It got reposted by a bunch of people and provoked a tremendous amount of discussion (for a math topic, anyway), much of which was somewhere in the continuum between merely wrong and psychedelically incoherent. It’s not a new subject – a version of the image got discussed on Stack Exchange last year – but it’s an interesting one and hey, it’s not all that often that the subtle properties of the set of real numbers get press on Facebook. Let’s do a taxonomy of the real numbers and see what we can figure out about pi and whether or not it has the properties stated in the picture.

The Natural Numbers and Integers

These are the counting numbers: 0, 1, 2, 3, 4… There’s an infinity of them, but there are gaps. If you have 5 dollars and you give half of them to your friend, you’re stuck. The number you need is not a natural number. If we want to be able to deal with ratios of natural numbers, we need more numbers so we can deal with those gaps between the natural numbers. We can include the set of natural numbers with negative signs in from of them, and we have what’s called the integers: …-3, -2, -1, 0, 1, 2, 3… Later on I won’t worry about explicitly discussing negative numbers, but of course all of the subsequent sets include negative numbers.

The Rational Numbers

These are the ratios of integers, or fractions. Divide 1 by 4 and you get the rational number 1/4. We can write it in decimal notation as 0.25. Divide 1 by 3 and you have the rational number 1/3 = 0.333… All rational numbers have a decimal representation that either terminates or repeats infinitely. In fact, it’s better to say that all rational numbers have a decimal representation that repeats infinitely: 1/4 = 0.25000000… and we just happen to have a notation that suppresses trailing zeros. Sometimes you have to go out quite a ways before the repeat happens, but it always does. 115/151= 0.761589403973509933774834437086092715231788079470198675496688741721854304635761589… All rationals have repeating decimal representations, and all repeating decimals represent rational numbers.

The rational numbers are dense. Between any two rational numbers, there is another rational number. Which immediately implies that between any two rational numbers, there are an infinite number of rational numbers. Pick any point on the number line, and you’re guaranteed that you can find a rational number as close as you want to it. But alas, you’re not guaranteed that every point on the number line is a rational number. Some of them aren’t.

The Irrational Numbers Part 1: The Algebraic Numbers

The square root of 2 is the most famous example of an irrational number. It’s the number which, when squared, gives exactly 2. It’s equal to 1.41421356237…, but the decimal representation never repeats. This is because there are no two integers A and B such that (A/B)² = 2. You can get as close as you want: 7/5 = 1.4 is kind of close, and 3363/2378 is much closer still, but you’ll never find a rational number whose square is exactly 2. This can be rigorously proven and means that the square root of 2 is irrational, and never repeats.

The square root of two is the solution to the equation . This is an example of a polynomial with integer coefficients. Another random example is , which happens to have the irrational number x = 1.84302… as one of its solutions. Numbers which are solutions to these kinds of polynomials are the algebraic numbers.

Does all this mean the decimal expansion of the square root of 2 includes any and every combination of digits?  Maybe. Maybe not.

The Irrational Numbers Part 2: The Transcendental Numbers

Not all irrational numbers can be written in terms of the solutions of polynomials with integer cofficients. The ones that can’t are called transcendental numbers. Pi is one of them. So is Euler’s number e = 2.71828… Transcendental numbers are all irrational.

In a precise but somewhat technical mathematical sense, “almost all” real numbers are irrational. Throw a dart at the real number line and you will hit an irrational number with probability 1. This makes some intuitive sense. If you just start mashing random digits after a decimal point, it seems reasonable that you won’t just happen to make an infinitely repeating sequence. It turns out that the same thing is true of the transcendental numbers. “Almost all” real numbers are transcendental. But at the present time, even with hundreds of years of brilliant mathematicians pouring unfathomable effort into the problem, our toolkit for dealing with transcendental numbers is pretty sparse. It’s very difficult to prove that specific numbers are transcendental, even if they pretty obviously seem to be. Is transcendental? Almost certainly, but nobody has proved it.

Here’s a number called Liouville’s constant which is proven to be transcendental: 0.110001000000000000000001000000… (It has 1s at positions corresponding to factorials, 0s elsewhere.) It was among the first numbers known to be transcendental and was in fact explicitly constructed as an example of a transcendental number. It’s irrational, of course. It is an “infinite, nonrepeating decimal”, as the Facebook picture puts it. But is my DNA in it? Heck no, my phone number’s not even in it. Infinite and nonrepeating is not synonymous with “contains everything”.

The Normal Numbers

A normal number is one whose decimal representation contains every string of digits on average as often as you’d expect them to occur by chance. So the digit 4 occurs 1/10th of the time, the digit string 39 occurs 1/100th of the time, the digit string 721 occurs 1/1000th of the time, and so on. All normal numbers are irrational. Normal numbers satisfy Takei’s criteria. Any finite string of digits occurs in the decimal representation of a normal number with probability 1.

Is pi a normal number? Nobody knows. If our toolkit is sparse for proving things about transcendental numbers, it’s almost completely empty for proving anything about normal numbers. There are a few contrived examples. The number 0.123456789101112131415… is normal in base 10 at least, and in fact it contains every finite string of digits, because it was constructed so that it would. It also satisfies the properties which Takei’s image ascribes to pi, though it also shows that these criteria aren’t especially profound. A string that contains all numbers turns out to contain all numbers, which is true but not all that impressive.

But is this specific number normal in other bases? Nobody knows. Are there numbers that are normal in every base? Yes – again, “almost all” of them. Can I actually write out the first few digits of one? Nope. As far as I can tell, while examples of absolutely normal numbers have been given at in terms of algorithms, there’s not yet been anyone who’s been able to start generating the digits of a provably absolutely normal number. [Edit: I think in the comments we've found in the literature an example of the first few digits of a provably absolutely normal number.]

Mathematicians love proof. I’m a physicist. I love proof too, but I’m a lot more willing to work with intuition and experiment. Do the billions of digits of pi that we’ve calculated act as though they’re distributed in the “random” way that the digits of an absolutely normal number ought to be distributed? Yes. Just about everyone suspects pi is absolutely normal. Same for e and the square root of 2 and the rest of the famous irrationals of math other than the ones that are obviously not normal. Numerical evidence is not dispositive though, and has misled mathematicians before.

If pi is absolutely normal, than Takei’s image is true. If you can prove this conjecture, you will have boldly gone where no one has gone before.

Let’s see if we can find some insight into a similar question: why are clouds white?

Clouds are made of water droplets. Pour yourself a glass of water. You’ll notice that the glass of water is not white. It is in fact perfectly clear. Well, glasses of water of big. Maybe tiny droplets are different. Dip a toothpick into the water and get the smallest droplet you can, and put it on a hard surface. You’ll be able to see the surface through the water with equal clarity. Even small drops of water are themselves clear.

So if clouds are water, and water is clear, why aren’t clouds clear? They look to us a lot like they reflect light. Light that shines on them bounces off, and when we’re looking at the bottom of clouds on a stormy day we see that most of the sunlight doesn’t make it through the clouds because it has reflected off the tops of the clouds.

Here’s a short New York Times piece attempting to answer the question. The answer given there is that droplets do scatter light through a process called Mie scattering, which is essentially just refraction. The direction of the incoming light gets bent and changed just as in the photograph of the droplet above. Crucially, Mie scattering is more or less independent of wavelength. If droplets scatter all colors of light, “all colors” is basically white.

That’s true, but not complete. Why should this make clouds reflect white? How is it that randomly-directed scattering can preferentially send the light back in the direction it came from?

