Are Real Numbers Essential in Scientific Measurements and Models?

[Dale]

@PeroK you keep mentioning precision, both in this thread and in the previous thread. Precision is not the issue in my mind.

Precision describes the spread of a probability density function. I am talking about the sample space. You could have two random variables with equal variance but one is over the rationals and another is over the reals.

Real valued measurement is not a synonym for infinite precision, in my mind. Hopefully that helps us not talk past each other.

 

[fresh_42]

The question about the number system does remind me of the problem with AC. If we say that $(0,1)$ has a maximal element then most people wouldn't understand it. If we say "let $\displaystyle{a\in \times_{\iota\in I}A_\iota}$" then most people would continue reading without noticing that they have just used the axiom of choice.

I do not see how the number system is relevant to physics. If we measure a length as the $192 703 382 559 127 383 007 402 699$ th atom from left on the edge of our ruler, or as $1.41 \pm 0.005$ inches or as $\sqrt{2}$ inches as the theory tells us - where is the difference? The choice of the number system only says: there is definitely a length (if we use $\mathbb{R}$) or there is a length close enough to the number of atoms (if we use $\mathbb{Q}$). I prefer the existence of that length over "close enough" although I know that we will never be able to measure $\sqrt{2}$ inches.
$$
\ddot x = G_0\, , \,x(0)=0\, , \,\dot x(0)=1
$$
spits out when exactly I will hit the ground due to the completeness of the real numbers, regardless of the fact that my input variable can only be measured up to eight digits or so. The result shouldn't depend on the number system. I wouldn't really like to read: I hit the ground after $0.45152364098573090445081112433814$ seconds but only if $G_0=9.81.$ Such a result would not make sense, and if I write $0.45 \pm 0.002$ seconds, then it is not what the theory says. The truth is that I have measured a real random variable
$$
T_0=\sqrt{\dfrac{2X(0)}{G_0}}
$$
consisting of the outcome of a random variable height $X(0)$ and the outcome of a random variable called local acceleration $G_0.$ This would be what I actually have done in reality. Whether $G_0,X(0),T_0$ are real or rational is irrelevant, they are neither. However, only $\mathbb{R}$ guarantees me that there is definitely a solution to my equation which makes sense since it hurts after half a second.

 

[Dale]

I do not see how the number system is relevant to physics.

Well, you cannot do calculus with the rationals. That seems important on the theory side.

 

[fresh_42]

Well, you cannot do calculus with the rationals. That seems important on the theory side.

I'm not so sure. If we put the measurements in the center of consideration, then arbitrary close is close enough. It helps that we have names for $\mathrm{e}$ and $\pi$ but where do we need all their digits? Rational Cauchy sequences will no longer converge, nevertheless, they are still Cauchy sequences.

It's a matter of convenience, in my mind, not a matter of physics.

 

[PeroK]

It's a matter of convenience, in my mind, not a matter of physics.

The question is whether we could ever confirm by measurement that the ratio of a circle's circumference to its diameter is $\pi$.

IMO, there are a number of problems with mapping the mathematics to a physical circle.

 

[fresh_42]

The question is whether we could ever confirm by measurement that the ratio of a circle's circumference to its diameter is $\pi$.

I came across the Leibniz series for $\pi/4$ lately. I think this contains every aspect of our discussion here.
$$
\sum_{n=1}^{\infty }{\dfrac {(-1)^{n-1}}{2n-1}}= 1-{\dfrac{1}{3}}+{\dfrac{1}{5}}-{\dfrac{1}{7}}+{\dfrac{1}{9}}-\dotsb ={\dfrac{\pi }{4}}
$$
Should we give up assigning a value because it is no longer rational? If we hand over the left-hand side to physicists in order to perform a measurement and they say that they cannot add infinitely often, then is it a problem of our theory, or reality in general?

Surely, we can never confirm by measurement that the assigned value is what the theory says. However, that doesn't stop us from assigning its value or from using the formula. That's why I said it is a matter of convenience. We deduced this formula logically and the physicists cannot measure a significant violation. Therefore, we conveniently accept it.

From a purely logical point of view, physics has already a significant problem with that apple. Just because it always fell down does not guarantee it always will! We accept that it will without further thought, but we cannot be sure. It's a convenience, a commitment, a theoretical assumption, but not a truth chiseled in stone. You can always say that a single measurement will not be sufficient. That was the origin of my idea to treat physical quantities as random variables by default.

