Are Real Numbers Essential in Scientific Measurements and Models?

[Dale]

TL;DR Summary

Discuss the use of different number systems (rationals vs reals) in science

A couple of weeks ago we had an interesting thread where a tangent developed discussing whether real-valued measurements were possible. I would like to generalize that discussion a bit in this one and discuss all scientific purposes, not just measurements.

1) What is a measurement anyway? Is it the physical interaction, or is it the number we assign to the interaction. For example, in a galvanometer, is the measurement the deflection of the needle or is it the number that we assign to the deflection? This is probably a matter of opinion, so it probably is important just to state one's opinion.

2) One thing that I found in my research on the topic is that there is only one axiom that the reals satisfy and the rationals do not. If you have a bounded set of reals then the bound is a real, but there are bounded sets of rationals whose bound is irrational. For example, the set of all rationals such that ##\left(\frac{p}{q}\right)^2<2##. This set is bounded from above, but the bound is ##\sqrt{2}## which is irrational. So, for measurements it is unclear how this applies. There are no infinite sets of measurements, and for any finite set of numbers the bound is an element of that set. So the one axiom that distinguishes reals and rationals doesn't apply to measurements.

3) Because precision is finite, measurements don't have limits in the epsilon-delta sense. Even means of measurements don't have such limits. So speaking of large sets of measurements in the limit as the number of measurements goes to infinity still doesn't give convergence, bounding, or limits even in principle.

4) Suppose that we have two scientific models, one based on real numbers and one based on rational numbers. Suppose further that whenever the real model predicts a value, the rational model predicts the rational value that is closest to that real value. There is no experimental measurement which can provide evidence for one of these models and against the other. Any measurement is compatible with an infinite number of reals and an infinite number of rationals.

5) There are theoretical reasons to prefer the reals over the rationals. For example, you cannot do integration of rational functions, or rather the resulting integrals can be irrational. If we prefer reals over rationals for theoretical reasons, and if our models are compatible with both, can we not declare our measurements to be reals?

6) What about hyperreals? We might make theoretical arguments favoring hyperreals. The same compatibility and preference arguments would apply. Can we declare our measurements to be hyperreals? What about surreals? Is there a line, and if so where do we draw it and why?

 

[Mark44]

5) There are theoretical reasons to prefer the reals over the rationals. For example, you cannot do integration of rational functions, or rather the resulting integrals can be irrational.

@Dale, I'm not sure what you mean here by a rational function. A rational function is the quotient of two polynomials; e.g., ##f(x) = \frac{p(x)}{q(x)}## where both p and q are polynomials. Some rational functions do have integrals that are themselves rational functions. An example is ##\int \frac{dx}{x^2}##, whose integral (or antiderivative) is ##\frac{-1}x + C##. I believe that what you meant by a rational function was one that evaluated to a rational number, but I'm not sure.

 

[Dale]

@Dale, I'm not sure what you mean here by a rational function. A rational function is the quotient of two polynomials; e.g., ##f(x) = \frac{p(x)}{q(x)}## where both p and q are polynomials. Some rational functions do have integrals that are themselves rational functions. An example is ##\int \frac{dx}{x^2}##, whose integral (or antiderivative) is ##\frac{-1}x + C##. I believe that what you meant by a rational function was one that evaluated to a rational number, but I'm not sure.

You are correct, I was being unclear. By "rational function" I had intended to mean a function mapping from the rationals to the rationals. The integral $$\int \frac{1}{x} dx = \ln (x) + C$$ is the main culprit I was thinking of. If ##x## is a non-zero rational number then ##1/x## is also rational, but ##\ln (x)## is not.

So it seems that integration cannot be defined strictly on the rationals. This is actually related to the bounding axiom that distinguishes rationals and reals. I suspect that standard trig functions also cannot be defined between the rationals.

 

[fresh_42]

I suspect that standard trig functions also cannot be defined between the rationals.

No. They are defined either by a series that needs the topological completion of the reals to guarantee a limit or by the exponential function which is already impossible over the rationals since ##e## isn't rational.

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.

On the other hand, physical quantities are always denoted as ##A \pm B.## Doesn't this mean that you cannot even get a rational measurement?

 

[Dale]

Doesn't this mean that you cannot even get a rational measurement?

I think that is correct. Due to finite precision any measurement is consistent with an infinite number of rationals. How can you claim that any one of those rationals "is" the measurement?

 

[fresh_42]

I think that is correct. Due to finite precision any measurement is consistent with an infinite number of rationals. How can you claim that any one of those rationals "is" the measurement?

Do we have to say that any physical quantity is only a distribution? And if so, isn't it meaningless whether we consider it over real numbers or a dense subset?

 

[Dale]

And if so, isn't it meaningless whether we consider it over real numbers or a dense subset?

I think it is meaningless, indeed. Although I am not sure that is anything even remotely approaching a consensus view.

I think then it is interesting to consider supersets instead of subsets. What about hyperreals? If we are just arbitrarily mapping distributions to dense sets, why not use sets even denser than the reals?

