Reading the Comics, July 22, 2017: Counter-mudgeon Edition


I’m not sure there is an overarching theme to the past week’s gifts from Comic Strip Master Command. If there is, it’s that I feel like some strips are making cranky points and I want to argue against their cases. I’m not sure what the opposite of a curmudgeon is. So I shall dub myself, pending a better idea, a counter-mudgeon. This won’t last, as it’s not really a good name, but there must be a better one somewhere. We’ll see it, now that I’ve said I don’t know what it is.

Rabbits at a chalkboard. 'The result is not at all what we expected, Von Thump. According to our calculations, parallel universes may exist, and we may also be able to link them with our own by wormholes that, in strictly mathematical terms, end up in a black top hat.'
Niklas Eriksson’s Carpe Diem for the 17th of July, 2017. First, if anyone isn’t thinking of that Pixar short then I’m not sure we can really understand each other. Second, ‘von Thump’ is a fine name for a bunny scientist and if it wasn’t ever used in the rich lore of Usenet group alt.devilbunnies I shall be disappointed. Third, Eriksson made an understandable but unfortunate mistake in composing this panel. While both rabbits are wearing glasses, they’re facing away from the viewer. It’s always correct to draw animals wearing eyeglasses, or to photograph them so. But we should get to see them in full eyeglass pelage. You’d think they would teach that in Cartoonist School or something.

Niklas Eriksson’s Carpe Diem for the 17th features the blackboard full of equations as icon for serious, deep mathematical work. It also features rabbits, although probably not for their role in shaping mathematical thinking. Rabbits and their breeding were used in the simple toy model that gave us Fibonacci numbers, famously. And the population of Arctic hares gives those of us who’ve reached differential equations a great problem to do. The ecosystem in which Arctic hares live can be modelled very simply, as hares and a generic predator. We can model how the populations of both grow with simple equations that nevertheless give us surprises. In a rich, diverse ecosystem we see a lot of population stability: one year where an animal is a little more fecund than usual doesn’t matter much. In the sparse ecosystem of the Arctic, and the one we’re building worldwide, small changes can have matter enormously. We can even produce deterministic chaos, in which if we knew exactly how many hares and predators there were, and exactly how many of them would be born and exactly how many would die, we could predict future populations. But the tiny difference between our attainable estimate and the reality, even if it’s as small as one hare too many or too few in our model, makes our predictions worthless. It’s thrilling stuff.

Vic Lee’s Pardon My Planet for the 17th reads, to me, as a word problem joke. The talk about how much change Marian should get back from Blake could be any kind of minor hassle in the real world where one friend covers the cost of something for another but expects to be repaid. But counting how many more nickels one person has than another? That’s of interest to kids and to story-problem authors. Who else worries about that count?

Fortune teller: 'All of your money problems will soon be solved, including how many more nickels Beth has than Jonathan, and how much change Marian should get back from Blake.'
Vic Lee’s Pardon My Planet for the 17th of July, 2017. I am surprised she had no questions about how many dimes Jonathan must have, although perhaps that will follow obviously from knowing the Beth nickel situation.

Jef Mallet’s Frazz for the 17th straddles that triple point joining mathematics, philosophy, and economics. It seems sensible, in an age that embraces the idea that everything can be measured, to try to quantify happiness. And it seems sensible, in age that embraces the idea that we can model and extrapolate and act on reasonable projections, to try to see what might improve our happiness. This is so even if it’s as simple as identifying what we should or shouldn’t be happy about. Caulfield is circling around the discovery of utilitarianism. It’s a philosophy that (for my money) is better-suited to problems like how ought the city arrange its bus lines than matters too integral to life. But it, too, can bring comfort.

Corey Pandolph’s Barkeater Lake rerun for the 20th features some mischievous arithmetic. I’m amused. It turns out that people do have enough of a number sense that very few people would let “17 plus 79 is 4,178” pass without comment. People might not be able to say exactly what it is, on a glance. If you answered that 17 plus 79 was 95, or 102, most people would need to stop and think about whether either was right. But they’re likely to know without thinking that it can’t be, say, 56 or 206. This, I understand, is so even for people who aren’t good at arithmetic. There is something amazing that we can do this sort of arithmetic so well, considering that there’s little obvious in the natural world that would need the human animal to add 17 and 79. There are things about how animals understand numbers which we don’t know yet.

Alex Hallatt’s Human Cull for the 21st seems almost a direct response to the Barkeater Lake rerun. Somehow “making change” is treated as the highest calling of mathematics. I suppose it has a fair claim to the title of mathematics most often done. Still, I can’t get behind Hallatt’s crankiness here, and not just because Human Cull is one of the most needlessly curmudgeonly strips I regularly read. For one, store clerks don’t need to do mathematics. The cash registers do all the mathematics that clerks might need to do, and do it very well. The machines are cheap, fast, and reliable. Not using them is an affectation. I’ll grant it gives some charm to antiques shops and boutiques where they write your receipt out by hand, but that’s for atmosphere, not reliability. And it is useful the clerk having a rough idea what the change should be. But that’s just to avoid the risk of mistakes getting through. No matter how mathematically skilled the clerk is, there’ll sometimes be a price entered wrong, or the customer’s money counted wrong, or a one-dollar bill put in the five-dollar bill’s tray, or a clerk picking up two nickels when three would have been more appropriate. We should have empathy for the people doing this work.

