Reading the Comics, May 3, 2016: Lots Of Images Edition


After the heavy pace of March and April I figure to take it easy and settle to about a three-a-week schedule around here. That doesn’t mean that Comic Strip Master Command wants things to be too slow for me. And this time they gave me more comics than usual that have expiring URLs. I don’t think I’ve had this many pictures to include in a long while.

Bill Whitehead’s Free Range for the 28th presents an equation-solving nightmare. From my experience, this would be … a great pain, yes. But it wouldn’t be a career-wrecking mess. Typically a problem that’s hard to solve is hard because you have no idea what to do. Given an expression, you’re allowed to do anything that doesn’t change its truth value. And many approaches might look promising without quite resolving to something useful. The real breakthrough is working out what approach should be used. For an astrophysics problem, there are some classes of key decisions to make. One class is what to include and what to omit in the model. Another class is what to approximate — and how — versus what to treat exactly. Another class is what sorts of substitutions and transformations turn the original expression into one that reveals what you want. Those are the hard parts, and those are unlikely to have been forgotten. Applying those may be tedious, and I don’t doubt it would be anguishing to have the finished work wiped out. But it wouldn’t set one back years either. It would just hurt.

Christopher Grady’s Lunar Babboon for the 29th I classify as the “anthropomorphic numerals” joke for this essay. Boy, have we all been there.

'Numbers are boring!' complains the audience. 'Not so. They contain high drama and narrative. Here's an expense account that was turned in to me last week. Can you create a *story* based on these numbers?' 'Once upon a time, a guy was fired for malfeasance ... ' 'If you skip right to the big finish, sure.'
Bill Holbrook’s On The Fastrack for the 29th of April, 2016. Spoiler: there aren’t any numbers in the second panel.

Bill Holbrook’s On The Fastrack for the 29th continues the storyline about Fi giving her STEM talk. She is right, as I see it, in attributing drama and narrative to numbers. This is most easily seen in the sorts of finance and accounting mathematics which the character does. And the inevitable answer to “numbers are boring” (or “mathematics is boring”) is surely to show how they are about people. Even abstract mathematics is about things (some) people find interesting, and that must be about the people too.

'Look, Grandpa! I got 100% on my math test! Do you know what that means? It means that out of ten questions, I got at least half of them correct!' 'It must be that new, new, new math.' 'So many friendly numbers!'
Rick Detorie’s One Big Happy for the 3rd of May, 2016. Ever notice how many shirt pockets Grandpa has? I’m not saying it’s unrealistic, just that it’s more than the average.

Rick Detorie’s One Big Happy for the 16th is a confused-mathematics joke. Grandpa tosses off a New Math joke that’s reasonably age-appropriate too, which is always nice to see in a comic strip. I don’t know how seriously to take Ruthie’s assertion that a 100% means she only got at least half of the questions correct. It could be a cartoonist grumbling about how kids these days never learn anything, the same way ever past generation of cartoonists had complained. But Ruthie is also the sort of perpetually-confused, perpetually-confusing character who would get the implications of a 100% on a test wrong. Or would state them weirdly, since yes, a 100% does imply getting at least half the test’s questions right.

Border Collies, as we know, are highly intelligent. 'Yup, the math confirms it --- we can't get by without people.'
Niklas Eriksson’s Carpe Diem for the 3rd of May, 2016. I’m a little unnerved there seems to be a multiplication x at the end of the square root vinculum on the third line there.

Niklas Eriksson’s Carpe Diem for the 3rd uses the traditional board full of mathematical symbols as signifier of intelligence. There’s some interesting mixes of symbols here. The c2, for example, isn’t wrong for mathematics. But it does evoke Einstein and physics. There’s the curious mix of the symbol π and the approximation 3.14. But then I’m not sure how we would get from any of this to a proposition like “whether we can survive without people”.

'What comes after eleven?' 'I can't do it. I don't have enough fingers to count on!' Tiger hands him a baseball glove. 'Use this.'
Bud Blake’s Tiger for the 3rd of May, 2016. How did Punkinhead get up to eleven?

Bud Blake’s Tiger for the 3rd is a cute little kids-learning-to-count thing. I suppose it doesn’t really need to be here. But Punkinhead looks so cute wearing his tie dangling down onto the floor, the way kids wear their ties these days.

Tony Murphy’s It’s All About You for the 3rd name-drops algebra. I think what the author really wanted here was arithmetic, if the goal is to figure out the right time based on four clocks. They seem to be trying to do a simple arithmetic mean of the time on the four clocks, which is fair if we make some assumptions about how clocks drift away from the correct time. Mostly those assumptions are that the clocks all started right and are equally likely to drift backwards or forwards, and do that drifting at the same rate. If some clocks are more reliable than others, then, their claimed time should get more weight than the others. And something like that must be at work here. The mean of 7:56, 8:02, 8:07, and 8:13, uncorrected, is 8:04 and thirty seconds. That’s not close enough to 8:03 “and five-eighths” unless someone’s been calculating wrong, or supposing that 8:02 is more probably right than 8:13 is.

