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Sorry, Feynman, whatever else he did, on probability he gets a grade of flat F. Here is why: In his book Lectures on Physics he states that a particle of unkn

Sorry, Feynman, whatever else he did, on probability he gets a grade of flat F. Here is why: In his book Lectures on Physics he states that a particle of unknown location has a probability density uniform over all of space. No it doesn't. No such density can exist. Done. Grade flat F. I tried to be a physics major but could not swallow all the daily really stupid mistakes such as this one by Feynman I got each physics lecture and I didn't have time both to learn the physics AND to clean up the sloppy math. So, I majored in math. As I learned the math, from some of the best sources, I came to understand just how just plain awful the math of the physics community is. Then in one of the Adams lectures on quantum mechanics at MIT I saw some of the reason: The physics community takes pride in doing junk math. They get by with it because they won't take the math seriously anyway, that is, they insist on experimental evidence. So, to them, the math can be just a heuristic, a hint for some guessing. Students need to be told this, in clear terms, early on. It went on this way: In one of the lectures from MIT a statement was that the wave functions were differentiable and also continuous. Of COURSE they are continuous -- every differentiable function is continuous. The lectures made a total mess out of correlation and independence. It looks like Adams does not understand the two or their difference clearly. There was more really sloppy stuff around Fourier theory. I got my Fourier theory from three of W. Rudin's books. It looks like at MIT they get Fourier theory from a comic book. I got sick, really sick, of the math in physics. Feynman on probability is just one example.

