Wednesday, November 17, 2021

How Long Does It Take to Do an Infinite Number of Things?

Infinite processes shred our intuition to Hell like nothing other than probabilities and relativity. This was the infernal currency of Zeno of Elea, who proposed devilish paradoxes in order to prove the conjecture that movement is impossible. Zeno would construct a simple scenario that would show that everyday aspects of life, like chasing a tortoise or shooting an arrow. We really only have accounts from other authors, basically Aristotle, who disagree with him. Some of these arguments intertwine ideas about space and time, the arrow argues that an object is motionless at any instant, and both the dichotomy and Achilles and the Tortoise make arguments about the impossibility of doing an infinite number of things. I will eventually move on to an example to calculate the total time that it takes for a ball to bounce an infinite amount of time, but I will first describe the dichotomy using Philocetes' Arrow as a story (rather than Aristotle's bare-boned description from the Physics [1]).

Philocetes looses an arrow from the Bow of Heracles at the Trojan prince, Paris. In order for the arrow to strike the Son of Priam, first, it must fly half the distance between the heroes. But, it is clear that in order for an arrow to travel half way to the midpoint before it can get to the midpoint. And it has to travel half way to the point before that, ad infinitum. Therefore, in order to move from one spot to another, no matter how close, you have to move an infinite number of times in a finite amount of time to get anywhere, so the arrow never flies and Trojan War never ends.

This is paradox because we do move, but at least the first time your hear the argument, you don't have a good reason why it is wrong. The arrow must move, but logically it cannot. And this is applicable to every kind of motion. Before you can eat your Wheaties in the morning, you have to get out of bed and get into the kitchen. But, you have to move through an infinite number of small separations to get there.

And you can't do an infinite number of things before breakfast.

Or can you?

Most people think that the invention of Calculus resolved Zeno's paradoxes. This is certainly true in the case of the dichotomy: Leibniz used and even dirtier trick with infinity than Zeno did. Leibniz built calculus out of the idea of an infinitesimal to align with his cosmological ideas. An infinitesimal is a chunk of the universe that is smaller than the smallest division, basically the reciprocal of infinity. The integral calculus would define the distance that the arrow must travel as a sum of all of the infinitesimal chunks of space between Philocretes and Paris. The nature of infinitesimals is that they are smaller than the smallest fraction, there are an infinite number of them between each rational number. The infinity of the infinitesimals is that of the real numbers (the continuum) and the infinity of the dichotomy is that of the rational numbers (countable). So, if you can construct a theory of motion that adds up all the infinitesimal points, it will automatically encompass the infinity of halves used by Zeno.

This solve the dichotomy, but it does so indirectly with an end around.

By subsuming the motion of the dichotomy into a single, continuous process that can be analyzed separately, we show that motion can exist. We solve the riddle by changing the problem, though. However, this leaves Zeno's premise unchallenged: an infinite number of processes take an infinite amount of time. What I'd like to do here is take on the premise that an infinite number of discrete, sequential processes needs to take an infinite amount of time. You could do the same thing with the dichotomy, as well, but since that is an arbitrary partition of a single, continuous process, which I feel is a little different.

The specific question I ask is: how long does it take a rubber ball to stop bouncing? The physics here is quite simple. It can be done with kinematics using the simplest of deflection theories: the coefficient of restitution. The model uses the simple rule that the velocity of the rebound of an object is proportional to its original speed, and that proportionality (the coefficient of restitution) remains the same after each bounce. The duration of the air time of the ball is given by uniformly accelerated motion. The sum of a sequence of such bounces will lead to an infinite series with a known sum (thanks again, calculus), and this sum will be finite.