IMO, there are a number of problems with mapping the mathematics to a physical circle.

That was the reason why I quoted my ODE professor: the real world is discrete.

 

[fresh_42]

I think I should link two papers that might make sense in this discussion. I haven't read them in detail - mainly because I still can't evaluate the seriousness of the discussion here or where it is supposed to got to - one is a classical paper from a mathematician, and the newer one is at least interesting and from a physicist.

Eugene Wigner, The Unreasonable Effectiveness of Mathematics in Natural Sciences, New York, 1960
(the original reference is on the first page, the link to the university server in Edinburgh, so hopefully no copyright issues)
https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf

A.N. Mitra, Mathematics: The Language of Science, Delhi, 2018
https://arxiv.org/pdf/1111.6560v3.pdf

 

[Dale]

I have a slightly different take on this. Suppose that we have a theory which uses real numbers and predicts that some value is exactly $\pi$. All measurements have some finite precision. If we make any measurement that includes $\pi$ in its uncertainty interval, then we have evidence supporting the theory and it is reasonable to say that you have measured the value to be the real number $\pi$ to within the experimental precision.

To me, that is how science should work. Measurements and experiments are indeed important, but so is theory. If the theory claims a real number value and the experiment is consistent with the theory then there is no reason to not claim that the measurement is a real number.

 

[PeroK]

I have a slightly different take on this. Suppose that we have a theory which uses real numbers and predicts that some value is exactly $\pi$. All measurements have some finite precision. If we make any measurement that includes $\pi$ in its uncertainty interval, then we have evidence supporting the theory and it is reasonable to say that you have measured the value to be the real number $\pi$ to within the experimental precision.

To me, that is how science should work. Measurements and experiments are indeed important, but so is theory. If the theory claims a real number value and the experiment is consistent with the theory then there is no reason to not claim that the measurement is a real number.

I don't totally disagree with this. It has a potential flaw in that the measurement outcome depends on the theory. The measurement, IMO, would be something like $3.14159 \pm 0.00005$, say. That's the result of the measurement. If the theory says that the precise value is $3.141592$, then you say that is the result of the measurement. And, if the theory says that the precise value is $\pi$, then you say that is the result the measurement. I'm not totally convinced by this. IMO, the result of the measurement is $3.14159 \pm 0.00005$ and let the theorists make of that what they will.

 

[PeroK]

I have a slightly different take on this. Suppose that we have a theory which uses real numbers and predicts that some value is exactly $\pi$. All measurements have some finite precision. If we make any measurement that includes $\pi$ in its uncertainty interval, then we have evidence supporting the theory and it is reasonable to say that you have measured the value to be the real number $\pi$ to within the experimental precision.

To me, that is how science should work. Measurements and experiments are indeed important, but so is theory. If the theory claims a real number value and the experiment is consistent with the theory then there is no reason to not claim that the measurement is a real number.

Actually, I have a more significant disagreement with this. If we replace the numbers on a clock with $2\pi, \frac {\pi} {6}, \frac {\pi} {3}, \frac{\pi}{4}, \frac{2\pi}{3} \dots$, then we get one of only twelve possible measurements for the position of a clock hand. They are all irrational. That's not the point. The point is there are only twelve possible outcomes.

In your experiment, there are only two possible outcomes: an interval containing $\pi$ and an interval not containing $\pi$.

There can never be an uncountable number of possible measurement outcomes.

 

[fresh_42]

There can never be an uncountable number of possible measurement outcomes.

This is tautological in my opinion. You can always only perform a finite number of measurements at all, hence getting a finite number of outcomes in finite time. The possible outcomes are also finite per the construction of our measurements based on counting. We simply cannot distinguish $\pi$ from a rational number, but this handicap does not mean we haven't measured $\pi$. Who is it to tell us the difference?

 

[PeroK]

We simply cannot distinguish $\pi$ from a rational number, but this handicap does not mean we haven't measured $\pi$. Who is it to tell us the difference?

The curved spacetime of GR? If space is not static and Euclidean, then the ratio may not be $\pi$. This is another aspect of confusing the mathematical model with the physical objects and phenomena themselves.

