God Plays Dice

A surprising fact: drill a straight tunnel through the earth, between any two points. Drop a burrito in at one end. Assuming that you could actually build the tunnel, and that there's no friction, the burrito comes out the other end in 42 minutes. This is called a gravity train and it's not hard to prove (the version I link to is due to Alexandre Eremenko of Purdue) that the time it takes the burrito to get from one end to the other is (3π/4)^-1/2 (G ρ)^-1/2, where G is the gravitational constant and ρ is the density of the Earth. Alternatively this can be written as (r³/Gm)^1/2, where r is the radius of the earth and m its mass.

Everyone's so surprised, when they see this, that the time doesn't depend on the distance between the two points! And this is interesting, but as a result you don't see the more subtle fact that the time doesn't depend on the size of the planet. If I make a super-Earth that is twice the radius but made of the same stuff, so it's eight times as massive, then the density stays the same. Somewhat surprisingly you can see this using dimensional analysis. It's "obvious" that this time, if it exists, can only depend on the mass of the earth, the radius of the earth, and the gravitational constant. The mass of the earth, m, has dimension M; the radius, r, has dimension L; the gravitational constant has dimension L³ M^-1 T^-2. The only combination m^α r^β G^γ that has units of time is G^-1/2 m^-1/2 r^3/2.

Of course I'm making a big assumption there -- that the constant time "42 minutes" is actually a constant! It seems perfectly reasonable that it could depend on the distance between the two termini. I'll handwave that away by saying that it depends on the angle formed by the two termini and the center of the earth. And angles are of course dimensionless.

(The Alameda-Weehawken burrito tunnel, being non-fictional, uses magnets to accelerate and decelerate the foil-wrapper burritos and takes 64 minutes instead of the theoretical 42.)

Chad Orzel, physicist, writes why I'd never make it as a mathematician. He calls himself a "swashbuckling experimentalist" and says that he doesn't like thinking too hard about questions of convergence and the like. This is in reference to Matt Springer's most recent Sunday function, which gives the paradox:

1 - 1/2 + 1/3 - 1/4 + ... = log 2

1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + ... = (log 2)/2

I find that I tend to act "like a physicist" in my more experimental work. Often I'm dealing with the coefficients of some complicated power series (usually a generating function) which I can compute (with computer assistance) and don't understand too well. Most of the time the things that "look true" are. This work is, in some ways, experimental, which is why it's tempting to act like a physicist.

Oh, yeah, I graduated today.

Also, here's an interesting tidbit, from this New York Times piece on SPF. SPF, or "sun protection factor", is the number on the sunscreen bottle; if a properly applied sunscreen lets through a fraction p of the UV rays it's meant to protect against, then that sunscreen has SPF 1/p. (The numbers in the article talk about the proportion of the UV rays which are blocked; in this case, if a fraction q of the UV rays are blocked, the sunscreen has SPF 1/(1-q).)

Anyway, you're supposed to apply some ridiculous amount of sunscreen to your body, about an ounce. This seems like a lot to most people, because that stuff is expensive! So a lot of people underapply sunscreen. (I'll include myself here.) The article quotes Darrell Rigel, NYU dermatologist, as saying that if you apply half the sunscreen you're "supposed" to, you have to take the square root of the SPF.

That sounds obvious once you think about it -- but I'll admit I'd never thought about it. Say I have a sunscreen that allows through one-sixteenth of the light which hits it when applied properly. Now imagine splitting it up into two coats, each of which allows through the same proportion of the light that hits it. One-fourth of the light makes it through the outer coat; one-fourth of that light makes it through to the skin.

Of course there are issues with this analysis, but according to this paper in the British Journal of Dermatology it appears to hold up. And applying twice the usual amount of sunscreen apparently squares the SPF. (The effect is actually a bit less than this, because sunscreens don't block all wavelengths equally, nor does the sun's spectrum contain all wavelengths equally.)