clircle 4y ago
What is the problem with density on space?
- graycat 4y ago
  Take one cubic inch. Let the probability the particle is in that cubic inch be p. Then take integer n >= 2 / p cubic inches. Then the probability that the particle is in those n cubic inches is np >= 2 > 1 greater than 1, a contradiction. Done. One well considered and informed explanation is that the physics community abuses its students.
  - vitus 4y ago
    One counterargument is that space is finite, and so your choice of n is greater than the volume of the universe. (And so sigma-additivity doesn't apply, since your choices of cubic inches are not disjoint.) But sure, if you're assuming an unbounded space with finite measure, a uniform density across that space must be identically zero everywhere.
    - graycat 4y ago
      > identically zero everywhere. Not a probability density.
  - hackandthink 4y ago
    Probability Theory without Measure Theory is advancing. Interestingly Tobias Fritz is a Physicist. "A synthetic approach to Markov kernels..." https://arxiv.org/abs/1908.07021 https://arxiv.org/abs/1908.07021
    - dchftcs 4y ago
      graycat's pedantic approach to mathematical formalism is quite defeatist in that it basically disallows any mathematical concept to advance from precise but limited language towards the edge of imagination. The kind that forbids sqrt(2) from existing in the Pythagorean days. A probability theory that accommodates the concept of "picking a random even number from all integers" can be valuable, isn't compatible with measure theory but is easy to grasp intuitively. When the mathematical tools aren't good enough you still want to be able to reason about concepts, which is why the tools are developed to begin with. Fortunately almost all things are conceptually tractable when dealing with finite space or quantities; if a statement can be transformed to something rigorous (if numerically imprecise) by forcing boundedness it's not too terrible to speak in unbounded terms when the physical world is what you want to model. I can understand why physicists don't want to be burdened with too much about rigor - they can afford the small risk they're wrong sometimes, but can't afford to slow down their search for new discoveries, when so many questions remain unanswered.
jbay808 4y ago
> a particle of unknown location has a probability density uniform over all of space. No it doesn't. In that case, in which locations is the density higher and in which is it lower?
q-big 4y ago
> Sorry, Feynman, whatever else he did, on probability he gets a grade of flat F. Here is why: In his book Lectures on Physics he states that a particle of unknown location has a probability density uniform over all of space. No it doesn't. No such density can exist. Done. Grade flat F. I would rather consider that because it seems that you "need" a uniform distribution for a particle of unknown location, it might makes sense for such applications from physics to weaken the property that a probability measure has to be σ-additive to that a probability measure has to be additive. Then it should be possible to define such a "uniform probability 'measure' over all space", perhaps similarly to the example given at > https://en.wikipedia.org/w/index.php?title=Sigma-additive_set_function&oldid=1099006176#An_additive_function_which_is_not_%CF%83-additive https://en.wikipedia.org/w/index.php?title=Sigma-additive_se...
kgwgk 4y ago
> he states that a particle of unknown location has a probability density uniform over all of space. No it doesn't. No such density can exist. Not such a particle can exist then. You’re assuming infinite space though. Did he?
- graycat 4y ago
  Of course, the space was not made explicit. With R the set of real numbers, the usual assumption in the physics is that the math is done in R^3 with the usual inner product, norm, metric, and topology.
  - kgwgk 4y ago
    The usual assumption in physics is that the observable universe is bounded. https://www.feynmanlectures.caltech.edu/I_05.html#Ch5-S6 https://www.feynmanlectures.caltech.edu/I_05.html#Ch5-S6 Edit: you may be talking about the paragraph "For an atom at rest," in https://www.feynmanlectures.caltech.edu/III_07.html#Ch7-S1 https://www.feynmanlectures.caltech.edu/III_07.html#Ch7-S1 The section starts saying that "We want now to talk a little bit about the behavior of probability amplitudes in time. [...] We are always in the difficulty that we can either treat something in a logically rigorous but quite abstract way, or we can do something which is not at all rigorous but which gives us some idea of a real situation—postponing until later a more careful treatment. With regard to energy dependence, we are going to take the second course. We will make a number of statements. We will not try to be rigorous—but will just be telling you things that have been found out, to give you some feeling for the behavior of amplitudes as a function of time. "
    - graycat 4y ago
      I put in a qualification: > ... the usual assumption in the physics is that the math ... Soooo, you got into the physics. I avoided getting into the physics. E.g., when Newton wrote out his second law or Maxwell wrote out his equations, it was just math and the assumption was as I mentioned R^3. Stokes theorem, the Navier-Stokes equations were implicitly in R^3.
hither_shores 4y ago
> really stupid mistakes such as this one by Feynman It's not a mistake, it's a "lie-to-children" fundamentally no different from an intro analysis class talking about "the" real numbers. Freshmen aren't ready for model theory, and they're not ready for rigged Hilbert spaces.
- graycat 4y ago
  The set of real numbers is commonly defined and their properties established in a first course in abstract algebra. Model theory is not required. Axiomatic set theory is not even required. Hilbert space? A complete inner product space where complete means every Cauchy convergent sequence converges. The real numbers serve as one example. With the usual inner product, R^3 (where R is the set of real numbers) is another example.
  - hither_shores 4y ago
    > The set of real numbers is commonly defined and their properties established in a first course in abstract algebra. A set modelling the real numbers is defined in introductory math courses. The process of quotienting out all the set-theoretic slag to arrive at the real numbers, singular, is generally skipped. As well it should be: it's too ugly to be an end in itself, and too subtle to be of use to freshmen. > Hilbert space? A complete inner product space where complete means every Cauchy convergent sequence converges. I said rigged Hilbert spaces. The proper setting for QM is a Hilbert space equipped with dense subspace P of test functions (for which the inclusion P -> H is continuous). This induces an inclusion of dual spaces H* -> P*, and Reisz-Frechet gives us a natural isomorphism H ~ H*, so we have the "Gelfand triple": P -> H -> P*. The view from 10,000 feet here is that P describes the space of possible measurements, P* describes the distributions, and the fact that the inner product <P, P*> factors through H is a compactness condition. So in particular, a plane wave |k> lives in P*, not H. But you can't teach this to introductory physics students. They won't have the necessary math background for several years. (Many of them will never have it). And yet they have to be taught.
    - graycat 4y ago
mturmon 4y ago
It's tough. I used to work in an area of applied probability where some statistical-mechanics principles were applicable. I'd read papers where authors were making analogies of a large neural network to a stat-mech system, using an applicable stat-mech approximation, and then differentiating that approximation to get a probability bound. It gave interesting results, and did show you something about the original problem that was hard to get by sticking to the original formalism. But at the end of the day, you really would not bet the farm on the truth of those approximations... On the other hand, Fourier analysis was originally doubted and scorned by mathematics, but (if I'm remembering the story correctly) ended up being used so much that theory was developed to explain in what sense the Fourier transform approximates the original function. Another example of the interplay between physics and mathematics is the percolation problem, where there was a kind of archipelago of physics-motivated results that probabilists have been trying to tidy up for decades now. E.g., sec. 1.2 of: https://www.unige.ch/~duminil/publi/2018ICM.pdf https://www.unige.ch/~duminil/publi/2018ICM.pdf
- graycat 4y ago
  There is both Fourier series and the Fourier transform. For each, need to establish that they exist. Then will usually want to know that the inverse transform DOES approximate the original function. Details are in 4 books by W. Rudin. One of these, on groups, I ignored. I did too much group theory, e.g., wrote my honors paper on it, and didn't want to do more. Principles does Fourier series and does not use measure theory. Real and Complex Analysis does the Fourier transform and uses the Lebesgue integral, that is, measure theory. Functional Analysis does Fourier theory with distributions. I had serious courses from Principles and R & CA. Also Royden and Neveu. All from a star student of Cinlar, long at Princeton. I read FA, quickly. I don't much care to bother with distributions. I got into applications via the fast Fourier transform and power spectral estimation as in Blackman and Tukey. There is an intuitive view that sort of works: The given function is a point in a vector space, with an inner product. The sine waves are coordinate axes. They are orthogonal. The Fourier things are projections of the point onto the coordinate axes. The inverse Fourier thing reconstructs the original function. If use only some of the coordinate axes, then get a least squares approximation of the original function. This all works exactly in ordinary linear algebra, e.g., as in Halmos, Finite Dimensional Vector Spaces, sometimes given to physics students studying quantum mechanics. But these Fourier things have infinitely many coordinate axes, countably infinite for Fourier series and uncountably infinite for the transform, and there the finite dimensional things don't always work. So, Rudin has to be very careful in presenting what does work and proving it -- so with the details it's not easy reading. What fails is a fairly general situation in a fully general Hilbert space. E.g., are not locally compact, but can get some help from a clever use of the parallolgram inequality (somewhat relevant in my startup). I'm no teacher. I do not now nor have I ever had any desire to be a prof. For more, get copies of Rudin's books and dig in. Wonder of wonders, not all physics profs have done that.
Gatsky 4y ago
You rage-quit physics, essentially.
- deleted 4y ago
  [deleted]
dchftcs 4y ago
>probability density uniform over all of space If space is bounded then such a density can exist.
deleted 4y ago
[deleted]