 

[Dale]

I don't totally disagree with this. It has a potential flaw in that the measurement outcome depends on the theory. The measurement, IMO, would be something like $3.14159 \pm 0.00005$, say. That's the result of the measurement. If the theory says that the precise value is $3.141592$, then you say that is the result of the measurement. And, if the theory says that the precise value is $\pi$, then you say that is the result the measurement.

Yes. I don't think that this is so much a flaw as an acknowledgement that some measurements will not be able to distinguish between two theories. That is a scientific fact. Such a measurement would indeed support both theories without providing evidence for one theory over the other.

However, if you do not allow measurements to represent real numbers then the theory that predicts $\pi$ is inherently contradicted by any measurement, no matter how precisely it agrees with the theory. If you simply assert that all measurements must be inherently rational or natural, then one thing you automatically know is that your theory saying it is $\pi$ is disproven by a measurement of $3.14159 \pm 0.00005$.

IMO, the result of the measurement is 3.14159±0.00005 and let the theorists make of that what they will.

I guess I have a more holistic view of science. I view both theory and experiment as co-essential to science.

They are all irrational. That's not the point.

Why not? Seems like a good point to me. We are trying to determine what number system to use, so it does seem important that when you change from $\mathrm{Hz}$ to $\mathrm{s^{-1}}$ at least one must be irrational.

The point is there are only twelve possible outcomes.

Counterexample: Rolex. Also, the phase of any frequency standard that produces a phase.

There can never be an uncountable number of possible measurement outcomes.

I disagree. I don't know the correct wording for quantum stuff, but my understanding is that there do exist quantum measurements that have a continuous spectrum. And clearly classically there can be an uncountable number of possible measurement outcomes. So on what theory do you base this theoretical claim?

 

[PeroK]

If you simply assert that all measurements must be inherently rational or natural, then one thing you automatically know is that your theory saying it is $\pi$ is disproven by a measurement of $3.14159 \pm 0.00005$.

You continue to confuse a finite set with a countable set with an uncountable set. You can have a finite subset of real numbers: $\{ \pi, e, \sqrt 2 \}$ is a finite set. You can choose from that set.

I view both theory and experiment as co-essential to science.

Who doesn't?

And clearly classically there can be an uncountable number of possible measurement outcomes. So on what theory do you base this theoretical claim?

No measuring device can produce an uncountably infinite number of possible outputs. Most real numbers would require an infinite amount of information to describe them, so cannot be the outcome of a measurement. These real numbers have only a mathematical existence.

Counterexample: Rolex.

Even a Rolex watch cannot have an uncountably infinite number of markings.

I disagree. I don't know the correct wording for quantum stuff, but my understanding is that there do exist quantum measurements that have a continuous spectrum. And clearly classically there can be an uncountable number of possible measurement outcomes. So on what theory do you base this theoretical claim?

We are never going to agree about this. You say you can produce any real number and I say you can only produce a number from a pre-defined finite set. You see measurement as a theoretical, mathemetical process; I see it as a practical, physical process.

A good test case would be to consider a clock that could stop at any instant and record any time interval. You say it's possible. I say that, ultimately, any clock can only have a (pre-defined) finite number of possible outputs. This is explicit in the most accurate clocks we have. So, although theoretically time is a continuous variable, in practice any measurement of time with any clock is a discrete variable. And how do you measure time without a clock?

 

[Dale]

You continue to confuse a finite set with a countable set with an uncountable set. You can have a finite subset of real numbers: {π,e,2} is a finite set. You can choose from that set.

I am not confusing them. A finite set of irrational numbers are still irrational numbers. To me that is an indication that measurements cannot be a priori restricted to rational numbers.

You see measurement as a theoretical, mathemetical process; I see it as a practical, physical process.

Instead of telling me that I am confusing things I am not confusing and instead of telling me how I see things that I don't see that way, why don't you stick to justifying your own view. I also see measurement as a practical physical process, I just disagree about some of your assertions about that physical process. I also think that it is important to consider both the practical physical things as well as the theoretical considerations. Both are needed for science.

I say you can only produce a number from a pre-defined finite set.

On what do you base that claim?

No measuring device can produce an uncountably infinite number of possible outputs.

I am not convinced that is true. On what basis do you make that claim? Both classically and in QM there are measurements with an uncountably infinite number of possible outputs.

However, consider also that not only do we want to make measurements with one device on one measurand, we also want to compare measurement results on the same consistent scale across devices and across measurands.