This all implies that if you want to compare prices of sunscreens, you should divide the cost of the sunscreen by the product of the bottle's volume and the logarithm of the SPF. Do sunscreen prices actually work this way?

I was singing in the shower, as I do.

I noticed that certain notes seemed to resonate with the shower more than others.

These were, in ascending order, Eb2, G2, C3, and G3, where C4 is middle C. (These may not be exactly right; I don't have perfect pitch. The intervals are right, though.)

Exercise for the reader: how large is my shower?

The blog's been slow. I've been off writing real mathematics, thinking for and preparing for the class I'm teaching this summer, and so on. But I'm still here!

And while I'm here, you should read Chad Orzel on the faulty thermodynamics of children's stories. In the story of Goldilocks and the three bears, one would expect that the papa bear is the largest, then the mama bear, and then the baby bear. Furthermore, you'd think that the larger the bear, the larger the bowl of porridge, and the slower it should cool off. But it doesn't seem to work that way! Read the comments come up with some interesting explanations.

Exercise for the scientifically-inclined reader: comment on the physical implications of the Three Little Pigs.

Exercise for the not-so-scientifically-inclined reader: what's with all the animals coming in threes?

The acceleration due to gravity in the Mario games is ridiculously high.

From 60 Minutes on Sunday, video here: the banking crisis is the fault of the mathematicians and physicists, who went to work on Wall Street and invented some complicated models. The "financial expert" says that "you can't model human behavior with math". That may be true with our current mathematics, but I suspect the fault doesn't lie with the modelers so much as with the people who went ahead and thought the models were perfect.

Although there are rumors that a lot of the models basically worked on the principle that defaults on mortgages were independent, which is so obviously false that you'd fail a freshman who said it. (Basically, when the economy gets bad, more people can't pay their mortgages. The effect is larger because there are adjustable-rate mortgages, and everybody suffers roughly the same adjustments.) I would not be surprised to learn that the problem here is that Conditional Probability Is Subtle; the quants may very well have known their models were flawed, but the suits didn't want to hear it.

I just hope this meme dies out quickly and doesn't gain traction. Our PR already isn't so good.

Trivium Mathématique, V. I. Arnold (in French) is a list of 100 problems that Arnold set as a list of mathematical problems that (university) students in physics ought to be able to solve. This is preceded by Arnold stating that the system of oral exams in Russia should be replaced with written exams, and that a system based around the ability to solve certain problems is superior to a system based around the ability to regurgitate certain theorems. He then goes on to give a list of such problems.

As someone who tends to prefer problem-solving to theory-building, I find the following particularly interesting (my translation):

A student who takes more than five minutes to calculate the average of sin ¹⁰⁰ x with a precision of 10% has no mastery of mathematics, even if he has studied nonstandard analysis, universal algebra, supervarieties, or plongement [I don't know this word] theorems.

Go ahead, try that problem! (Curiously, it's not one on the list.) More generally, it's an interesting bunch of questions in various branches of mathematics, biased towards but by no means exclusively in calculus and differential geometry; this is no surprise as it's meant for physicists.

(Why the French version, you ask? The article was originally in Russian; there's an English translation but it's not free. Links to free versions in Russian or English, if they exist, would be appreciated.)

Edit, 12:07 am Sunday: An anonymous commentator provides the English version, and Dmitri Pavlov the Russian.

I'm currently watching Science Saturday at bloggingheads.tv, which this week features Peter Woit (Not Even Wrong) and Sabine Hossenfelder (Backreaction).

When the video started, I thought "hmm, Woit has an awful lot of yellow books behind him for a physicist".

Woit, it turns out, works in the mathematics department at Columbia, as I was reminded when he started talking about the different job situations in physics and mathematics about fifteen minutes in. Basically, Woit says that jobs in physics are scarcer compared to PhDs in physics than the analogous situation in mathematics, so physicists feel more pressure to "do everything right" -- in his view this means they feel unnatural pressure to work in string theory, which Woit sees as a bad thing. After all, what if string theory's wrong? Physics as a discipline should diversify.

