Mathematics is built out of arguments. These are normally logical arguments, sequences of things which we say are true. We know they’re true because either they start from something we assume to be true or because they follow from logical deduction from things we assumed were true. Even calculations are a string of arguments. We start out with an expression we’re interested in, and do things which change the way it looks but which we can prove don’t change whether it’s true.

A fallacy is an argument that isn’t deductively sound. By deductively sound we mean that the premises we start with are true, and the reasoning we follow obeys the rules of deductive logic (omitted for clarity). if we’ve done that, then the conclusion at the end of the reasoning is — and must be — true.

I’m sorry to have fallen silent the last few days; it’s been a bit busy and I’ve been working on follow-ups to a couple of threads. Fortunately Comic Strip Master Command is still around and working to make sure I don’t disappear altogether, and I have a selection of comic strips which at least include a Jumble world puzzle, which should be a fun little diversion.

Tony Rubino and Gary Markstein’s Daddy’s Home (March 23) asks what seems like a confused question to me, “if you believe in infinity, does that mean anything is possible?” As I say, I’m not sure I understand how belief in infinity comes into play, but that might just reflect my background: I’ve been thoroughly convinced that one can describe collections of things that have infinitely many elements — the counting numbers, rectangles, continuous functions — as well as that one can subdivide things — like segments of a number line — infinitely many times — as well as of quantities that are larger than any finite number and so must be infinitely large; so, what’s to not believe in? (I’m aware that there are philosophical and theological questions that get into things termed “potential” and “actual” infinities, but I don’t understand the questions those terms are meant to address.) The phrasing of “anything is possible” seems obviously flawed to me. But if we take it to mean instead “anything not logically inconsistent or physically prohibited is possible” then we seem to have a reasonable question, if that hasn’t just reduced to “anything not impossible is possible”. I guess ultimately I just wonder if the kid is actually trying to understand anything or if he’s just procrastinating.

I had assumed it was a freak event last time that there weren’t any Comics Kingdom strips with mathematical topics to discuss, and which comics I include as pictures here because I don’t know that the links made to them will work for everyone arbitrarily far in the future. Apparently they’re just not in a very mathematical mood this month, though. Such happens; I’m sure they’ll reappear soon enough.

John Zakour and Scott Roberts’ Working Daze (October 22, a “best of” rerun) brings up one of my very many peeves-regarding-pedantry, the notion that you “can’t give more than 100 percent”. It depends on what 100 percent means. The metaphor of “giving 110 percent” is based on the one-would-think-obvious point that there is a standard quantity of effort, which is the 100 percent, and to give 110 percent is to give measurably more than the standard effort. The English language has enough illogical phrases in it; we don’t need to attack ones that are only senseless if you go out to pick a fight with them.

Mark Anderson’s Andertoons (October 23) shows a student attacking a problem with appreciable persistence. As the teacher says, though, there’s no way the student’s attempts at making 2 plus 2 equal 5 is ever not going to be wrong, at least unless we have different ideas about what is meant by 2, plus, equals, and 5. It’s easy to get from this point to some pretty heady territory: since it’s true that two plus two can’t equal five (using the ordinary definitions of these words), then this statement is true not just everywhere in this universe but in all possible universes. This — indeed, all — arithmetic would even be true if there were no universe. But if something can be true regardless of what the universe is like, or even if there is no universe, then how can it tell us anything about the specific universe that actually exists? And yet it seems to do so, quite well.

Tim Lachowski’s Get A Life (October 23) is really an accounting joke, or really more a “taxes they so mean” joke, but I thought it worth mentioning that, really, the majority of the mathematics the world has done have got to have been for the purposes of bookkeeping and accounting. I’m sorry that I’m not better-informed about this so as to better appreciate what is, in some ways, the dark matter of mathematical history.

Keith Tutt and Daniel Saunders’s chipper Lard’s World Peace Tips (October 23) recommends “be a genius” as one of the ways to bring about world peace, and uses mathematics as the comic shorthand for “genius activity”, not to mention sudoku as the comic shorthand for “mathematics”. People have tried to gripe that sudoku isn’t really mathematics; while it’s not arithmetic, though — you could replace the numerals with letters or with arbitrary symbols not to be repeated in one line, column, or subsquare and not change the problem at all — it’s certainly logic.

John Graziano’s Ripley’s Believe It or Not (October 23) besides giving me a spot of dizziness with that attribution line makes the claim that “elephants have been found to be better at some numerical tasks than chimps or even humans”. I can believe that, more or less, though I notice it doesn’t say exactly what tasks elephants are so good (or chimps and humans so bad) at. Counting and addition or subtraction seem most likely, though, because those are processes it seems possible to create tests for. At some stages in human and animal development the animals have a clear edge in speed or accuracy. I don’t remember reading evidence of elephant skills before but I can accept that they surely have some.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (October 24) applies the tools of infinite series — adding up infinitely many of a sequence of terms, often to a finite total — to parenting and the problem of one kid hitting another. This is held up as Real Analysis — – the field in which you learn why Calculus works — and it is, yeah, although this is the part of Real Analysis you can do in high school.

