Saturday, March 5, 2011

Intuitively, this shouldn't be happening...

Believe it or not, I've still been receiving the odd comment on the gun control posts I wrote before Christmas. This rather extraordinary one was left by Epsilon Given a day or two ago, although to begin with you might be hard-pressed to spot any connection with the subject in hand...

"Just for the record, I'd like to clarify just what "counter-intuitive" means to a mathematician.

Intuition is a funny thing. It generally leads us to truth, and a good intuition helps us to discover new theorems, even entirely new lines of thought.

When something we expect to be false, or nonsensical even, turns out to be true, we call such things "counter-intuitive". As we encounter these things, we sometimes adopt them into our intuition--but there is no shortage of things to surprise us!

Historically, we have encountered many counter-intuitive ideas--ideas that often met fierce resistance every step of the way to acceptance. The square root of -1, for example. Or a number for "nothing". Or the irrationality of the square root of 2.

And sometimes, proving something counter-intuitive can lead to your death. The person who first proved that the square root of 2 was irrational, for example, was drowned at sea by his fellow Pythagorians.

I'll give several more counter-intuitive examples that I'd expect to go over your head without deep explanation, unless you are already acquainted with them. You can cut up a unit sphere into nine pieces, and re-assemble them to make two unit spheres (Banach-Tarski Paradox). You can have two intersecting lines both parallel to a third line (hyperbolic geometry). There are more irrational numbers in the interval between zero and one, than there are integers (Georg Cantor's diagonal argument). When we "generalize" the factorial function, the factorial of 1/2 is 1/pi (Gamma function). e^(i*pi)+1=0 (Euler's Formula).

Heck, I spent many hours trying to trisect an angle using a compass and straightedge, and I spent hours trying to integrate e^(-x^2) in terms of "elementary functions", because I was told that these were impossible--and this conflicted with what I was taught in our culture, that "nothing is impossible". Later, I actually read the proof that demonstrated, once and for all, that the former is impossible, and more recently I learned that the latter proof is done along the same lines, so to me, these are no longer counter-intuitive.

The idea "More guns, less crime" is, like all these other things, counter-intuitive. Our intuition tells us that getting rid of guns will decrease violence. But we can find statistics that show when we ban guns, violence increases. And we can find statistically that, internationally, there is no correlation between gun laws, gun ownership, the murder rate, and the suicide rate, from country to country. As I have examined these things closely, I've come to the conclusion that honest people with guns is a good thing, because they can defend their lives against dishonest people determined to do harm.

I have seen the statistics and the studies. I have thought about the philosophies behind the two positions. In all this, I have come to the conclusion that more guns really do mean less crime! And that, more importantly, if you value liberty and security, then you shouldn't depend on the State for protection--it is your duty to protect yourself and those you love--and nothing can change that. Indeed, the State, more often than not, cannot protect you--and occasionally, the State is even your enemy.

And that those who believe otherwise, might as well believe that the moon is made of cheese, or that imaginary numbers really are imaginary, for all the truth that their position holds."

Words fail me. Epsilon, this is not the first time that I've had to make the elementary point to you that I do - honestly, truly - know what "counter-intuitive" means, and have done all along. Your apparent belief that if I take issue with anything you say it must mean that I've simply misunderstood the terminology you use (and require a detailed explanation to bring me up to speed) is...well, a touch bemusing, to say the least. I'll simply reiterate what I said to you before - some things that are counter-intuitive turn out to be true. Many don't. I note that you confirm that at the outset yourself, so your point is...?

The latter part of your comment essentially boils down to this - you've generously had a little think about the statistical evidence on our behalf, you've selectively culled the bits that suit you, you've declared your side of the argument the winner on that basis, and thus regard yourself entitled to feel morally superior to those of us who don't "do our duty" by "protecting" ourselves with a gun.

Well, let me put this to you. Here is the compelling evidence (as opposed to intuition) that those of us on the other side of the argument in Scotland look towards. It suggests overwhelmingly that our approach has succeeded in protecting the public. Might I suggest therefore that we are equally entitled to think - valuing security and liberty as we do - that it would be a disgraceful dereliction of our collective "duty" if we were to suddenly decide to indulge the wishes of a minority to own handguns, and thus compromise the safety and liberty of everyone?