In his paper The Mathematical Universe, Max Tegmark explores the physics implications of what he calls the External Reality Hypothesis, namely that there exists observer-independent reality. From this he argues for what he calls a Mathematical Universe Hypothesis, namely that the universe is mathematics. This is in the same sense that, say, Newtonian gravitation is just the study of curves in R⁴ (that is, three space dimensions and one time dimension) with minimum action.

It's interesting, but at thirty pages kind of long; Tegmark has also written a three-page article on the same topics called Shut Up and Calculate!.

I am led to wonder if Tegmark is deliberately referring to the Flying Spaghetti Monster here, in his discussion of Newtonian gravitation. (The "frog" sees only the universe as it exists right now; the "bird" sees the universe "as a whole" and in particular sees all time at once.)

If the frog sees a particle moving with constant velocity, the bird sees a straight strand of uncooked spaghetti. If the frog sees a pair of orbiting particles, the bird sees two spaghetti strands intertwined like a double helix. To the frog, the world is described by Newton’s laws of motion and gravitation. To the bird, it is described by the geometry of the pasta, obeying the mathematical relations corresponding to minimizing the Newtonian action.

Also of interest is the claim that the Mathematical Universe Hypothesis "banishes... the classical notion of randomness", essentially because all probability can be recast as measure theory. The randomness of quantum mechanics, according to Tegmark, is essentially "epistemological" -- we perceive what looks like randomness because we do not have perfect knowledge. This, of course, is not believed by all physicists; the Copenhagen interpretation of quantum physics does fundamentally include randomness.

And why aren't there other interpretations of quantum mechanics named after cities?

Word problems take place in a graded ring, from (the recently relocated) Mathematics under the Microscope (Alexandre Borovik), via The Unapologetic Mathematician (John Armstrong).

In short, Borovik claims that elementary school word problems take place over $\mathbb{Q}[x_1, x_1^{-1}, x_2, x_2^{-1}, \ldots, x_n, x_n^{-1}],ドル where the x_i represent different things that could be added. In this formalism, it makes sense to add, say, apples and oranges, going against the usual rule that you're only allowed to add quantities with the same "dimension". (Indeed, Borovik illustrates the idea with an example of this nature.)

I'm reminded of "dimensional analysis" as taught in, say, introductory physics classes, where we only allow monomials to have meaning, namely that the monomial x m^a kg^b sec^c measures some physical quantity with dimensions L^aM^bT^c where L, M, T stand for length, mass, and time. (For example, in the case of speed, a = 1, b = 0, c = -1.) I can't think of situations in physics where one deals with a quantity of the form, say, a kg + b m. Is this because they don't exist, or because I don't know as much physics as some people?

Eugene Wigner's famous essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences is available online.

Today (well, technically, yesterday) I came across Concentration of Measure for the Analysis of Randomised Algorithms, a draft of a monograph by Alessandro Panconesi and Devdatt Dubhashi. I'm giving an expository seminar talk on, basically, the concept of negative dependence next week. There are certain sets of random variables that are inversely correlated, in the sense that if one of these variables gets larger (for example, if the variables are indicator variables, the indicated event gets more likely) then the other variables get smaller, and this can be made precise. The prototypical example occurs when balls are thrown into bins, and there's one random variable for each bin, which is 0 when the bin is empty and 1 when the bin is occupied. (Bins can be multiply occupied in the example of thinking of; the case where only single occupation is allowed should also show negative dependence, but it's not as interesting.)

Not surprisingly, if you have a large collection of such random variables, and you add them all up, the distributions you get tend to be pretty tightly concentrated -- intuitively you think they should be more concentrated than they would be if you add up a bunch of independent random variables. I don't know of formal results in this direction, but at the very least they're not less concentrated than a sum of independent random variables. (In the extreme cases, the sum is constant, so we get concentration on a point.) This particular result is a Chernoff bound for negatively dependent random variables.