Where can this go wrong? Well, it's not in the assumption of uniform acceleration. Yes, it's not quite true, but it's pretty accurate at low velocities for short times, which a rubber ball acts in. If we complicate the problem by adding in air resistance, that will give us a slightly more accurate estimate at the cost of an annoying integral (no thanks, calculus). This accuracy will give us a time that is strictly smaller than the uniform acceleration version by giving us a factor similar to the coefficient of restitution itself. The significant assumption that would break this analysis, if it were relevant, would be that the time of the bounce itself will be the same each time the ball hits the ground if the bounce is modeled on an elastic restoring force, which is probably the best model available. Even though this will be small, at some point it will be larger than the air time per bounce, and since it remains the same, adding an infinite number of them would create an infinite time for the bounce.

But, to answer the basic question, can an infinite number of processes be completed in a finite amount of time, eliminating the time of the bounce is justifiable.

So what happens in this case? Well, from basic kinematics, we find that the time of an individual flight is proportional to the initial speed of that bounce.* Since the initial speed of each process is the coefficient of restitution is just the initial speed of the previous process, the duration of the subsequent process is scaled down by the sane proportionality,

tn = r tn-1 = rn t0

which means that flight is scaled down by a power of the coefficient of restitution.
When these are summed, we find an infinite series in powers of the coefficient of restitution that has a known sum: the inverse of one less the coefficient [ 1/(1-r) ]. So, the total time the ball bounces is finite if r < 1 (which is must be unless it is gaining energy from the environment somehow).

So, an infinite number of bounces takes a finite amount of time

t = 1/(1-r) t0.

This is a reasonable answer because if r = 1 the bouncing goes on forever and if r = 0 it stops after the first flight. This should be the same result you'd find if you were to sum the time to travel each segment of the arrow's path, but here we have distinctive processes represented by the flights between bounces. Our hero Leibniz has defeated Zeno of Elea's Satanic dichotomy.

So the Trojan War terminates, and you can do an infinite number of things before breakfast.


* The proportionality constant is 2/g.

[1] All of my Aristotle is missing. Most of what I know about this comes from Sainsbury's Paradoxes, >although I was using the Stanford Encyclopedia of Philosophy.

Wednesday, November 10, 2021

Critical Thinking vs. Problem Solving I - How Are They Different?

I've spent a lot of time thinking about critical thinking and problem solving. Nominally, my course is required to include a critical thinking component by the university so that it order to satisfies a distribution requirement. Over the past five years, I have never really been able to discern the difference between critical thinking and problem solving, a more common term for what we "teach" in physics. Some papers explicitly say that problem solving is a kind of critical thinking, [1] others say that it's a separate skill. [2] I don't really know the proportion. I originally just went for a rather strict problem solving format from the University of Minnesota* that I feel shows me how a student is thinking about problems,and based on Force Concept Inventory scores, I implemented it well (gain around 35-45%).

The students hated it, but my job isn't to be adored.

However, one spring day in 2020, I was informed that my course was to be reviewed for its critical thinking component.** When reviewing a course, all I have to do is to submit a product for each student that shows their critical thinking skills. This product can be a test question, a homework problem, a paper. I have a section on the tests where students individually provide an explanation of how to solve a problem, and I felt that this would do. It is a little stilted. What do you want to do? "Find the velocity." How are you going to do it? "Use conservation of energy." How do you represent that? "1/2 m v^2 = 1/2 m u^2 + mgh." It shows exactly what the student is thinking and how they utilize the data. I felt it would be a good way to show the development of the students' problem solving capabilities over the semester.

But I was given a rubric, and the rubric told me that I was wrong.

These are some highlights from the rubric:

Explanation of Issues. Problem is stated clearly and described comprehensively, including all relevant information.
Evidence. Viewpoints of experts are questioned thoroughly.
Context. Thoroughly analyzes assumptions and carefully evaluates the relevance of contexts when presenting a position.
Student's Position. Position is imaginative and other's views are synthesized within it.
Conclusions. Conclusions are logical and reflect the evidence and perspectives in priority order.

Try that with conservation of energy.

For some of this, expanding the selection from just he planning phase of the problem solving process would probably do. For others, it seems irrelevant. In fact, some of the categories seem to be completely irrelevant to the course ("Influence of Context and Assumptions" is the full title). But, looking at the rubric for the curriculum component, I feel at minimum it requires a term paper, and probably a thesis. The school implicitly takes the side that problem solving is not a part of critical thinking.