  1. As a supplement to that James, you and your readers may be interested in the data available on Scots gun crimes, beyond homicide statistics.

  2. Thanks, LPW - I don't know how you do it!

  3. "Words fail me. Epsilon, this is not the first time that I've had to make the elementary point to you that I do - honestly, truly - know what "counter-intuitive" means, and have done all along."

    Yes, words *do* fail you. You *clearly* don't know what "counter-intuitive" means. I will have to break it down:

    Something is intuitive if it is something we naturally expect to be true. We expect 2+2 to be 4, for example, because that fits our sense of what addition is, and what integers are.

    Something is counter-intuitive if it something that is true, but goes against intuition. I have *never* heard the phrase to mean something that we expect to be false, but is false--and that's because such a thing fits our intuition, by the definition of intuition. I would not say 2+2=5 is "counter-intuitive", because it simply. isn't. true.

    But 2+2=1 *is* counter-intuitive, when we encounter "arithmetic mod 3" for the first time. We *expect* it to be 4, but in this case, it's the remainder of division by three that counts--thus, it *is* true that the answer is 1.

    I'm sorry if you are annoyed by my ranting on this definition--but you are rubbing against the grain of *years* of mathematical learning when you say "something counter-intuitive can be false".

    Now, to move on to that "absolute proof" that you provided: I found it amusing that this "proof" was immediately neutered by the first commenter to that blog post. I have seen enough studies on the matter that a single new one isn't going to sway me. At this point, we're just flinging studies at each other, like monkeys flinging their poo--which particularly ironic, when both of us have admitted that neither of us really trust studies and statistics, especially when it comes to defending our freedoms.

    But it astounds me--doesn't it astound you, too?--how small the numbers invariably are in these studies. How many murders are we discussing? We're discussing numbers like "324", "404" "79", "13", "10", and "60". How do these numbers compare to the populations involved? "6,595.558" and "5,194,400". Do you *really* expect me to believe that gun laws alone are going to affect these types of numbers? *Especially* when these numbers are largely canceled out by looking at knife deaths, as the commenter Dan Sutton did immediately?

    If you want to convince me that "more guns cause more crime", you will have to explain to me why, philosophically, humans are only prone to violence when they are in possession of a gun, and why, if they are so prone, that police officers and soldiers should be allowed near guns while the "civilian" population should be strictly forbidden from access.

    Oh, and here's another statistic I would like to see: gun death per gun--counting *all* guns, legal, illegal, military, and police--in both Scotland and Arizona. I'll have to see if I could drudge that one up myself...

  4. Epsilon, this joke is wearing a bit thin. What part of the sentence "I HAVE UNDERSTOOD ALL ALONG WHAT COUNTER-INTUITIVE MEANS" are you struggling with? I must admit, though, for the first time I'm beginning to wonder whether you've understood what it means yourself. Here are a few definitions -

    "contrary to what common sense would suggest"

    "counter to what intuition would lead one to expect"

    "The word 'counterintuitive' literally means counter to intuition, and so it essentially means that something does not seem right or correct."

    None of these definitions are conditional on the "something" turning out to be true. The mistake you appear to be making is that, because there have been limitations in the way you personally have always used the term, you assume those limitations must also apply to the literal meaning of that term. They don't. Now for all our sakes, I hope you will now accept that and won't try to "explain" the concept to me for a 17th time, because for the moment you're doing a rather good job of proving my point that if someone dares to disagree with you, in your mind they must simply have "misunderstood" the words you are using.

    "If you want to convince me that "more guns cause more crime", you will have to explain to me why, philosophically, humans are only prone to violence when they are in possession of a gun"

    No, I do not have to explain that, because I am not suggesting that. What I am suggesting is that people have a rather greater capacity for inflicting violence when they are in possession of a gun. That, as I've noted many times, is implicitly and hypocritically accepted in one context by the gun lobby, otherwise there could be no legitimate basis for claiming that people "need" guns to defend themselves with.

