[Dating] Calculus

When I was in high school, my math teacher warned against the foolishness of projecting a line from only two points. Two points, she explained, could easily lie on a curve. But you’d never know unless you had at least a third point plotted.

When I got older, I applied to Procter & Gamble for an internship. To determine whether you’re smart enough to work for them, they give you a reasoning test with questions like this (only harder):

It flummoxed me the first time I saw them and I flunked. But I went back and studied the questions and I realized they were testing only one skill: the ability to predict a complex pattern based on only two datapoints. And, like most mass-produced standardized tests, there’s a limit to how many permutations can arise, so it was easy to study for. I passed the second time.

So now I can say that I’m smart because I can project a line from only two points.

It might seem to follow that it would be even smarter to project a line from a single point. Indeed, this is (sort of) what is known as calculus.

When you take the derivative of an equation at a given point, you are essentially saying, “If the line proceeded straight from this point, it would have this slope.” In this example, the two derivatives taken are lines with slopes that continue forever downward.

But the function doesn’t really behave that way. It bends. At zero, the derivative suggests a flat line.

That isn’t how it happens either. You probably recognize the function as y=x^2, and it’s parabolic.

It isn’t exactly projecting a line from a single point, because the point itself must be on a line. But it’s as close as you can reasonably get.

Derivatives are very handy tools when you want to know the behavior of a system at an exact point. But projecting those lines and assuming they hold in the future is clearly unsafe. It is also the stuff of Malthusian doomsayers.

Malthus was the 18th century philosopher who took a look at population growth and found that it looked something like this:

Then he examined food production capacity, and discovered that it plotted something like this:

And of course he freaked out. Because it seemed patently clear that the world was going to starve in just a few generations.

Of course Malthus was wrong. He didn’t take into account that family size reduces with prosperity and that scientific farming would spur huge leaps in agricultural output. But that has not stopped other doom-and-gloom prophets of the past from foreseeing us all dying of famine, drought, resistant bacteria, superviruses, and various energy-related disasters by the year 2000.

It is also the sort of mental math used by social doomsayers who gloomily predict that at current rates of moral degeneration we will be bonobos in a mere generation or two. They look at the trend from the Victorian era, project a line from the most recent point, and get something steeply negative. Clearly these people are unfamiliar with the Restoration period in England. Or almost any other non-Victorian era in world history. We’re relatively chaste and exceptionally ethical by comparison. Morality, I would posit, is more of a sine curve.

And this is because people are not (gasp) equations. We have the ability to self-correct based on feedback from our environment. (Including, for example, doom-and-gloom predictions.)

And I was going to connect this to dating, but I’m at my word limit, so to be continued.

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17 thoughts on “[Dating] Calculus

  1. Family size might have temporarily reduced with prosperity, but natural selection is going to take care of that trend pretty quick. In a couple of centuries, assuming we haven’t all become robots yet, we’ll all be Mormons, Catholics, Muslims, Orthodox Jews, communists, and others who believe in having lots of children. We can observe this happening before our eyes in Israel: Arabs and charedim steadily climb as a percentage of the population, while chilonim drop. The Israeli Central Bureau of Statistics found that 42% of Israeli Jews identified as chiloni in 2009, compared to 44% in 2005.

    As for these silly “spot the pattern” questions, I like this one: 4, 14, 34, 42, 59, what comes next? It’s no more arbitrary than some of the others you see in this vein, and less useless if anything.

  2. TA, will there be no self-correction for chareidim who have lots of children and very few decent jobs to support them? Do you assume that the non-chareidi Jews of America and Europe, and the working taxpayers of Israel, will continue to support them for a couple of centuries?

    And most Catholics and Mormons aren’t having lots of kids these days.

  3. Note to self: Apply at P&G, they have easy exams.

    All seriousness, I really enjoyed this post. My math nerd self from childhood was so happy…

  4. Wow, maybe if you’d been my calculus teacher I would’ve actually understood class instead of having to struggle to teach things to myself…
    In a different vein…
    Who is morris and what is with his Muslim videos?