clircle 4y ago

What is the problem with density on space?

graycat 4y ago
Take one cubic inch. Let the probability the particle is in that cubic inch be p. Then take integer n >= 2 / p cubic inches. Then the probability that the particle is in those n cubic inches is np >= 2 > 1 greater than 1, a contradiction. Done. One well considered and informed explanation is that the physics community abuses its students.
- vitus 4y ago
  One counterargument is that space is finite, and so your choice of n is greater than the volume of the universe. (And so sigma-additivity doesn't apply, since your choices of cubic inches are not disjoint.) But sure, if you're assuming an unbounded space with finite measure, a uniform density across that space must be identically zero everywhere.
  - graycat 4y ago
    > identically zero everywhere. Not a probability density.
- hackandthink 4y ago
  Probability Theory without Measure Theory is advancing. Interestingly Tobias Fritz is a Physicist. "A synthetic approach to Markov kernels..." https://arxiv.org/abs/1908.07021 https://arxiv.org/abs/1908.07021
  - dchftcs 4y ago
    graycat's pedantic approach to mathematical formalism is quite defeatist in that it basically disallows any mathematical concept to advance from precise but limited language towards the edge of imagination. The kind that forbids sqrt(2) from existing in the Pythagorean days. A probability theory that accommodates the concept of "picking a random even number from all integers" can be valuable, isn't compatible with measure theory but is easy to grasp intuitively. When the mathematical tools aren't good enough you still want to be able to reason about concepts, which is why the tools are developed to begin with. Fortunately almost all things are conceptually tractable when dealing with finite space or quantities; if a statement can be transformed to something rigorous (if numerically imprecise) by forcing boundedness it's not too terrible to speak in unbounded terms when the physical world is what you want to model. I can understand why physicists don't want to be burdened with too much about rigor - they can afford the small risk they're wrong sometimes, but can't afford to slow down their search for new discoveries, when so many questions remain unanswered.

