I trust nobody’s too upset that I postponed the big Reading the Comics posts of this week a day. There’s enough comics from last week to split them into two essays. Please enjoy.
Scott Shaw! and Stan Sakai’s Popeye’s Cartoon Club for the 22nd is one of a yearlong series of Sunday strips, each by different cartoonists, celebrating the 90th year of Popeye’s existence as a character. And, I’m a Popeye fan from all the way back when Popeye was still a part of the pop culture. So that’s why I’m bringing such focus to a strip that, really, just mentions the existence of algebra teachers and that they might present a fearsome appearance to people.
Lincoln Pierce’s Big Nate for the 22nd has Nate seeking an omen for his mathematics test. This too seems marginal. But I can bring it back to mathematics. One of the fascinating things about having data is finding correlations between things. Sometimes we’ll find two things that seem to go together, including apparently disparate things like basketball success and test-taking scores. This can be an avenue for further research. One of these things might cause the other, or at least encourage it. Or the link may be spurious, both things caused by the same common factor. (Superstition can be one of those things: doing a thing ritually, in a competitive event, can help you perform better, even if you don’t believe in superstitions. Psychology is weird.)
But there are dangers too. Nate shows off here the danger of selecting the data set to give the result one wants. Even people with honest intentions can fall prey to this. Any real data set will have some points that just do not make sense, and look like a fluke or some error in data-gathering. Often the obvious nonsense can be safely disregarded, but you do need to think carefully to see that you are disregarding it for safe reasons. The other danger is that while two things do correlate, it’s all coincidence. Have enough pieces of data and sometimes they will seem to match up.
Norm Feuti’s Gil rerun for the 22nd has Gil practicing multiplication. It’s really about the difficulties of any kind of educational reform, especially in arithmetic. Gil’s mother is horrified by the appearance of this long multiplication. She dubs it both inefficient and harder than the way she learned. She doesn’t say the way she learned, but I’m guessing it’s the way that I learned too, which would have these problems done in three rows beneath the horizontal equals sign, with a bunch of little carry notes dotting above.
Gil’s Mother is horrified for bad reasons. Gil is doing exactly the same work that she was doing. The components of it are just written out differently. The only part of this that’s less “efficient” is that it fills out a little more paper. To me, who has no shortage of paper, this efficiency doens’t seem worth pursuing. I also like this way of writing things out, as it separates cleanly the partial products from the summations done with them. It also means that the carries from, say, multiplying the top number by the first digit of the lower can’t get in the way of carries from multiplying by the second digits. This seems likely to make it easier to avoid arithmetic errors, or to detect errors once suspected. I’d like to think that Gil’s Mom, having this pointed out, would drop her suspicions of this different way of writing things down. But people get very attached to the way they learned things, and will give that up only reluctantly. I include myself in this; there’s things I do for little better reason than inertia.
People will get hung up on the number of “steps” involved in a mathematical process. They shouldn’t. Whether, say, “37 x 2” is done in one step, two steps, or three steps is a matter of how you’re keeping the books. Even if we agree on how much computation is one step, we’re left with value judgements. Like, is it better to do many small steps, or few big steps? My own inclination is towards reliability. I’d rather take more steps than strictly necessary, if they can all be done more surely. If you want speed, my experience is, it’s better off aiming for reliability and consistency. Speed will follow from experience.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd builds on mathematical physics. Lagrangian mechanics offers great, powerful tools for solving physics problems. It also offers a philosophically challenging interpretation of physics problems. Look at the space made up of all the possible configurations of the system. Take one point to represent the way the system starts. Take another point to represent the way the system ends. Grant that the system gets from that starting point to that ending point. How does it do that? What is the path in this configuration space that goes in-between this start and this end?
We can find the path by using the Lagrangian. Particularly, integrate the Lagrangian over every possible curve that connects the starting point and the ending point. This is every possible way to match start and end. The path that the system actually follows will be an extremum. The actual path will be one that minimizes (or maximizes) this integral, compared to all the other paths nearby that it might follow. Yes, that’s bizarre. How would the particle even know about those other paths?
This seems bad enough. But we can ignore the problem in classical mechanics. The extremum turns out to always match the path that we’d get from taking derivatives of the Lagrangian. Those derivatives look like calculating forces and stuff, like normal.
Then in quantum mechanics the problem reappears and we can’t just ignore it. In the quantum mechanics view no particle follows “a” “path”. It instead is found more likely in some configurations than in others. The most likely configurations correspond to extreme values of this integral. But we can’t just pretend that only the best-possible path “exists”.
Thus the strip’s point. We can represent mechanics quite well. We do this by pretending there are designated starting and ending conditions. And pretending that the system selects the best of every imaginable alternative. The incautious pop physics writer, eager to find exciting stuff about quantum mechanics, will describe this as a particle “exploring” or “considering” all its options before “selecting” one. This is true in the same way that we can say a weight “wants” to roll down the hill, or two magnets “try” to match north and south poles together. We should not mistake it for thinking that electrons go out planning their days, though. Newtonian mechanics gets us used to the idea that if we knew the positions and momentums and forces between everything in the universe perfectly well, we could forecast the future and retrodict the past perfectly. Lagrangian mechanics seems to invite us to imagine a world where everything “perceives” its future and all its possible options. It would be amazing if this did not capture our imaginations.
Bil Keane and Jeff Keane’s Family Circus for the 24th has young Billy amazed by the prospect of algebra, of doing mathematics with both numbers and letters. I’m assuming Billy’s awestruck by the idea of letters representing numbers. Geometry also uses quite a few letters, mostly as labels for the parts of shapes. But that seems like a less fascinating use of letters.
The second half of last week’s comics I hope to post here on Wednesday. Stick around and we’ll see how close I come to making it. Thank you.