I don't want to go into the details here. (I'll probably try to make a blog post summarizing the talk, but the usual procedure for these things seems to be that the post should follow the talk, and the talk isn't quite ready yet.) But I just want to give a quote (edited for typos) from the draft monograph I linked to above:

In more sophisticated forms, the phenomenon of the concentration of measure underlies much of our physical world. As we know now, the world is made up of microscopic particles that are governed by probabilistic laws – those of quantum and statistical physics. The reason that the macroscopic properties determined by these large ensembles of particles nevertheless appear deterministic when viewed on our larger scales is precisely the concentration of measure: the observed possibilities are concentrated into a very narrow range.

In three words? God plays dice.

I'm reading Random sampling of plane partitions, by Olivier Bodini, Eric Fusy, and Carine Pivoteau. (arXiv:0712.0111v1). (Note: if you follow posts where I mention new things I've come across in the arXiv, you'll find that I'm currently reading papers I come across there at about a six-week lag.)

The authors give a way to sample from plane partitions of size n uniformly at random. The method is not as efficient as one might like -- it begins by generating a plane partition of size approximately n, where "approximately" means something like "within o(n) of", from a distribution which is uniform when restricted to partitions of exactly n, by a method known as "Boltzmann sampling", and then throws out those partitions which are not of size n. Still, a plane partition of size n can be chosen uniformly at random in time O(n^4/3). (Note that uniformity is important here; if we didn't care about the distribution, we could just write down a bunch of numbers that sum up to n and be done! More seriously, uniform sampling of this sort of combinatorial object with a bunch of highly interdependent parts tends to be tricky.)

But the idea I really like here is that of Boltzmann sampling, the inspiration for which I assume comes from physics. Namely, given a combinatorial class, we give each object a weight xⁿ where n is its size, where x is a parameter between 0 than 1; then we pick objects with probabilities proportional to their weights. It turns out to be routine to give a Boltzmann sampler -- that is, an algorithm which picks a member of the class according to this distribution -- for any combinatorial class we can specify. (This is according to the papers of Duchon et al. and Flajolet et al. I've listed below, which I haven't actually read yet.) It reminds me of the partition function of statistical mechanics (the fact that this has "partition" in the name is a coincidence, as one could do this for any combinatorial class). Say a certain sort of physical system can occupy states labeled 1, 2, ..., N. Let E_j be the energy of state j. Then give each state the weight exp(-β E_j), where β = 1/(kT) is the "inverse temperature"; the probabilities that the system occupies various states are proportional to these weights. Replacing energy by size and exp(-β) by x gives the combinatorial Boltzmann sampling. So the parameter x is some sort of temperature.

References

Olivier Bodini, Eric Fusy, and Carine Pivoteau. Random sampling of plane partitions. arXiv:0712.0111v1.

P. Duchon, P. Flajolet, G. Louchard, and G. Schaeffer. Boltzmann samplers for the random generation of combinatorial structures. Combinatorics, Probability and Computing, 13(4-5):577-625, 2004. Special issue on Analysis of Algorithms.

P. Flajolet, E. Fusy, and C. Pivoteau. Boltzmann sampling of unlabelled structures. In Proceedings of the 4th Workshop on Analytic Algorithms and Combinatorics, ANALCO'07 (New Orleans), pages 201-211. SIAM, 2007.

Guess what, folks? Probability and cosmology are weird when they interact. (Big Brain Theory: Have Cosmologists Lost Theirs?, by Dennis Overbye, January 15, 2008 NY Times.)

Basically, it appears to be more likely that we are some sort of naked brain living in an illusion of a world than that we live in the actual world we perceive. Roughly speaking, this occurs if we assume that the universe is infinite -- and thus everything that can occur does occur -- because a naked brain is supposedly much more likely to form by chance than the reality we think surrounds us does.