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A Leap Day 2016 Mathematics A To Z: Wlog


Wait for it.

Wlog.

I’d like to say a good word for boredom. It needs the good words. The emotional state has an appalling reputation. We think it’s the sad state someone’s in when they can’t find anything interesting. It’s not. It’s the state in which we are so desperate for engagement that anything is interesting enough.

And that isn’t a bad thing! Finding something interesting enough is a precursor to noticing something curious. And curiosity is a precursor to discovery. And discovery is a precursor to seeing a fuller richness of the world.

Think of being stuck in a waiting room, deprived of reading materials or a phone to play with or much of anything to do. But there is a clock. Your classic analog-face clock. Its long minute hand sweeps out the full 360 degrees of the circle once every hour, 24 times a day. Its short hour hand sweeps out that same arc every twelve hours, only twice a day. Why is the big unit of time marked with the short hand? Good question, I don’t know. Probably, ultimately, because it changes so much less than the minute hand that it doesn’t need the attention of length drawn to it.

But let our waiting mathematician get a little more bored, and think more about the clock. The hour and minute hand must sometimes point in the same direction. They do at 12:00 by the clock, for example. And they will at … a little bit past 1:00, and a little more past 2:00, and a good while after 9:00, and so on. How many times during the day will they point the same direction?

Well, one easy way to do this is to work out how long it takes the hands, once they’ve met, to meet up again. Presumably we don’t want to wait the whole hour-and-some-more-time for it. But how long is that? Well, we know the hands start out pointing the same direction at 12:00. The first time after that will be after 1:00. At exactly 1:00 the hour hand is 30 degrees clockwise of the minute hand. The minute hand will need five minutes to catch up to that. In those five minutes the hour hand will have moved another 2.5 degrees clockwise. The minute hand needs about four-tenths of a minute to catch up to that. In that time the hour hand moves — OK, we’re starting to see why Zeno was not an idiot. He never was.

But we have this roughly worked out. It’s about one hour, five and a half minutes between one time the hands meet and the next. In the course of twelve hours there’ll be time for them to meet up … oh, of course, eleven times. Over the course of the day they’ll meet up 22 times and we can get into a fight over whether midnight counts as part of today, tomorrow, or both days, or neither. (The answer: pretend the day starts at 12:01.)

Hold on, though. How do we know that the time between the hands meeting up at 12:00 and the one at about 1:05 is the same as the time between the hands meeting up near 1:05 and the next one, sometime a little after 2:10? Or between that one and the one at a little past 3:15? What grounds do we have for saying this one interval is a fair representation of them all?

We can argue that it should be fairly enough. Imagine that all the markings were washed off the clock. It’s just two hands sweeping around in circles, one relatively fast, one relatively slow, forever. Give the clockface a spin. When the hands come together again rotate the clock so those two hands are vertical, the “12:00” position. Is this actually 12:00? … Well, we’ve got a one-in-eleven chance it is. It might be a little past 1:05; it might be that time something past 6:30. The movement of the clock hands gives no hint what time it really is.

And that is why we’re justified taking this one interval as representative of them all. The rate at which the hands move, relative to each other, doesn’t depend on what the clock face behind it says. The rate is, if the clock isn’t broken, always the same. So we can use information about one special case that happens to be easy to work out to handle all the cases.

That’s the mathematics term for this essay. We can study the one specific case without loss of generality, or as it’s inevitably abbreviated, wlog. This is the trick of studying something possibly complicated, possibly abstract, by looking for a representative case. That representative case may tell us everything we need to know, at least about this particular problem. Generality means what you might figure from the ordinary English meaning of it: it means this answer holds in general, as opposed to in this specific instance.

Some thought has to go in to choosing the representative case. We have to pick something that doesn’t, somehow, miss out on a class of problems we would want to solve. We mustn’t lose the generality. And it’s an easy mistake to make, especially as a mathematics student first venturing into more abstract waters. I remember coming up against that often when trying to prove properties of infinitely long series. It’s so hard to reason something about a bunch of numbers whose identities I have no idea about; why can’t I just use the sequence, oh, 1/1, 1/2, 1/3, 1/4, et cetera and let that be good enough? Maybe 1/1, 1/4, 1/9, 1/16, et cetera for a second test, just in case? It’s because it takes time to learn how to safely handle infinities.

It’s still worth doing. Few of us are good at manipulating things in the abstract. We have to spend more mental energy imagining the thing rather than asking the questions we want of it. Reducing that abstraction — even if it’s just a little bit, changing, say, from “an infinitely-differentiable function” to “a polynomial of high enough degree” — can rescue us. We can try out things we’re confident we understand, and derive from it things we don’t know.

I can’t say that a bored person observing a clock would deduce all this. Parts of it, certainly. Maybe all, if she thought long enough. I believe it’s worth noticing and thinking of these kinds of things. And it’s why I believe it’s fine to be bored sometimes.