Friday, January 22, 2010

The Suspension of Belief

You may think I've mistitled that, but no, not really. Suppose I put to you two ways to say a common sentiment:
  1. All that glitters is not gold.
  2. Not all that glitters is gold.
Now, put aside all notions of poetic rhythm or provenance. (Or that the original version in Shakespeare's Merchant of Venice had "glisters" instead of "glitters." The former comes from Dutch, while the latter comes from Norse. In our day, the Norse version has entirely displaced the Dutch version, but in Shakespeare's day, they both had currency. Or at least so Shakespeare would have us believe.) Does either of these seem "righter" to you than the other?

I've put little quizlets like this to various people and they seem to fall mostly into two groups. One group of people can't see anything at all to recommend one over the other. Moreover, when the particular distinguishing feature is pointed out, they either don't see it or can't see why anyone would care. (You might, if you fall into this group, see if you can figure out before reading on what this distinguishing feature is, if you don't already know.)

The second group, of course, sees a logical distinction between the two and what's more, they're irritated that there's a mismatch between intent and wording. What's still more, they're irritated that the first group doesn't acknowledge this. To this group, the above two sentences are logically equivalent to the following:
  1. All glittery things are non-gold.
  2. Some glittery things are non-gold.
A quick glance at the script for Merchant of Venice indicates that Shakespeare chose the first wording ("All that glitters...") but his meaning is clearly the second. Does this bother you?

OK, that's not really all that important, as we all know what Shakespeare meant. Here's another one:
  1. I don't believe we have a coherent plan for the Middle East.
  2. I believe we don't have a coherent plan for the Middle East.
Obviously, when it's presented this baldly, it's clear what the difference between this two (especially, I hope, in light of the previous example), but I can't count the number of times that people have interpreted #1 (or minor variations thereof) as #2. And honestly, I don't think it's because they can't think logically. I think it's because they're impatient with disbelief.

Nowhere is this more evident than in politics. It's practically a cliché to demand politicians give their position on some issue or another, to the point that it's considered a weakness if they can't immediately spit one out. While I'm all for politicians being prepared for new situations (and as a by-product, for questions from the press), is having a response for all such questions really preferable to being able to suspend belief when the situation warrants? We've seen the dangers that feigned certainty can bring. And it's not as though suspension of belief necessarily means suspension of action. We can act rationally on uncertainty just as well as we can act on strong belief.

As prominent as it is in politics, though, this rejection of uncertainty permeates our whole world, including science, where it has no business. Political truths may last for a generation or two (think about how long the Democratic party has been on the side of civil rights), but scientific truths, once verified, last essentially for eternity, subject only to occasional refinement. Given that, what's the rush to judgment? Why not suspend belief until we know for sure? Impatience with uncertainty is fine as long as it motivates us to reduce it, but not if it forces belief before we're ready.

Monday, January 11, 2010

Cutting Your Losses

I was standing at the vending machine at work today, buying some chips with lots of small coins (nickels and dimes). And as I often do, I carefully inserted the nickels first, then the dimes; if I had used any quarters, they'd have come last.

You may—assuming you've read this far—wondered why this is. To be fair, having done this for a long time, I wondered myself for a moment. And then I remembered.

See, when I first started doing this, I was in college. I was living in the dorms. The dorms had vending machines, which were balky, much like anything in the dorms. They would, occasionally, find something objectionable about your change. They were even particular about the way you inserted your change; sometimes, it would take six or seven tries for you to get it to accept a specific dime. I would bring extra change just in case, if I had any, but sometimes even that would run out. So there I would be standing, with 45 cents that the machine was refusing to take, and more money back in the dorm room that I could try out on the Keeper of the Fizzies. But in order to get that money, I'd actually have to back to the dorm room. Away from the vending machine.

I'd run downstairs, get the change, run back upstairs, and hope that in the meantime, no dormitory Grinch had decided to get a 30-cent discount on his Coke.

Because, as it happens, sometimes they would. I'd get back and there would be no credit at all in the vending machine. You might suppose that Whoever It Was would at least leave the credit they had benefited from in change on the side, but noooooo.

That's when this business with inserting change in ascending order of value started. It was a way of cutting my losses. You might think that it would be simpler for me to just push the coin return and withdraw my change before heading downstairs, but in the first place, the coin return lever was balky, like everything else, and in the second place, it had often taken me lots of effort to get those coins in and I was reluctant to relinquish those hard-won gains.

Eventually, I managed to obtain a small dorm fridge and thereafter bought my drinks at the market. But this was before all that. Just the same, I continued my coin-sorting practice even to the present day, where (I daresay) my co-workers are far less likely to stiff me out of a handful of change than my dormmates were.

You know me, always looking for something mathy about the situation, so here's the question: Suppose that I only used n nickels and d dimes (no quarters), that I foolishly brought exact change, and that the vending machine refuses to take exactly one coin, randomly and uniformly selected from all the coins. On average, how much less money did I place at risk going nickels first than I did going dimes first?

The answer: The average reduction in risk was equal to the value of the nickels multiplied by the fraction of coins that were dimes.

I had thought to try to tie this story to something deeper, but I just can't bring myself to do it.