Are you sure that there are a finite number of possible outputs across all possible measuring devices measuring all possible measurands? And furthermore are you sure that those different devices and measurands are all rationally related?

If I have two clocks and one measures in a fixed fraction of $\mathrm{Hz}$, and the other measures in a fixed fraction of $\mathrm{rad/s}$ then even though each individually gives a pre-defined finite set of outcomes, you cannot express them both on the same consistent scale using only rational numbers.

So even if it were true that no single measuring device can produce an uncountably infinite number of possible outputs, that still does not imply that the rationals are sufficient for representing measurements in general.

 

[PeroK]

I am not convinced that is true. On what basis do you make that claim? Both classically and in QM there are measurements with an uncountably infinite number of possible outputs.

If I were an experimental physicist, I would not say that an experiment could produce any one of an uncountable infinity of outcomes. If I were using modern digital, computerized equipment that claim would be patently false.

Similarly, if I were asked to produce a number, then whatever process I devise would ultimately choose from a predefined finite set - with the specific set dependent on the process. I'm not convinced that, for example, "throwing a dart at a dart board" is an uncountable random number generator.

That doesn't stop you believing you can generate any real number at random or from an experiment. Since we know that most real numbers are not describable, you would at least have to admit that generating a number or measurement outcome excludes actually providing the number explicitly or reporting the result of that measurement. I'm not convinced by an argument that says "I've generated a real number, but I can't say what it is". Or, "I've made a measurement, but I can't say what the outcome actually is."

Perhaps we just have a difference of opinion on what constitutes (the outcome of) a measurement.

 

[Vanadium 50]

I think this contains every aspect of our discussion here.

You mean it converges incredibly slowly?

 

[fresh_42]

You mean it converges incredibly slowly?

I didn't know the metaphor was that good!

 

[Dale]

Perhaps we just have a difference of opinion on what constitutes (the outcome of) a measurement

Perhaps. Because I do see a measurement as a physical process, I tend to think that the connection between measurements and numbers is artificial. Something that we choose as a convention, not something forced on us by nature.

If I were using modern digital, computerized equipment that claim would be patently false.

Measurements existed long before computers.

There is a dial or a gauge or a display, but is that the measurement or is the measurement the physical process that occurs on the other side of the instrument? Maybe calling the physical process a measurement is impractical.

if I were asked to produce a number, then whatever process I devise would ultimately choose from a predefined finite set - with the specific set dependent on the process

This brings me back to the consistent scale. Suppose we have two processes, each with their respective predefined finite sets, that are to be measured on the same consistent scale. Can you guarantee that they are always rationally related? If not, then wouldn’t you find it more convenient to use real numbers?

Since we know that most real numbers are not describable, you would at least have to admit that generating a number or measurement outcome excludes actually providing the number explicitly or reporting the result of that measurement

Yes, I do admit that. And I don’t have a counter argument for it.

 

[Structure seeker]

I am not confusing them. A finite set of irrational numbers are still irrational numbers. To me that is an indication that measurements cannot be a priori restricted to rational numbers.

If I get a measurement of $3.14159 \pm 0.0001$ that doesn't contradict the theory that contains transcedental numbers in reality, right? Still the measurement returns rational numbers.

 

[Structure seeker]

I think @PeroK is thinking in the context of quantum physics, where the eigenvalues of a Hamiltonian have a countable spectrum of values and you measure one of these. Since any system is a countable heap of quantum systems, this applies to all measurements.

 

[martinbn]

Sorry for the off topic but i have to say that I find this discussion very interesting, but I have no idea what it is about.

 

[Dale]

I think @PeroK is thinking in the context of quantum physics, where the eigenvalues of a Hamiltonian have a countable spectrum of values and you measure one of these. Since any system is a countable heap of quantum systems, this applies to all measurements.

Possibly, although I am pretty sure that there are quantum spectra that are continuous. And there may also be spectra where the individual values, though countable, are irrationally spaced on a consistent scale.

 

[PeroK]

Possibly, although I am pretty sure that there are quantum spectra that are continuous. And there may also be spectra where the individual values, though countable, are irrationally spaced on a consistent scale.

Yes, but that's not the issue. The question is whether you could design a spectrometer that could produce an uncountable number of measurement outcomes.