John Zakour and Scott Roberts’s Maria’s Day (October 25) picks up on the Math Wiz Monster in Maria’s closet mentioned last time I did one of these roundups. And it includes an attack on the “Common Core” standards, understandably: it’s unreasonable to today’s generation of parents that mathematics should be taught any differently from how it was taught to them, when they didn’t understand the mathematics they were being taught. Innovation in teaching never has a chance.

Dave Whamond’s Reality Check (October 25) reminds us that just because stock framing can be used to turn a subtraction problem into a word problem doesn’t mean that it can’t jump all the way out of mathematics into another field.

I haven’t included any comics from today — the 26th of October — in my reading yet but really, what are the odds there’s like a half-dozen comics of obvious relevance with nice, juicy topics to discuss?

Since my last roundup of mathematics-themed comic strips there’s been a modest drizzle of new ones, and I’m not sure that I can find any particular themes to them, except that Zach Weinersmith and the artistic collective behind Eric the Circle apparently like my attention. Well, what the heck; that’s easy enough to give.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 29) hopes to be that guy who appears somewhere around the fourth comment of every news article ever that mentions a correlation being found between two quantities. A lot of what’s valuable about science is finding causal links between things, but it’s only in rare and, often, rather artificial circumstances that such links are easy to show. What’s more often necessary is showing that as one quantity changes so does another, which allows one to suspect a link. Then, typically, one would look for a plausible reason they might have anything to do with one another, and look for ways to experiment and prove whether there is or is not.

But just because there is a correlation doesn’t by itself mean that one thing necessarily has anything to do with another. They could be coincidence, for example, or they could be influenced by some other confounding factor. To be worth mention in a decent journal, a correlation is probably going to be strong enough that it’s hard to believe it’s just coincidence, but there could yet be some confounding factor. And even if there is a causal link, in the complicated mess that is reality it can be difficult to discern which way the link flows. This is summarized in deductive logic by saying that correlation does not imply causation, but that uses deductive logic’s definition of “imply”.

In deductive logic to say “this implies that” means it is impossible for “this” to be true and “that” false simultaneously. It is perfectly permissible for both “this” and “that” to be true, and permissible for “this” to be false and “that” false, and — this is the point where Intro to Logic students typically crash — permissible for “this” to be false and “that” true. Colloquially, though, “imply” has a different connotation, something more along the lines of “this” and “that” have to both be false or both be true together. Don’t make that mistake on your logic test.

When a logician says that correlation does not imply causation, she is saying that it is imaginable for the correlation to be true while the causation is false. She is not saying the causation is false; she is just saying that the case is not proved from the fact of a correlation being true. And that’s so; if we just knew two things were correlated we would have to experiment to find whether there is a causal link. But finding a correlation one of the ways to start finding casual links; it’d be obviously daft not to use them as the start of one’s search. Anyway, that guy in about the fourth comment of every news report about a correlation just wants you to know it’s very important he tell you he’s smarter than journalists.

Mikael Wulff and Anders Morgenthaler’s Truth Facts (September 30) — a panel strip that’s often engaging in showing comic charts — gives a guide to what the number of digits you’ve memorized says about you. (For what it’s worth, I peter out at “897932”.) I’m mildly delighted to find that their marker for Isaac Newton is more or less correct: Newton did work out pi to fifteen decimal places, by using his binomial theorem and a calculation of the area within a particular wedge of the circle. (As I make it out Wulff and Morgenthaler put Newton at fourteen decimal points, but they might have read references to Newton working out “fifteen decimal points” as meaning something different to what I do.) Newton’s was not the best calculation of pi in the 1660s when he worked it out — Christoph Grienberger, an Austrian Jesuit astronomer, had calculated 38 decimal places a generation earlier — but I can’t blame Wulff and Morgenthaler for supposing Newton to be a more recognizable name than Grienberger. I imagine if Einstein or Stephen Hawking had done any particularly unique work in calculating the digits of pi they’d have appeared on the chart too.

John Graziano’s Ripley’s Believe It or Not (October 1) — and don’t tell me that attribution doesn’t look weird — shares a story about the followers of the Ancient Greek mathematician, philosopher, and mystic Pythagoras, that they were forbidden to wear wool, eat beans, or pick up things they had dropped. I have heard the beans thing before and I think I’ve heard the wool prohibition before, but I don’t remember hearing about them not being able to pick up things before.