 

[fresh_42]

I think it is meaningless, indeed. Although I am not sure that is anything even remotely approaching a consensus view.

I think then it is interesting to consider supersets instead of subsets. What about hyperreals? If we are just arbitrarily mapping distributions to dense sets, why not use sets even denser than the reals?

I'm not sure whether the hyperreals are a valid alternative. I think it is a bit like putting the cart before the ox. If we agree that physics is a collection of distributions, then we will need a calculus of random variables. I don't think that we can call what we have, central limit theorem, 1-2-3 theorem, and rules like that already an entire calculus. The approach of analysis by Borel algebras and measure theory (pun not intended) seems more promising. It could create a new perspective.

 

[PeroK]

I don't particularly see the relevance of different number systems. If we take a measurement to be a physical process, then (IMO) there is always some finite limit to the precision of any given measurement. It wouldn't make sense, for example, to give a winning time in a 100m race as an arbitrary real number. Any measurement of the time can only be one of a finite number of possible values, depending on the measurement apparatus.

It's a moot point whether the time itself is a real number. Any measurement of that time can only be chosen from a finite number of possibilities for any given event.

One reason this topic arises is that the Internet is littered with web pages saying: if a person chooses a real number uniformly from the interval ##[0,1]##, then ... That, IMO is not a physically realisable process. It can only be done through the mathematics of continuous probability distributions. But, there is no way that someone can choose equally from anything but a finite set of possibilities.

 

[Dale]

The approach of analysis by Borel algebras and measure theory

I am vaguely aware of measure theory, but not Borel algebras. I will take a look.

 

[fresh_42]

I am vaguely aware of measure theory, but not Borel algebras. I will take a look.

It's only the set of measurable volumes so that we can integrate. It's what we need for the Lebesgue measure, the integral we use anyway. Hewitt / Stromberg, Real and Abstract Analysis, GTM 25 is written along those lines.

 

[Mark44]

By "rational function" I had intended to mean a function mapping from the rationals to the rationals. The integral $$\int \frac{1}{x} dx = \ln (x) + C$$ is the main culprit I was thinking of. If ##x## is a non-zero rational number then ##1/x## is also rational, but ##\ln (x)## is not.

I thought you might have that one in mind, which is why I chose the integral I showed as a counterexample.

 

[Dale]

I thought you might have that one in mind, which is why I chose the integral I showed as a counterexample.

Yes, there are many rational to rational functions that do have an integral, but there are some that do not. So I don't think that you can generally say that you can do integration with the rationals.

In other words, my point is that it cannot always be done, not that it can never be done.

 

[Mark44]

Yes, there are many rational to rational functions that do have an integral, but there are some that do not. So I don't think that you can generally say that you can do integration with the rationals.

No, I wasn't claiming that.

 

[Dale]

OK, I think we are in agreement then?

 

[Dale]

I don't particularly see the relevance of different number systems. If we take a measurement to be a physical process, then (IMO) there is always some finite limit to the precision of any given measurement. It wouldn't make sense, for example, to give a winning time in a 100m race as an arbitrary real number. Any measurement of the time can only be one of a finite number of possible values, depending on the measurement apparatus.

If we take a measurement to be a physical process then I don't think that I agree that any measurement can only be one of a finite number of possible values.

I mentioned the galvanometer as an example. If the measurement is the physical process, the deflection of the needle, then I disagree that it can take only one of a finite number of possible values. Position is continuous in both classical and quantum mechanics.

 

[fresh_42]

Another example is the good old analog oscilloscope. The exact frequency and even more the amplitude of a sine wave can be any number close to what the theory says it will be.

 

[PeroK]

If we take a measurement to be a physical process then I don't think that I agree that any measurement can only be one of a finite number of possible values.

I mentioned the galvanometer as an example. If the measurement is the physical process, the deflection of the needle, then I disagree that it can take only one of a finite number of possible values. Position is continuous in both classical and quantum mechanics.

The needle on a galvonometer - even on a macroscopic scale - is distinctly finite in size. Eventually, at a small enough scale, the point of a needle is no longer point like. And, eventually, even in a classical model the point of the needle is just a vacuum between atoms.

Moreover, QM has the HUP (Heisenberg Uncertainty Principle), which disallows a precise measurement to an arbitrary degree.

So, both practically and theoretically (as far as QM is concerned), the point of a needle cannot be determined to arbitrary precision.

 

[Dale]

Eventually, at a small enough scale, the point of a needle is no longer point like. And, eventually, even in a classical model the point of the needle is just a vacuum between atoms.

I have no objection to that.

Moreover, QM has the HUP (Heisenberg Uncertainty Principle), which disallows a precise measurement to an arbitrary degree.

I am not disagreeing with that either.

So, both practically and theoretically (as far as QM is concerned), the point of a needle cannot be determined to arbitrary precision.

I am not talking about precision. I think that we are talking past each other.

If the measurement is the physical process, rather than the number we get from the physical process, then it is a state of some physical system. In both classical mechanics and QM there are not a finite number of states that a galvanometer needle can be in. I am not talking about how precisely we can put it into a state nor how precisely we can determine what state it is in, just how many states our physics models say it can be in.