If we consider QM then we have to tackle the measurement problem. If our device is macroscopic, then it has a predefined scale of resolution. And, if the device itself becomes too small, then it behaves like part of the QM system that it's supposed to be measuring. And no measurement takes place.

But, even at the microscopic scale, things have a finite scale. Whether we have atoms or EM radiation as part of our device, these things cannot be used to probe indefinitely down to the infinite resolution required to distinguish real numbers. Or, even a countable infinity of rational numbers within a fixed range.

That's the issue, IMO. Or, at least, one of the issues.

 

[Dale]

Yes, but that's not the issue.

I disagree. I think that is fundamental to the issue. If we can get a measurement with a continuous spectrum then ...

The question is whether you could design a spectrometer that could produce an uncountable number of measurement outcomes.

even if a single measuring device can only produce a finite number of outcomes, two different measuring devices may have irrationally related outcomes.

 

[fresh_42]

... even if a single measuring device can only produce a finite number of outcomes, two different measuring devices may have irrationally related outcomes.

Not if they cannot produce irrational measurements. If two devices produce a measurement with finitely many digits, then everything, including their ratio is rational.

But why is this even a question? Are we talking about ...

• measurements in general?
• empiric versus deductive sciences?
• the continuum in reality?
• something else that I haven't seen yet?

Maybe I am too much accustomed to algebra. I have no problems with concepts that make life easier (AC, CH, nonconstructive proofs). I leave it to the philosophers and logicians to think about their validity. And maybe this is a fundamental difference between mathematicians and physicists. We both had our crisis in the early 20th century, but mathematicians somehow shrugged their shoulders and kept going. Physicist on the other hand ... https://www.physicsforums.com/forums/quantum-interpretations-and-foundations.292/

Why is it important to concentrate on the number system if the outcome of every single measurement is always a random interval? Shouldn't the randomness be in focus rather than the numbers?

 

[vanhees71]

Yes, but that's not the issue. The question is whether you could design a spectrometer that could produce an uncountable number of measurement outcomes.

You cannot even store the outcome of an uncountable number of measurement outcomes. Thus That's overly speculative.

If we consider QM then we have to tackle the measurement problem. If our device is macroscopic, then it has a predefined scale of resolution. And, if the device itself becomes too small, then it behaves like part of the QM system that it's supposed to be measuring. And no measurement takes place.

If experimentalists had waited until the philosophers have solved their pseudo-measurement-problem, they'd not have measured anything since 1926 ;-)). Measurements simply work in practice, as does their analysis and interpretation in terms of QT.

But, even at the microscopic scale, things have a finite scale. Whether we have atoms or EM radiation as part of our device, these things cannot be used to probe indefinitely down to the infinite resolution required to distinguish real numbers. Or, even a countable infinity of rational numbers within a fixed range.

That's the issue, IMO. Or, at least, one of the issues.

Of course, any measurement has a finite accuracy, and there are always statistical and systematical errors. Take time measurements which I think one can consider the most accurate measurements one can do nowadays. Even with an ideal clock, there's a finite uncertainty due to the finite line width of any transition of a quantum system you might use to define your time unit. You'll never be able to pin down a time or frequency at an accuracy where it makes a difference whether you use real or rational numbers. The right thing to do is to give the finite number of significant digits for the "measurement result" with as good an estimate of the "error" of this result.

The debate, whether you measure real or rational numbers of some quantity is entirely academic.

 

[PeroK]

The debate, whether you measure real or rational numbers of some quantity is entirely academic.

The issue is the cardinality of the set of measurement outcomes. You can measure an angle, which can be converted into a fraction of $\pi$. But, only finitely many angles are possible from any given measurement device.

A simple protractor may return transcendental numbers as measurement outcomes. But there can only be finitely many markings on a given protractor.

 

[vanhees71]

The possible outcomes of measurements for angles are the real numbers in an interval of length $2 \pi$ (according to quantum theory) (where the definition of an angle as and observable in quantum theory is not so easy, but that's another subject).

 

[Dale]

Not if they cannot produce irrational measurements. If two devices produce a measurement with finitely many digits, then everything, including their ratio is rational.

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.

Why is it important to concentrate on the number system if the outcome of every single measurement is always a random interval? Shouldn't the randomness be in focus rather than the numbers?

I agree, that probably should be more important.

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.