I’m not sure I can believe it, though: Pythagoras was a strange fellow, so far as the historical record is clear. It’s hard to be sure just what is true about him and his followers, though, and what is made up, either out of devoted followers building up the figure they admire or out of critics making fun of a strange fellow with his own little cult. Perhaps it’s so, perhaps it’s not. I would like to see a primary source, and I don’t think any exist.

I’m sorry to admit that I can’t think of a unifying theme for the most recent round of comic strips which mention mathematical topics, other than that this is one of those rare instances of nobody mentioning infinite numbers of typing monkeys. I have to guess Comic Strip Master Command sent around a notice that summer vacation (in the United States) will be ending soon, so cartoonists should start practicing their mathematics jokes.

Tom Toles’s Randolph Itch, 2 a.m. (August 22, rerun) presents what’s surely the lowest-probability outcome of a toss of a fair coin: its landing on the edge. (I remember this as also the gimmick starting a genial episode of The Twilight Zone.) It’s a nice reminder that you do have to consider all the things that might affect an experiment’s outcome before concluding what are likely and unlikely results.

It also inspires, in me, a side question: a single coin, obviously, has a tiny chance of landing on its side. A roll of coins has a tiny chance of not landing on its side. How thick a roll has to be assembled before the chance of landing on the side and the chance of landing on either edge become equal? (Without working it out, my guess is it’s about when the roll of coins is as tall as it is across, but I wouldn’t be surprised if it were some slightly oddball thing like the roll has to be the square root of two times the diameter of the coins.)

Doug Savage’s Savage Chickens (August 22) presents an “advanced Sudoku”, in a puzzle that’s either trivially easy or utterly impossible: there’s so few constraints on the numbers in the presented puzzle that it’s not hard to write in digits that will satisfy the results, but, if there’s one right answer, there’s not nearly enough information to tell which one it is. I do find interesting the problem of satisfiability — giving just enough information to solve the puzzle, without allowing more than one solution to be valid — an interesting one. I imagine there’s a very similar problem at work in composing Ivasallay’s Find The Factors puzzles.

Phil Frank and Joe Troise’s The Elderberries (August 24, rerun) presents a “mind aerobics” puzzle in the classic mathematical form of drawing socks out of a drawer. Talking about pulling socks out of drawers suggests a probability puzzle, but the question actually takes it a different direction, into a different sort of logic, and asks about how many socks need to be taken out in order to be sure you have one of each color. The easiest way to apply this is, I believe, to use what’s termed the “pigeon hole principle”, which is one of those mathematical concepts so clear it’s hard to actually notice it. The principle is just that if you have fewer pigeon holes than you have pigeons, and put every pigeon in a pigeon hole, then there’s got to be at least one pigeon hole with more than one pigeons. (Wolfram’s MathWorld credits the statement to Peter Gustav Lejeune Dirichlet, a 19th century German mathematician with a long record of things named for him in number theory, probability, analysis, and differential equations.)

Dave Whamond’s Reality Check (August 24) pulls out the old little pun about algebra and former romantic partners. You’ve probably seen this joke passed around your friends’ Twitter or Facebook feeds too.

Julie Larson’s The Dinette Set (August 25) presents some terrible people’s definition of calculus, as “useless math with letters instead of numbers”, which I have to gripe about because that seems like a more on-point definition of algebra. I’m actually sympathetic to the complaint that calculus is useless, at least if you don’t go into a field that requires it (although that’s rather a circular definition, isn’t it?), but I don’t hold to the idea that whether something is “useful” should determine whether it’s worth learning. My suspicion is that things you find interesting are worth learning, either because you’ll find uses for them, or just because you’ll be surrounding yourself with things you find interesting.

Shifting from numbers to letters, as are used in algebra and calculus, is a great advantage. It allows you to prove things that are true for many problems at once, rather than just the one you’re interested in at the moment. This generality may be too much work to bother with, at least for some problems, but it’s easy to see what’s attractive in solving a problem once and for all.

Mikael Wulff and Anders Morgenthaler’s WuMo (August 25) uses a couple of motifs none of which I’m sure are precisely mathematical, but that seem close enough for my needs. First there’s the motif of Albert Einstein as just being so spectacularly brilliant that he can form an argument in favor of anything, regardless of whether it’s right or wrong. Surely that derives from Einstein’s general reputation of utter brilliance, perhaps flavored by the point that he was able to show how common-sense intuitive ideas about things like “it’s possible to say whether this event happened before or after that event” go wrong. And then there’s the motif of a sophistic argument being so massive and impressive in its bulk that it’s easier to just give in to it rather than try to understand or refute it.

