## Reading the Comics, July 20, 2019: What Are The Chances Edition

The temperature’s cooled. So let me get to the comics that, Saturday, I thought were substantial enough to get specific discussion. It’s possible I was overestimating how much there was to say about some of these. These are the risks I take.

Paige Braddock’s Jane’s World for the 15th sees Jane’s niece talk about enjoying mathematics. I’m glad to see. You sometimes see comic strip characters who are preposterously good at mathematics. Here I mean Jason and Marcus over in Bill Amend’s FoxTrot. But even they don’t often talk about why mathematics is appealing. There is no one answer for all people. I suspect even for a single person the biggest appeal changes over time. That mathematics seems to offer certainty, though, appeals to many. Deductive logic promises truths that can be known independent of any human failings. (The catch is actually doing a full proof, because that takes way too many boring steps. Mathematicians more often do enough of a prove to convince anyone that the full proof could be produced if needed.)

Alexa also enjoys math for there always being a right answer. Given her age there probably always is. There are mathematical questions for which there is no known right answer. Some of these are questions for which we just don’t know the answer, like, “is there an odd perfect number?” Some of these are more like value judgements, though. Is Euclidean geometry or non-Euclidean geometry more correct? The answer depends on what you want to do. There’s no more a right answer to that question than there is a right answer to “what shall I eat for dinner”.

Jane is disturbed by the idea of there being a right answer that she doesn’t know. She would not be happy to learn about “existence proofs”. This is a kind of proof in which the goal is not to find an answer. It’s just to show that there is an answer. This might seem pointless. But there are problems for which there can’t be an answer. If an answer’s been hard to find, it’s worth checking whether there are answers to find.

Art Sansom and Chip Sansom’s The Born Loser for the 16th builds on comparing the probability of winning the lottery to that of being hit by lightning. This comparison’s turned up a couple of times, including in Mister Boffo and The Wandering Melon, when I learned that Peter McCathie had both won the lottery and been hit by lightning.

Pab Sungenis’s New Adventures of Queen Victoria for the 17th is maybe too marginal for full discussion. It’s just reeling off a physics-major joke. The comedy is from it being a pun: Planck’s Constant is a number important in many quantum mechanics problems. It’s named for Max Planck, one of the pioneers of the field. The constant is represented in symbols as either $h$ or as $\hbar$. The constant $\hbar$ is equal to $\frac{h}{2 \pi}$ and might be used even more often. It turns out $\frac{h}{2 \pi}$ appears all over the place in quantum mechanics, so it’s convenient to write it with fewer symbols. $\hbar$ is maybe properly called the reduced Planck’s constant, although in my physics classes I never encountered anyone calling it “reduced”. We just accepted there were these two Planck’s Constants and trusted context to make clear which one we wanted. It was $\hbar$. Planck’s Constant made some news among mensuration fans recently. The International Bureau of Weights and Measures chose to fix the value of this constant. This, through various physics truths, thus fixes the mass of the kilogram in terms of physical constants. This is regarded as better than the old method, where we just had a lump of metal that we used as reference.

Jonathan Lemon’s Rabbits Against Magic for the 17th is another probability joke. If a dropped piece of toast is equally likely to land butter-side-up or butter-side-down, then it’s quite unlikely to have it turn up the same way twenty times in a row. There’s about one chance in 524,288 of doing it in a string of twenty toast-flips. (That is, of twenty butter-side-up or butter-side-down in a row. If all you want is twenty butter-side-up, then there’s one chance in 1,048,576.) It’s understandable that Eight-Ball would take Lettuce to be quite lucky just now.

But there’s problems with the reasoning. First is the supposition that toast is as likely to fall butter-side-up as butter-side-down. I have a dim recollection of a mid-2000s pop physics book explaining why, given how tall a table usually is, a piece of toast is more likely to make half a turn — to land butter-side-down — before falling. Lettuce isn’t shown anywhere near a table, though. She might be dropping toast from a height that makes butter-side-up more likely. And there’s no reason to suppose that luck in toast-dropping connects to any formal game of chance. Or that her luck would continue to hold: even if she can drop the toast consistently twenty times there’s not much reason to think she could do it twenty-five times, or even twenty-one.

And then there’s this, a trivia that’s flawed but striking. Suppose that all seven billion people in the world have, at some point, tossed a coin at least twenty times. Then there should be seven thousand of them who had the coin turn up tails every single one of the first twenty times they’ve tossed a coin. And, yes, not everyone in the world has touched a coin, much less tossed it twenty times. But there could reasonably be quite a few people who grew up just thinking that every time you toss a coin it comes up tails. That doesn’t mean they’re going to have any luck gambling.

Thanks for waiting for me. The weather looks like I should have my next Reading the Comics post at this link, and on time. I’ll let you know if circumstances change.

## Reading the Comics, March 2, 2019: Process Edition

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.

Dan Thompson’s Brevity for the 28th is a nice simple anthropomorphic figures joke. It would’ve been a good match for the strips I talked about Sunday. I’m just normally reluctant to sort these comic strips other than by publication date.

And there were some comic strips I didn’t think worth making paragraphs about. Chris Giarrusso’s G-Man Webcomics for the 25th of February mentioned negative numbers and built a joke on the … negative … connotations of that word. (And inaugurates a tag for that comic strip. This fact will certainly come back to baffle me some later day.) Art Sansom and Chip Sansom’s The Born Loser for the 2nd of March has a bad mathematics report card. Tony Rubino and Gary Markstein’s Daddy’s Home for the 2nd has geometry be the subject parents don’t understand. Bill Amend’s FoxTrot Classics for the 2nd has a mathematics-anxiety dream.