Critical thinking, as described by the rubric, is really separate from problem solving.

However, I still think that there should be some overlap. I think in come coming posts, I'll talk about what I think problem solving and critical thinking are, possibly in several posts each, and then I'll talk about some specific problem-solving tools for first year physics students.


* That, I think, they don't use it any more.

** And just after I wrote this, they told me that this was the evaluation of the engineering students' "written communication."


[1] Willingham, D., "Critical Thinking: Why Is It So Hard to Teach?" American Educator (2007).

[2] Pasquinelli, E., M. Farina, A. bedel, and R. Casati, "Naturalizing Critical THinking: Consequences for Education, Blueprint for Future Research in Cognitive Science." Mind, brain, and Education 15, 168 (2021).

Saturday, May 2, 2020

Alain Aspect's Quantum Optics on Coursera

I finally got done with finals, and as a reward, I decided to see if I could find a Coursera course to play with. I had been looking to see if there was a quantum computing MOOC available (and there are many), but among the search results was Alain Aspect's Quantum Optics course on Coursera. This is, he says, the first of two MOOCs based on his textbook (at right*). It's a short course (4 weeks, basically four lectures), and so, just right for me.

The subject is interesting (and very similar to a special topics course I took as an undergraduate called "The Quantum Mechanics of the Laser" -- I wish I'd kept those notes when I moved), but the lectures are dense. They do go over a lot of the material in Sakurai's Modern Quantum Mechanics*, which I worked through two summers ago, but of course with a focus on the meaning in terms of quantum optics. Already, some things I haven't heard of before, some that relate to experimental design (quantization volume), some straightforward interpretations of mathematical expressions (the energy of a single photon). The understand in terms of experimental parameters is particularly helpful to me (since I understand things in terms of experiments, due to my training).

The course, however, is not for those who are afraid of mathematics. Aspect's discussion in mathematically dense. Really dense. My students think I use too much mathematics in university physics classes, but this is all math. And Aspect expects you to have seen it all before: many times he references your prior knowledge. He doesn't quite say that you're an uneducated ignoramus if you can't recall trivialities like the photon energy or the uncertainty relations (he calls them dispersion relations, an aspect of his philosophy -- it's nice to hear an expert talk explain the mechanics of physics in a way that makes it clear he has opinions). And the homework is tough. Not as tough as it sounds when you read it, but pretty tough.[1] Even on the internet, you're expected to know your stuff.

I have found that I had to speed his lectures up. I don't know if it's him, if it's because he's European, or because Coursera makes everyone speak like someone shilling their latest book in a bad TED talk, but he talks slowly. Last summer, I took Werner Krauth's MOOC,[2] from the same school but a different country, and he spoke with the same cadence. I found I had to speed up the lecture to 1.5x so that they spoke at a normal speed.

This minor technical problem aside, I certainly am enjoying the break this provides before I start preparing for my summer courses (How did I let myself get roped into summer courses? At least they're on-line so I can get a lot of the work out early).

[1] I didn't pay the $49.99, or whatever, it costs in order to get it graded, but I did work it. And it reinforced the advice I give to my students: try the homework before class, then the class will be more useful to you. [2] Which was serendipitous, since I'd begun setting up to work through the book it was based on, Statistical Mechanics: Algorithms and Computation,* when Coursera sent me an e-mail about it. I get the feeling there's as much shilling on Coursera as there is at TED talks. But it couldn't be more: a TED talk is just an advertisement for a book. If you're lucky, there's more to the book than just the TED talk. Obviously, though, there's more to a physics textbook than eight hours of lecture. Hell, there's more to a physics textbook than the forty hours of lecture in a semester.

* Note: These links are to Amazon pages. Purchases on those pages from the links will give me a commission (at least for now -- every time I've tried to use the Amazon Associates program they've kicked me off for not selling anything, but I do like having the links in the show notes so that you can pick up the books we might reference in a discussion).