The obvious rebuttal, if one is wedded to this particular model of cosmology, is an evolutionary one -- maybe naked brains aren't so likely after all, because brains are produced (or so we think) by evolutionary processes, so is one really so likely to find a brain just sitting there without the biology in which it evolved? Overbye's article only mentions physicists; I wonder what (if anything) the biologists have to say. And I don't think our probabilistic understanding of evolution is quite to the point where the first sentence of this paragraph can be made rigorous. (On this point, I'd love to be told I'm wrong!)

edit: Sean at Cosmic Variance has written about this much more insightfully than I, and with links to a lot of the relevant research.

The game Gravity Pods is a very fun game, and it vividly illustrates the principle of sensitive dependence on initial conditions. You're asked to fire a projectile in an area which includes some gravitational attractors, in an attempt to hit a given target. (At later levels you can move some of the attractors around. At even later levels there are repulsive objects, a nice twist. There might be even more tricks later, but I haven't gotten that far.)

It was an even more vivid illustration of sensitivity before, because in earlier incarnations of the game there were levels that could not be beaten -- it would be possible to fire at an angle of 10 degrees from the horizontal, or 10.1 degrees (0.1 degrees is the finest control you get over the angle at which the projectile is fired), but 10.05 degrees (say) is what you really needed. Similarly, the movable attractors and repellers can only be placed to the nearest pixel. I don't think this problem exists any more, but there are still annoying (although perhaps fascinating as well) levels in which sub-pixel positioning abilities would be nice.

From Walter Lewin's 8.01 Lecture 14, about 14 minutes in:

"What is power? Power is work that is done in a certain amount of time. dw/dt, if w is the work, is the instantaneous power at time t. We also know power in terms of political power. That's very different. Political power, you can do no work at all in a lot of time and you have a lot of power. Here in physics, life is not that easy."

This reminds me of the classic claim that, say, bricklayers do more work than lawyers, because work is force times distance, and bricklayers are exerting much larger forces when they lift things, since bricks are heavier than pieces of paper. This is a sort of translingual pun, the two languages being English and scientific jargon.

A friend of mine, not long ago, noticed that it was snowing up outside her office window. (When I've seen this happening, it's usually due to something like an internal courtyard in a building, so the air currents get sufficiently weird to push the snow upwards. Don't worry, physics isn't broken.)

This raises the (somewhat silly, I must admit) question: what if gravity had actually reversed itself outside my friends' office? Three scenarios are possible:

• gravity instantaneously reverses itself;


• the downward-pointing vector of gravitational acceleration decreases in magnitude, goes through zero, and then becomes an upward-pointing vector;


• the gravitational acceleration vector always has the same magnitude, but swings around through different angles to point up instead of down.

I claim that the third of these is not reasonable. Basically, this is for reasons of symmetry. Imagine gravity that neither points straight down nor straight up. How would gravity know in which of the many possible "sideways" directions to point? There are only two points that could possibly have any effect on the direction of gravitational acceleration, namely the point that you are at and the center of the Earth. The direction of gravity must be invariant under any isometry which fixes those two points -- thus it must point straight towards the center of the Earth or straight away from it.

Apparently I am more attached to gravity being a central force than to its particular strength or even to the fact that it is attractive, not repulsive.

(I'm not sure whether the first or second of the models suggested above is reasonable. This post is silly enough as it is.)

There's a classic argument, supposedly due to Galileo, for why mice and elephants are shaped differently, and there are no organisms that have the general mammalian body plan but are a hundred feet tall.