    "But it astounds me--doesn't it astound you, too?--how small the numbers invariably are in these studies. How many murders are we discussing? We're discussing numbers like "324", "404" "79", "13", "10", and "60". How do these numbers compare to the populations involved? "6,595.558" and "5,194,400"."

    No, Epsilon. Let's instead zone in on the stark comparison of numbers that you appear to be trying to muddy. LPW's closing summary -

    "In brief, in 2009: Arizona: 6,595,778 people, 198 gun deaths. Scotland: 5,194,000 people, 2 gun deaths."

    To answer your next question, yes we are expected to believe that gun laws are affecting those numbers. Given the extent of the disparity, it would be astounding - to use your own word - if they were not.

    "Oh, and here's another statistic I would like to see: gun death per gun--counting *all* guns, legal, illegal, military, and police--in both Scotland and Arizona. I'll have to see if I could drudge that one up myself..."

    Feel free. I haven't the faintest idea what you think you'll prove by it. One of the fundamental propositions of the American gun lobby is that gun bans are totally ineffective at preventing criminals from getting their hands on guns, and using them to murder people. I'm afraid an absolute figure of two gun deaths in 2009 in a handgun-banning jurisdiction of five million people does rather blow that proposition out of the water, no matter what statistical contortions you go into of "death per gun".

  5. No guns may not mean no violence, but it certainly has a fairly serious impact on gun-violence.

    Gun violence seems much more likely to result in death than a bout of fisti-cuffs, a bottling, clubbing, or chibbing. It also seems a wee bit less dangerous for the perpetrator to stand a few metres away from his victim and unleash that explosive propulsion of lead than to actually engage with the victim - no real need to worry if intended victim might be a tidy operator in the ring himself.

    Does epsilon propose that instead of trying to fight knife crime in Scotland, we should all carry knifes? or is he suggesting that the measures required to address our knife crime statistics is the introduction of handguns for all?

    Hypothetically - Do you think Pythagorus should have simply shot Hippasus when he pointed out the square root of 2 couldn't be defined as a ratio of 2 integers or was the drowning something more symbolic? Would Pythagarus have been more willing to listen to Hippasus had the proof been delivered at the end of a gun barrel?

  6. "or is he suggesting that the measures required to address our knife crime statistics is the introduction of handguns for all?"

    That's pretty close to what he's saying, Jim - he said on the previous thread that the state is incapable of protecting the public (ie. through gun bans and the like) and therefore it was an individual's "duty" to "protect" themselves and their family with a gun. The only snag is, of course, that his premise is spectacularly wrong. It's somewhat ironic that he claimed the commenter Dan Sutton had "neutered" the statistics in LPW's post, because in fact what Mr Sutton actually did was broaden the point out and demonstrate that even if you take non-gun crime into account, Arizona's homicide rate (both absolute and per capita) is far, far higher than Scotland's. If Epsilon's proposition was correct, the opposite should be happening - the legality of guns for 'protection' in Arizona should be making murders of all types less likely, not more so.

  7. While I'm thinking of it, I should just amplify the point about Dan Sutton's comment, because Epsilon has totally misrepresented what he said -

    "*Especially* when these numbers are largely canceled out by looking at knife deaths, as the commenter Dan Sutton did immediately?"

    No, he didn't, Epsilon. While he acknowledges that Scotland's relationship with knives is somewhat comparable to Arizona's relationship with guns, the conclusions he draws are the complete opposite to what you appear to think they are -

    "Firstly, if we had the same murder rate as Arizona we would have had about 258 murders compared to the 79 we did have.

    Secondly, if Scotland had the same population as Arizona but the same murder rate we could have expected almost exactly 100 homicides in 2009 compared to 327...
    [Note - this sentence could potentially be misconstrued. The "same murder rate" refers to the one actually experienced in Scotland, not to Arizona's, and the 327 figure is Arizona's number of homicides in 2009.]

    Our per capita homicide rate is 0.00015% compared to Arizona’s 0.005%. You are some 3.33 times more likely to be the victim of a homicide in Arizona than you are in Scotland."