Let’s look at the process in a little more detail. When light hits a drop, it gets redirected through refraction and scattering – mostly in the forward direction, but after this redirected light hits more drops the randomness of the orientations of the light and the drops washes out all information about the original direction of the incoming light. At this point, the direction of the light is random and unrelated to the direction of the incoming light. It seems paradoxical that this would end up causing the light to leave the cloud in the same direction it came in. The answer is the random walk.

Imagine you’re walking down the sidewalk and you flip a coin. Heads you take a step forward, tails you take a step back. You keep up this process for many flips of a coin. Your position over a thousand steps might look like this:

Or like this[1]:

The point of an unbiased random walk is that you’re equally likely to go one way or another. Should you end up ten steps forward of where you started (call it y = 10), you’re equally likely to end up at position y = 0 and y = 20 at a given number of steps in the future. If the end of the sidewalk is at y = 10,000, and you’re sitting at y = 10, it is much, much more probable for you to end up back at y = 0 before you wobble your way to y = 10,000. For instance, here’s a 100,000 step random walk:

It returns to 0 several times in the first 20,000 steps, and then by the 100,000th step has only managed to wander off to around y = 500. Should we keep stepping, we’re still much more likely to wobble back down the 500 steps to 0 than we are to wander the 9,500 steps up to y = 10,000.

And that’s why clouds are white. Each drop, or at least each several drops, are essentially a step of a random walk for the incoming light waves[2]. If the cloud is very large compared to the size of the step (which is on the order of a few times the distance between drops), then the light is much more likely to wander back out to the same side of the cloud it came in than it is to wander all the way to the other side.

This concept generalizes to all kinds of light scattering objects. White sand is mostly clear silica particles, but this same scattering process bounces the light back diffusely. In an astrophysical context, reflection nebula work in similar ways:

Sometimes there’s a lot of insight to be gained by asking kid-style questions, even if the path to the answer is kind of random.

[1] You’re not alone if you think these look like stock market charts. There’s a pretty large body of academic literature that models financial markets as random walks with varying levels of bias.

[2] There’s a temptation to talk in terms of photons bouncing around. This temptation ought to be resisted. The process of scattering is entirely a classical wave phenomenon.

Pick up a comb, rub it with your hair and you have got some electric charge. Now shake it and you are generating an electromagnetic wave. Am I right?

Yes indeed. So why don’t we see light emitted when we brush our hair? Let’s run some numbers. If you wiggle around an electric point charge, electromagnetic radiation is emitted. The power carried by this radiation is given by the Larmor formula:

Well, a comb isn’t a point charge. But if we’re just interested in an order-of-magnitude estimate, we can pretend it is. How much charge is on a comb? It’s probably a substantial overestimate, but the human body has a capacitance of around a hundred picofarads, which is part of why you can get slightly shocked when you rub your feet on carpet and touch a doorknob. For a purpose-built capacitor in an electronic circuit that’s pretty small, but a comb isn’t a purpose-built capacitor either so it’s not unreasonable to say that as an order of magnitude it has a capacitance of 100 picofarads. That doesn’t tell us how much charge it holds though, we also need to know the voltage. The voltage required to zap you with static electricity when you touch a doorknob is a surprisingly high number on the order of 10,000 volts, so we’ll say the comb is charged to that potential since a comb can hold enough charge to produce a spark. 100 picofarads multiplied by 10kV gives 1 microcoulomb.

That gives us the charge e for the Larmor formula. How about the acceleration a? Earth’s surface gravity is about 9.8 m/s^2, and I think a person can probably fling a comb around faster than that. Let’s be generous and call it 100 m/s^2. Plug all that into the formula:

So around a trillionth of a trillionth of a watt. That’s why combs don’t glow when you shake them. Well, the first of two reasons.

Say you then went to Wal-Mart, bought a bucket of electrons, and dumped a million coulombs worth of charge on your comb. (This would in fact blast the comb to bits, but let’s pretend.) Now you’ve got a pretty bright 2.2 watts, but it would in fact still be invisible. You’re waving the comb around a few times per second, and the resulting electromagnetic wave will tend to have a similar frequency. These are extremely long-wavelength radio electromagnetic waves, which are invisible to our eyes.

Nonetheless, this is more or less how actual radio devices work. Waves are generated by moving electrons back and forth within the metal wire antenna. However, the quantity of charge that’s being moved around is large (the bazillion conduction electrons in the metal are all moving back and forth), and the acceleration that can be produced on an individual conduction electron by an applied electric field is pretty large as well.

A car driver going at some speed v suddenly finds a wide wall at a distance r. Should he apply brakes or turn the car in a circle of radius r to avoid hitting the wall?

My first thought was that surely the question wasn’t doable without more information, but it turns out that we do have enough to give a straightforward answer. Let’s take the “turns in a circle” and “slams on brakes” scenarios one at a time.

Turns in a circle:

Velocity is a vector whose magnitude is the speed and whose direction is the direction of travel. If you turn, your speed remains the same but your direction of travel changes. So the velocity is changing even if the speed isn’t. A changing velocity is by definition an acceleration, and one of the key equations of first semester physics is the acceleration required to produce uniform circular motion. It turns out to be a function of the speed and the radius of the circle:

Since we don’t have any numbers to plug in or really anywhere else to go with this, we’re done with this part. The required acceleration to avoid the wall is equal to the square of the speed divided by the radius of the circle, which is just the initial distance to the wall.

Slams on brakes:

This one is a little more involved. The direction of the velocity is not changing, but the speed is. Another of the key equations of freshman physics is the formula for position in uniformly accelerated motion. It’s:

where a is the acceleration, v0 is the initial velocity, x0 is the initial position, and t is the elapsed time. In this case we’d like to solve for a at the point where x = r (we define our coordinates such that x0 = 0). But we don’t know how much time has elapsed by the time the car reaches the wall, so we need the formula for velocity in uniformly accelerated motion, which we might write from memory or find by differentiating the position equation if we know calculus:

Now I’ll start subscripting the letter f on the specific time when the car reaches the wall. We know that we’ve come to a stop at at that time, so we have:

Which means

Don’t worry about the negative sign. a is itself negative (we’re decelerating), so tf will be positive as well. Now that we know how much time has elapsed when the motion is complete, we can plug that into our position formula:

Remembering that at the wall, x = r and that we defined x0 = 0. You can do the algebra to solve for a, and you’ll find that

Which is (ignoring the minus sign that just tells us which way the acceleration is pointed) just half the acceleration we found for the turning scenario. So purely from a standpoint of the acceleration car tires can produce, braking works better than swerving.

From Back to the Future – Biff shoulda braked…

After writing this post, I came across a ScienceBlogs post on Dot Physics a few years ago on the same subject. He approaches the problem in a different way, and I think it’s well worth reading both solution methods.

The Theoretical Minimum: What You Need to Know to Start Doing Physics

When this book appeared in my mailbox I judged it by its cover and was a little concerned. The problem with the cover is the name of one of the authors: Leonard Susskind. He’s an extremely talented physicist and writer, to be sure, but he’s a string theorist. Worse, he’s one of the major names behind the string theory landscape idea. Though not a high-energy physicist myself and thus not really being terribly qualified to judge, I tend to classify the string theory landscape as somewhere between speculative and pseudoscience.

Beyond the cover, I am happy to report that my initial worries were absolutely incorrect. This is a charming and erudite instance of a genre with very few members – a pop-physics book with partial differential equations on a good fraction of the pages. The goal of the book according to the forward by Susskind (a physicist) and Hravovsky (an engineer) is to give a substantive but not-textbook-detailed introduction to physics. Not just to teach about physics, as is the typical pop-physics book’s goal, but to actually teach physics.