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Some End-Of-August Mathematics Reading


I’ve found a good way to procrastinate on the next essay in the Why Stuff Can Orbit series. (I’m considering explaining all of differential calculus, or as much as anyone really needs, to save myself a little work later on.) In the meanwhile, though, here’s some interesting reading that’s come to my attention the last few weeks and that you might procrastinate your own projects with. (Remember Benchley’s Principle!)

First is Jeremy Kun’s essay Habits of highly mathematical people. I think it’s right in describing some of the worldview mathematics training instills, or that encourage people to become mathematicians. It does seem to me, though, that most everything Kun describes is also true of philosophers. I’m less certain, but I strongly suspect, that it’s also true of lawyers. These concentrations all tend to encourage thinking about we mean by things, and to test those definitions by thought experiments. If we suppose this to be true, then what implications would it have? What would we have to conclude is also true? Does it include anything that would be absurd to say? And is are the results useful enough we can accept a bit of apparent absurdity?

New York magazine had an essay: Jesse Singal’s How Researchers Discovered the Basketball “Hot Hand”. The “Hot Hand” phenomenon is one every sports enthusiast, and most casual fans, know: sometimes someone is just playing really, really well. The problem has always been figuring out whether it exists. Do anything that isn’t a sure bet long enough and there will be streaks. There’ll be a stretch where it always happens; there’ll be a stretch where it never does. That’s how randomness works.

But it’s hard to show that. The messiness of the real world interferes. A chance of making a basketball shot is not some fixed thing over the course of a career, or over a season, or even over a game. Sometimes players do seem to be hot. Certainly anyone who plays anything competitively experiences a feeling of being in the zone, during which stuff seems to just keep going right. It’s hard to disbelieve something that you witness, even experience.

So the essay describes some of the challenges of this: coming up with a definition of a “hot hand”, for one. Coming up with a way to test whether a player has a hot hand. Seeing whether they’re observed in the historical record. Singal’s essay writes about some of the history of studying hot hands. There is a lot of probability, and of psychology, and of experimental design in it.

And then there’s this intriguing question Analysis Fact Of The Day linked to: did Gaston Julia ever see a computer-generated image of a Julia Set? There are many Julia Sets; they and their relative, the Mandelbrot Set, became trendy in the fractals boom of the 1980s. If you knew a mathematics major back then, there was at least one on her wall. It typically looks like a craggly, lightning-rimmed cloud. Its shapes are not easy to imagine. It’s almost designed for the computer to render. Gaston Julia died in March of 1978. Could he have seen a depiction?

It’s not clear. The linked discussion digs up early computer renderings. It also brings up an example of a late-19th-century hand-drawn depiction of a Julia-like set, and compares it to a modern digital rendition of the thing. Numerical simulation saves a lot of tedious work; but it’s always breathtaking to see how much can be done by reason.

Why Stuff Can Orbit: Why It’s Waiting


I can’t imagine people are going to be surprised to hear this. But I have to put the “Why Stuff Can Orbit” series. It’s about central forces and what circumstances make it possible for something to have a stable orbit. I mean to get back to it. It’s just that the Theorem Thursday posts take up a lot of thinking on my part. They end up running quite long and detailed. I figure to get back to it once I’ve exhausted the Theorem Thursday topics I have in mind, which should be shortly into August.

It happens I’d run across a WordPress blog that contained the whole of the stable-central-orbits argument, in terse but legitimate terms. I wanted to link to that now but the site’s been deleted for reasons I won’t presume to guess. I have guesses. Sorry.

But for some other interesting reading, here’s a bit about Immanuel Kant:

I have long understood, and passed on, that Immanuel Kant had the insight that the laws of physics tell us things about the geometry of space and vice-versa. I haven’t had the chance yet to read Francisco Caruso and Roberto Moreira Xavier’s On Kant’s First Insight into the Problem of Space Dimensionality and its Physical Foundations. But the abstract promises “a conclusion that does not match the usually accepted interpretation of Kant’s reasoning”. I would imagine this to be an interesting introduction to the question, then, and to what might be controversial about Kant and the number of dimensions space should have. Also we need to use the word “tridimensionality” more.

Reading the Comics, November 27, 2015: 30,000 Edition


By rights, if this installment has any title it should be “confident ignorance”. That state appears in many of the strips I want to talk about. But according to WordPress, my little mathematics blog here reached its 30,000th page view at long last. This is thanks largely to spillover from The Onion AV Club discovering my humor blog and its talk about the late comic strip Apartment 3-G. But a reader is a reader. And I want to celebrate reaching that big, round number. As I write this I’m at 30,162 page views, because there were a lot of AV Club-related readers.

Bob Weber Jr’s Slylock Fox for the 23rd of November maybe shouldn’t really be here. It’s just a puzzle game that depends on the reader remembering that two rectangles put against the other can be a rectangle again. It also requires deciding whether the frame of the artwork counts as one of the rectangles. The commenters at Comics Kingdom seem unsure whether to count squares as rectangles too. I don’t see any shapes that look more clearly like squares to me. But it’s late in the month and I haven’t had anything with visual appeal in these Reading the Comics installments in a while. Later we can wonder if “counting rectangles in a painting” is the most reasonable way a secret agent has to pass on a number. It reminds me of many, many puzzle mysteries Isaac Asimov wrote that were all about complicated ways secret agents could pass one bit of information on.