And this closes out my mathematics comics for the week. Come Sunday I should have a fresh post with more comics, and I thank you for considering reading that.

## Reading the Comics, February 16, 2019: The Rest And The Rejects

I’d promised on Sunday the remainder of last week’s mathematically-themed comic strips. I got busy with house chores yesterday and failed to post on time. That’s why this is late. It’s only a couple of comics here, but it does include my list of strips that I didn’t think were on-topic enough. You might like them, or be able to use them, yourself, though.

Niklas Eriksson’s Carpe Diem for the 14th depicts a kid enthusiastic about the abilities of mathematics to uncover truths. Suppressed truths, in this case. Well, it’s not as if mathematics hasn’t been put to the service of conspiracy theories before. Mathematics holds a great promise of truth. Answers calculated correctly are, after all, universally true. They can also offer a hypnotizing precision, with all the digits past the decimal point that anyone could want. But one catch among many is whether your calculations are about anything relevant to what you want to know. Another is whether the calculations were done correctly. It’s easy to make a mistake. If one thinks one has found exciting results it’s hard to imagine even looking for one.

You can’t use shadow analysis to prove the Moon landings fake. But the analysis of shadows can be good mathematics. It can locate things in space and in time. This is a kind of “inverse problem”: given this observable result, what combinations of light and shadow and position would have caused that? And there is a related problem. Johannes Vermeer produced many paintings with awesome, photorealistic detail. One hypothesis for how he achieved this skill is that he used optical tools, including a camera obscura and appropriate curved mirrors. So, is it possible to use the objects shown in perspective in his paintings to project where the original objects had to be, and where the painter had to be, to see them? We can calculate this, at least. I am not well enough versed in art history to say whether we have compelling answers.

Art Sansom and Chip Sansom’s The Born Loser for the 16th is the rare Roman Numerals joke strip that isn’t anthropomorphizing the numerals. Or a play on how the numerals used are also letters. But yeah, there’s not much use for them that isn’t decorative. Hindu-Arabic numerals have great advantages in compactness, and multiplication and division, and handling fractions of a whole number. And handling big numbers. Roman numerals are probably about as good for adding or subtracting small numbers, but that’s not enough of what we do anymore.

And past that there were three comic strips that had some mathematics element. But they were slight ones, and I didn’t feel I could write about them at length. Might like them anyway. Gordon Bess’s Redeye for the 10th of February, and originally run the 24th of September, 1972, has the start of a word problem as example of Pokey’s homework. Mark Litzler’s Joe Vanilla for the 11th has a couple scientist-types standing in front of a board with some mathematics symbols. The symbols don’t quite parse, to me, but they look close to it. Like, the line about $l(\omega) = \int_{-\infty}^{\infty} l(x) e$ is close to what one would write for the Fourier transformation of the function named l. It would need to be more like $L(\omega) = \int_{-\infty}^{\infty} l(x) e^{\imath \omega x} dx$ and even then it wouldn’t be quite done. So I guess Litzler used some actual reference but only copied as much as worked for the composition. (Which is not a problem, of course. The mathematics has no role in this strip beyond its visual appeal, so only the part that looks good needs to be there.) The Fourier transform’s a commonly-used trick; among many things, it lets us replace differential equations (hard, but instructive, and everywhere) with polynomials (comfortable and familiar and well-understood). Finally among the not-quite-comment-worthy is Pascal Wyse and Joe Berger’s Berger And Wyse for the 12th, showing off a Venn Diagram for its joke.

Next Sunday should be a fresh Reading the Comics post, which like all its kind, should appear at this link.

## Reading the Comics, December 8, 2018: Sam and Son Edition

That there were twelve comic strips making my cut as mention-worthy this week should have let me do three essays of four comics each. But the desire to include all the comics from the same day in one essay leaves me one short here. So be it. Three of the four cartoonists featured here have a name of Sansom or Samson, so, that’s an edition title for you. No, Sam and Silo do not appear here.

Art Sansom and Chip Sansom’s Born Loser for the 6th uses arithmetic as a test of deference. Will someone deny a true thing in order to demonstrate loyalty? Arithmetic is full of things that are inarguably true. If we take the ordinary meanings of one, plus, equals, and three, it can’t be that one plus one equals three. Most fields of human endeavor are vulnerable to personal taste, or can get lost in definitions and technicalities. Or the advance of knowledge: my love and I were talking last night how we remembered hearing, as kids, the trivia that panda bears were not really bears, but a kind of raccoon. (Genetic evidence has us now put giant pandas with the bears, and red pandas as part of the same superfamily as raccoons, but barely.) Or even be subject to sarcasm. Arithmetic has a harder time of that. Mathematical ideas do evolve in time, certainly. But basic arithmetic is pretty stable. Logic is also a reliable source of things we can be confident are true. But arithmetic is more familiar than most logical propositions.

Samson’s Dark Side of the Horse for the 8th is the Roman Numerals joke for the week. It’s also a bit of a wordplay joke, although the music wordplay rather tha mathematics. Me, I still haven’t heard a clear reason why ‘MIC’ wouldn’t be a legitimate Roman numeral representation of 1099. I’m not sure whether ‘MIC’ would step on or augment the final joke, though.

Pab Sungenis’s New Adventures of Queen Victoria for the 8th has a comedia dell’arte-based structure for its joke. (The strip does that, now and then.) The comic uses a story problem, with the calculated answer rejected for the nonsense it would be. I suppose it must be possible for someone to eat eighty apples over a long enough time that it’s not distressing, and yet another twenty apples wouldn’t spoil. I wouldn’t try it, though.

This and my other Reading the Comics posts should all be available at this link.