The title refers to a slightly notorious requirement the great Soviet physicist Lev Landau put on his students before they could join his group. There was a level of knowledge of physics he called the “theoretical minimum”, which for him meant exhaustive mastery of theoretical physics. In the more limited goal of this book, the theoretical minimum is to understand physics as it actually works mathematically – beyond just the Scientific American level. Not to the level where you’re actually solving graduate textbook problems, but to the level where you know what the concept of a Lagrangian actually entails.

More impressive still is that the book entirely resists the temptation to skip to the good stuff – quantum mechanics and so on. This is a book which is purely about classical mechanics. More volumes are planned on electromagnetism and quantum mechanics, but for now this is the true basics. These basics of course turn out to be built into the fabric of electrodynamics and quantum mechanics, aside from the minor fact of the vast importance of classical mechanics in the world of practical problems.

The succeeds admirably in its goal. It presents classical mechanics in all its glory, from forces to Hamiltonians to symmetry and conservation laws, in a casual but detailed style.

Hawking famously suggested that each equation halved the sales of a book, so the question here is whether or not you might be interested in reading The Theoretical Minimum if you haven’t learned calculus or don’t remember it. It’s a judgement call. I suspect you won’t get the whole experience if you haven’t at least seen calculus at some point in your life. But even a half-remembered course years ago is probably good enough – there’s a pretty substantial bit of mathematical refresher material presented in a visual and intuitive way. If in doubt, give it a try. On the other hand, a reader without any calculus background could probably pick up some of the flavor of the physics but I don’t think I recommend starting with this book.

I’m looking forward to the rest of the books in this series. They address a niche that sees very few solid attempts to fill.

[Standard disclosure: the publisher sent me a free copy of the book to review. I am not otherwise compensated for this review.]

Here’s the intensity (formally: power per area per unit solid angle per unit wavelength – whew!) of the radiation emitted by an object with the temperature of the sun, plotted as a function of wavelength in nanometers according to Planck’s law:

Spectral radiance (W/(sr m^3)) vs. wavelength (nm)

You’ll notice it also peaks around the same place as the spectral response of the human eye. Optimization!

Or is it? That previous equation was how much light the sun dumps out per nanometer of bandwidth at a given wavelength. But nothing stops us from plotting Planck’s law in terms of the frequency of the light:

Spectral radiance (W/(sr m^2 Hz)) vs. frequency (Hz)

In this case what’s on the y axis is power per area per unit solid angle per frequency. Ok, great. But notice it’s not just the previous graph with f given by c/λ. It’s a different graph, with different units. To see the difference, let’s see this radiance per frequency graph with the x-axis labeled in terms of wavelength:

Spectral radiance (W/(sr m^2 Hz)) vs. wavelength (nm)

Well. This is manifestly not the same graph as the radiance per nanometer. Its peak is lower, in the near infrared and outside the sensitivity curve of the human eye. This makes some sense – there’s not much frequency difference between light with wavelength of 1 kilometer and light with wavelength of 1 kilometer + 1 nanometer. But light of 100 nanometer wavelength has a frequency about 3 x 10¹³ Hz more than light with wavelength 101 nanometers.

So what gives? Is the eye most sensitive where the sun emits the most light or not? The simple fact of the matter is there’s no such thing as an equation that just gives “how much light the sun puts out at a given wavelength”. That’s simply not a well-defined quantity. What is well defined is how much light the sun puts out per nanometer or per hertz. In this sense our eye isn’t optimized so that its response peak matches the sun’s emission peak, because “the sun’s peak” isn’t really a coherent concept. The sensitivity of our eyes is probably more strongly determined by the available chemistry – long-wavelength infrared light doesn’t have the energy to excite most molecular energy levels, and short-wavelength ultraviolet light is energetic enough to risk destroying the photosensitive molecules completely.

This wavelength/frequency distribution function issue isn’t just a trivial point – it’s one of those things that actually gets physicists in trouble when they forget that one isn’t the same thing as the other. For a detailed discussion, I can’t think of a better one than this AJP article by Soffer and Lynch. Enjoy, and be careful out there with your units!

Mark and I have been conducting a debate/discussion over gun control in the United States. For the first round, here’s his post and my response. Here’s his second round post, and this post is my response.

First, let me summarize where the debate stands. We have four main topics as set forth in Mark’s posts: gun violence in “ordinary” crime, gun violence in the context of mass shootings, suggestions for gun control, and miscellaneous ancillary arguments. Most of the points in the ancillary category were fairly comprehensively covered, and I think both of us are pretty satisfied with what has been said. The exception is the “good guys with guns” argument, which we’ll continue.

Mark classifies my responses to the ordinary crime and mass shooting topics as “no problem” arguments. This is incorrect. I am trying to quantify the problem, and to quantify the impact of the proposed solutions. If it turns out that both these quantities are so small as to be classified as “no problem” in the mind of the reader, well, the numbers are what they are. I myself reject the idea that there is no problem. But I also reject the idea that argument from anecdote is an effective guide to the truth. We want to ask whether or not there is a problem which is caused by the prevalence of guns, and if so whether or not gun control could do anything to ameliorate it.

Let’s dive right in to the general gun crime topic.

Mark quotes the Institute of Medicine in comparing the US to similar industrialized countries in terms of life expectancy found that our homicide rate is far in excess of comparable OECD countries, and significantly affects our life expectancy. The IOM study found our homicide rate to be 6.9 times higher than the other OECD countries, our gun homicide rate 19.5 times higher, and of the 23 countries in the study, the US was responsible for 80% of all firearm deaths.

There are two obvious questions. First, is the US comparable to those other OECD countries? Second, how much does gun control actually have to do with this?

The answer to the first question is an obvious no, and to demonstrate this we need look no farther than the very study linked. The US has higher than average death rates in almost every category from car accidents to disease, the highest rates of adolescent pregnancy, sexually transmitted diseases, diabetes, and so forth. (But not suicide, incidentally.) In fact, in the words of the study,

On nearly all indicators of mortality, survival, and life expectancy, the United States ranks at or near the bottom among high-income countries.

I’m not trying to insult my country – it’s a great place, much better in most of these categories than most of the rest of the world. However, comparisons to these 16 other top OECD nations are untenable. We aren’t comparable. We are different in almost every measurable respect involving health and mortality.

Well ok, guns obviously don’t give people diabetes or make teens pregnant, but “lots of guns, lots of violence” vs “not many guns, not much violence” might look less like correlation and more like causation. (At least relative to the not-very-comparable top of the OECD.) This conclusion is unwarranted and probably false. Here’s some reasons, some of which I have mentioned in my last post.

1. US vs. OECD entirely aside, we can’t even easily compare US vs. US over time without running into extreme confounding variables. Our murder rate has been precipitously falling over the last few decades even as gun laws have become much looser (I do not claim a causal relationship). The last time our murder rate was as low as it is now, we were literally in the Leave It To Beaver era.

2. Murder rates vary wildly within the US under identical gun control regimes. White Americans, for instance, kill each other at roughly OECD rates (albeit on the high end), and well below the rates of eastern Europe and the Baltics. I shouldn’t have to point out that epidermis reflectivity doesn’t have squat to do with this. It does, however, show that socioeconomic and cultural variables overwhelmingly determine rates of violence.