'The painting (of interlocking rectangles) is really a secret message left by an informant. It reveals the address of a house where stolen artwork is being stashed. The title, Riverside, is the street name, and the total amount of rectangles is the house number. Where will Slylock Fox find the stolen artwork?
Bob Weber Jr’s Slylock Fox for the 23rd of November, 2015. I suppose the artist is lucky they weren’t hiding out at number 38, or she wouldn’t have been able to make such a compellingly symmetric diagram.

Ryan North’s Dinosaur Comics for the 23rd of November is a rerun from goodness knows when it first ran on Quantz.com. It features T Rex thinking about the Turing Test. The test, named for Alan Turing, says that while we may not know what exactly makes up an artificial intelligence, we will know it when we see it. That is the sort of confident ignorance that earned Socrates a living. (I joke. Actually, Socrates was a stonecutter. Who knew, besides the entire philosophy department?) But the idea seems hard to dispute. If we can converse with an entity in such a way that we can’t tell it isn’t human, then, what grounds do we have for saying it isn’t human?

T Rex has an idea that the philosophy department had long ago, of course. That’s to simply “be ready for any possible opening with a reasonable conclusion”. He calls this a matter of brute force. That is, sometimes, a reasonable way to solve problems. It’s got a long and honorable history of use in mathematics. The name suggests some disapproval; it sounds like the way you get a new washing machine through a too-small set of doors. But sometimes the easiest way to find an answer is to just try all the possible outcomes until you find the ones that work, or show that nothing can. If I want to know whether 319 is a prime number, I can try reasoning my way through it. Or I can divide it by all the prime numbers from 2 up to 17. (The square root of 319 is a bit under 18.) Or I could look it up in a table someone already made of the prime numbers less than 400. I know what’s easier, if I have a table already.

The problem with brute force — well, one problem — is that it can be longwinded. We have to break the problem down into each possible different case. Even if each case is easily disposed of, the number of different cases can grow far too fast to be manageable. The amount of working time required, and the amount of storage required, can easily become too much to deal with. Mathematicians, and computer scientists, have a couple approaches for this. One is getting bigger computers with more memory. We might consider this the brute force method to solving the limits of brute force methods.

Or we might try to reduce the number of possible cases, so that less work is needed. Perhaps we can find a line of reasoning that covers many cases. Working out specific cases, as brute force requires, can often give us a hint to what a general proof would look like. Or we can at least get a bunch of cases dealt with, even if we can’t get them all done.

Jim Unger’s Herman rerun for the 23rd of November turns confident ignorance into a running theme for this essay’s comic strips.

Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 24th of November has a similar confient ignorance. This time it’s of the orders of magnitude that separate billions from trillions. I wanted to try passing off some line about how there can be contexts where it doesn’t much matter whether a billion or a trillion is at stake. But I can’t think of one that makes sense for the Man At The Business Company Office setting.

Reza Farazmand’s Poorly Drawn Lines for the 25th of November is built on the same confusion about the orders of magnitude that Bottomliners is. In this case it’s ants that aren’t sure about how big millions are, so their confusion seems more natural.

The ants are also engaged in a fun sort of recreational mathematics: can you estimate something from little information? You’ve done that right, typically, if you get the size of the number about right. That it should be millions rather than thousands or hundreds of millions; that there should be something like ten rather than ten thousand. These kinds of problems are often called Fermi Problems, after Enrico Fermi. This is the same person the Fermi Paradox is named after, but that’s a different problem. The Fermi Paradox asks if there are extraterrestrial aliens, why we don’t see evidence of them. A Fermi Problem is simpler. Its the iconic example is, “how many professional piano tuners are there in New York?” It’s easy to look up how big is the population of New York. It’s possible to estimate how many pianos there should be for a population that size. Then you can guess how often a piano needs tuning, and therefore, how many full-time piano tuners would be supported by that much piano-tuning demand. And there’s probably not many more professional piano tuners than there’s demand for. (Wikipedia uses Chicago as the example city for this, and asserts the population of Chicago to be nine million people. I will suppose this to be the Chicago metropolitan region, but that still seems high. Wikipedia says that is the rough population of the Chicago metropolitan area, but it’s got a vested interest in saying so.)

Mark Anderson’s Andertoons finally appears on the 27th. Here we combine the rational division of labor with resisting mathematics problems.

Reading the Comics, October 1, 2015: Big Questions Edition


I’m cutting the collection of mathematically-themed comic strips at the transition between months. The set I have through the 1st of October is long enough already. That’s mostly because the first couple strips suggested some big topics at least somewhat mathematically-based came up. Those are fun to reason about, but take time to introduce. So let’s jump into them.

