So imagine a wire, made of protons that stay still and electrons that drift rightward; that drift is what we call a current. And imagine a nearby charged particle—call it Fred—also traveling rightward.

Now relativity tells us that Fred is allowed to think of himself as stationary, and the protons (along with you and me) as drifting off to the left. Relativity also tells us that if passengers on a moving train say the cars are 100 feet apart, then an observer at the station will say they’re closer than that. In this case (according to Fred) you and I are the passengers moving with the train of protons, and if we say they’re an angstrom apart, then Fred says they’re closer. That means Fred sees more positive charge per inch of wire than we do. If Fred himself happens to be negatively charged, he’ll be drawn toward the wire.

As far as Fred is concerned, that’s a purely electrical force, but it’s a force that you and I can’t account for on electrical grounds. So you and I call it magnetism.

At the same time, Fred sees the electrons in the wire as slower-moving, and therefore farther apart, than you and I do, so he sees less negative charge per inch of wire than you and I do. According to Fred, then, the gap between positive and negative charge in the wire is even greater, which means he’s pulled in even harder, which you and I call even more magnetism.

If you’re geeky enough to care, it’s a nice exercise in relativity theory to show that the magnetic force is proportional both to the current (that is, the number of electrons per inch, times the speed of the electrons, as measured by you and me) and to Fred’s velocity. It just now took me three tries to get this right, but it’s very nice when it finally works.

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The brain teasers were all solved by multiple commenters; I’ll summarize their answers at the end of this post. The special relativity problem proved trickier; here it is again:

A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?

Now to the answer.

First, “At some point, the train accelerates” is ambiguous. Presumably it means that each part of the train accelerates at the same time, but of course “at the same time” means something different to a train passenger than it does to you and me (the observers stationary relative to the track).

But let’s resolve this ambiguity in the natural way by assuming that the entire train starts moving at the same time as measured by you and me. In that case, we do not see the train shrink. How could we? The front and back ends of each car have, at every moment (as measured by our watches) been moving forward at identical speeds. Given that, the distance between those front and back ends (a measured by our meter sticks) cannot change. Ditto for any couplings between the cars.

What just became of relativity? If the cars are in motion shouldn’t they appear smaller to us than to our friend Jeeter, who’s riding on the train? Sure. But that doesn’t mean we have to see Jeeter’s train car get smaller. In this case, it means that Jeeter sees his car get bigger—because by his watch, the front of his car started moving before the back did, so his train car got stretched out.

But that doesn’t mean Jeeter sees the entire train get bigger. Yes, his car got stretched when the front started moving before the back. But the car opposite him (180 degrees around the track) got shrunk when its back end started moving (according to Jeeter’s watch) before its front.

So nobody has to see the train change size and nobody has to believe the train leaves the track—which is good, because the train doesn’t leave the track.

I have ignored the fact that Jeeter is not in an inertial frame, which complicates the calculation of exactly what he experiences, but I’m nearly sure that the above captures everything important. If you want, replace the circular track with a nearly square track (with slightly rounded corners if you like) so that most of the train passengers are in inertial frames at any given moment.

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Now to the brain teasers: For #1, choose a number randomly c from (say) a normal distribution on the real numbers and compare it to the number x that I’ve just revealed to you. If x is greater than c, guess that x is the larger of my numbers; if x is less than c, guess that x is the smaller of my numbers. Your chance of winning is 1 if c is between my two numbers and 1/2 otherwise; this makes your overall chance of winning greater than 1/2.

For #2, Jon Shea’s answer is perfect.

For #3, New Mexico/Colorado is one of many good answers.

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So: A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?

Yes, the train is moving in a circle and is therefore not in an inertial frame. No, that has nothing to do with resolving this. I won’t spoil the fun by posting the answer just yet. I had a very “d’oh!” moment when I got it.

I’m not sure whether I’m a geek for figuring this out or a dork for not seeing it instantly.

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Although Belief in God in an Age of Science is a very short book, it is too long to review in a single blog post. Fortunately, though, much of the non-lunatic content is concentrated in roughly the first ten pages, so I’ll comment here only on those.

Polkinghorne begins in awe. He is awestruck by the extent to which our Universe seems to have been fine-tuned to support life; this is the subject matter of the much-discussed anthropic cosmological principle. To take just one example (which Polkinghorne does not mention): The very existence of elements other than hydrogen and helium depends on the fact that it’s possible, in the interior of a star, to smoosh three helum atoms together and make a carbon atom; everything else is built from there. But it’s not enough to make that carbon atom; you’ve also got to make it stick together long enough for a series of other complicated reactions to occur. Ordinarily, that doesn’t happen, but now and then it does. And the reason it happens even occasionally is that the carbon atom happens to have an energy level of exactly 7.82 million electron volts. In fact, this energy level was predicted (by Fred Hoyle and Edwin Salpeter) before it was observed, precisely on the basis that without this energy level, there could be no stable carbon, no higher elements, and no you or me.

That energy level is only one of many (apparent) cosmic coincidences that make us possible; change any of the fundamental physical constants (like, say, the strength of gravity) by a little bit in either direction, and the Universe would, as far as we can tell, become completely inhospitable to life. So one does tend to feel that there’s something here that needs explaining.

Some have attempted to dismiss the issue by turning the direction of causality on its head: Here we are, so of course the laws of physics must allow for our existence. Case closed. Douglas Adams, for example, offers this brief and brilliant parable:

Imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in, an interesting hole I find myself in, fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’

But I have some sympathy for Professor Polkinghorne’s refusal to accept this dismissal. Instead, he takes his stand with the philosopher John Leslie:

The fine tuning is evidence, genuine evidence, of the following fact: that God is real, and/or there are many and varied universes.