3. Sharp changes in gun laws haven’t done anything significant to the homicide rates of other countries. The best-studied case is post-Port Arthur Australia. The effect on overall homicide rates was somewhere between negligible and nonexistent. The effect on gun homicide rates was similar. Let’s take a look at the study Mark cites:

Additional research, readily available suggests a significant drop in the rate of gun violence after the ban. This suggests to me, both in the specific intervention, and overall given their tight regulation of handguns, that Australia is quite a strong example of gun control working.

I will reproduce a few of the graphs from this paper, unedited. First, gun homicides and non-gun homicides:

The statisticians in the audience who have not died of heart attacks at the statistical illiteracy of the pre- and post- trend lines will of course notice that the overall decline in violence and gun violence continued just as it was doing before the gun control was implemented. In fact, the rate of non-gun violence displays a much more dramatic (though also statistically spurious) change. And this is Australia, the best possible scenario for the success of of gun control. Gun control did nothing to the overall homicide rate. It didn’t even do anything to the gun homicide rate. (More graphs from the paper here, about accidental deaths and suicides, if you’re curious.)

4. Trying to account for confounding variables is extraordinarily difficult in this context, but a number of studies have attempted to do so. One study compares the prairie provinces of Canada with their bordering US states. In this case,

Patterns of homicide in the United States and Canada were examined with a view to finding out whether the availability of firearms affects the homicide rate independently of the other social, demographic and economic factors in play.  If this is the case, then low-homicide areas, which generally have fewer social and economic problems but the same access to firearms, should have a higher proportion of their homicides by firearms.  This is not the case for the four border states.

Other studies (commenter LH pointed out these two) have come to similar conclusions. Now I strongly suggest that you not read too much into these results – while if they are accurate they support my point, attempts to disentangle confounding variables are fraught with danger even when the result happens to land on my side.

In short, there is no good evidence that gun availability causes increased crime rates. There is extremely good evidence that socioeconomic variables are far and away the primary drivers of crime rates. Violence in general and gun violence in particular are real problems in the US, but gun control as a solution is so ill-supported as to verge on superstition.

While Mark and I are mostly focused on numerical metrics as to what effects gun control actually produces, it’s probably worth looking briefly at the practical problems of implementing it as well. Mark quotes former Australian prime minister John Howard writing on the Port Arthur gun control measures:

In the end, we won the battle to change gun laws because there was majority support across Australia for banning certain weapons.

Howard is right. In Australia, gun control was implemented with the overwhelming support of the population. This is not the case in the US. The change in support for gun control after Sandy Hook is marginal[1], and those opposed to it are very opposed to it and are voting with their wallets. The single week of December 17-23 likely saw almost a million new guns sold. Over the last month I’ve had occasion to be in five gun stores, and every one of them was completely sold out of every AR-15, every semi-automatic rifle of any description for that matter, every magazine holding >10 rounds, and every box of .223 ammo. Every online retailer I’ve checked is in the same boat. I personally have an outstanding parts order with Rock River Arms, and they’re backordered so badly they won’t even provide estimated lead times.

On to mass shootings. Both Mark and I as scientists run into some trouble here in that there is very little available systematic data of any kind. Trying to disentangle ordinary crime statistics from their confounding variables is hard enough, but the small-N statistics of mass murder are much harder still. We have noted that the Wikipedia lists of mass killings are similar in size in the US and Europe, and the US’s is slightly larger (119 vs. 100). This is worse on a per-capita basis because Europe has a higher population. But it is clear that confounding cultural and socioeconomic factors are in play as well. Mexico, for instance, has a homicide rate about 4 times that of the US but as far as I can tell has apparently never had a school shooting. (There have been a few “ordinary” murders at schools, but I have not been able to find any examples of a school shooting of the crazed-gunman variety.) Australia seems to have some success with their gun control regime in the specific case of mass violence, but their success is probably not replicable in the US which is (as I have pointed out) a very different place with 10 times the population and historically much higher levels of violence (gun and non-gun alike), to say nothing of the fact that we’re starting with gun ownership rates which are higher by a factor of 10.

We have a problem with mass violence. It’s a staggeringly rare problem, rarer than lightning strikes, but a dramatic and tragic one and one that deserves our best efforts to fix. The place to start is not a massive and likely completely ineffective reconstruction of a fundamental right exercised by nearly half the population of the country. As both Mark and I have both pointed out, government overreactions to tragedy tend not to turn out well in this country. We know for a fact that the last iteration of the assault weapons ban failed to prevent Columbine or to do anything significant to either ordinary or mass violence during the ten years it was in effect.

Instead, we should start with the obvious basics. Physical security of the entrances to schools would be my focus if I were a principal. Improved accessibility of mental health treatment is also a good idea (though this is a tall order and the verdict on its effectiveness is still out). The occasional presence of resource officers and/or the elimination of the silly “gun free zone” designation could also be a good deterrent. This last point we’ll discuss separately at the end of the post, as it’s quite controversial.

Mark makes a few suggestions for tighter gun laws. His primary suggestion is:

…since magazine-fed semi-automatic weapons are the weapons of choice in the last few dozen of these shootings that before sale the purchaser should get a bit more eyeball by authorities. Specifically in regards to the VT shooter, the Aurora Shooter, or the Giffords shooter, I suggested increased scrutiny for these purchases, law-enforcement taught training and competence testing for their use, and I also suggested the Canadian voucher system (as did Kristof immediately after Sandy Hook), which would require two other people to stand up for you and say you are responsible enough to possess such a machine.

As I pointed out last time, “magazine-fed semi-automatic weapons” is a near-synonym for “all guns”. Most shootings involve semi-automatic firearms because most firearms are semi-automatic. But that’s a side point, and doesn’t really affect his argument too much. (He’s not advocating a ban, but more on this later.)

Let’s start with the idea of a voucher system. If I want to buy a gun, I have to find two people who are willing to put their name to paper asserting that I’m not an obvious nut. Let me give three reasons I think this might be a bad idea, and two reasons I think it might work. First, even the most scuzzy two-bit crooks can round up two scuzzy two-bit friends to sign for them. Second, anyone’s good-faith assessment of another’s character could prove to be wrong. Third, it could be prone to abuse – are there exorbitant filing fees involved? Can New York decide a person needs twenty signatures? I would suggest that if you object to, say, voter ID laws then you can see how such a voucher system might be problematic. But there are a few reasons it might work in some cases. While Bugsy Siegel wouldn’t have a problem getting signatures, obvious dead-eyed psychopaths like James Eagan Holmes or Seung-Hui Cho might have found it a hurdle. Secondly, the second amendment does talk in terms of civic purpose. While the right to bear arms is obviously individual an individual right[2] and US law defines the militia as all able-bodied males between 17 and 45, the civic purpose of the second amendment might suggest that something like a voucher system in an otherwise permissive regulatory regime might fit the bill. I’d have to chew on the voucher idea for a while longer before deciding if I really think it’s a good idea, but on its face it seems much more in the spirit of the reason behind the right to keep and bear arms than do some other gun control suggestions.

Mark has also suggested greater scrutiny such as background checks for the private sale of guns. I’m much less sanguine about this. It would certainly accomplish nothing to prevent mass shootings – these weapons are usually purchased legally or stolen – but in the context of keeping guns out of the hands of crooks it seems like a reasonable place to start thinking. So we should ask ourselves what we might gain by implementing such a scheme. It’s an old staple of this debate to assert that criminals inherently aren’t inclined to have a lot of respect for gun laws. This can be countered by asserting that their respect for the law is irrelevant if there were no guns in the first place, but in terms of doing paperwork on transfers this response doesn’t work so well. Bugsy buys a gun for convicted felon Mugsy, cops trace the serial and ask Bugsy how Mugsy got the gun: “I dunno officer, he musta stole it”. In the mean time law-abiding gun owners are effectively forced into a registry and have to deal with the expensive bureaucratic morass of the FFL system. Maybe this could be sidestepped by some clever way of opening NICS to private parties other than FFLs, and such proposals ought to be heard out. Once somebody proposes one, anyway.