It’s fair of the strip to present Einstein as beginning with questions about how one perceives the universe, though: his relativity work in many ways depends on questions like “how can you tell whether time has passed?” and “how can you tell whether two things happened at the same time?” These are questions which straddle physics, mathematics, and philosophy, and trying to find answers which are logically coherent and testable produced much of the work that’s given him such lasting fame.

I can tell the school year is getting near the end: it took a full week to get enough mathematics-themed comic strips to put together a useful bundle of them this time. I don’t know what I’m going to do this summer when there’s maybe two comic strips I can talk about per week and I have to go finding my own initiative to write about things.

Jef Mallet’s Frazz (June 6) is a pun strip, yeah, although it’s one that’s more or less legitimate for a word problem. The reason I have to say “more or less” is that it’s not clear to me whether, per Caulfield’s specification, the amount of ore lost across each Great Lake is three percent of the original cargo or three percent of the remaining cargo. But writing a word problem so that there’s only the one correct solution is a skill that needs development no less than solving word problems is, and probably if we imagine Caulfield grading he’d realize there was an ambiguity when a substantial number of of the papers make the opposite assumption to what he’d had in his mind.

Ruben Bolling’s Tom the Dancing Bug (June 6, and I believe it’s a rerun) steps into some of the philosophically heady waters that one gets into when you look seriously at probability, and that get outright silly when you mix omniscience into the mix. The Supreme Planner has worked out what he concludes to be a plan certain of success, but: does that actually mean one will succeed? Even if we assume that the Supreme Planner is able to successfully know and account for every factor which might affect his success — well, for a less criminal plan, consider: one is certain to toss heads at least once, if one flips a fair coin infinitely many times. And yet it would not actually be impossible to flip a fair coin infinitely many times and have it turn up tails every time. That something can have a probability of 1 (or 100%) of happening and nevertheless not happen — or equivalently, that something can have a probability of 0 (0%) of happening and still happen — is exactly analogous to how a concept can be true almost everywhere, that is, it can be true with exceptions that in some sense don’t matter. Ruben Bolling tosses in the troublesome notion of the multiverse, the idea that everything which might conceivably happen does happen “somewhere”, to make these impossible events all the more imminent. I’m impressed Bolling is able to touch on so much, with a taste of how unsettling the implications are, in a dozen panels and stay funny about it.

Bud Grace’s The Piranha Club (June 9) gives us Enos cheating with perfectly appropriate formulas for a mathematics exam. I’m kind of surprised the Pythagorean Theorem would rate cheat-sheet knowledge, actually, as I thought that had reached the popular culture at least as well as Einstein’s E = mc^{2} had, although perhaps it’s reached it much as Einstein’s has, as a charming set of sounds without any particular meaning behind them. I admit my tendency in giving exams, too, has been to allow students to bring their own sheet of notes, or even to have open-book exams, on the grounds that I don’t really care whether they’ve memorized formulas and am more interested in whether they can find and apply the relevant formulas. But that doesn’t make me right; I agree there’s value in being able to identify what the important parts of the course are and to remember them well, and even more value in being able to figure out the area of a triangle or a trapezoid from thinking hard about the subject on your own.

Jason Poland’s Robbie and Bobbie (June 10) is looking for philosophy and mathematics majors, so, here’s hoping it’s found a couple more. The joke here is about the classification of logical arguments. A valid argument is one in which the conclusion does indeed follow from the premises according to the rules of deductive logic. A sound argument is a valid argument in which the premises are also true. The reason these aren’t exactly the same thing is that whether a conclusion follows from the premise depends on the structure of the argument; the content is irrelevant. This means we can do a great deal of work, reasoning out things which follow if we suppose that proposition A being true implies B is false, or that we know B and C cannot both be false, or whatnot. But this means we may fill in, Mad-Libs-style, whatever we like to those propositions and come away with some funny-sounding arguments.

So this is how we can have an argument that’s valid yet not sound. It is valid to say that, if baseball is a form of band organ always found in amusement parks, and if amusement parks are always found in the cubby-hole under my bathroom sink, then, baseball is always found in the cubby-hole under my bathroom sink. But as none of the premises going into that argument are true, the argument’s not sound, which is how you can have anything be “valid but not sound”. Identifying arguments that are valid but not sound is good for a couple questions on your logic exam, so, be ready for that.

