Last week was busy enough in mathematically themed comic strips. Some of these are pretty slight topics. But including them lets me do one of my favorite things, to have an essay that’s all comics from a single day. It’s my blog, I can use it to amuse myself.
Marcus Hamilton and Ron Ferdinand’s Dennis the Menace for the 27th shows the kind of slightness I’m dealing with. ‘Statistic’ has some nasty connotations in this sense. It suggests something dehumanizing has happened. But the word was maybe doomed to that. The word came about in the 18th century, to describe the systematic collection and study of information about whole populations. They started out being the gathering of information about the state.
But gathering information about a whole state implies, first, that the thing one finds interesting about a people are some measured and recorded aspect. Not the whole of their person-hood. Second, it implies that you wish to approximate the diversity of a whole people with some smaller set of numbers. There’s compelling reasons for a state to want to have statistics. They make it more plausible to know what the state can do. They make it plausible to forecast the results of a policy. Ideally, this encourages wisdom in policy-making. If the tools are used well.
Jan Eliot’s Stone Soup Classics for the 27th is the slightest of the comic strips I’m featuring this week. Really it should have been just a mention, but I wanted to have at least three comics shown for today’s essay. Making and counting change is constantly held up as the supreme purpose of teaching arithmetic. This though most any shop has a cash register that will calculate change faster and more accurately than even someone skilled in arithmetic will. I understand the crankiness of people who give the cashier $15.13 for their $12.38 bill, and get the thirteen cents handed back to them before it’s rung up. It’s not evidence that civilization is collapsing. It’s loose change.
Paul Trap’s Thatababy for the 27th continues the strip’s thread of turning geometry figures into jokes. This one is less useful than the comic featured Tuesday, which might help one remember what a scalene triangle or a rhombus looks like. Still might be fun.
One is that, as we’ve thought of counting numbers, there is always “one more”. This doesn’t have to be. We could work with perfectly good number systems that have a largest number. We do, in fact. Every computer programming language has some largest integer that it will deal with. If you need a larger number, you have to do something clever. Your clever idea will let you address some range of bigger numbers, but it too will have a maximum. We’ve set those limits large enough that, usually, they’re not an inconvenience. They’re still there.
But those limits are forced on us by the many failings of matter. What when we get just past Plato’s line’s division, into the reasoning of pure mathematics? There we can set up counting numbers. The standard way to do this is to suppose there is a number “1”. And to suppose that, for any counting number we have, there is a successor, a number one-plus-that. If Joey were to ask why there has to be, all Dennis could do is shrug. This makes an axiom out of there always being one more. If you don’t like it, make some other arithmetic. Anyway we only understand any of this using fallible matter, so good luck.
This progression can be heady, though. The counting numbers are probably the most understandable infinitely large set there is. Thinking about them seriously can induce the sort of dizzy awe that pondering Deep Time or the vastness of space can do. That seems a bit above Dennis’s age level, but some people are stricken with the infinite sooner than others are.
Charles Schulz’s Peanuts Begins rerun for the 2nd has Charlie Brown dismiss arithmetic as impractical. It fits the motif of mathematics as an unworldly subject. There’s the common joke that pure mathematics even dreams of being of no use to anyone. Arithmetic, though, has always been a practical subject. It introduces us to many abstract ideas, particularly group theory. This subject looks at what we can do with systems that work like arithmetic without necessarily having numbers, or anything that works with numbers.
And a couple of comic strips mentioned mathematics, although in too slight a way to discuss. Dana Simpson’s Phoebe and her Unicorn on the 30th of April started a sequence in which doodles on Phoebe’s homework came to life. That it’s mathematics homework was mostly incidental. I’m open to the argument that mathematics encourages doodling in a way that, say, spelling does not. I’d also be open to the argument you aren’t doing geometry if you don’t doodle. Anyway. Dan Thompson’s Brevity for the 2nd of May features Sesame Street’s Count von Count. It’s a bit of wordplay on the use of “numbers” for songs. And, of course, the folkloric tradition of vampires as compulsive counters.
There were a handful of comic strips from last week which I didn’t already discuss. Two of them inspire me to write about how we know how to do things. That makes a good theme.
Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 27th gets into deep territory. How does we could count to a million? Maybe some determined soul has actually done it. But it would take the better part of a month. Things improve some if we allow that anything a computing machine can do, a person could do. This seems reasonable enough. It’s heady to imagine that all the computing done to support, say, a game of Roller Coaster Tycoon could be done by one person working alone with a sheet of paper. Anyway, a computer could show counting up to a million, a billion, a trillion, although then we start asking whether anyone’s checked that it hasn’t skipped some numbers. (Don’t laugh. The New York Times print edition includes an issue number, today at 58,258, at the top of the front page. It’s meant to list the number of published daily editions since the paper started. They mis-counted once, in 1898, and nobody noticed until 1999.)
Anyway, allow that. Nobody doubts that, if we put enough time and effort into it, we could count up to any positive whole number, or as they say in the trade, “counting number”. But … there is some largest number that we could possibly count to, even if we put every possible resource and all the time left in the universe to that counting. So how do we know we “could” count to a number bigger than that? What does it mean to say we “could” if the circumstances of the universe are such that we literally could not?
Counting up to a number seems uncontroversial enough. If I wanted to prove it I’d say something like “if we can count to the whole number with value N, then we can count to the whole number with value N + 1 by … going one higher.” And “We can count to the whole number 1”, proving that by enunciating as clearly as I can. The induction follows. Fine enough. That’s a nice little induction proof.
But … what if we needed to do more work? What if we needed to do a lot of work? There is a corner of logic which considers infinitely long proofs, or infinitely long statements. They’re not part of the usual deductive logic that any mathematician knows and relies on. We’re used to, at least in principle, being able to go through and check every step of a proof. If that becomes impossible is that still a proof? It’s not my field, so I feel comfortable not saying what’s right and what’s wrong. But it is one of those lectures in your Mathematical Logic course that leaves you hanging your jaw open.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th is a joke about algorithms. These are the processes by which we know how to do a thing. Here, Hansel and Gretel are shown using what’s termed a “greedy algorithm” to follow pebbles back home. This kind of thing reflects trying to find an acceptable solution, in this case, finding a path somewhere. What makes it “greedy” is each step. You’re at a pebble. You can see other pebbles nearby. Which one do you go to? Go to some extreme one; in this case, the nearest. It could instead have been the biggest, or the shiniest, the one at the greatest altitude, the one nearest a water source. Doesn’t matter. You choose your summum bonum and, at each step, take the move that maximizes that.
The wicked mother knows something about this sort of algorithm, one that promises merely a solution and not the best solution. And that is that all these solutions can be broken. You can set up a problem that the algorithm can’t solve. Greedy algorithms are particularly vulnerable to this. They’re called “local maximums”. You find the best answer of the ones nearby, but not the best one you possibly could locate.
Why use an algorithm like this, that can be broken so? That’s because we often want to do problems like finding a path through the woods. There are so many possible paths that it’s hard to find one of the acceptable ones. But there are processes that will, typically, find an acceptable answer. Maybe processes that will let us take an acceptable answer and improve it to a good answer. And this is getting into my field.
Actual persons encountering one of these pebble rings would (probably) notice they were caught in a loop. And what they’d do, then, is suspend the greedy rule: instead of going to the nearest pebble they could find, they’d pick something else. Maybe simply the nearest pebble they hadn’t recently visited. Maybe the second-nearest pebble. Maybe they’d give up and strike out in a random direction, trusting they’ll find some more pebbles. This can lead them out of the local maximum they don’t want toward the “global maximum”, the path home, that they do. There’s no reason they can’t get trapped again — this is why the wicked mother made many loops — and no reason they might not get caught in a loop of loops again. Every algorithm like this can get broken by some problem, after all. But sometimes taking the not-the-best steps can lead you to a better solution. That’s the insight at the heart of “Metropolis-Hastings” algorithms, which was my field before I just read comic strips all the time.
