-5

The realistic definition of transition probability in physics is well defined and constrains the probability to rational numbers. The abstract definition of probability in mathematics is also well defined, but it allows probability to be an arbitrary real number element of [0,1], whether rational or irrational. The difference is enormous and the conclusions diverge widely. The question is who do we track and when? Additional clarification required: We can provide two specific published examples among many others,

  1. Numerical resolution of the 3D PDE of heat diffusion as a function of time in its most general case using the physical definition of probability.

  2. Solve the statistical numerical integration for an arbitrary number of free nodes using the physical definition of probability. The trapezoidal ruler and the FDM-based Simpson ruler would be just a special case.

Qmechanic
  • 201,751
  • 1
    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 31 '22 at 13:25
  • 12
    Why do you think physics cares about rational vs irrational numbers? $1/ \pi$ is a perfectly reasonable number in physics. – Jon Custer Oct 31 '22 at 13:36
  • Can the value of potentials be an irrational number? – mmesser314 Oct 31 '22 at 13:46
  • 2
    I've never heard of rational numbers having anything special regarding probabilities… All I can think of is during a beginner-level course where the focus is kept on discrete probability, but it's nothing special really. – Miyase Oct 31 '22 at 14:16
  • 2
    The realistic definition of transition probability in physics is well defined and constrains the probability to rational numbers. What definition are you talking about? I don’t know any definition that constrains probability to rational numbers. But if there was one that did, what would it matter? The rationals are dense in the reals. Any real number can be approximated by a rational to whatever precision you would like, say a googleplex of decimal digits. – Ghoster Oct 31 '22 at 16:34
  • The OP is clearly mistaking frequencies for probabilities. That is a beginner's mistake. – FlatterMann Oct 31 '22 at 22:30
  • @FlatterMann A true frequentist approach to probability defines probabilities in terms of frequencies. It is not a beginner's mistake. – GiorgioP-DoomsdayClockIsAt-90 Oct 31 '22 at 23:09
  • GiorgioP I agree with you 100% that the frequentist interpretation of physical probability is not the best. The right definition of physical probability emerges from the 4 rules of natural physical probability that underlie the construction of well-defined chains called B-transition matrices in 4D(x,y,z,t). –  Oct 31 '22 at 23:46
  • FlatterMann I agree with you 100% that the frequentist interpretation of physical probability is not the best. The right definition of physical probability emerges from the 4 rules of natural physical probability that underlie the construction of well-defined chains called B-transition matrices in 4D(x,y,z,t). –  Oct 31 '22 at 23:47
  • Jon Custer We believe that physics cares about rational numbers versus irrational numbers because one of the 4 axioms of physics is that the total probability of the entire rhythm must add up to 1. This is impossible for Sqrt(2) or 1/π ... etc. 1/Pie is a perfectly reasonable number in a continuous real-time domain where, as in a numerical solution, physical probability operates in a dimensionless time domain. –  Nov 01 '22 at 00:04
  • 1
    This is impossible No, it’s not. For example a probability of $1/\pi$ and a probability of $1-1/\pi$ add up to a probability of $1$. I can’t understand what you are thinking. – Ghoster Nov 01 '22 at 02:22
  • @GiorgioP I don't know what a true frequentist is, I am afraid. Is that like a true Scotsman? I only know people who use Kolmogorov on paper and who count frequencies in the real world. That's the only thing that makes sense from a practical perspective. – FlatterMann Nov 01 '22 at 06:18
  • @Lionheart My point was simply that nature doesn't give us anything other than frequencies. We may ask for a probability distributions pony, but it is always frequency socks for Christmas. There is nothing wrong with that. Frequencies can keep us nice and warm, as long as there are plenty of them. I also don't do rules. I only do observations. – FlatterMann Nov 01 '22 at 06:22
  • FlatterMann Thank you very much for your very helpful comment, but believe me my theory is that you can count on nature to give you everything and even surprising facts and mathematical-physical proofs. Indeed it is my passion. –  Nov 01 '22 at 08:11
  • Maybe you happen to have only done physics problems where the probabilities are rational, but there are others where they aren't. For example, a particle in a $1$-dimensional box satisfies$$P\left(x\in\left[0,,\frac{L}{4}\right]\right)=\frac14-\frac{1}{2\pi}$$in the ground state. – J.G. Nov 01 '22 at 08:52
  • J.G. Thank you ,I must say I love this comment. i-your comment is self-explanatory and its answer is contained in the comment itself. The existence, stability and uniqueness of the sol is ensured using the definition of physical probability while in the mathematical definition the first thing to look for is whether the sol. exists stably and uniquely or not. So what. ii. To applyof the phy. definition of prob to QM, in particular to a subatomic particle in a 3D potential box, is still a matter of speculation. The nature of Sch Equ with its fluid interpretation of Bohr-Copenhagen. –  Nov 02 '22 at 13:09

1 Answers1

3

I would not say there is a difference between the definition of probability in Mathematics and Physics. Probably, what you are referring to as the definition in Physics is the interpretation of probability theory.