John Hambrock’s The Brilliant Mind of Edison Lee (June 11) has the brilliant yet annoying Edison trying to prove his genius by calculating precisely where the baseball will drop. This is a legitimate mathematics/physics problem, of course: one could argue that the modern history of mathematical physics comes from the study of falling balls, albeit more of cannonballs than baseballs. If there’s no air resistance and if gravity is uniform, the problem is easy and you get to show off your knowledge of parabolas. If gravity isn’t uniform, you have to show off your knowledge of ellipses. Either way, you can get into some fine differential equations work, and that work gets all the more impressive if you do have to pay attention to the fact that a ball moving through the air loses some of its speed to the air molecules. That said, it’s amazing that people are able to, in effect, work out approximate solutions to “where is this ball going” in their heads, not to mention to act on it and get to the roughly correct spot, lat least when they’ve had some practice.

The Maths History feed on Twitter reminded me that the second of November is the birthday of George Boole, one of a handful of people who’s managed to get a critically important computer data type named for him (others, of course, include Arthur von Integer and the Lady Annabelle String). Reminded is the wrong word; actually, I didn’t have any idea when his birthday was, other than that it was in the first half of the 19th century. To that extent I was right (it was 1815).

He’s famous, to the extent anyone in mathematics who isn’t Newton or Leibniz is, for his work in logic. “Boolean algebra” is even almost the default term for the kind of reasoning done on variables that may have either of exactly two possible values, which match neatly to the idea of propositions being either true or false. He’d also publicized how neatly the study of logic and the manipulation of algebraic symbols could parallel one another, which is a familiar enough notion that it takes some imagination to realize that it isn’t obviously so.

Boole also did work on linear differential equations, which are important because differential equations are nearly inevitably the way one describes a system in which the current state of the system affects how it is going to change, and linear differential equations are nearly the only kinds of differential equations that can actually be exactly solved. (There are some nonlinear differential equations that can be solved, but more commonly, we’ll find a linear differential equation that’s close enough to the original. Many nonlinear differential equations can also be approximately solved numerically, but that’s also quite difficult.)

His MacTutor History of Mathematics biography notes that Boole (when young) spent five years trying to teach himself differential and integral calculus — money just didn’t allow for him to attend school or hire a tutor — although given that he was, before the age of fourteen, able to teach himself ancient Greek I can certainly understand his supposition that he just needed the right books and some hard work. Apparently, at age fourteen he translated a poem by Meleager — I assume the poet from the first century BCE, though MacTutor doesn’t specify; there was also a Meleager who was briefly king of Macedon in 279 BCE, and another some decades before that who was a general serving Alexander the Great — so well that when it was published a local schoolmaster argued that a 14-year-old could not possibly have done that translation. He’d also, something I didn’t know until today, married Mary Everest, niece of the fellow whose name is on that tall mountain.

John Zakour and Scott Roberts’s Maria’s Day (September 12) tells the basic “not understanding fractions” joke. I suspect that Zakour and Roberts — who’re pretty well-steeped in nerd culture, as their panel strip Working Daze shows — were summoning one of those warmly familiar old jokes. Well, Sydney Harris got away with the same punch line; why not them?

Brett Koth’s Diamond Lil (September 14) also mentions fractions, but as an example of one of those inexplicably complicated mathematics things that’ll haunt you rather than be useful or interesting or even understandable. I choose not to be offended by this insult of my preferred profession and won’t even point out that Koth totally redrew the panel three times over so it’s not a static shot of immobile talking heads.

The slightly dirty secret, though, is that it isn’t. It’s built around logical arguments, certainly, and the more rigorous the argument the better-proven a thing is usually considered to be. But you don’t get results proven with perfectly rigorously airtight deductive reasoning, at least not in the journals and monographs that report interesting new results, because it turns out this requires so much work that it takes forever. What you typically see is enough of an argument to be convincing that anything elided over could be filled in, if required. This is part of why huge results professing major new accomplishments, like a proof of Goldbach’s Conjecture, take time to verify: not only is there a lot that’s there, but suddenly the question of whether the elided steps really are secure has to be filled in.

Most of the big gaps-to-be-filled in basic mathematics were filled in a century ago. Pasch was among the people who found some points in Euclidean geometry where physical intuition about real-world things was assumed into mathematical arguments without it being explicitly stated. This didn’t mean any geometric results were wrong or counterintuitive or anything; just that there were assumptions in the system that Euclid — and everybody else — had made without saying they were making them, which is pretty impressive considering that Euclid thought to mention that he was assuming all right angles were congruent.

One of those discovered spots gets called now Pasch’s Axion, and it gives a good example of the kind of thing which can go centuries being assumed without drawing attention to itself: suppose you have a triangle connecting the points we label A, B, and C. And suppose you have a line which enters the triangle through the leg connecting points A and B, and which doesn’t pass through the point C. Then the line exits the triangle either through the leg between points B and C or through the leg between points C and A.

Obvious? Perhaps, but not more obvious than the axiom that a line segment can be drawn between any two different points, and it’s a special insight to notice these things are assumptions.