Lincoln Pierce’s Big Nate: First Class for the 27th of September was originally published the 22nd of September, 1991. Nate and Francis trade off possession of the basketball, and a strikingly high number of successful shots in a row considering their age, in the infinitesimally sliced last second of the game. There’s a rather good Zeno’s-paradox-type-question to be made out of this. Suppose the game started with one second to go and Nate ahead by one point, since it is his strip. At one-half second to go, Francis makes a basket and takes a one point lead. At one-quarter second to go, Nate makes a basket and takes a one point lead. At one-eighth of a second to go, Francis repeats the basket; at one-sixteenth of a second, Nate does. And so on. Suppose they always make their shots, and suppose that they are able to make shots without requiring any more than half the remaining time available. Who wins, and why?

Tim Rickard’s Brewster Rockit for the 27th of September is built on the question of whether the universe might be just a computer simulation, and if so, how we might tell. Being a computer simulation is one of those things that would seem to explain why mathematics tells us so much about the universe. One can make a probabilistic argument about this. Suppose there is one universe, and there are some number of simulations of the universe. Call that number N. If we don’t know whether we’re in the real or the simulated universe, then it would seem we have an estimated probability of being in the real universe of one divided by N plus 1. The chance of being in the real universe starts out none too great and gets dismally small pretty fast.

But this does put us in philosophical difficulties. If we are in something that is a complete, logically consistent universe that cannot be escaped, how is it not “the real” universe? And if “the real” universe is accessible from within “the simulation” then how can they be separate? The question is hard to answer and it’s far outside my realm of competence anyway.

Mark Leiknes’s Cow and Boy Classics for the 27th of September originally ran the 15th of September, 2008. And it talks about the ideas of zero-point energy and a false vacuum. This is about something that seems core to cosmology: how much energy is there in a vacuum? That is, if there’s nothing in a space, how much energy is in it? Quantum mechanics tells us it isn’t zero, in part because matter and antimatter flutter into and out of existence all the time. And there’s gravity, which is hard to explain quite perfectly. Mathematical models of quantum mechanics, and gravity, make various predictions about how much the energy of the vacuum should be. Right now, the models don’t give us really good answers.

Some suggest that there might be more energy in the vacuum than we could ever use, and that if there were some way to draw it off — well, there’d never be a limit to anything ever again. I think this an overly optimistic projection. The opposite side of this suggests that if it is possible to draw energy out of the vacuum, that means it must be possible to shift empty space from its current state to a lower-energy state, much the way you can get energy out of a pile of rocks by making the rocks fall. But the lower-energy vacuum might have different physics in ways that make it very hard for us to live, or for us to exist. I think this an overly pessimistic projection. But I am not an expert in the fields, which include cosmology, quantum mechanics, and certain rather difficult tinkerings with the infinitely many.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 28th of September is a joke in the form of true, but useless, word problem answers. Well, putting down a lower bound on what the answer is can help. If you knew what three times twelve was, you could get to four times twelve reliably, and that’s a help. But if you’re lost for three times twelve then you’re just stalling for time and the teacher knows it.

Paul Gilligan’s Pooch Cafe for the 28th of September uses the monkeys-on-keyboards concept. It’s shifted here to cats on a keyboard, but the principle is the same. Give a random process enough time and you can expect it to produce anything you want. It’s a matter of how long you can wait, though. And all the complications of how to make something that’s random. Cats won’t do it.

Mel Henze’s Gentle Creatures for the 29th of September is a rerun. I’m not sure when it was first printed. But it does use “ability to do mathematics” as a shorthand for “is intelligent at all”. That’s flattering to put in front of a mathematician, but I don’t think that’s really fair.

Paul Trap’s Thatababy for the 30th of September is a protest about using mathematics in real life. I’m surprised Thatababy’s Dad had an algebra teacher proclaiming differential equations would be used. Usually teachers assert that whatever they’re teaching will be useful, which is how we provide motivation.

How To Build Infinite Numbers


I had missed it, as mentioned in the above tweet. The link is to a page on the Form And Formalism blog, reprinting a translation of one of Georg Cantor’s papers in which he founded the modern understanding of sets, of infinite sets, and of infinitely large numbers. Although it gets into pretty heady topics, it doesn’t actually require a mathematical background, at least as I look at it; it just requires a willingness to follow long chains of reasoning, which I admit is much harder than algebra.

Cantor — whom I’d talked a bit about in a recent Reading The Comics post — was deeply concerned and intrigued by infinity. His paper enters into that curious space where mathematics, philosophy, and even theology blend together, since it’s difficult to talk about the infinite without people thinking of God. I admit the philosophical side of the discussion is difficult for me to follow, and the theological side harder yet, but a philosopher or theologian would probably have symmetric complaints.

The translation is provided as scans of a typewritten document, so you can see what it was like trying to include mathematical symbols in non-typeset text in the days before LaTeX (which is great at it, but requires annoying amounts of setup) or HTML (which is mediocre at it, but requires less setup) or Word (I don’t use Word) were available. Somehow, folks managed to live through times like that, but it wasn’t pretty.

Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition


So, the mathematics comics ran away from me a little bit, and I didn’t have the chance to write up a proper post on Thursday or Friday. So I’m writing what I probably would have got to on Friday had time allowed, and there’ll be another in this sequence sooner than usual. I hope you’ll understand.