The argument goes as follows: assume that all animals have roughly similar skeletons. Consider the femur (thigh bone), which supports most of the weight of the body. The length of the femur, l, is proportional to the animal's linear size (height or length), S. The animal's mass, m, is proportional to the cube of the size, S³. Now, consider a cross-section of the bone; say the bone has cross-sectional area d. The pressure in the bone is proportional to the mass divided by the cross-sectional area, or l³/d². Bones are probably not much thicker than they have to be (that's how evolution works), and have roughly the same composition across organisms. So l³/d² is a constant, and so d is proportional to l^3/2 or to s^3/2. SSimilar arguments apply to other bones. So the total volume of the skeleton, if bones are asusmed to be cylindrical, scales like S^7/2 -- and so the proportion of the animal taken up by the skeleton scales like S^1/2. What matters here is that if S is large enough then the entire animal must be made up of bone! And that just can't happen.

Unfortunately, the 3/2 scaling exponent isn't true, as I learned from Walter Lewin's 8.01 (Physics I) lecture available at MIT's OpenCourseWare. It's one of the canonical examples of dimensional analysis... but it turns out that it just doesn't hold up. I suspect, although I can't confirm, that this is because elephant bones are actually substantially different from mouse bones. It looks like for actual animals, d scales with l (or perhaps like l^α for some α slightly greater than 1), not with l^3/2. Lewin uses this as an example of dimensional analysis; he also predicts that the time that it takes an apple to fall from a height is proportional to the square root of that height, which is true, but that's such a familiar result that it seems boring.

(Watching the 8.01 lecture is interesting. The videotaped lectures available on OCW are from the fall of 1999, which is two years before I entered MIT; occasionally the camera operator pans to the crowd of students, and a few of them look vaguely familiar.)

P. S. By the way, this blog is six months old. Thanks for reading!

Edited, Wednesday, 9:14 AM: Mark Dominus points out that the argument is in fact due to Galileo, and can be found in his Discourses on Two new Sciences. This is available online in English translation.

Language Log's Mark Liberman wrote this morning about The biggest typo in history, from this New York Times article, which referred to 10500 instead of 10⁵⁰⁰. (They've fixed the mistake now; I wonder if they found it independently or if someone pointed out the Language Log post to them?) The number here refers to the size of the string theory landscape, which I don't pretend to understand.

The Times made a similar mistake in The Death of Checkers, December 9, writing:

The brute-force method is slow, which is its big limit. Schaeffer says he suspects you couldn’t use it to solve chess, because that game — with between 1040 and 1050 possible arrangements of pieces — is far more complicated than checkers, which has 5 × 1020 positions. “Chess won’t be solved in my lifetime,” he predicts. “We need some new breakthrough in technology to do that.

(James Kushner, a reader of this blog, pointed this out to me; I meant to mention it at the time but didn't.)

This particular mistake is actually so common that I barely notice it any more, especially because I usually have a decent idea of what the number should be. But to someone who doesn't know, it could make a big difference in how they perceive the article. For one thing, if you interpret the checkers quote literally, you get the idea that checkers is more complicated than chess: after all, 5 × 1020 = (削除) 5010 (削除ここまで) 5100.

Another paper that's been known to make these mistakes is The Tech, MIT's student newspaper. The difference here is that The Tech ought to know better; their entire staff is MIT students, who ought to be familiar with exponents. The Times at least has an excuse. (Disclaimer: I went to MIT as an undergraduate.)

Now if only I could get the Times to write about tilings of the plane, where they could say that an 80 by 80 rectangle has "10800" domino tilings. (That's "ten thousand eight hundred", as opposed to the correct number, which is around "ten to the eight hundredth".) Or just in general to write about statistical mechanics, which is where numbers like this come up.

(The Times article which provoked all this -- on the question of whether the laws of nature are inherently mathematical or whether we've just gotten lucky -- is interesting as well. The way I see it, even if a nonmathematical physics is possible, the mathematical physics is so far ahead that we might as well just act as if physics is ultimately governed by mathematics. But I don't think a nonmathematical physics that has the same explanatory power as the physics we have is possible. That might just be a matter of faith.)