I’m sorry to have fallen silent so long; I was away from home and thought I’d be able to put up a couple of short pieces along the way, and turned out to be rather busy doing other things instead. It’s given me at least one nice problem with dramatic photographs to use in a near-future entry, though, so not all is lost (although I’m trying to think of a way to re-do the work in it that doesn’t involve quite so much algebra; I’m afraid of losing my readers and worse of making a hash of the LaTeX involved). Meanwhile, it’s been surprisingly close to a month since the last summary of comic strips with mathematical themes — I imagine the cartoonists are taking a break on Students In Classroom setups what with it being summer vacation across so much of the United States — so let me return to that.

I’m sorry to have fallen quiet for so long; the week has been a busy one and I haven’t been able to write as much as I want. I did want to point everyone to Geoffrey Brent’s elegant solution of my puzzle about loose change, and whether one could have different types of coin without changing the total number of value of those coins. It’s a wonderful proof and one I can’t see a way to improve on, including an argument for the smallest number of coins that allow this ambiguity. I want to give it some attention.

The proof that there is some ambiguous change amount is a neat sort known as an existence proof, which you likely made it through mathematics class without seeing. In an existence proof one doesn’t particularly care whether one finds a solution to the problem, but instead bothers trying to show whether a solution exists. In mathematics classes for people who aren’t becoming majors, the existence of a solution is nearly guaranteed, except when a problem is poorly proofread (I recall accidentally forcing an introduction-to-multivariable-calculus class to step into elliptic integrals, one of the most viciously difficult fields you can step into without requiring grad school backgrounds), or when the instructor wants to see whether people are just plugging numbers into formulas without understanding them. (I mean the formulas, although the numbers can be a bit iffy too.) (Spoiler alert: they have no idea what the formulas are for, but using them seems to make the instructor happy.)

Parker Glynn-Adey here speaks some about the Pigeon Hole Principle, which is one of those little corners of mathematics whose name alone brings a smile to people’s faces. There are a couple of ways of stating the principle. The version I remember from time immemorial is that if one has N pigeons and a smaller number M of pigeon-holes, then if we’ve put all the pigeons somewhere, there must be at least one pigeon-hole with more than one pigeon.

Glynn-Adey starts from a more general way of describing this situation, and goes through a couple of equivalent versions of the idea, before launching into some of the neat little puzzles that follow directly from this idea. Some of them are nicely surprising and I recommend any of the exercises as a fun pastime.

I admit that when I first learned of the Pigeon-Hole Principle it was in a class that also needed the idea of keeping pigeons on purpose explained to it. We’d have thought more naturally of cubby-holes, but hadn’t ever encountered a cubby.

I can’t blame them for wanting to make sure people go through paths they control — and, pay for, at least in advertising clicks — but I can fault them for doing a rotten job of it. They’re just not very good web masters, and end up serving strips — you may have seen them if you’ve gone to the comics page of your local newspaper — that are tiny, which kills plot-heavy features like The Phantom or fine-print heavy features like Slylock Fox Sunday pages, and loaded with referrer-based and cookie-based nonsense that makes it too easy to fail to show a comic altogether or to screw up hopelessly loading up several web browser tabs with different comics in them.

For now that hasn’t happened, at least, but I’m warning that if it does, I might not necessarily read all the King Features strips — their advertising claims they have the best strips in the world, but then, they also run The Katzenjammer Kids which, believe it or not, still exists — and might not be able to comment on them. We’ll see. On to the strips for the middle of September, though:

Richer Ramblings here presents a rather attractive Venn diagram showing the possible combinations of eleven distinct sets. It’s a neat picture and one of the things that people who insist mathematics can be artistic are thinking of when they say it.

Venn diagrams are fairly good ways to visualize data, particularly the ways in which things can be parts of one or more sets simultaneously (or maybe part of no set). I find them most useful, in teaching, in doing probability questions, because so many questions about how probable something is amount to “how many ways can a described outcome happen”, and a nice, clean diagram can show just which outcomes fit which description. (“Coin comes up heads and the first child is a girl; coin comes up heads and the second child is a girl; coin comes up tails and the die roll is a prime number”, etc).

For that, though, I find their use kind of limited: if there are too many things happening (coin, child’s gender, die being rolled, goat behind door number two) the problem becomes one students’ eyes glaze over rather than try solving and I lose the thread of the question too. Worse, if there are too many possibilities, the number of lumpy circles I need to draw becomes smaller than the number of lumpy circles I can draw.

This picture does pretty completely away with the lumpy circles and goes in for much more involved curves. Some of the details are kind of small, but, this covers — at least if it was done correctly and I admit not testing — all the different ways that something can belong or not belong to eleven distinct sets simultaneously.

