Are Real Numbers Essential in Scientific Measurements and Models?

[vanhees71]

A Rolex, or even a sundial, or just the analog voltage in a LC oscillator. If a measurement is a physical process then all of those physical processes are continuous.

Sure, time evolution in quantum theory is continuous, also position and momentum observables are continuous. That doesn't mean that it makes sense to discuss, whether the measurements deliver real or rational numbers. You anyway always have a finite resolution for any continuous observable, even in principle, as the most simple example of the position and momentum uncertainty relation show. Time is somewhat special since it's not an observable but a parameter in QT (inherited from our classical space-time concepts). Also here you have, however, an energy-time uncertainty relation (with the careful analysis of its meaning given by, e.g., Tamm). The most accurate clocks are based on transitions between atomic states, used to define the unit second in the SI based on measurements of the transition frequencies. Any transition line, however, has a finite "natural line width" you cannot avoid in principle. So also here the question, whether time is measured with real or rational numbers is mute.

There is also the galvanometer I mentioned in the OP, and other classical analog measurements. And there are many other QM measurements with continuous spectrums.

A galvanometer reading after all is based on position measurements of its pointer and again at least you have the position-momentum uncertainty relation, i.e., there's always a principle minimal limit of accuracy. This quantum limit is of course very hard to reach (although it's possible as the example of the LIGO mirrors shows). Macroscopic positions are much less accurate and the main source of noise is thermodynamical, but on the other hand that's accurate enough in the macro world, and that's the reason why macroscopic objects appear to behave according to classical physics. Again given this level of accuracy the question, whether you measure currents or voltages as real or rational numbers with your galvanometer is pretty meaningless.

One unresolved issue is whether you consider the position of the galvanometer needle to be the measurement, or whether you consider the number that you write down to be the measurement. I am still somewhat ambivalent although I tend toward the first, but the choice does have consequences. In the first case, the measurement is continuous, but cannot be easily written down. In the second case the measurement is not continuous, but it is more than just the physical process.

A measurement of course means to get "a number with an estimate of its accuracy" out. The (macroscopic) position already is in a sense the measurement, because it indeed consists of averaging over macroscopically small but microscopically large space-time intervals thus averaging out all the thermal (and of course also quantum) fluctuations.

I am not accepting that onus. I have never made any claims about photon detectors that I would have any onus to either defend or retract.

Of course photon detectors have, as any detector for "particles", a finite resolution of position, e.g., the pixels of a Si-pixel detector. You can only say that a photon was detected within a space-time interval of finite extent.

A finite number of particles may still have an infinite number of possible arrangements or states.

Here is my current thinking. In QM there are measurements with continuous spectra and in classical mechanics there are system properties that vary continuously and which can be measured. So, if a measurement is the physical process, then those are continuous. On the other hand, if a measurement is the number obtained from a physical process then there is more than just the physical process involved.

I don't understand the latter statement. Measurement devices as any piece of matter obey the physical laws and their use for measurements needs a construction based on these physical laws.

 

[Dale]

I don't understand the latter statement. Measurement devices as any piece of matter obey the physical laws and their use for measurements needs a construction based on these physical laws.

Obviously measurement devices are matter and are based on physical laws. The point is that numbers are not part of nature. I can run a given current through a given galvanometer. That will produce some amount of deflection. The number that is generated by that deflection is not set by nature, but is a matter of convention. You could choose different units, you could choose a different dimensionality, or even a different quantity entirely.

 

[vanhees71]

Sure? So?

 

[Dale]

Sure? So?

So what I said earlier.

 

[fresh_42]

I also would find that very interesting. If you ever do run into such a thing in the future, please post it!

Report #2:

A second, closer look and a question on MO resulted in:

• what do you mean?
• hopeless
• symbolism: https://en.wikipedia.org/wiki/Algebra_of_random_variables
• M.D. Springer, 1979, The Algebra of Random Variables
  https://www.amazon.de/dp/0471014060/

Seems nobody wanted to deal with the problem of how to get a hold of the dependencies. The book is unfortunately copyright-protected (and ridiculously expensive) so I cannot see what Springer did. Symbolism is of course not satisfactory, and the other answers were only an admission of lack of imagination. I see the difficulties, too, but one should expect a few more theoretical results on processes we countlessly perform every single day. I thought of velocity as an example of the quotient of distance and time randomness, coupled by the object we assign velocity to. We measure it all the time in our cars, and unfortunately, police officers do the same. I expected a bit more substance than $\pm 5km/h$ and Doppler.