The title for this entry is basically thanks to Zach Weinersmith, because his comics over the past week gave me reasons to talk about Georg Cantor and Bernard Riemann. These were two of the many extremely sharp, extremely perceptive German mathematicians of the 19th Century who put solid, rigorously logical foundations under the work of centuries of mathematics, only to discover that this implied new and very difficult questions about mathematics. Some of them are good material for jokes.

Eric and Bill Teitelbaum’s Bottomliners panel (February 14) builds a joke around everything in some set of medical tests coming back negative, as well as the bank account. “Negative”, the word, has connotations that are … well, negative, which may inspire the question why is it a medical test coming back “negative” corresponds with what is usually good news, nothing being wrong? As best I can make out the terminology derives from statistics. The diagnosis of any condition amounts to measuring some property (or properties), and working out whether it’s plausible that the measurements could reflect the body’s normal processes, or whether they’re such that there just has to be some special cause. A “negative” result amounts to saying that we are not forced to suppose something is causing these measurements; that is, we don’t have a strong reason to think something is wrong. And so in this context a “negative” result is the one we ordinarily hope for.

Continue reading “Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition”

A bit more about Thomas Hobbes


You might remember a post from last April, Thomas Hobbes and the Doing of Important Mathematics, timed to the renowned philosopher’s birthday. I talked about him because a good bit of his intellectual life was spent trying to achieve mathematical greatness, which he never did.

Recently I’ve had the chance to read Douglas M Jesseph’s Squaring The Circle: The War Between Hobbes And Wallis, about Hobbes’s attempts to re-build mathematics on an intellectual foundation he found more satisfying, and the conflict this put him in with mainstream mathematicians, particularly John Wallis (algebra and calculus pioneer, and popularizer of the ∞ symbol). The situation of Hobbes’s mathematical ambitions is more complicated than I realized, although the one thing history teaches us is that the situation is always more complicated than we realized, and I wanted to at least make my writings about Hobbes a bit less incomplete. Jesseph’s book can’t be fairly reduced to a blog post, of course, and I’d recommend it to people who want to really understand what the fuss was all about. It’s a very good idea to have some background in philosophy and in 17th century English history going in, though, because it turns out a lot of the struggle — and particularly the bitterness with which Hobbes and Wallis fought, for decades — ties into the religious and political struggles of England of the 1600s.

Hobbes’s project, I better understand now, was not merely the squaring of the circle or the solving of other ancient geometric problems like the doubling of the cube or the trisecting of an arbitrary angle, although he did claim to have various proofs or approximate proofs of them. He seems to have been interested in building a geometry on more materialist grounds, more directly as models of the real world, instead of the pure abstractions that held sway then (and, for that matter, now). This is not by itself a ridiculous thing to do: we are almost always better off for having multiple independent ways to construct something, because the differences in those ways teaches us not just about the thing, but about the methods we use to discover things. And purely abstract constructions have problems also: for example, if a line can be decomposed into nothing but an enormous number of points, and absolutely none of those points has any length, then how can the line have length? You can answer that, but it’s going to require a pretty long running start.

Trying to re-build the logical foundations of mathematics is an enormously difficult thing to do, and it’s not surprising that someone might fail to do so perfectly. Whole schools of mathematicians might be needed just to achieve mixed success. And Hobbes wasn’t able to attract whole schools of mathematicians, in good part because of who he was.

Hobbes achieved immortality as an important philosopher with the publication of Leviathan. What I had not appreciated and Jesseph made clear was that in the context of England of the 1650s, Hobbes’s views on the natures of God, King, Society, Law, and Authority managed to offend — in the “I do not know how I can continue to speak with a person who holds views like that” — pretty much everybody in England who had any strong opinion about anything in politics, philosophy, or religion. I do not know for a fact that Hobbes then went around kicking the pet dogs of any English folk who didn’t have strong opinions about politics, philosophy, or religion, but I can’t rule it out. At least part of the relentlessness and bitterness with which Wallis (and his supporters) attacked Hobbes, and with which Hobbes (and his supporters) attacked back, can be viewed as a spinoff of the great struggle between the Crown and Parliament that produced the Civil War, the Commonwealth, and the Restoration, and in that context it’s easier to understand why all parties carried on, often quibbling about extremely minor points, well past the point that their friends were advising them that the quibbling was making themselves look bad. Hobbes was a difficult person to side with, even when he was right, and a lot of his mathematics just wasn’t right. Some of it I’m not sure ever could be made right, however many ingenious people you had working to avoid flaws.

An amusing little point that Jesseph quotes is a bit in which Hobbes, making an argument about the rights that authority has, asserts that if the King decreed that Euclid’s Fifth Postulate should be taught as false, then false it would be in the kingdom. The Fifth Postulate, also known as the Parallel Postulate, is one of the axioms on which classical Greek geometry was built and it was always the piece that people didn’t like. The other postulates are all nice, simple, uncontroversial, common-sense things like “all right angles are equal”, the kinds of things so obvious they just have to be axioms. The Fifth Postulate is this complicated-sounding thing about how, if a line is crossed by two non-parallel lines, you can determine on which side of the first line the non-parallel lines will meet.