Thinking about the number of different subsets and shades that are needed — go on, how many are needed to give every distinct combination its own color (which isn’t what’s done here)? — makes me appreciate how choroplethy isn’t my thing.

“If you think Venn diagrams are just a bunch of interlocking circles, think again. Pushing this iconic branch of mathematics to its limits reveals just how varied – and beautiful – these diagrams can be. This gallery showcases some of the wilder possibilities, including the most recent breakthrough in Venn geometry – the first simple, symmetric diagram to encompass a whopping 11 sets.” (New Scientist)

The picture above is the said first simple, symmetric diagram to encompass 11 sets, and yes, it is beautiful. “One of the sets is outlined in white, and the colours correspond to the number of overlapping sets. The team called their creation Newroz, Kurdish for “the new day”. The name also sounds like “new rose” in English, reflecting the diagram’s flowery appearance.“

“Notes On Mathematics” here presents a lovely, visual proof of the Pythagorean Theorem, that bit about the squares of the lengths of two sides of a right triangle adding up to the square of the hypotenuse’s length. I find particularly lovely about this that it can be done without words or explanatory text, which one can’t often get away with.

I think anyone staring at the two pictures would come away convinced of the theorem, but might still ask whether this is actually a logically rigorous proof. One can draw pictures showing all sorts of things which look like they’re so but actually aren’t. I’m thinking here of that puzzle where a grid of 49 squares is cut apart into polygons and rearranged and what do you know but there’s one missing square.

Generally, appeals to “just look at the picture” are a touch suspicious, since that carries along a lot of assumptions about what we see versus what we think, and the eye is pretty much in a continuous state of being fooled about everything we think it sees. I exaggerate but not that much.

But the argument presented here could be written out entirely in prose, without appealing to the physical intuition of what ought to happen if we move triangular blocks around. If you look at that one long enough, or work it out, you might get the same grin of cheerful accomplishment that this pair of pictures provides.

I’m also surprised to find it’s been about a month since my last roundup of mathematics-themed comic strips, but that’s about how it worked out. There was a long stretch of not many syndicated comics touching on any subjects at all and then a rush as cartoonists noticed that summer vacation is on the verge of ending. (I understand in some United States school districts it already has ended, but I grew up in a state where school simply never started before Labor Day, so the idea of school in August feels fundamentally implausible.)

I intend to be back to regular mathematics-based posts soon. I had a fine idea for a couple posts based on Sunday’s closing of the Diaster Transport roller coaster ride at Cedar Point, actually, although I have to technically write them first. (My bride and I made a trip to the park to get a last ride in before its closing, and that lead to inspiration.) But reviews of math-touching comic strips are always good for my readership, if I’m readin the statistics page here right, so let’s see what’s come up since the last recap, going up to the 14th of July.

An interesting parallel’s struck me between nonexistent things and the dead: you can say anything you want about them. At least in United States law it’s not possible to libel the dead, since they can’t be hurt by any loss of reputation. That parallel doesn’t lead me anywhere obviously interesting, but I’ll take it anyway. At least it lets me start this discussion without too closely recapitulating the previous essay. The important thing is that at least in a logic class, if I say, “all the coins in this purse are my property”, as Lewis Carroll suggested, I’m asserting something I say is true without claiming that there are any coins in there. Further, I could also just as easily said “all the coins in this purse are not my property” and made as true a statement, as long as there aren’t any coins there.

The modern interpretation of what we mean by a statement like “all unicorns are one-horned animals” is that we aren’t making the assertion that any unicorns exist. If any did happen to exist, sure, they’d be one-horned animals, if our proposition is true, but we’re reserving judgement about whether they do exist. If we don’t like the way the natural-language interpretation of the proposition leads us, we might be satisfied by saying it’s equivalent to saying, “there are no non-one-horned animals which are unicorns”, and that doesn’t feel quite like it claims unicorns exist. You might not even come away feeling there ought to be non-one-horned animals from that sentence alone.

Midway through “What Lewis Carroll Says Exists That I Don’t” I put forth an example of claiming a property belongs to something which clearly doesn’t exist. The problem — and Carroll was writing this bit, in Symbolic Logic, at a time when it hadn’t reached the current conclusion — is about logical propositions. If you assert it to be true that, “All (something) have (a given property)”, are you making the assertion that the thing exists? Carroll gave the example of “All the sovereigns in that purse are made of gold” and “all the sovereigns in that purse are my property”, leading to the conclusion, “some of my property is made of gold”, and pointing out that if you put that syllogism up to anyone and asked if she thought you were asserting there were sovereigns in that purse, she’d say of course. Carroll has got the way normal people talk in normal conversations on his side here. Put that syllogism before anyone and point out that nowhere is it asserted that there are any coins in the purse and you’ll get a vaguely annoyed response, like when the last chapter of a murder cozy legalistically parses all the alibis until nothing makes sense.