Training and competence testing? I’m all for people being trained and competent, but that has nothing to do with crime and violence and formal training is pretty expensive. I’d hate to see it made into an effective “no poor people need apply” restriction. Safe storage? Fantastic, especially for people with kids, but the same caveats apply.

Finally, we should discuss the ban vs. paperwork hoops issue:

Every time you talk gun regulation at all it seems to become a ban in the pro-gun side’s mind. However, at no point, for any currently available weapon, have I suggested a ban. Just paperwork. It’s not the end of the world people.

This is true, and fair enough as it goes. We gun-rights types are justifiably a bit jumpy about this sort of thing. It would be nice if Mark were the one writing the various laws being proposed in congress and various state legislatures. Unfortunately it’s people like Dianne “Turn ‘em all in” Feinstein and Carolyn “Shoulder thing that goes up” McCarthy and Andrew “Confiscation could be an option” Cuomo. It’s great for the two of us to discuss our Platonic ideals of the way things ought to be, but we also have to remember that we’re dealing with members of the world’s second oldest (and least reputable) profession. Since their stated intent is to take a mile, I’m not very willing to give them any free inches without an airtight case as to effectiveness and respect for the rights of the law-abiding.

Finally let’s return to the idea of stopping shootings via “good guys with guns”. Quoting Mark:

In the vast majority of cases, mass shootings are stopped when the perpetrator is shot…by themselves. Do we have evidence of police or armed citizens interrupting even one of the mass shootings in the last 20 years? Do we have any evidence of good guys with guns making a dent except after the shooting is done? Nope.

The “Nope” is a link to a Mother Jones article which actually lists five cases in which good guys with guns did just that. Mother Jones’ point is that each of the five cases listed magically don’t count because the citizens involved were current or former law enforcement or military, not (say) some dentist who just decided to get a concealed carry permit. I’m not sure that this tells us much more than that people with experience are more likely to get permits and that ordinary citizens’ permits are not generally valid in the places where mass shootings occur, but in any case it kills the argument that armed citizens can’t possibly accomplish anything positive. While active uniformed police haven’t actually shot many mass killers, it is probably more than suspicious coincidence that the perpetrators tend to shoot themselves right when police arrive (Lanza and Cho are prominent examples). This is also alleged to have happened in the Clackamas shooting when a citizen with a concealed carry permit drew his weapon, but as this is not independently verifiable Mark (not unreasonably) dismisses it and I won’t try to build a case around it. Mark also mentions the fact that an officer was present at the initial stage of the Columbine attack but failed to stop the shooting. This is roughly as out-of-date as insisting that passenger resistance to hijackers is futile because it failed to stop 9/11 – at the time it was generally believed that these were hostage situations, and that the proper response was to wait until it was all sorted out much later. This mistake is no longer made.

It is possible, and it has happened, that in the process of trying to stop a mass killer a person carrying could get themselves killed. As Mark says

It’s not as easy as it looks in the movies, and the usual creepy fantasist gun lover who buys into this myth is not John McCain, he’s Walter Mitty.

Ok, ok, I can’t resist: Walter Mitty would probably fantasize about being Die Hard hero John McClane, not the senior senator from Arizona. But I’m at a loss to see how this is an argument against resistance. Am I extra-dead if I get killed while trying and failing to resist? All that’s being asked is that the situation be an improvement on an unopposed mass shooter, who is at any rate hardly Hans Gruber either. (Neither are ordinary criminals. See here for an example which is simultaneously horrifying and hilarious.) Same thing for the Mother Jones hysterics here:

They also make it more difficult for law enforcement officers to do their jobs. “In a scenario like that,” McMenomy told me recently, “they wouldn’t know who was good or who was bad, and it would divert them from the real threat.”

In the billions of man hours that millions of permit holders spend carrying ever year, this has literally never happened. This should not be a surprise. Defensive shootings almost exclusively take place at very short ranges and are over in seconds. As I said in the last post, it’s not possible for me to claim that police or armed citizens are a panacea. The statistical data is badly inadequate. But what data we do have indicates that the concept is plausible in principle.

All right, it’s about time to conclude this Part 2. In two-sentence summary: Gun violence is bad. Gun laws have very little to do with it.

[1] A policy is not automatically good or bad based on how it polls, of course. And sometimes public opinion doesn’t make a lot of internal sense anyway. The assault weapons ban polls rather poorly (sub 50% in the Gallup poll), but universal background checks poll very well even though none of the mass shooters in recent years acquired their weapons through private sale. Go figure.

[2] Even the four dissenting justices in DC v. Heller agree. They disagree as to the scope of this right, but agree that it is an individual right. The first lines of the dissent:

The question presented by this case is not whether the Second Amendment protects a “collective right” or an “individual right.” Surely it protects a right that can be enforced by individuals. But a conclusion that the Second Amendment protects an individual right does not tell us anything about the scope of that right.

One of the major reasons for this is the fact that toys are pretty simple. They have just a few moving parts for the computer to keep track of during the rendering process. People have many more. Every hair on a person’s head really consists of many moving parts, since it can bend anywhere along its length to various degrees. And there are around a hundred thousand hairs which interact with each other, applying force to the other hairs they touch. Since it’s pretty much impossible to place every computer-animated hair by hand, if you want convincing hair in your animated films you need a good model for how the physics of hair works.

By the time Toy Story 3 came around, faster computers and better knowledge of how hair behaves made the problem of animating humans a lot more tractable:

Now we’re one step farther in our understanding of hair, with an interesting research article in Physical Review Letters:

Phys. Rev. Lett. 108, 078101 (2012)
Raymond E. Goldstein, Patrick B. Warren, and Robin C. Ball
Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles

They’ve attempted to create a model of why ponytails have the shape they do. They postulate that the energy of a bundle of hair is a function of the average curvature of the hairs (ie, the overall shape of the ponytail), the potential energy due to the gravitational field of the earth, and and average force per length due to the statistical properties of the individual hairs – their points of contact, waviness, split ends, whatever. The equation from the paper is:

Where the κ term is the average curvature, the φ term is the gravitational potential, and the u term is an average over the statistics of the individual hairs. How well does it work? Pretty well:

Pretty good, especially for a model which is effectively two-dimensional.

You might wonder why anyone would bother researching this. (This particular work even won an Ig Nobel prize.) Like many seemingly weird bits of science, there’s actually quite a bit of practical point to it. Many-body problems are hard, and any advance that allows you to avoid having to do a full-blown simulation has the potential to be extremely useful. I mentioned computer animation as a flashy example, but this research is useful in any kind of bundled-fiber system from fiber-optics telecommunications to the medical treatment of the fiber bundles in your body.

For a five-second hairstyle, that’s not a bad day’s work.

As I visit on different days or on different times on the same day, the links and their order changes. This keeps the site fresh and news-y, at least if you like your news full of cat memes. It’s pretty clear that the ordering of these links is both a function of when they were submitted by the users and of the votes they receive, but how exactly does this work?