  8. On "counter-intuitive": I'm sorry, but I've tried to bend my mind around to your understanding of this word. I admit a bias is caused by my experience with how this term is used in mathematics, and even in computer science (user interfaces, in particular). I've even looked up the definition, from several different sources:


    In each case, the concept of "something that seems false, but is true" is preserved.

    I apologize for being such a stickler on this particular point, though.

  9. On "Statistics": In looking more closely at the first comment, I've had to conclude that you are right. I misunderstood his point. But I *still* don't find this single interpretation of statistics all that compelling, for many reasons:

    -- Scotland and Arizona, besides having different populations, they also have different cultures, histories, population densities, flora and fauna, mental health institutions, etc. That is, it is not entirely clear that it's justified to compare the two populations (despite both being "common law" institutions).

    -- I'd like to see the claim that knives, or even fists, are somehow "safer" than guns justified: determine the number of violent encounters per 100,000, and compare Scotland's to Arizona's. If knives really are less deadly that guns, then there should be comparable numbers of violent encounters, but less murders, in Scotland.

    -- Let's *really* make sure that the "homicide" figure in Arizona is strictly "murder". American crime reporting has this nasty habit of including justifiable homicide with murder--in part, because justifiable homicide is often determined by a jury.

    -- How do these statistics change over time? If violence increases after guns are banned in Scotland, then it's disingenuous to compare Arizona and Scotland, and then say "See, gun laws work!" It would be equally disingenuous to compare these numbers, if violence *decreases* after a "shall issue" gun permit law is passed in Arizona.

    -- Why should I trust *any* data? Whenever I try to look up this data on my own, I always run into obstacles obtaining it. The data *I* want never seems to be available--admittedly, I can't look too deeply, because of time constraints--but, in the end, why should I trust data other people select as "important"? A Stupid Example: I requested "guns per gun death"--in part, because I wanted to test the claim "more guns means more death"--but Arizona gun stats are almost non-existent, as is the category of illegal gun stats for Scotland. Even if these stats were available, though, I would have to understand that a statistic like "illegal guns" is, at best, and educated guess, and so my analysis (and conclusions) of such data would probably be meaningless anyway.

    -- In my half-hearted attempt to obtain these numbers, I learned that Scotland had a significant drop in violence this last year. Are we comparing a "bad" Arizona year to a "good" Scotland year? How can *that* be fair?!?

  10. Although I really wish I had the time to gather this data, and crunch these numbers, why should we trust any of this as "meaningful"? Societies change all the time. When we look over all the countries of the world, we see no correlation between "strictness of gun control" and murder rates and even suicide rates. And, when all things are considered, we are talking about very. small. numbers.

    Numbers so small, in fact, that the single greatest terrorist attack in America would count as a 10% increase in America's gun death rate (counting both murders and suicides), if those deaths in America had been committed with guns.

    And to further complicate things, in America, we have *counter-intuitive* results: Every time a change in gun law is considered, there is a claim that passage of such laws will result in an increase of gun violence. Yet, with Florida's "shall-issue" permits, and lately, the Heller decision that affected Washington D.C., violence--both gun and non-gun--*dropped*. And these aren't just isolated incidents. It happens time and time again, to the point that the debate among criminologists in the United States *isn't* "Do guns cause crime?"; it's "What effect, if any, do guns have in lowering crime?".

    I would also add that the Washington, D.C. drop perplexes even gun rights advocates. D.C. hasn't done all that much to loosen gun laws, yet the drop occurred anyway. Because there are ebbs and flows to these sorts of things, the drop may not be due to gun freedom--but, since we're talking about complex systems we call "society", we can't entirely rule out this change in law as a cause, either.

    Which is yet another reason I don't like statistics all that much!

