

Now imaginary numbers are useful constructs too and everything with practical engineering usecases too but the reason that such discussion is happening in my opinion is that they aren't intuitive because they aren't between two real numbers but rather they have a completely new line of axis/imaginary line because they don't lie anyone in the real number plane.

It's kind of scary for me to imagine what the first person who thought of imaginary numbers to be a line perpendicular to real numbers think.

It literally opened up a new dimension for mathematics and introduced plane/graph like properties and one can imagine circles/squares and so many other shapes in now pure numbers/algebra.

e^(pi * i) = -1 is one of the most (if not the most) elegant equation for a reason.

The so called Planck system of units, proposed by him in 1899, when he computed what is now called Planck's constant, is just an example of how a system of fundamental units must not be defined. To explain exactly the mistakes done by Planck then requires a longer space than here.

Unfortunately, probably because most textbooks of physics do an extremely poor job in explaining the foundation of physics, which is the theory of the measurement of the physical quantities, most people are not aware that the Planck system of units is completely bogus, like also a few other similar attempts, like the Stoney system of units.

Thus far too often one can see on Internet people talking about the "Planck units" as if they would mean something.

Unlike with the "Planck units", there are fundamental constants that really mean something. For instance, the so called "constant of fine structure", a.k.a. Sommerfeld's constant, is the ratio between the speed of an electron and the speed of light, when the electron moves on the orbit corresponding to the lowest total energy around a nucleus of infinite mass.

This "constant of fine structure" is a measure of the strength of the electromagnetic interaction, like the Newtonian constant of gravitation is a measure of the strength of the gravitational interaction. The Planck length and time are derived from the Newtonian constant of gravitation, and they are so small because the gravitational interaction is much weaker, but they do not correspond to any quantities that could characterize a physical system.

For now, there exists no evidence whatsoever of some minimum value for length or time, i.e. there exists no evidence that time and length are not indefinitely divisible.