The algorithm itself is explained in this very informative post by Amir Salihefendic. In short, every post is assigned a number given by the function:

Here n is the net number of upvotes. For 10 votes up and 0 votes down, n = 10. For 50 votes up and 40 votes down, n also equals 10. Next, t is the time in seconds after an arbitrary moment that happens to be in 2005. The choice of that arbitrary moment doesn’t matter – what matters are the differences in scores. This function f(n, t) is calculated for each link, and they are sorted in order from the greatest to the least value of f. I’ve slightly simplified the equation by dropping a coefficient that makes no difference for positive n.

Ok, great. Now what does this all mean? Amir’s post gives some examples, but I want to dig a little bit into the interpretation of this equation. In physics it’s very often the case that an equation isn’t just some abstract mathematical machine, but rather it’s a natural statement which has an intuitive interpretation we can understand. For instance, is an abstract vector calculus statement, but physicists see that equation and understand it intuitively as the idea that electric field lines diverge outward from sources of electric charge. That’s a more useful way of thinking of it than “Ok, now we have to solve some horrible partial differential equation before we can know anything at all.” Intuition gives us a qualitative picture, and from there we can do the hard work to get a numerical answer when required.

Since Reddit’s equation is just used to generate an ordering, an overall multiplicative factor doesn’t matter. If a score of 20 is ranked ahead of 15, then 200 will be ranked ahead of 150. So let’s multiply Reddit’s equation by 45000 seconds.

Effectively this just means the posts are sorted in order by t, the time they were posted. Newer posts are higher. But there’s that log(n) term – it moves the posts forward in time. Newer posts are listed first, and a post becomes even newer by getting votes. If n = 10, then log(10) = 1 and the post is moved forward 45000 seconds, or 12.5 hours. If n = 100, then log(100) = 2 and the post is moved forward 90000 seconds, or 25 hours. We can plot this for more and more net upvotes:

Hours added as a function of net upvotes received

The returns are diminishing. Logarithms are slowly increasing functions, so each additional upvote moves the post forward in time by a smaller and smaller amount. Even with thousands of votes, a post has only moved about two days into the future, which is why posts never last more than a day or so on the front page. After that it gets overtaken by any new posts, even ones with few upvotes.

In politics we often hear that every vote counts. In Reddit, we can actually figure out how much each vote counts. If I upvote or downvote a post, how far does my individual vote move that post in time? For large n, it’s a very accurate to approximate the change in log(n) (for each additional vote) by its derivative:

Well that 0.434 is a little annoying but hey, I didn’t chose to use base 10 logarithms. (Had they used base e = 2.718… then it would just be 1/n.) What this means is that if a post has 10 votes, your upvote will add about 45000*0.434/10 = 1737.2 seconds, or about 29 minutes. A downvote would move it backwards by that same amount. If a post has 50 votes, your upvote (or downvote) will move it forward or backward by about 5.7 minutes. For a 4700 vote post like one of the ones in the screenshot above, each vote makes a mere 3.7 seconds difference.

Seconds added (or subtracted) for each additional upvote (or downvote), as a function of net upvotes received so far

This might suggest an improvement on the “subscribe” and “unsuscribe” system – if there’s a subreddit you’re interested in but not that interested in (/r/aww maybe?), you could give it a handicap by having Reddit subtract (say) a 6 hour penalty on every post from that subreddit. This would require a /r/aww post to get about 3 times as many votes to overtake an unpenalized post which was originally made at the same time. (Homework: given a h hour penalty, how many times more votes does the penalized post require to overtake a simultaneously-posted unpenalized post?) Correspondingly, you could give a bonus for subeddits you want to see more of. Unfortunately this is probably not a feasible suggestion. Separately sorting huge lists for millions of users would probably melt the servers. But it would be a nice feature.

All right, better wrap this one up. As far as user-vote-based ranking goes, Reddit’s is unusually interesting from a mathematical standpoint. For what it’s worth, I give it my upvote.

[Update: Fixed a mistake in the calculations.]

By the rude bridge that arched the flood,
Their flag to April’s breeze unfurled,
Here once the embattled farmers stood,
And fired the shot heard round the world.
- Ralph Waldo Emerson, “Concord Hymn”

American independence began with a shot fired from an anonymous man’s personal musket. In the centuries since, the gun has become embedded into the American mythos from the rifles of the pioneers to the revolvers of the cowboys. This culture has not diminished in the present day. There are roughly as many firearms in the US as people, and though self-reported data is notoriously unreliable, somewhere between one third and one half of households own at least one firearm. The reasons for this choice are as diverse as the country itself. For some, it’s a marker of self-reliance, a sign that in this country power flows from the people in a way that’s not merely symbolic. For others, it’s shouldering the responsibility for the defense of self and family in a way that doesn’t rely on the vagaries of police patrol density and response time. Still others see it as a vital backstop against the the breakdown of civil order in the event of natural or man-made disaster. More still use their firearms to put food on the table in a natural way that doesn’t rely on grim stainless steel factory farms. Any of these are good reasons for a responsible person to consider firearms ownership, and their choice to do so puts them in a long and unique tradition of keeping and bearing arms.

Yet this culture of firearms ownership has its opponents. In the US in 2011, some 8,583 people were murdered by assailants using firearms. In 2012, mass murders at a theater and an elementary school shocked the nation in a way that hasn’t been felt in many years. Both were committed with firearms. In the wake of these attacks, debate over guns in the United States has reignited. Mark has penned his own contribution, in which he argues along four lines. First, that gun control will reduce acts of mass violence. Second, that gun control will reduce violent crime generally. Third, that specific gun control methods he enumerates ought to be put in place in view of the previous two points. Fourth, that a number of ancillary pro-gun arguments are spurious and ought not have a place in serious discourse. I will respond to each of them in turn.

In its coverage of the recent attacks, the strongly anti-gun publication Mother Jones has compiled a list of mass shootings since 1980. Though my disagreement with their editorial conclusions can hardly be overstated, I both admire their resolve to stick to a data-driven analysis and am glad to have a data source which can’t easily be accused of bias in my favor. The first key point is the extreme rarity of mass violence. In the 32 years since 1980, there have been some 513 people killed in 62 incidents of mass shootings under the Mother Jones definition (perpetrators generally included). This amounts to an average of 16 per year. Over the 12 years for which the CDC has available data, the average death rate due to lightning strikes has been 45 per year. All other things being equal, you are almost three times more likely to be killed by lightning than you are to be killed by a mass shooter. Now any preventable cause of even a single death should be prevented, and while mass murder shocks the conscience in a way that the anonymous and impersonal forces of nature cannot, this ought to cause us to pause and consider whether what is being proposed will actually do any good. The choices we make in response to these tragedies will have consequences that we foresee and consequences we don’t. These consequences may well include the failure of new laws to save anyone in the future. This concern is not hypothetical – we’re well over a decade into our government’s frantic response to 9/11, and we may well be less safe than we were on 9/10.

We could begin by looking at whether or not semi-automatic rifles are actually a particularly heinous implement of death (we will discuss handguns in the “crime generally” section). In 2011, the United States experienced a total of 2,437,163 deaths. Of these, 12,664 were victims of murder. So for every 192 people who died in the United States, one was a victim of murder. Of those 12,664 murder victims, 323 were killed by a rifle. Thus for every 40 people who were victims of murder, one was shot by a rifle. This comes in well behind knives or blunt objects or even bare hands. In terms of death toll, rifles are roughly on par with falling off ladders (which killed 404 in the year 2010). And this category comprises all rifles, from bolt-action deer rifles to AR-15s to .50-caliber Barretts.