It wouldn’t be really understood or accepted for another two centuries, but, you can suppose the Fifth Postulate to be false. This gives you things named “non-Euclidean geometries”, and the modern understanding of the universe’s geometry is non-Euclidean. In picking out an example of something a King might decree and the people would have to follow regardless of what was really true, Hobbes picked out an example of something that could be decreed false, and that people could follow profitably.

That’s not mere ironical luck, probably. A streak of mathematicians spent a long time trying to prove the Fifth Postulate was unnecessary, at least, by showing it followed from the remaining and non-controversial postulates, or at least that it could be replaced with something that felt more axiomatic. Of course, in principle you can use any set of axioms you like to work, but some sets produce more interesting results than others. I don’t know of any interesting geometry which results from supposing “not all right angles are equal”; supposing that the Fifth Postule is untrue gives us general relativity, which is quite nice to have.

Again I have to warn that Jesseph’s book is not always easy reading. I had to struggle particularly over some of the philosophical points being made, because I’ve got only a lay understanding of the history of philosophy, and I was able to call on my love (a professional philosopher) for help at points. I imagine someone well-versed in philosophy but inexperienced with mathematics would have a similar problem (although — don’t let the secret out — you’re allowed to just skim over the diagrams and proofs and go on to the explanatory text afterwards). But for people who want to understand the scope and meaning of the fighting better, or who just want to read long excerpts of the wonderful academic insulting that was current in the era, I do recommend it. Check your local college or university library.

Reading the Comics, September 8, 2014: What Is The Problem Edition


Must be the start of school or something. In today’s roundup of mathematically-themed comics there are a couple of strips that I think touch on the question of defining just what the problem is: what are you trying to measure, what are you trying to calculate, what are the rules of this sort of calculation? That’s a lot of what’s really interesting about mathematics, which is how I’m able to say something about a rerun Archie comic. It’s not easy work but that’s why I get that big math-blogger paycheck.

Edison Lee works out the shape of the universe, and as ever in this sort of thing, he forgot to carry a number.
I’d have thought the universe to be at least three-dimensional.

John Hambrock’s The Brilliant Mind of Edison Lee (September 2) talks about the shape of the universe. Measuring the world, or the universe, is certainly one of the older influences on mathematical thought. From a handful of observations and some careful reasoning, for example, one can understand how large the Earth is, and how far away the Moon and the Sun must be, without going past the kinds of reasoning or calculations that a middle school student would probably be able to follow.

There is something deeper to consider about the shape of space, though: the geometry of the universe affects what things can happen in them, and can even be seen in the kinds of physics that happen. A famous, and astounding, result by the mathematical physicist Emmy Noether shows that symmetries in space correspond to conservation laws. That the universe is, apparently, rotationally symmetric — everything would look the same if the whole universe were picked up and rotated (say) 80 degrees along one axis — means that there is such a thing as the conservation of angular momentum. That the universe is time-symmetric — the universe would look the same if it had got started five hours later (please pretend that’s a statement that can have any coherent meaning) — means that energy is conserved. And so on. It may seem, superficially, like a cosmologist is engaged in some almost ancient-Greek-style abstract reasoning to wonder what shapes the universe could have and what it does, but (putting aside that it gets hard to divide mathematics, physics, and philosophy in this kind of field) we can imagine observable, testable consequences of the answer.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 5) tells a joke starting with “two perfectly rational perfectly informed individuals walk into a bar”, along the way to a joke about economists. The idea of “perfectly rational perfectly informed” people is part of the mathematical modeling that’s become a popular strain of economic thought in recent decades. It’s a model, and like many models, is properly speaking wrong, but it allows one to describe interesting behavior — in this case, how people will make decisions — without complications you either can’t handle or aren’t interested in. The joke goes on to the idea that one can assign costs and benefits to continuing in the joke. The idea that one can quantify preferences and pleasures and happiness I think of as being made concrete by Jeremy Bentham and the utilitarian philosophers, although trying to find ways to measure things has been a streak in Western thought for close to a thousand years now, and rather fruitfully so. But I wouldn’t have much to do with protagonists who can’t stay around through the whole joke either.

Marc Anderson’s Andertoons (September 6) was probably composed in the spirit of joking, but it does hit something that I understand baffles kids learning it every year: that subtracting a negative number does the same thing as adding a positive number. To be fair to kids who need a couple months to feel quite confident in what they’re doing, mathematicians needed a couple generations to get the hang of it too. We have now a pretty sound set of rules for how to work with negative numbers, that’s nice and logically tested and very successful at representing things we want to know, but there seems to be a strong intuition that says “subtracting a negative three” and “adding a positive three” might just be different somehow, and we won’t really know negative numbers until that sense of something being awry is resolved.

Andertoons pops up again the next day (September 7) with a completely different drawing of a chalkboard and this time a scientist and a rabbit standing in front of it. The rabbit’s shown to be able to do more than multiply and, indeed, the mathematics is correct. Cosines and sines have a rather famous link to exponentiation and to imaginary- and complex-valued numbers, and it can be useful to change an ordinary cosine or sine into this exponentiation of a complex-valued number. Why? Mostly, because exponentiation tends to be pretty nice, analytically: you can multiply and divide terms pretty easily, you can take derivatives and integrals almost effortlessly, and then if you need a cosine or a sine you can get that out at the end again. It’s a good trick to know how to do.