  5. Hm. I scheduled this a while ago and couldn’t check up on it due to working through someone’s internet filter. I don’t know why I chose that example as it does not illustrate my point at all (there are four data points there, not two). Most of the questions are better approximated by these: http://www.psychometric-success.com/practice-papers/Psychometric%20Success%20Abstract%20Reasoning%20-%20Practice%20Test%201.pdf
    Those are somewhat harder; there are one or two toward the end that I still don’t know the answer to.

  6. Thanks, Cheep. Of course, this is only the most basic stuff, so I won’t let it get to my head.
    morris seems to be spamming. Not appreciated, mister.

  7. Anon, charedim get jobs when they need to. The article I linked to says that as of 2009, 52% of charedi men and 61% of charedi women have jobs (as opposed to about 90% of secular men and women). In America, where they don’t get as much government money, the figures are probably higher. As charedim become more numerous, more of them will have to get jobs, but it doesn’t follow they’ll have fewer children. To the contrary, charedim who have to get jobs will be able to support lots of children more easily than they can now — the government payments they get are enough to live on, but not as much as you’d make from a real job. Muslims demonstrate that there’s nothing incompatible about having jobs and also lots of children.

    Catholics and Mormons aren’t having nearly as many kids these days as Muslims or charedim, but they’re still having a lot more than the general population. Most developed nations today are reproducing at below replacement rate, some drastically below. Countries like Japan or Germany have a birth rate at not much over half replacement rate, which means that their populations would halve every generation if not for immigrants. (Israel, incidentally, has the highest birth rate of any developed nation.)

    So anyway, predominantly Christian nations that don’t let in lots of Muslim immigrants — like the United States, say — will see proportional increases over time in Christian groups that encourage having children, like Catholics and Mormons. Maybe the average Catholic family will only have four kids, and we don’t consider that a lot, but it’s more than the average secular family has.

  8. In case you are interested in more moderation today, there’s another one of his videos in the post titled Thoughts upon Graduating (May 30th).

    Anyway, getting back to the beginning of your post, suppose you had three points. What if the three points were located on the linear part of a piecewise function, then what?

    Or, perhaps your teacher was unaware of projective geometry.

  9. Double Trip – Presumably, you’d know it was a piecewise function. They’re written differently so you can tell where your points lie. At least, the ones I’ve come across. Anyway, she said at least three points, not that you need three points. Very few people are satisfied with three points anyway. Except experimentally when you get three values for a single point, and that’s not really the same thing.

  10. Why would you know that it was a piecewise function? All you stated was that you were given at least three points (you can have 300 if you want as well), and you implied that you could conclude that they were part of a linear function. In other words, I just wanted to warn you about the foolishness of projecting a line from an infinite number of points. By the way, what did you get your degree in?

  11. Re the 4, 14, 34, 42, 59 pattern…

    I’ve known this one since I was a kid, but you are missing 2 points. The pattern should have 4, 14, 23, 34, 42, 50, 59

    They are stops in Manhattan on the F train in the NYC subway system and the next point is 63.

    Of course, it could also be the stops on the A train or the C train and the next point would be 72.

  12. Pinny, I was doing the A, so the next point is 125. If it were the C, I’d be missing 23 and 50. I like the A best because jumping from 59 to 125 so completely fails to match the pattern up to that point.

  13. Double Trip – sorry, I got confused because the context of the high school teacher’s statement was different and a little muddled in my head. But she was talking about plugging a number into an equation to determine what sort it was. In that case, the function would be defined as broken up.
    If you’re doing an experiment, obviously you don’t get a defined equation, and then yes, you want a great deal more than three points unless you are merely confirming well-known results.
    If you’re talking theoretical mathematics, forget it. I do practical stuff.

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