One might be tempted to speculate that while the absolute numbers of rifle fatalities are small, their combat effectiveness makes them disproportionately dangerous in the context of random acts of mass violence. Mark, for instance, notes that a recent knife attack in China wounded many children but killed none. This conclusion, though tempting, is numerically dubious. A number of other Chinese knife/hammer attacks at schools since 2010 have indeed between them exceeded the Connecticut death toll, but far and away the most effective weapons of mass murder are bombs and, for the less technically capable psychopaths, the kitchen match. The Bath School bombing in 1927 killed more than Sandy Hill and Columbine combined. Julio González killed 87 in the Bronx in 1990 with a match and some gasoline. The most lethal female mass killer in America used a Lincoln Continental. The unknown perpetrators of the Black Saturday fires in Australia killed many times more people than the Port Arthur massacre that touched off that nation’s much stricter gun laws. Now a list of anecdotes is not dispositive, but it is surprisingly hard to find comprehensive academic lists of rampage killings for the purpose of comparing across borders and between gun and non-gun attacks. At any rate the Wikipedia compendium of rampage killings internationally is striking (if not rigorous) reading – it lists 119 mass killings in the Americas and 100 in Europe.

Though non-gun massacres have been extremely lethal, one might speculate that at least gun control could prevent those that are gun-related. This has not seemed to be the case as regards firearms-related mass murder in other countries. The United Kingdom, a country with 1/5 of the US population, has experienced several mass shootings despite very strict gun control. The Dunblane school massacre in 1996 killed 17 in an environment more restrictive than even what Mark advocates. The further tightening of those laws – to the point of completely banning civilian possession of handguns – failed to prevent the Cumbria shootings in 2010. The same holds elsewhere in Europe. Of course the Norway shooting is well-known, but many others are less so. As I mentioned earlier I am unaware of a comprehensive list of European mass murders generally, which would allow us to make per-capita comparison. I would like to see one, should any of you know where to find a rigorous systematic study.  The best example on the pro-gun-control side is that of Australia, in which guns are regulated under a regime which has been a near ban since the Port Arthur massacre in 1996. There has nonetheless been at least one school shooting since then, though fortunately it only resulted in two deaths. A mass shooting of 4 or more victims has not occurred since 1996. The total rate of mass killings by arbitrary methods did not change significantly, and overall homicide rates were unaffected[1]. It is a matter of conjecture as to what extent this quite modest success in a island nation of a very different culture and just 10% of the US population can be extrapolated to the US. (Here’s a data point for the difference in firearms culture: the Australian buyback program after Port Arthur brought in 631,000 firearms. US citizens bought almost three million last month alone.) In summary, mass violence, despite its considerable press, is exceptionally rare, not particularly reliant on guns, and not particularly preventable by gun control.

Now it’s worth taking a look at violent crime generally. While an uncommon cause of death in percentage terms, murder is a significant source of mortality in the United States. Of murders in the United States in 2011, 8,583 (about two thirds) were committed with firearms. Of firearms murders, the overwhelming majority were committed with handguns (6,220 with certainty, and likely most of the 1,587 “type not specified”). The rate of homicides (by all methods) is about 4.8 per 100,000, which is high compared to Australia (1.0) the UK (1.2), or Canada (1.6). On the other hand, it’s low when compared to most nations outside the highly developed first world, such as Russia (10.2) and Mexico (22.7). The highest, Honduras, is a staggering 91.6.

Which comparisons are fair, and what are the causal factors? Here’s a cautionary tale – one very similar nation has a homicide rate of 9.8, more than twice that of the US. That nation was the US, in 1991. What changed? Sociologists differ, but it was certainly not due to stricter gun laws. (One frequently-mentioned and surprisingly robust possibility is reduced childhood lead exposure after lead was phased out of gasoline.) Simultaneous declines took place in many other developed nations to various extents. If non-gun factors can change the murder rate within the same country by more than a factor of two, it is extremely challenging to say anything about the effects of gun laws in nations with very different cultures, histories, economies, and demographics. And we shouldn’t kid ourselves, the culture, history, economy, and demographics of the US are very different than most of the rest of the small class of highly-developed western nations. For that matter, the nations of that class with lower murder rates than ours have always have had lower murder rates, even in the eras when their gun laws (and our own) were quite different. The homicide rate in England has not been substantially north of 1 per 100,000 since the year 1800. Mark dismisses any comparison to Mexico, but the comparison is more instructive than it first appears. The population of the Texas border city of El Paso is some 75% Mexican (and more than 80% Hispanic), but its homicide rate is a relatively bucolic 0.8 per 100,000. Its adjoining Mexican neighbor of Juarez is one of the most violent places on the planet, peaking at a hideous 130 per 100,000 in 2009. Do El Paso’s gun-friendly Texas laws make it so much more peaceful than Juarez? I doubt it. But it does put to bed the idea that gun laws are the primary or even a significant driver of the crime rates.

Within the US, violence in general and gun violence specifically are not evenly distributed. To enormous racial gap is well-known – the number of white murder victims[2] divided by the census number of white Americans gives a rate of 2.6 per 100,000. (This would be at the high end of western European homicide rates. Finland at 2.2 is the highest in western Europe. Outside of western Europe the rate is higher. The Baltics contain typical examples: Latvia is around 3, Lithuania and Estonia are around 5 to 6.) The equivalent rate for black Americans is 16.3 per 100,000. Gun laws are of course the same for both groups within a given state. As demonstrated by the El Paso vs. Juarez comparison, skin color means nothing by itself, but as a proxy for deeper cultural and socioeconomic factors it shows how little difference gun laws make. Since most murder victims have criminal records, one more possibly more parsimonious explanation for some of the murder rate in Mexico and many American urban cores is violence directly or indirectly related to the saturation of organized crime including the drug trade. Please note that I am not blaming the victim – the perpetrator is the person with sole moral responsibility. But to the extent that statistical patterns in the victims exist, we should examine them to see how they might be broken. The complete disconnect between gun laws and violent crime indicates that this is an ineffective way to try to break the pattern. It will be interesting to see if the recently-relaxed marijuana laws in some states result in reduced violence associated with the fact that criminal organizations no longer have exclusive control over its sale. While I am already generally opposed to the drug war as currently constituted, its end or at least reform could potentially be an effective anti-violence measure.

At this point we move on to Mark’s specific suggestions as to new gun control laws. The first is “significant restrictions on civilian ownership of magazine-fed semi-automatic weapons”. Here we hit the first problem. This category amounts to all firearms which fire one shot per trigger pull, which is itself very close to “all firearms”. The only modern firearms which are not semi-automatic and are sold in any meaningful quantity are bolt-action rifles, pump-action shotguns, and revolvers. Revolvers are functionally semi-automatic in that they fire one shot per trigger pull though they aren’t classified as semi-automatics for technical reasons[3]. If you include them, you’ve come close to banning civilian handgun ownership. This is politically impossible, legally dubious, and not likely to put a dent in mass shootings, much less mass killings generally.