jbay808 4y ago

> a particle of unknown location has a probability density uniform over all of space. No it doesn't. In that case, in which locations is the density higher and in which is it lower?

q-big 4y ago

> Sorry, Feynman, whatever else he did, on probability he gets a grade of flat F. Here is why: In his book Lectures on Physics he states that a particle of unknown location has a probability density uniform over all of space. No it doesn't. No such density can exist. Done. Grade flat F. I would rather consider that because it seems that you "need" a uniform distribution for a particle of unknown location, it might makes sense for such applications from physics to weaken the property that a probability measure has to be σ-additive to that a probability measure has to be additive. Then it should be possible to define such a "uniform probability 'measure' over all space", perhaps similarly to the example given at > https://en.wikipedia.org/w/index.php?title=Sigma-additive_set_function&oldid=1099006176#An_additive_function_which_is_not_%CF%83-additive https://en.wikipedia.org/w/index.php?title=Sigma-additive_se...

kgwgk 4y ago

> he states that a particle of unknown location has a probability density uniform over all of space. No it doesn't. No such density can exist. Not such a particle can exist then. You’re assuming infinite space though. Did he?

graycat 4y ago
Of course, the space was not made explicit. With R the set of real numbers, the usual assumption in the physics is that the math is done in R^3 with the usual inner product, norm, metric, and topology.
- kgwgk 4y ago
  The usual assumption in physics is that the observable universe is bounded. https://www.feynmanlectures.caltech.edu/I_05.html#Ch5-S6 https://www.feynmanlectures.caltech.edu/I_05.html#Ch5-S6 Edit: you may be talking about the paragraph "For an atom at rest," in https://www.feynmanlectures.caltech.edu/III_07.html#Ch7-S1 https://www.feynmanlectures.caltech.edu/III_07.html#Ch7-S1 The section starts saying that "We want now to talk a little bit about the behavior of probability amplitudes in time. [...] We are always in the difficulty that we can either treat something in a logically rigorous but quite abstract way, or we can do something which is not at all rigorous but which gives us some idea of a real situation—postponing until later a more careful treatment. With regard to energy dependence, we are going to take the second course. We will make a number of statements. We will not try to be rigorous—but will just be telling you things that have been found out, to give you some feeling for the behavior of amplitudes as a function of time. "
  - graycat 4y ago
    I put in a qualification: > ... the usual assumption in the physics is that the math ... Soooo, you got into the physics. I avoided getting into the physics. E.g., when Newton wrote out his second law or Maxwell wrote out his equations, it was just math and the assumption was as I mentioned R^3. Stokes theorem, the Navier-Stokes equations were implicitly in R^3.

hither_shores 4y ago

> really stupid mistakes such as this one by Feynman It's not a mistake, it's a "lie-to-children" fundamentally no different from an intro analysis class talking about "the" real numbers. Freshmen aren't ready for model theory, and they're not ready for rigged Hilbert spaces.