Saturday, September 1, 2018

Quantum Sense and Nonsense by Jean Bricmont

I don't really know why I picked up Quantum Sense and Nonsense or when. I'm pretty sure it was in the last year when I was looking for some popular books to read after I finished The Wave Function, and this one, written by Jean Bricmont and published by Springer, stood out. The cover, and likely the description, seems a little misleading since it seems to say that the book will focus on crazy and unfounded assertions of psychic and mystical properties attributed to quantum mechanics (and as Bricmont has published with Sokal, that's exactly what you'd think), but instead the book focuses on two experiments (double slit experiments and EPR-type experiments, both of which seem to be recurring themes on Physics Frontiers) and the interpretation of each. Bricmont follows Bell in asserting that EPR-experiments like the Aspect experiment show that there is some kind of non-locality at play in quantum mechanics and that the best way to interpret the meaning of the wave function (that is, what the wave function, itself, is) is to look toward an interpretation like the de Broglie-Bohm vision of the wave function (see Bell's, Speakable and Unspeakable in Quantum Mechanics).

Despite being much different than what I thought the book would be, this made the Quantum Sense and Nonsense an excellent read.

The double slit experiment as performed by Thomas Young in the first decade of the 19th century showed that coherent light from the sun interfered with itself, showing that light is, in fact, a wave -- and brought about the belief in a mysterious Ether in which the light waves propagated. When Einstein showed the photoelectric effect requires quantization of light,* this made the interpretation more difficult. And de Broglie's prediction that electrons, really all material objects, have a wavelength and the subsequent discovery of electron diffraction, brought the same problem to all matter. And the interference is so strong that when a single photon or a single electron is sent through the slits, and the results of the experiments accumulated, the interference fringes are still seen. Material objects interfere with themselves.

A very strange property, and one that leads to many strange interpretations of quantum mechanics, is that if you set up a detector at one of the slits in the double slit experiment to see which slit the particle passes through, then the interference fringes will disappear. This leads to the idea that observation causes a change in the wave function, what is termed the wave function collapse. Many strange ideas come out of this, even from physicists (Bricmont's target). People use this idea to give consciousness a role in the measurement of quantum systems, Bricmont uses quotes from the following physicists to show the sloppy thinking on these points: d'Espagnat, Wheeler, and Mermin (to name only those I've heard of): they all give some role to the human mind in the collapse of the wave function. To be fair, understanding the collapse is impossible in the standard "Copenhagen" interpretation of quantum mechanics, which is what Schroedinger's cat was intended to show.

The EPR experiments, violations of Bell's theorem, are the second cause of sloppy thinking because they show one of two things: either (1) quantum mechanics is non-local or (2) quantum mechanics is non-causal. Those are the two assumptions that Bell uses to derive his inequalities beyond ordinary statistics and quantum theory. If you have to choose one of the two assumptions to invalidate, (1) is the more likely (although we recently published a podcast on retrocausality and Yakir Aharonov has a different version of a locality-preserving assumption, presented in his Quantum Paradoxes book as well as old papers). But once you remove locality from your assumptions about the world, people start babbling about telepathy and similar nonsense.

As befits someone of Bricmont's station, the descriptions of these experiments are exemplary, and Quantum Sense and Nonsense would be worth a read if only they were presented here. However, he does us another service by giving us a rich, logical and convincing description and defense of the de Broglie-Bohm pilot wave theory of quantum mechanics. In this theory, the wave and particle are broken up into two objects, an oscillation in space time that drives the motion of an otherwise deterministic particle. The randomness of quantum mechanics then ceases to be the mystical randomness associated with Bohr and Heisenberg and Copenhagen in general and becomes the deterministic randomness of statistical mechanics.** Bricmont goes so far to say that because of this and the fact that it can be mathematicised, de Broglie-Bohm is the only interpretation of quantum mechanics;*** the others (including statistical, Copenhagen, and many-worlds) don't meet that bar. Obviously, it doesn't mean that Bricmont is right, since he's delved into philosophy or worse in the comparison of interpretations by their linguistic characterizations, but it is a good way of thinking about the interpretations.

