Are Real Numbers Essential in Scientific Measurements and Models?

[fresh_42]

Not so. If one device measures a frequency as $1.000 \ 10^6 \mathrm{\ Hz}$ and another device measures a frequency as $6.283 \ 10^6 \mathrm{\ rad/s}$ then their ratio is $$\frac{6283}{2000 \pi}$$ which is irrational. You cannot place both measurements on the same scale with the rationals. Either the $\mathrm{Hz}$ measurements must be converted to an irrational number of $\mathrm{rad/s}$ or vice versa.

This isn't true irrationality. It is merely printed on the device.

I guess I don't like the assertion that all measurements are rationals primarily because, if we take it seriously, we would need to scrap the use of calculus in our theories. That seems like too high a price to pay.

And here is where we disagree. To me, it is a matter of convenience, whereas you seem to search for a fundamental truth behind it. I'm comfortable leaving this to the Wittgensteins, Russells, and Gödels in the world.

The world is discrete and all we can ever do in it is finite per definition. It is a bit like eating with chopsticks. One can do it, but what's wrong with a fork?

 

[Structure seeker]

Shouldn't the randomness be in focus rather than the numbers?

Agree if you take the entire density matrix rather than just probabilities.

 

[PeroK]

I guess I don't like the assertion that all measurements are rationals primarily because, if we take it seriously, we would need to scrap the use of calculus in our theories. That seems like too high a price to pay.

This is not, and never has been the debate. The theory has continuous variables, but measurement devices do not have a continuous (or even infinite) set of possible outcomes.

This is, in any case, an invalid argument. Although it has been established that matter consists of a finite number of atoms, that does not prevent a macroscopic object being modelled as a continuous mass distribution. Nor a fluid being modelled using the Navier-Stokes equations. What it means is that water on the atomic scale cannot be modelled by the Navier-Stokes equations (instead of QM).

But, back to the point. What I'm saying is that you, or even @vanhees71, cannot design a clock that measures time continuously, without a finite set of discrete ticks. That is the point at issue here. You say you can make continuous time measurements and I say you cannot. (I'm still waiting for you to specify a device which, even in theory, could measure time continuously).

Another example would be the detection of photons. Any detector can only be made of a finite number of photon-detection cells. So, although you can claim that the detector occupies a continuous region of space, it can only measure photon detection events in one of a finite number of cells. Again, the onus is on you to design a photon detector that could detect a photon as infinitely many different points in space. Even a countably infinite number of cells is, I claim, impossible.

My claim is that you cannot do either of those things. The same applies to any other measurements - ultimately you have only a finite number of particles to work with.

 

[fresh_42]

Agree if you take the entire density matrix rather than just probabilities.

I thought of the distribution function, yes.

 

[Structure seeker]

I thought of the distribution function, yes

But the density matrix also describes the correlations, mixedness and so on. If you focus merely on the randomness, there's no reason quantum mechanics should be different from general statistics. In fact the phase should also be included.

 

[fresh_42]

But the density matrix also describes the correlations, phase and so on. If you focus merely on the randomness, there's no reason quantum mechanics should be different from general statistics.

Yes, that's where a calculus comes into play. Physical quantities as distribution functions are only the basic entities, just like variables in ordinary calculus are. x alone doesn't mean a thing. Of course, we have to consider all kinds of functions and dependencies. My suggestion was to reconsider analysis from the point of view of pdf.

Whether our numbers are rational or real is indeed academic as @vanhees71 said, or a matter of convenience which is my point of view. A calculus based on pdf, however, would be - as far as I know - a new perspective. Measure theory comes close, but it focuses too much on measurability and not on the calculus part. But it could also be that there is such a calculus based on pdf and I simply do not know it.

 

[Dale]

you, or even @vanhees71, cannot design a clock that measures time continuously

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.

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.

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.

the onus is on you to design a photon detector that could detect a photon as infinitely many different points in space

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.

The same applies to any other measurements - ultimately you have only a finite number of particles to work with.

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.

 

[Dale]

A calculus based on pdf, however, would be - as far as I know - a new perspective. Measure theory comes close, but it focuses too much on measurability and not on the calculus part. But it could also be that there is such a calculus based on pdf and I simply do not know it

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

This isn't true irrationality. It is merely printed on the device

If a measurement is the number you get from a physical process then the printing on the device is an essential part of the measurement.

 

[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 #1 (started searching):

Looks as if Gian Carlo Wick has thought in that direction and developed some results. It looks more promising than I first thought. E.g. I found a paper on "stochastic calculus in discrete time" but that was just a generalization of the continuous world (master thesis).

 

[Dale]

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.

I just realized that this doesn’t matter except for clarity of communication. Either way there is a physical interaction and the generation of a number. Whether you use the word measurement to refer only to the first step or to both steps is only semantics.

Either way, the physical interaction may be continuous. Either way, the number generated is a matter of convention and convenience.

The number generated by a single measurement is typically (always?) from a discrete finite set, but it is convenient to treat it as a real number. So we do. Because we can. And because doing so can be necessary for comparing other measurements on the same scale.

 

[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.