graycat 4y ago
The set of real numbers is commonly defined and their properties established in a first course in abstract algebra. Model theory is not required. Axiomatic set theory is not even required. Hilbert space? A complete inner product space where complete means every Cauchy convergent sequence converges. The real numbers serve as one example. With the usual inner product, R^3 (where R is the set of real numbers) is another example.
- hither_shores 4y ago
  > The set of real numbers is commonly defined and their properties established in a first course in abstract algebra. A set modelling the real numbers is defined in introductory math courses. The process of quotienting out all the set-theoretic slag to arrive at the real numbers, singular, is generally skipped. As well it should be: it's too ugly to be an end in itself, and too subtle to be of use to freshmen. > Hilbert space? A complete inner product space where complete means every Cauchy convergent sequence converges. I said rigged Hilbert spaces. The proper setting for QM is a Hilbert space equipped with dense subspace P of test functions (for which the inclusion P -> H is continuous). This induces an inclusion of dual spaces H* -> P*, and Reisz-Frechet gives us a natural isomorphism H ~ H*, so we have the "Gelfand triple": P -> H -> P*. The view from 10,000 feet here is that P describes the space of possible measurements, P* describes the distributions, and the fact that the inner product <P, P*> factors through H is a compactness condition. So in particular, a plane wave |k> lives in P*, not H. But you can't teach this to introductory physics students. They won't have the necessary math background for several years. (Many of them will never have it). And yet they have to be taught.
  - graycat 4y ago

mturmon 4y ago

It's tough. I used to work in an area of applied probability where some statistical-mechanics principles were applicable. I'd read papers where authors were making analogies of a large neural network to a stat-mech system, using an applicable stat-mech approximation, and then differentiating that approximation to get a probability bound. It gave interesting results, and did show you something about the original problem that was hard to get by sticking to the original formalism. But at the end of the day, you really would not bet the farm on the truth of those approximations... On the other hand, Fourier analysis was originally doubted and scorned by mathematics, but (if I'm remembering the story correctly) ended up being used so much that theory was developed to explain in what sense the Fourier transform approximates the original function. Another example of the interplay between physics and mathematics is the percolation problem, where there was a kind of archipelago of physics-motivated results that probabilists have been trying to tidy up for decades now. E.g., sec. 1.2 of: https://www.unige.ch/~duminil/publi/2018ICM.pdf https://www.unige.ch/~duminil/publi/2018ICM.pdf

graycat 4y ago
There is both Fourier series and the Fourier transform. For each, need to establish that they exist. Then will usually want to know that the inverse transform DOES approximate the original function. Details are in 4 books by W. Rudin. One of these, on groups, I ignored. I did too much group theory, e.g., wrote my honors paper on it, and didn't want to do more. Principles does Fourier series and does not use measure theory. Real and Complex Analysis does the Fourier transform and uses the Lebesgue integral, that is, measure theory. Functional Analysis does Fourier theory with distributions. I had serious courses from Principles and R & CA. Also Royden and Neveu. All from a star student of Cinlar, long at Princeton. I read FA, quickly. I don't much care to bother with distributions. I got into applications via the fast Fourier transform and power spectral estimation as in Blackman and Tukey. There is an intuitive view that sort of works: The given function is a point in a vector space, with an inner product. The sine waves are coordinate axes. They are orthogonal. The Fourier things are projections of the point onto the coordinate axes. The inverse Fourier thing reconstructs the original function. If use only some of the coordinate axes, then get a least squares approximation of the original function. This all works exactly in ordinary linear algebra, e.g., as in Halmos, Finite Dimensional Vector Spaces, sometimes given to physics students studying quantum mechanics. But these Fourier things have infinitely many coordinate axes, countably infinite for Fourier series and uncountably infinite for the transform, and there the finite dimensional things don't always work. So, Rudin has to be very careful in presenting what does work and proving it -- so with the details it's not easy reading. What fails is a fairly general situation in a fully general Hilbert space. E.g., are not locally compact, but can get some help from a clever use of the parallolgram inequality (somewhat relevant in my startup). I'm no teacher. I do not now nor have I ever had any desire to be a prof. For more, get copies of Rudin's books and dig in. Wonder of wonders, not all physics profs have done that.

Gatsky 4y ago

You rage-quit physics, essentially.

deleted 4y ago
[deleted]

dchftcs 4y ago

>probability density uniform over all of space If space is bounded then such a density can exist.

deleted 4y ago

[deleted]