 

[Dale]

Report #2:

A second, closer look and a question on MO resulted in:

• what do you mean?
• hopeless
• symbolism: https://en.wikipedia.org/wiki/Algebra_of_random_variables
• M.D. Springer, 1979, The Algebra of Random Variables
  https://www.amazon.de/dp/0471014060/

Seems nobody wanted to deal with the problem of how to get a hold of the dependencies. The book is unfortunately copyright-protected (and ridiculously expensive) so I cannot see what Springer did. Symbolism is of course not satisfactory, and the other answers were only an admission of lack of imagination. I see the difficulties, too, but one should expect a few more theoretical results on processes we countlessly perform every single day. I thought of velocity as an example of the quotient of distance and time randomness, coupled by the object we assign velocity to. We measure it all the time in our cars, and unfortunately, police officers do the same. I expected a bit more substance than $\pm 5km/h$ and Doppler.

I found this on Wiki but I got lost pretty quickly https://en.wikipedia.org/wiki/Itô_calculus

 

[fresh_42]

I found this on Wiki but I got lost pretty quickly https://en.wikipedia.org/wiki/Itô_calculus

This reminds me of my motto: "Look where the money goes!" If someone deals with randomness and wants to handle margins and risks, then it is finance.

 

[AndreasC]

This discussion reminds me of my professor in my ODE class who said: "The real world is discrete!" The rationals are already unphysical because they are dense, and the real world, well, let's stop at the nucleus size or for the idealists at Planck length, is discrete.

But spacetime is not discrete (to the best of our knowledge).

 

[fresh_42]

But spacetime is not discrete (to the best of our knowledge).

We cannot even decide this question. Or tell "what" it is! Why should the manifold spacetime be continuous and the manifold living room table be discrete?

 

[martinbn]

We cannot even decide this question. Or tell "what" it is! Why should the manifold spacetime be continuous and the manifold living room table be discrete?

A discrete spacetime will have a different group of symmetries compare to a continuous one, and different representations. This may result in different set of elementary particles.

 

[AndreasC]

We cannot even decide this question. Or tell "what" it is! Why should the manifold spacetime be continuous and the manifold living room table be discrete?

Discrete spacetime messes up a bunch of symmetries we generally know to be true. It also can't be modeled with our current mathematical tools, which is in contrast to things such as a table being modelled as "discrete".

 

[martinbn]

Discrete spacetime messes up a bunch of symmetries we generally know to be true. It also can't be modeled with our current mathematical tools, which is in contrast to things such as a table being modelled as "discrete".

Why can't it be modeled with current maths tools?

 

[AndreasC]

Why can't it be modeled with current maths tools?

Wellll I probably phrased it kinda badly, you could model it with different mathematical apparati but you can't just use the same ones we use right now, because currently spacetime is formulated using the theory of smooth manifolds and if you make it discrete it's not smooth any more, so you lose things such as derivatives etc. Of course you could come up with a discrete theory (and I believe such theories do exist) but you would then have a completely different theory, that uses different sorts of mathematical constructs. I think loop quantum gravity does something like that, but I don't know much about it, and my understanding is that spacetime isn't exactly discrete even there.

 

[apostolosdt]

If by numbers and measurements the posts in this thread mean experimental practice in a physics lab, they surely don’t reflect what I have experienced in those lab years of mine.

 

[Paul Colby]

There are infinitely many spin states for a spin 1/2 particle, but only two results for a measurement.

but but but, while any given apparatus will yield one of two spin states, one only knows the direction of this apparatus to some finite precision.

 

[cyboman]

Way over my head but gonna toss some laymen barstool talk in the mix. Feel free to ignore me if this is useless. This reminds me of something I find myself coming back to when contemplating esoteric mathematical ideas: Notions of a finite but unbounded universe. You can have a set of real numbers on the number line and those are very useful for calculus and predicting reality in the Newtonian 3D world we've evolved in, they are not incorrect. Let's call them localized approximations. But when bigger questions are asked and you start to zoom out to the galaxy scale you end up facing a situation where looking through binoculars reveals looking at the back of your own head. And then you have to let go of preconceptions, and perhaps imagine new descriptions and theories which I think Dale might be suggesting. As a laymen, he seems to be questioning established notions of valuing one number system over the other and that perhaps this favoritism is entirely convenient and arbitrary.

In the QM world, measurement is problematic. And this is where real vs rational numbers perhaps seems to breakdown. B/c before one even gets into the weight of appropriate mathematical symbolism and describing that phenomena the entire system breaks down by the very act of measuring itself.