So I would recommend this book. I do think that it is a little too popular for most people that would read this, and he often refers to his own, more technical Making Sense of Quantum Mechanics quite a bit for more quantitative details. He also says that this is only "slightly" more rigorous and would probably point you to P. Holland's The Quantum Theory of Motion for a really rigorous treatment. I haven't read either of those two books, however, so I can't recommend them to you.


I wrote this review a little faster than I'd like because I'd just finished the book yesterday and Google sent me a "news story" on my phone today, which I read over my morning coffee. It was a rather infantile post by LuboŇ° Motl, someone I've never heard of, who calls himself a "freelance string theorist" (but who has a reasonably impressive pedigree) reviewing books by science journalists. It makes me sad when a physicist does as bad a job of presenting science as a science journalist does.

The blog does a good job of showing two very bad ways to think about the interpretations of quantum mechanics. The first is from the book he reviews (or really, the blog post that he reviews of the book that it reviews). In that case, the science journalist author, whose name is of no importance, suggests that all interpretations are valid. This seems quite odd to me, especially when most of them are logically contradictory: if you believe in a wave function collapse, then you can't coherently believe in the universal wave function of Everett. You can make up a pretty complex and silly rationale if you want to, but it will always end up being incoherent somewhere (and I'm not going to read it to find out where). The reason you would want to hold multiple conceptions in your head is to find out places where they disagree -- and then to find an experiment that distinguishes them.

Motl himself presents to us the second version, which is to deny all interpretations. But that is clearly unsatisfactory. Although it is called the Copenhagen interpretation (by some, what is meant by that changes from philosopher to philosopher, physicist to physicist), you still have to have some interpretation. You have to have some ontological vision of the wave function to assert that information cannot travel faster than light during its collapse, for example, or to state that it would be impossible to ever use it for long distance communication. That you refuse to examine your beliefs doesn't mean that they're not there.

Bricmont does a good job of showing how to deal with interpretations without getting so dogmatic that his assertions become meaningless, just the opposite of Motl


* Planck's experiments don't show this. Since the quantized electromagnetic waves are coming out of an enclosed chamber, black body radiation could be interpreted as having something to do with standing waves in the oven.

** Interestingly, though, the roles of randomness are reversed. In statistical mechanics we measure macroscopic parameters associated with microstates. In quantum mechanics, and especially in the de Broglie-Bohm interpretation, the wave function is the microstate and the measurement is of the particle, or the microstate.

*** I should mention that de Broglie-Bohm is not excessively popular among physicists. Reading The Wave Function, however, I came out of it thinking it was extremely popular among professional philosophers of science.

Thursday, August 2, 2018

Physics Frontiers: Most Popular Episodes

All Time
1. Physics Frontiers 45: Loop Quantum Gravity
2. Physics Frontiers 46: Wigner's Friend
3. Physics Frontiers 44: Spooky Action at a Distance
4. Physics Frontiers 38: The Dimensionality of Space-Time
5. Physics Frontiers 53: Electromagntic-Gravitational Repulsion

1. Physics Frontiers 53: Electromagnetic-Gravitational Repulsion
2. Physics Frontiers 55: Multiversality
3. Physics Frontiers 48: The Gerstenstein Effect

1. Physics Frontiers 45: Loop Quantum Gravity
2. Physics Frontiers 46: Wigner's Friend
3. Physics Frontiers 44: Spooky Action at a Distance

1. Physics Frontiers 38: The Dimensionality of Space-Time
2. Physics Frontiers 33: The String Theory Landscape
3. Physics Frontiers 40: The Octonions

1. Physics Frontiers 17: The Physics of Time Travel
2. Physics Frontiers 9: f(R) Theories of Gravity
3. Phyiscs Fronteirs 12: A Graviational Arrow of Time

Physics Frontiers Index

[Edited 12/8/2020]