I’m sorry to go another day without following up the essay I meant to follow up, but it’s been a frantically busy week on a frantically busy month and something has to give somewhere. But before I return the Symbolic Logic book to the library — Project Gutenberg has the first part of it, but the second is soundly in copyright, I would expect (its first publication in a recognizable form was in the 1970s) — I wanted to pick some more stuff out of the second part.

I mean to return to the subject brought up Monday, about the properties of things that don’t exist, since as BunnyHugger noted I cheated in talking briefly about what properties they have or don’t have. But I wanted to bring up a nice syllogism whose analysis I’d alluded to a couple weeks back, and which it turns out I’d remembered wrong, in details but not in substance.

I borrowed from the library Symbolic Logic, a collection of an elementary textbook — intended for children, and more fun than usual because of that — on logic by Lewis Carroll, combined with notes and manuscript pages which William Warren Bartley III found toward the second volume in the series. The first part is particularly nice since it’s text that not only was finished in Carroll’s life but went through several editions so he could improve the unclear parts. In case I do get to teaching a new logic course I’ll have to plunder it for examples as well as for this rather nice visual representation Carroll used for sorting out what was implied by a set of propositions regard “All (something) are (something else)” and “Some (something) are (this)” and “No (something) are (whatnot)”. It’s not quite Venn diagrams, although you can see them from there. Oddly, Carroll apparently couldn’t; there’s a rather amusing bit in the second volume where Carroll makes Venn diagrams out to be silly because you can make them terribly complicated.

haggisthesheep here offers a pleasant report about a math-oriented event in the National Museum of Scotland recently. The theme was “A Night In Wonderland”, which is probably almost inevitable, since mathematicians really, really like that (a) people have heard of Lewis Carrol, (b) he can be fairly described as a mathematician, and (c) people aren’t scared of mathematics when it’s presented as Lewis Carroll clowning around.

I recall flipping through one of his logic books and finding a delightful demonstration of how the conclusion may be true while the arguments are not — I believe it was “if a person is late for a train, then he will be running” and “the man is running”, doesn’t prove he’s late for the train, because he might be being chased by a tiger. A good tiger chase livens up any discussion of logic.

On 18th May I was lucky enough to get involved with my first RBS Museum Lates at the National Museum of Scotland. These events happen about 3 times a year and are a chance for the (over 18) public to come back into the museum after hours and to get cosy with the exhibits with a cocktail and live band. It’s also a chance for science (and arts!) communicators like me to run an activity and get some surreptitious education into the evening.

The theme for this month’s Museum Late was “A Night in Wonderland”, so there were lots of top hats, white rabbits and red queens! (See lots of photos of the event on the Museum’s Flickr page.) Knowing that Lewis Carroll (real name Charles Dodgson) was a mathematician and logician as well as nonsense-poem writer, it seemed wrong for there not to be a mathematical component to…

I wanted to talk about drawing graphs that represent something, and to get there have to say what kinds of things I mean to represent. The quick and expected answer is that I mean to represent some kind of equation, such as “y = 3*x – 2” or “x^{2} + y^{2} = 4”, and that probably does come up the most often. We might also be interested in representing an inequality, something like “x^{2} – 2 y^{2} ≤ 1”. On occasion we’re interested just in the region where something is not true, saying something like “y ≠ 3 – x”. (I’ve used nice small counting numbers here not out of any interest in these numbers, or because larger ones or non-whole numbers or even irrational numbers don’t work, but because there is something pleasantly reassuring about seeing a “1” or a “2” in an equation. We strongly believe we know what we mean by “1”.)

Anyway, what we’ve written down is something describing a relationship which we are willing to suppose is true. We might not know what x or y are, and we might not care, but at least for the length of the problem we will suppose that the number represented by y must be equal to three times whatever number is represented by x and minus two. There might be only a single value of x we find interesting; there might be several; there might be infinitely many such values. There’ll be a corresponding number of y’s, at least, so long as the equation is true.

Sometimes we’ll turn the description in terms of an equation into a description in terms of a graph right away. Some of these descriptions are like as those of a line — the “y = 3*x – 2” equation — or a simple shape — “x^{2} + y^{2} = 4” is a circle — in that we can turn them into graphs right away without having to process them, at least not once we’re familiar and comfortable with the idea of graphing. Some of these descriptions are going to be in awkward forms. “x + 2 = – y^{2} / x + 2 y /x” is really just an awkward way to describe a circle (more or less), but that shape is hidden in the writing.