Jeff Harris’s Shortcuts children’s activity panel (September 9) is a page of stuff about “Geometry”, and it’s got some nice facts (some mathematical, some historical), and a fair bunch of puzzles about the field.

Morrie Turner’s Wee Pals (September 7, perhaps a rerun; Turner died several months ago, though I don’t know how far ahead of publication he was working) features a word problem in terms of jellybeans that underlines the danger of unwarranted assumptions in this sort of problem-phrasing.

Moose has trouble working out 15 percent of $8.95; Jughead explains why.
How far back is this rerun from if Moose got lunch for two for $8.95?

Craig Boldman and Henry Scarpelli’s Archie (September 8, rerun) goes back to one of arithmetic’s traditional comic strip applications, that of working out the tip. Poor Moose is driving himself crazy trying to work out 15 percent of $8.95, probably from a quiz-inspired fear that if he doesn’t get it correct to the penny he’s completely wrong. Being able to do a calculation precisely is useful, certainly, but he’s forgetting that in tis real-world application he gets some flexibility in what has to be calculated. He’d save some effort if he realized the tip for $8.95 is probably close enough to the tip for $9.00 that he could afford the difference, most obviously, and (if his budget allows) that he could just as well work out one-sixth the bill instead of fifteen percent, and give up that workload in exchange for sixteen cents.

Mark Parisi’s Off The Mark (September 8) is another entry into the world of anthropomorphized numbers, so you can probably imagine just what π has to say here.

In the Overlap between Logic, Fun, and Information


Since I do need to make up for my former ignorance of John Venn’s diagrams and how to use them, let me join in what looks early on like a massive Internet swarm of mentions of Venn. The Daily Nous, a philosophy-news blog, was my first hint that anything interesting was going on (as my love is a philosopher and is much more in tune with the profession than I am with mathematics), and I appreciate the way they describe Venn’s interesting properties. (Also, for me at least, that page recommends I read Dungeons and Dragons and Derrida, itself pointing to an installment of philosophy-based web comic Existentialist Comics, so you get a sense of how things go over there.)

https://twitter.com/saladinahmed/status/496148485092433920

And then a friend retweeted the above cartoon (available as T-shirt or hoodie), which does indeed parse as a Venn diagram if you take the left circle as representing “things with flat tails playing guitar-like instruments” and the right circle as representing “things with duck bills playing keyboard-like instruments”. Remember — my love is “very picky” about Venn diagram jokes — the intersection in a Venn diagram is not a blend of the things in the two contributing circles, but is rather, properly, something which belongs to both the groups of things.

https://twitter.com/mathshistory/status/496224786109198337

The 4th of is also William Rowan Hamilton’s birthday. He’s known for the discovery of quaternions, which are kind of to complex-valued numbers what complex-valued numbers are to the reals, but they’re harder to make a fun Google Doodle about. Quaternions are a pretty good way of representing rotations in a three-dimensional space, but that just looks like rotating stuff on the computer screen.

Daily Nous

John Venn, an English philosopher who spent much of his career at Cambridge, died in 1923, but if he were alive today he would totally be dead, as it is his 180th birthday. Venn was named after the Venn diagram, owing to the fact that as a child he was terrible at math but good at drawing circles, and so was not held back in 5th grade. In celebration of this philosopher’s birthday Google has put up a fun, interactive doodle — just for today. Check it out.

Note: all comments on this post must be in Venn Diagram form.

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Where Does A Plane Touch A Sphere?


Recently my dear love, the professional philosopher, got to thinking about a plane that just touches a sphere, and wondered: where does the plane just touch the sphere? I, the mathematician, knew just what to call that: it’s the “point of tangency”, or if you want a phrasing that’s a little less Law French, the “tangent point”. The tangent to a curve is a flat surface, of one lower dimension than the space has — on the two-dimensional plane the tangent’s a line; in three-dimensional space the tangent’s a plane; in four-dimensional space the tangent’s a pain to quite visualize perfectly — and, ordinarily, it touches the original curve at just the one point, locally anyway.

But, and this is a good philosophical objection, is a “point” really anywhere? A single point has no breadth, no width, it occupies no volume. Mathematically we’d say it has measure zero. If you had a glass filled to the brim and dropped a point into it, it wouldn’t overflow. If you tried to point at the tangent point, you’d miss it. If you tried to highlight the spot with a magic marker, you couldn’t draw a mark centered on that point; the best you could do is draw out a swath that, presumably, has the point, somewhere within it, somewhere.

This feels somehow like one of Zeno’s Paradoxes, although it’s not one of the paradoxes to have come down to us, at least so far as I understand them. Those are all about the problem that there seem to be conclusions, contrary to intuition, that result from supposing that space (and time) can be infinitely divided; but, there are at least as great problems from supposing that they can’t. I’m a bit surprised by that, since it’s so easy to visualize a sphere and a plane — it almost leaps into the mind as soon as you have a fruit and a table — but perhaps we just don’t happen to have records of the Ancients discussing it.