Well, you could try restrictions that don’t amount to a ban. Mark suggests requiring a “legitimate reason” for ownership. While I would certainly object to self-defense not being a reason (and that reason applies to everyone), we already have good evidence that this type of “good reason” scheme is futile. California has a “may-issue” concealed carry system, where the issue of permits is at the discretion of the local county’s evaluation of your stated reason. As a result, some counties issue to effectively everyone, and some counties issue to no one, or only to the famous or politically well-connected. Or rich, as has been pointed out in New York. Just the paperwork to possess a handgun in NYC costs north of $400. A huge fee merely to ask for some bureaucrat’s personal fiat is the antithesis of equal protection under the law, and unworkable in a society that respects the rule of law. It’s evadable too – Anders Breivik jumped through Norway’s considerable hoops without trouble. Conversely, I’m with Mark in favor of safe gun storage, and in fact most new guns in the US are automatically sold with locks. Many states already have criminal penalties for not adequately securing guns against children. Criminal penalties for having your guns misused more generally are not reasonable. Locks and safes are breakable, often with something as simple as a sledgehammer (of course locks and safes are still a good idea, but far from infallible), and it is unacceptable to put people in jail for being the victim of a crime. Limits on magazine capacity are also unlikely to accomplish anything positive. I have seen no evidence that ordinary crime either requires or typically involves more than one or two shots. Mass shootings by definition involve more, but the numbers involved indicate that the success of magazine prohibition is dubious. Of the 62 shootings listed by Mother Jones, I count only two (the Giffords shooting and the Thurston High School shooting) which were stopped when the gunman was tackled during reloading. One occurred during a ban on >10-round magazines, one didn’t. The one that did actually involved more people being hit.

Mark’s second main thrust is toward better policing of guns moving from legal owners to prohibited ones. Currently buying a gun from a dealer means undergoing a background check through the federal NICS system. Private sale within the same state is not prohibited federally, so in principle a prohibited person could procure a firearm by buying one from a private individual who is unaware of the buyer’s criminal history. Alternately and more commonly, a prohibited person could entice a friend, romantic partner, or relative with a clean record to purchase a gun on the criminal’s behalf. It is illegal to serve as a proxy buyer (a “straw purchase”) even for a person with a clean record, but it does happen and seems to be the most common source of guns for criminals. Can this be prevented or mitigated? Probably so. It is possible to trace serial numbers to the original dealer, and from there to the original buyer via the Form 4473 records kept at the dealer’s location. For that matter there are thousands of criminal attempts at gun stores to purchase firearms which are stopped by NICS every year, but only a few dozen convictions. This dismal rate could be improved immensely under current law. While I have serious privacy concerns with opening or requiring NICS for private sales, it is conceivable that it could be done and might have some good effect.

Finally, Mark enumerates eight pro-gun arguments which he believes are not worthy of serious attention. I’ll briefly comment on each, sometimes to agree, sometimes to disagree, sometimes to clarify. (The arguments themselves are Mark’s words. His own responses to them are at his post.)

1. The only thing that stops gun violence is “good guys” with guns – the argument we should arm teachers, arm principles, or place armed guards to prevent mass shootings in school.  Or the even more obnoxious “when seconds count” argument.

In vast majority of cases, mass shootings are stopped when the perpetrator is shot, either by suicide or police. But we have almost no data on the possibility of concealed carry permit holders stopping mass shootings. This is not surprising. Mass shootings are extremely rare, concealed carry is rare in percentage terms (around 2% have a permit in Texas), and mass shootings almost exclusively occur in places where concealed carry is prohibited by statute or the property owner. On the other hand, in terms of raw numbers millions of permit holders rack up billions of man-hours carrying every year. Permit holders outnumber police 7 to 1 in Texas, for instance. It is unreasonable to expect that extension of concealed carry to schools will result in catastrophic movie shootouts when it has not done so anywhere else – including many college campuses. I would not suggest concealed carry in schools as a panacea in view of its lack of a track record and the small percentage who would actually do it, but I would also not suggest we automatically discount it.

2.  You can kill someone with a frozen banana, hence assault weapons shouldn’t be banned.

If assault weapons were actually a problem, and if banning them would actually accomplish anything, then maybe. But as it is, choking on food kills around three times as many people as assault weapons.

3.  The 2nd amendment protects us from tyranny.

I understand where you’re coming from. The American revolution can’t be exactly duplicated in the modern world. But I don’t dismiss this for two reasons. First, the long view. In living memory most of Europe fell to two different varieties of hideous tyranny, of a kind that no one could have seen coming even a few decades earlier. Fascism and communism are not descending in the US, but we would be foolish to think it’s always and forever impossible. And if it did, we shouldn’t assume that rifles are hopeless against stealth bombers and tanks. Unfortunately the pre-medieval primitives in the Taliban didn’t make that assumption either, and they’ve more or less succeeded in forcing a US withdrawal. Second, and more mundanely, the 2nd amendment actually talks about the security of the state, not just overthrowing it. There are times and places where the state can suddenly fail, and at that point the people are on their own until normality can be restored. I was less than 80 miles from landfall when Katrina temporarily eliminated the entire governmental apparatus in New Orleans, and saw the need for this firsthand.

4. It’s crazy people that’s the problem, we need to track them, institutionalize them etc.

Many – but not all – mass shooters are nuts. But the gargantuan, overwhelming majority of the mentally ill are completely harmless. If we did have a way to better separate out the tiny fraction of the dangerous mentally ill, we would be better off. Now it’s true that “better mental health treatment” is much easier said than done, though better health care is always worth working toward. There are already some easy ways we could do better, as the NICS failure to disqualify an already legally-adjudicated-ineligible Seung-Hui Cho demonstrates. Beyond that, we should tread carefully.

5. They already had an assault weapons ban, it didn’t work.

As you say, it was a pointless exercise that banned cosmetic features. But as I hope to have demonstrated above, even a much more comprehensive ban is not likely to accomplish anything.

6. It’s unconstitutional!

Well, there’s the Constitution as the Platonic ideal of its text (or original public meaning, if you prefer), and there’s the Constitution as actually interpreted by the supreme court. The proposed Feinstein ban is unconstitutional in the first sense in my opinion, but probably wouldn’t be in the opinion of the nine people whose votes matter.

7. But Israel lets everyone carry guns and they don’t have school shootings.

Israel is indeed a very different country. As I have argued, extrapolating from other nations in this context doesn’t generally work. I agree that it doesn’t work here either, in either direction.

8.  It’s because we don’t have school prayer, the students should have rushed the gunman, it’s because God isn’t in schools, it’s video games, it’s feminists, it’s doctors, it’s anti-depressants etc.

Yes, these are all dumb arguments.

To conclude: gun control is very unlikely to accomplish anything positive even in the most rationally-designed best-case scenario, and the proposals being floated in DC are not even in the same zip code as rational.

[1] Of course I prefer to cite linkable material, but unfortunately academic publications are generally behind paywalls. My statement about homicides is fairly well-documented and not especially controversial in the literature. Wikipedia’s article about gun politics in Australia links many typical studies by various groups on this issue of homicides in Australia. My statement about mass killings generally is taken from Evaluating Gun Policy: Effects on Crime and Violence, P 121-156, 2003, Philip J. Cook and Jens Ludwig, eds.

[2] I use victims rather than perpetrators because in a large percentage of homicides the race of the perpetrator is unknown. The great majority of homicide is intra-racial rather than inter-racial, so it’s not a bad first approximation. But it’s not a perfect approximation. The two major problems are the aforementioned failure to account for inter-racial homicide, and that the US is vastly more complicated than the simplistic FBI categories of “white”, “black”, and “other”.

[3] From a technical standpoint, semi-automatic means that a single trigger pull fires a round, removes the spent cartridge from the chamber, and inserts a new cartridge into the chamber. This is almost the same thing as saying “one shot per trigger pull until you run out of ammo”. The principle exception is revolvers, which are still one shot per pull, but each cartridge has its own separate chamber and the spent cartridge remains with the chamber after firing. While we’re being technical, there is also a difference between a clip and a magazine. A magazine is a device that holds and feeds cartridges into a chamber. A clip is a device that feeds cartridges into a magazine. But like “invite” as a noun, nobody worries about it too much.