The Universe of Discourse

Sun, 30 Nov 2014

I don't have impostor syndrome about programming, advanced mathematics, or public speaking. I cheerfully stand up in rooms full of professional programmers and authoritatively tell them what I think they should do.

However, when I put up shelves in the bathroom back in May, I was a psychological mess. For every little thing that went wrong—and there were quite a lot—I got all stressed out and wondered why I dared to perform this task. The outcome was good, but I had a lot of stress getting there.

I put in one plexiglass shelf, for which I had bought heavy-duty wall anchors in case the kids leaned on it, and two metal shelves higher up, which came with their own screws and anchors.

Here's a partial list of things that worried me:

1. The two upper shelves came with a paper template that I held up to the wall to mark where the holes should be drilled. What if the two shelves were slightly different and their templates were different and I needed to use both templates on the wall instead of using the same template twice?
2. When I putting the heavy-duty wall anchors into the drywall, big divots of plaster fell out of the wall around the anchors.
3. Then I filled in the holes with filler, and got filler in the screw holes in the wall anchors, and stressed about this. What if the filler in the sockets somehow prevented the screws from going into the anchors or caused some other unforeseeable problem?
4. The filler looked sloppy and I worried that it would look absurdly ugly to everyone who came into the bathroom. (The shelf would have hidden the ugly screw job from a normal view, except that it was made of plexiglass, so the filled holes were visible through it.)
5. I didn't know how big to drill the holes for the smaller wall anchors and stressed about it, examining the wall anchor packaging for some hint. There was none.
6. I wanted to insert the wall anchors into the holes with my rubber mallet. Where the hell is it? Then I stressed about using a claw hammer instead and maybe squishing the anchors, and spent a while looking for a piece of wood or something to soften the hammer blows. Eventually I gave up looking, wondering if I was dooming the project.
7. I guessed how big to make the hole for the anchor, and was wrong; my hole was too small. I didn't realize this until I had the first anchor halfway in. Then I stressed that I might ruin it when I pulled it back out of the wall.
8. Then I stressed about the size of the holes again when I drilled larger holes. What if I make the hole too big, and then have to fill all the holes and re-measure and re-drill the whole thing?
9. The anchors didn't go into two of the holes. I needed to yank them back out, then redrill the holes, with the outer end a little messy, or the anchors wouldn't go all the way into the holes. Again I worried about spoiling the anchors.
10. When I drilled the holes, sometimes the drill suddenly went all the way into the wall and the rotating chuck left a circular scar on the paint.
11. Also, two of the holes didn't drill easily; I had to lean on the drill really hard to get it to go through. For a while I was concerned that there was some undrillable metal thing in the wall just where I wanted my hole, and I would have to fill in all the holes and remeasure and redrill the whole thing.
12. Even though I had marked the wall for the lower shelf by holding the shelf against the wall and then poking a pencil through the actual holes, when time came to put the bolts in place, I found that the two holes were slightly too far apart. Somehow this worked itself out.

On review, I see that several of these worries could have been completely avoided if I had had a supply of extra wall anchors.

Stuff that could have worried me but (rightly or wrongly) didn't:

1. I knew enough to go to the store to buy wall anchors and screws for the bottom shelf, which did not come with its own hardware. There are a lot of different kinds of anchors, and I did not worry too much that I was getting the wrong thing.

2. I was concerned (although not worried) that the screws holding the bottom shelf to the wall might stress the plastic too much and cause it to crack, either immediately or over time. Obvious solution: insert washers between the screw heads and the shelf. I went to the hardware store to get nylon washers; they didn't have any. So I got thin metal washers instead. I did not worry about this; I was sure (perhaps wrongly) that metal washers would do the job.

3. When I asked the hardware people for plastic washers, they looked totally blank. “Plastic... washers?” they asked, as if this were a heretofore unimaginable combination. I could have felt like an idiot, but instead I felt, correctly I think, that they were idiots.

4. For some reason, I was not even slightly worried about properly leveling the marks for the holes. I used a spirit level, which I consider pretty fancy.

5. I was careful not to over-tighten the screws holding the plexiglass shelf in place, so as to avoid cracking them, but I was at no time afraid that I would somehow crack them anyway.

[Added in July: I have reread this article for the first time. I can report that all the worries I had about whether the shelves would look good have come to nothing; they all look just fine and I had forgotten all the things I was afraid would look bad. But I really do need to buy a couple of boxes of plastic wall anchors so I can stop worrying about spoiling the four I have.]

[The shelves look crooked in the picture, but that is because I am holding the camera crooked; in real life they look great.]

[ A later visit to a better hardware store confirmed that plastic washers do exist, and I did not hallucinate them. The rubber mallet still has not come to light.]