I agree with that (with the proviso that evidence is not proof). I agree with it to exactly the same extent that I agree with this:

The fine tuning is evidence, genuine evidence of the following fact: Either invisible pink bunny rabbits, created at the time of the Big Bang, fine tuned the physical constants in order to make the Universe hospitable to lettuce, and/or there are many and varied universes.

Or, more succinctly:

The fine tuning is evidence, genuine evidence of the following fact: There are many and varied universes.

Polkinghorne wants to reject this second horn of Leslie’s dilemma, but he manages to do so, I think, only by taking too crabbed a view of what those many and varied Universes might be. First, we have the parallel worlds promised to us by the many-worlds interpretation of quantum theory; Polkinghorne is absolutely right to say these can’t be the worlds we’re looking for, because they all obey the same basic laws of nature. Higher on what Polkinghorne calls the “scale of bold speculation” we have suggestions from quantum cosmology that Universes are bubbling up all the time as quantum fluctuations in some universal substrate. But again, Polkinghorne is right to say that this only pushes the mystery back a bit—why do those fluctuations obey laws that have even a chance of producing a habitable Universe? Where do the laws come from?

This is the point where Polkinghorne gives up and falls back on God. But it seems to me that he has given up just one level of abstraction too soon. A Universe is fundamentally a mathematical object—it’s an abstract pattern that might or might not contain subpatterns that might or might not be sufficiently complex in just the right away to achieve an awareness of their surroundings, and might or might perceive those surroundings as physical objects. And of course there are many Universes, because there are many mathematical patterns, including, as just one of a dazzling infinity of examples, the Universe in which we live.

That, in any event, is the best explanation I can come up with, and it’s an explanation that feels completely right to me (which admittedly proves nothing). In The Big Questions, I’ve elaborated on what I mean by all this, how it can be true, and why it is entirely consistent with mainstream physics and the stated views of many mainstream physicists.

Now, Professor Polkinghorne might or might not buy this vision, but my point is that he never even contemplates it. He makes the leap to theism by considering and rejecting all of the weakest alternatives, but ignoring the only one that makes sense. This oversight is all the more remarkable because Polkinghorne devotes his closing pages to a rousing defense of the independent reality of mathematical objects, in clear and convincing language that had me wishing I’d written these pages myself.

The rest of the book is far worse. I might come back to that in a later post.

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We know, as certainly as we know anything in science, that [evolution] is the process that has generated life on our own planet.

Now, I would be thunderstruck if the theory of evolution turned out to be fundamentally wrong, but not nearly so thunderstruck as if arithmetic turned out to be inconsistent. In fact, I can think of quite a few things I’m more sure about than evolution. For example:

1. The consistency of arithmetic. (This amounts to saying that a single arithmetic problem can’t have two different correct answers.)

2. The existence of conscious beings other than myself.

3. The fact that the North won the American Civil War. (That is, historians are not universally mistaken about this. I am not interested in quibbling about what constitutes a “win”; I mean to assert that the North won in the everyday sense of the word, as reported in all the history texts.)

4. The consistency of higher mathematics. (The math geeks in the audience can take this to mean the consistency of Zermelo-Frankel set theory.)

5. The special theory of relativity. (The science geeks in the audience can take this to mean that the laws of physics are locally Lorentz invariant.)

6.The efficiency of the price system. (The econ geeks in the audience can interpret this as the truth and appropriate applicability of the first and second fundamental theorems of welfare economics.)

And then somewhere down the list—though still way above anything I significantly doubt—we have:

7. The theory of evolution. (That is, all—or nearly all—living things evolved from simpler things, largely through some process involving reproduction, mutation and selection.)

I’m not at all sure I’m right about this ordering, and I’d probably have chosen a different ordering five minutes ago or five minutes from now.

What would your ordering be?

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But the physicist Peter Landsberg (no relation!) observes that sometimes the miracle runs in the opposite direction, and offers a curious use of physical reasoning to reveal a purely mathematical truth!

The mathematical truth in question concerns “arithmetic means” and “geometric means”, so let start by telling you what those are. Start with, say, 4 numbers; say 1, 3, 5, and 6. The arithmetic mean is just what you used to call the mean back in high school, and the average back in elementary school: To compute it, you add the numbers and then divide by 4 (or 5 or 6 or 7 if you started with 5 or 6 or 7 numbers). In this case, that gives you (1+3+5+6)/4 = 3.75. To compute the geometric mean, you multiply the numbers and then take the 4th root (or the 5th or 6th or 7th root if you started with 5 or 6 or 7 numbers): The fourth root of 1 x 3 x 5 x 6 is about 3.08.

Now here’s a mathematical truth: the arithmetic mean is always at least as big as the geometric mean. For example, 3.75 is larger than 3.08.

And here’s how you could discover that fact if you knew a little physics: Start with several buckets of water, all at different temperatures. Bring them together and let them sit until they all reach a single new temperature.

The laws of thermodynamics tell us two things: First, energy is conserved. That means the new temperature is the arithmetic mean of the original temperatures.

Second, entropy can only increase. If you write down the formula for the change in entropy, you’ll see that for entropy to increase, that new temperature must exceed the geometric mean of the original temperatures. Voila: A truth of pure mathematics directly accessible from established principles of physical science.

A tip of the hat to my colleague Bob Knox , who put me on to this.

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