I assume that what you mention as the Mathematical definition is the theory as summarized by Kolmogoroff's axioms (or equivalent). That is a special chapter of Measure Theory in Mathematics. However, we have to notice two important things on the mathematical side.

  1. Kolmogoroff's theory is a general scheme consistent with many interpretations of the theory. As an axiomatic theory, it does not define the basic objects it works on; like in axiomatic geometry, we do not define what a point, a line, or a plane are. Then, for application to the real world, we have to provide an interpretation fixing how we can assign probabilities to the events. It turns out that many interpretations can be used in connection and consistent with Kolmogoroff's axioms (frequentistic, classical "a priori," subjective Bayesian, ...).
  2. There are other mathematical frameworks accommodating probability theories quite different from Kolmogoroff's version. It is possible to find an extended, even though not up-to-date, report in the classical book by T.L. Fine Theories of Probability, Academic Press (1973). Comparative Probability, Complexity-based Probability, and Carnap's Theory of Logical Probability are a few of them.

In Physics, the axioms by Kolmogoroff are usually used. In many cases, the frequentistic approach prevails regarding the interpretation, although, in different contexts, the Bayesian point of view has been advocated as the most appropriate.

After this summary of the situation, the point about rational or irrational values for probability seems to raise a marginal issue.

First, in a frequentistic approach, one can assume that frequencies are ideally measured over infinite sequences of experiments. But, even without stressing this point of view, one can safely use a theory based on real numbers to approximate rational-number-based measures satisfactorily. That is common in Physics and any application of Mathematics to the real world. Strictly speaking, even measuring a length in the real world can only provide rational number outcomes. This fact does not prevent from using geometrical relations based on real numbers.

GiorgioP-DoomsdayClockIsAt-90
  • 34,706
  • The Bayesian approach simply accepts bias as a valid method in statistics. That might work well in political and social sciences, but in physics it will kill you in no time flat. I have been in sessions of experimental physics collaborations that lasted entire days where people were talking about nothing else than what does and what does not constitute bias in the data analysis pipeline. In practice we don't have anything other than frequencies. Complex experiments with non-standard (and unknown distributions) will usually compare frequentist theory (Monte Carlo) against frequentist data. – FlatterMann Oct 31 '22 at 22:36
  • 3
    @FlatterMann The nice thing separating axiomatic formalism and interpretation is that many points of view can coexist. I do not think the frequentist interpretation is as unavoidable as twentieth-century Physics seemed to assume. Laplace and Jaynes were not frequentists, and they were using different interpretations. Still, their contributions to Physics stay with us. I am convinced that in most cases, the precise choice of interpretation is irrelevant to the physical conclusions. – GiorgioP-DoomsdayClockIsAt-90 Oct 31 '22 at 23:07
  • FlatterMann I agree with you 100% that the frequentist interpretation or

    The Bayesian approach to physical probability is not the best. The right definition of physical probability emerges from the 4 rules of natural physical probability that underlie the construction of well-defined chains called B-transition matrices in 4D(x,y,z,t).

     –  Oct 31 '22 at 23:52
  • @GiorgioP I am skeptical about the "interpreations" stuff. Frequencies are what we get from observations. Probability distributions are mathematical idealizations. Sometimes the two connect and then we are happy campers because life is somewhat easy and sometimes they don't and then it sucks. Just ask the HEP data analysis folks how happy they are about having to run a million MC sims a month. It's all about using the right tool at the right time. – FlatterMann Nov 01 '22 at 06:31
  • 1
    @FlatterMann Bayesian inference works the way that real physical research works. You come in with some knowledge of the phenomenon you're studying, and Bayesian methods update that knowledge. However, doing this right is really hard: you have to maintain a likelihood function (usually multidimensional) and update it for each observation. Then, the result is a likelihood function. Taking the peak as a point estimate (which is often what you want to report) is a bit perilous, and doing a meta-analysis on such point estimates leads to nonsense. – John Doty Nov 01 '22 at 12:53