 

[PeroK]

I have no objection to that.

I am not disagreeing with that either.

I am not talking about precision. I think that we are talking past each other.

If the measurement is the physical process, rather than the number we get from the physical process, then it is a state of some physical system. In both classical mechanics and QM there are not a finite number of states that a galvanometer needle can be in. I am not talking about how precisely we can put it into a state nor how precisely we can determine what state it is in, just how many states our physics models say it can be in.

In QM, there is a clear distinction between a state and a measurement of that state. There are infinitely many spin states for a spin 1/2 particle, but only two results for a measurement.

No piece of apparatus can establish a measurement of position to arbitrary precision. There is also something strange about a measurement process that does not and cannot yield a result.

 

[Dale]

In QM, there is a clear distinction between a state and a measurement of that state. There are infinitely many spin states for a spin 1/2 particle, but only two results for a measurement.

Good point. My QM is not good, so perhaps I have this wrong. This is known as the spectrum of the eigenvectors, right? Is it the eigenvectors of the state or the operator or both?

No piece of apparatus can establish a measurement of position to arbitrary precision. There is also something strange about a measurement process that does not and cannot yield a result.

Again, I am not talking about precision.

 

[hutchphd]

I am reminded that Eddington defined physics as comparing pointer readings. In this context isn't this argument simply academic?
All experimental measurement is ratiometric. It is why and how we calibrate our experimental equipment.
Is there a "measurement" that is somehow divorced from actual experiment?

 

[PeroK]

Is there a "measurement" that is somehow divorced from actual experiment?

I.e. the rough analogy of non constructive mathematics. In a previous thread I suggested something like realising the fixed point mapping theorem as a real valued measurement of sorts.

 

[hutchphd]

This seems an exercise in omphalloskepsis to me. I understand this tobe a minority view! (Perhaps occasioned by my seeming incapacity to do rigorous mathematics)

 

[Structure seeker]

It's clear to me that in digital data, we only have integer values actually, with multiplication altered to have a subdivision of 1 in as many parts as the precision goes and rounding results. For measurements this is often with an error bar or assumed to be a normal distribution and a standard deviation is provided.

Regardless, there are many required theoretical aspects of modern science that were inspired by continuity (trigonometry, probability density, topology, Lie groups), algebraicity (complex numbers, galois theory and its isomorphisms), infinity (projective spaces, renormalization) and probably much more. IMO science should not be limited to describe only what is measurable, there should be space to expand theoretically and perhaps sometime also experimentally.

 

[PeroK]

IMO science should not be limited to describe only what is measurable, there should be space to expand theoretically and perhaps sometime also experimentally.

It's not. We use mathematics, which is not physically constrained. The simplest example is instantaneous velocity, which cannot be measured. It's a mathematical construction.

 

[fresh_42]

This thread drifted into strange parts. Measurements, precision, and even HUP has been mentioned. This has little to do with ...

Discuss the use of different number systems (rationals vs reals) in science

... and this thread is quickly on its way to vacuity, faster than usual. We already know that the measurement discussion leads nowhere. Heck, we even have a separate forum for it. We felt the need to lock such discussions away in the padded cell. It is a bottomless barrel.

I, too, think that the question ...

Discuss the use of different number systems (rationals vs reals) in science

... reverses the natural order. Which number system we use is an a posteriori question, after the decision has been made to use ordinary calculus. The question shouldn't be whether we assume the existence of limits that can neither be measured nor be written down, or restrict ourselves to finite decimals, i.e. a dense subset. Both, completeness and density are unphysical terms. The choice between reals and rationals is without substance in my mind, and hyperreals even more. These are all a posteriori considerations and only affect convenience and techniques. It is a similar discussion as to whether we believe in AC or not.

In my mind, we should take a step backward and ask what we want to deal with. If we take the perspective of measurements as the fundamental object of physics, then in my opinion, we should replace our variables which don't always represent numbers anyway, by probability distribution functions. This would be a new perspective with the potential to unify the physical calculus. And it sets the measurement in the center of consideration. Whether the related random variables are real, rational, or even complex, waves or operators is more or less not important.

 

[hutchphd]

and this thread is quickly on its way to vacuity

My comment was meant to imply that the all measurement (at least in the present structure of our science) is ratiometric. I do not consider that unimportant. Vacuity is in the eye of the beholder, but the slope is indeed slippery.

 

[pbuk]

4) ... Any measurement is compatible with an infinite number of reals and an infinite number of rationals.

This, and IMHO there is nothing more to consider.

 

[Dale]

These are all a posteriori considerations and only affect convenience and techniques. ... If we take the perspective of measurements as the fundamental object of physics, then in my opinion, we should replace our variables which don't always represent numbers anyway, by probability distribution functions

I like that as it would handle precision “natively” and the same framework would suffice for qualitative measurements, discrete valued measurements, and continuous valued measurements.

I think we are not too far from that already.

 

[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.