Two of the four comic strips I mean to feature here have credits that feel unsatisfying to me. One of them is someone’s pseudonym and, yeah, that’s their business. One is Dennis the Menace, for which I find an in-strip signature that doesn’t match the credentials on Comics Kingdom’s web site, never mind Wikipedia. I’ll go with what’s signed in the comic as probably authoritative. But I don’t like it.
R Ferdinand and S Ketcham’s Dennis the Menace for the 16th is about calculation. One eternally surprising little thing about calculators and computers is that they don’t do anything you can’t do by hand. Or, for that matter, in your head. They do it faster, typically, and more reliably. They can seem magical. But the only difference between what they do and what we do is the quantity with which they do this work. You can take this as humbling or as inspirational, as fits your worldview.
Ham’s Life on Earth for the 16th is a joke about the magical powers we attribute to mathematics. It’s also built on one of our underlying assumptions of the world, that it must be logically consistent. If one has an irrefutable logical argument that something isn’t so, then that thing must not be so. It’s hard to imagine how an illogical world would work. But it is hard not to wonder if there’s some arrogance involved in supposing the world has to square with the rules of logic that we find sensible. And to wonder whether we perceive world consistent with that logic because our expectations frame what we’re able to perceive.
In any case, as we frame logic, an argument’s validity shouldn’t depend on the person making the argument. Or even whether the argument has been made. So it’s hard to see how simply voicing the argument that one doesn’t exist could have that effect. Except that mathematics has got magical connotations, and vice-versa. That’ll be good for building jokes for a while yet.
Mark Anderson’s Andertoons for the 17th is the Mark Anderson’s Andertoons for the week. It’s wordplay, built on the connotation that division is a bad thing. It seems less dire if we think of division as learning how to equally share something that’s been held in common, though. Or if we think of it as learning what to multiply a thing by to get a particular value. Most mathematical operations can be taken to mean many things. Surely division has some constructive and happy interpretations.
Paul Gilligan’s Pooch Cafe for the 17th is a variation of the monkeys-on-keyboards joke. If what you need is a string of nonsense characters then … well, a cat on the keys is at least famous for producing some gibberish. It’s likely not going to be truly random, though. If a cat’s paw has stepped on, say, the ‘O’, there’s a good chance the cat is also stepping on ‘P’ or ‘9’. It also suggests that if the cat starts from the right, they’re more likely to have a character like ‘O’ early in the string of characters and less likely at the end. A completely random string would be as likely to have an ‘O’ at the start as at the end of the string.
And even if a cat on the keyboard did produce good-quality randomness, well. How likely a randomly-generated string of characters is to match a thing depends on the length of the thing. If the meaning of the symbols doesn’t matter, then ‘Penny Lane’ is as good as ‘*2ft,2igFIt’. This is not to say you can just use, say, ‘asdfghjkl’ as your password, at least not for anything that would hurt you if it were cracked. If everyone picked all passwords with no regard for what the symbols meant, these would be. But passwords that seem easy to think get used more often than they should be. It’s not that they’re easier to guess, but that guessing them is more likely to be correct.
Later this week I’ll host this month’s Playful Mathematics Blog Carnival! If you know of any mathematics that teaches or delights or both please share it with me, and we’ll let the world know. Also this week I should finally start my 2018 Mathematics A To Z, explaining words from mathematics one at a time.
It was looking like another slow week for something so early in the (United States) school year. Then Comic Strip Master Commend sent a flood of strips in for Friday and Saturday, so I’m splitting the load. It’s not a heavy one, as back-to-school jokes are on people’s minds. But here goes.
Mark Anderson’s Andertoons for the 3rd is the Mark Anderson installment for this week, so I’m glad to have that. It’s a good old classic cranky-students setup and it reminds me that “unlike fractions” is a thing. I’m not quibbling with the term, especially not after the whole long-division mess a couple weeks back. I just hadn’t thought in a long while about how different denominators do make adding fractions harder.
Jeff Harris’s Shortcuts informational feature for the 3rd I couldn’t remember why I put on the list of mathematically-themed comic strips. The reason’s in there. There’s a Pi Joke. But my interest was more in learning that strawberries are a hybrid created in France from a North American and a Chilean breed. Isn’t that intriguing stuff?