  11. Something is counter-intuitive if we get a result that we don't expect to come about. The example I just gave was "increased gun ownership decreases gun crimes"--and there are statistics that seem to demonstrate that this indeed may be the case. Often, counter-intuitive conclusions are the results of incorrect assumptions. Here are some reasons why increased gun ownership might not result in increased gun violence:

    -- Guns aren't as lethal as many people think, and alternatives--knives, fists, rocks, baseball bats, bottles, even rope--are more lethal than many people realize. Making guns difficult to obtain and carry may neutralize the first "threat", but these other threats are still there, and much more difficult to control besides! Additionally, it's easier to counter these other threats with a gun, than it is to counter these other threats with like threats; hence, when you have a legitimate need for self defense, you're better off with a gun.

    -- "Shall Issue" permits include a background check for past mental and legal problems. The people who would apply to such things would be law-abiding, and generally healthy mentally. Such people who carry guns aren't likely to go on crime sprees!

    -- Those who take self defense seriously--whether or not they carry a gun--will learn techniques to recognize potential assaults. They'll be more aware of their surroundings, they'll watch out for characters "scoping out" parking lots, and they'll avoid groups of scary-looking people loitering about an alleyway. And those who choose to carry guns are more likely to take self defense seriously.

    -- Those who choose to carry a gun are more likely to be mentally prepared if something bad happens. While it's impossible to know in advance how you'll react if you are in a violent situation, if you've thought about what you'll do, you're more likely to do something that will help protect yourself and those around you.

    -- Those who choose to carry a gun are more likely to recognize the responsibility that lethal force represents, and thus less likely to heatedly argue, and more likely to peacefully pull themselves from an argument before it comes to fighting.

    -- Criminals don't want to get shot. If they have reason to believe their victims will be armed, they are less likely to engage in behavior that can end with them in the hospital, or dead.

    -- The police can't respond to events in a timely manner. When a citizen is carrying a gun, he or she becomes the "first responder" to a criminal situation. In much the same way that knowing first-aid skills and having access to a first aid kit, can save a life while waiting for an ambulance, having self-defense skills an access to a weapon can save a life while waiting for the police.

    Now, I *don't* think that everyone in Scotland needs to carry a knife, or even a gun. Crime--even when it is high--is rare enough, that any given individual isn't likely to get mugged or murdered. Heck, even Mafia members can have surprisingly long lifespans! But the numbers are small enough *either way*, that it is counter-productive to forbid law-abiding citizens the choice to carry a weapon, and to train in the use of that weapon.

    And, yes, I have concluded that every responsible person should know how to use a gun, should carry one, and should be prepared to use it in self-defense--but I have concluded that every responsible person should also know what to do when someone gets injured or sick, or when there's an earthquake, or there is flooding, or a tidal wave, or a forest fire, or a hurricane, or even plague and famine and general societal collapse. Being prepared for danger is a moral responsibility that all of us should accept.

  12. This is my last post, and I mean it this time. I'll hang around to see how you'll pick all this apart; at this point, I don't care, because we have dragged this on for far too long. I should have ignored the last comment James made on that other post; I apologize for having not done so.

    In any case, this has been a spirited debate, and I appreciate your willingness to take on the issue! :-)

  13. "In each case, the concept of "something that seems false, but is true" is preserved."

    EG, you've just provided four links that entirely bear out what I've been saying all along - all four definitions are very straightforward, and not one of them contains even the slightest hint of the conditionality you're idiosyncratically ascribing to the term.

    You suggested that we break down the term earlier - OK, let's do that. "Intuitive" means something that intuition would lead you to expect to be true. I trust we could both agree that on some occasions that something will indeed turn out to be true, and sometimes it won't. Adding the "counter" prefix simply reverses the meaning - ie. it now means something that intuition would lead you to expect to be false, again without saying anything one way or another about whether that something will indeed turn out to be false.

    I'm really struggling to see how the addition of "counter" also has the magical effect of adding a huge degree of conditionality that no-one would suggest applies to "intuitive" itself. Indeed, perhaps the best way of satisfying yourself that your (dare I say it) intuition about the meaning of this term is wrong is simply to look at the definition of the "counter" prefix in its own right.

    Anyway, the next time I have a month to spare, I'll try to respond to the rest of your comments.

  14. I've responded to Epsilon's more substantive points in a fresh post here.