Sat, 22 Nov 2014

When opportunity permits, I have been trying to teach my ten-year-old daughter Katara rudiments of algebra and group theory. Last night I posed this problem:

Mary and Sue are sisters. Today, Mary is three times as old as Sue; in two years, she will be twice as old as Sue. How old are they now?

I have tried to teach Katara that these problems have several phases. In the first phase you translate the problem into algebra, and then in the second phase you manipulate the symbols, almost mechanically, until the answer pops out as if by magic.

There is a third phase, which is pedagogically and practically essential. This is to check that the solution is correct by translating the results back to the context of the original problem. It's surprising how often teachers neglect this step; it is as if a magician who had made a rabbit vanish from behind a screen then forgot to take away the screen to show the audience that the rabbit had vanished.

Katara set up the equations, not as I would have done, but using four unknowns, to represent the two ages today and the two ages in the future:

\begin{align} MT & = 3ST \\ MY & = 2SY \\ \end{align}

(!!MT!! here is the name of a single variable, not a product of !!M!! and !!T!!; the others should be understood similarly.)

“Good so far,” I said, “but you have four unknowns and only two equations. You need to find two more relationships between the unknowns.” She thought a bit and then wrote down the other two relations:

\begin{align} MY & = MT + 2 \\ SY & = ST + 2 \end{align}

I would have written two equations in two unknowns:

\begin{align} M_T & = 3S_T\\ M_T+2 & = 2(S_T + 2) \end{align}

but one of the best things about mathematics is that there are many ways to solve each problem, and no method is privileged above any other except perhaps for reasons of practicality. Katara's translation is different from what I would have done, and it requires more work in phase 2, but it is correct, and I am not going to tell her to do it my way. The method works both ways; this is one of its best features. If the problem can be solved by thinking of it as a problem in two unknowns, then it can also be solved by thinking of it as a problem in four or in eleven unknowns. You need to find more relationships, but they must exist and they can be found.

Katara may eventually want to learn a technically easier way to do it, but to teach that right now would be what programmers call a premature optimization. If her formulation of the problem requires more symbol manipulation than what I would have done, that is all right; she needs practice manipulating the symbols anyway.

She went ahead with the manipulations, reducing the system of four equations to three, then two and then one, solving the one equation to find the value of the single remaining unknown, and then substituting that value back to find the other unknowns. One nice thing about these simple problems is that when the solution is correct you can see it at a glance: Mary is six years old and Sue is two, and in two years they will be eight and four. Katara loves picking values for the unknowns ahead of time, writing down a random set of relations among those values, and then working the method and seeing the correct answer pop out. I remember being endlessly delighted by almost the same thing when I was a little older than her. In The Dying Earth Jack Vance writes of a wizard who travels to an alternate universe to learn from the master “the secret of renewed youth, many spells of the ancients, and a strange abstract lore that Pandelume termed ‘Mathematics.’”

“I find herein a wonderful beauty,” he told Pandelume. “This is no science, this is art, where equations fall away to elements like resolving chords, and where always prevails a symmetry either explicit or multiplex, but always of a crystalline serenity.”

After Katara had solved this problem, I asked if she was game for something a little weird, and she said she was, so I asked her:

Mary and Sue are sisters. Today, Mary is three times as old as Sue; in two years, they will be the same age. How old are they now?

“WHAAAAAT?” she said. She has a good number sense, and immediately saw that this was a strange set of conditions. (If they aren't the same age now, how can they be the same age in two years?) She asked me what would happen. I said (truthfully) that I wasn't sure, and suggested she work through it to find out. So she set up the equations as before and worked out the solution, which is obvious once you see it: Both girls are zero years old today, and zero is three times as old as zero. Katara was thrilled and delighted, and shared her discovery with her mother and her aunt.

There are some powerful lessons here. One is that the method works even when the conditions seem to make no sense; often the results pop out just the same, and can sometimes make sense of problems that seem ill-posed or impossible. Once you have set up the equations, you can just push the symbols around and the answer will emerge, like a familiar building approached through a fog.

But another lesson, only hinted at so far, is that mathematics has its own way of understanding things, and this is not always the way that humans understand them. Goethe famously said that whatever you say to mathematicians, they immediately translate it into their own language and then it is something different; I think this is exactly what he meant.

In this case it is not too much of a stretch to agree that Mary is three times as old as Sue when they are both zero years old. But in the future I plan to give Katara a problem that requires Mary and Sue to have negative ages—say that Mary is twice as old as Sue today, but in three years Sue will be twice as old—to demonstrate that the answer that pops out may not be a reasonable one, or that the original translation into mathematics can lose essential features of the original problem. The solution that says that !!M_T=-2, S_T=-1 !! is mathematically irreproachable, and if the original problem had been posed as “Find two numbers such that…” it would be perfectly correct. But translated back to the original context of a problem that asks about the ages of two sisters, the solution is unacceptable. This is the point of the joke about the spherical cow.