We can work out a good deal of information about the tangent point, and staying on firm ground all the way to the end. For example: imagine the sphere sliced into a big and a small half by a plane. Imagine moving the plane in the direction of the smaller slice; this produces a smaller slice yet. Keep repeating this ad infinitum and you’d have a smaller slice, volume approaching zero, and a plane that’s approaching tangency to the sphere. But then there is that slice that’s so close to the edge of the sphere that the sphere isn’t cut at all, and there is something curious about that point.

I Know Nothing Of John Venn’s Diagram Work


My Dearly Beloved, the professional philosopher, mentioned after reading the last comics review that one thing to protest in the Too Much Coffee Man strip — showing Venn diagram cartoons and Things That Are Funny as disjoint sets — was that the Venn diagram was drawn wrong. In philosophy, you see, they’re taught to draw a Venn diagram for two sets as two slightly overlapping circles, and then to black out any parts of the diagram which haven’t got any elements. If there are three sets, you draw three overlapping circles of equal size and again black out the parts that are empty.

I granted that this certainly better form, and indispensable if you don’t know anything about what sets, intersections, and unions have any elements in them, but that it was pretty much the default in mathematics to draw the loops that represent sets as not touching if you know the intersection of the sets is empty. That did get me to wondering what the proper way of doing things was, though, and I looked it up. And, indeed, according to MathWorld, I have been doing it wrong for a very long time. Per MathWorld (which is as good a general reference for this sort of thing as I can figure), to draw a Venn diagram reflecting data for N sets, the rules are:

  1. Draw N simple, closed curves on the plane, so that the curves partition the plane into 2N connected regions.
  2. Have each subset of the N different sets correspond to one and only one region formed by the intersection of the curves.

Partitioning the plane is pretty much exactly what you might imagine from the ordinary English meaning of the world: you divide the plane into parts that are in this group or that group or some other group, with every point in the plane in exactly one of these partitions (or on the border between them). And drawing circles which never touch mean that I (and Shannon Wheeler, and many people who draw Venn diagram cartoons) are not doing that first thing right: two circles that have no overlap the way the cartoon shows partition the plane into three pieces, not four.

I can make excuses for my sloppiness. For one, I learned about Venn diagrams in the far distant past and never went back to check I was using them right. For another, the thing I most often do with Venn diagrams is work out probability problems. One approach for figuring out the probability of something happen is to identify the set of all possible outcomes of an experiment — for a much-used example, all the possible numbers that can come up if you throw three fair dice simultaneously — and identify how many of those outcomes are in the set of whatever you’re interested in — say, rolling a nine total, or rolling a prime number, or for something complicated, “rolling a prime number or a nine”. When you’ve done this, if every possible outcome is equally likely, the probability of the outcome you’re interested in is the number of outcomes that satisfy what you’re looking for divided by the number of outcomes possible.

If you get to working that way, then, you might end up writing a list of all the possible outcomes and drawing a big bubble around the outcomes that give you nine, and around the outcomes that give you a prime number, and those aren’t going to touch for the reasons you’d expect. I’m not sure that this approach is properly considered a Venn diagram anymore, though, although I’d introduced it in statistics classes as such and seen it called that in the textbook. There might not be a better name for it, but it is doing violence to the Venn diagram concept and I’ll try to be more careful in future.

The Mathworld page, by the way, provides a couple examples of Venn diagrams for more than three propositions, down towards the bottom of the page. The last one that I can imagine being of any actual use is the starfish shape used to work out five propositions at once. That shows off 32 possible combinations of sets and I can barely imagine finding that useful as a way to visualize the relations between things. There are also representations based on seven sets, which have 128 different combinations, and for 11 propositions, a mind-boggling 2,048 possible combinations. By that point the diagram is no use for visualizing relationships of sets and is simply mathematics as artwork.

Something else I had no idea bout is that if you draw the three-circle Venn diagram, and set it so that the intersection of any two circles is at the center of the third, then the innermost intersection is a Reuleaux triangle, one of those oddball shapes that rolls as smoothly as a circle without actually being a circle. (MathWorld has an animated gif showing it rolling so.) This figure, it turns out, is also the base for something called the Henry Watt square drill bit. It can be used as a spinning drill bit to produce a (nearly) square hole, which is again pretty amazing as I make these things out, and which my father will be delighted to know I finally understand or have heard of.

In any case, the philosophy department did better teaching Venn diagrams properly than whatever math teacher I picked them up from did, or at least, my spouse retained the knowledge better than I did.

The Big Zero


I want to try re-proving the little point from last time, that the chance of picking one specific number from the range of zero to one is actually zero. This might not seem like a big point but it can be done using a mechanism that turns out to be about three-quarters of all the proofs in real analysis, which is probably the most spirit-crushing of courses you take as a mathematics undergraduate, and I like that it can be shown in a way that you can understand without knowing anything more sophisticated than the idea of “less than or equal to”.

So here’s my proposition: that the probability of selecting the number 1/2 from the range of numbers running from zero to one, is zero. This is assuming that you’re equally likely to pick any number. The technique I mean to use, and it’s an almost ubiquitous one, is to show that the probability has to be no smaller than zero, and no greater than zero, and therefore it has to be exactly zero. Very many proofs are done like this, showing that the thing you want can’t be smaller than some number, and can’t be greater than that same number, and we thus prove that it has to be that number.

Showing that the probability of picking exactly 1/2 can’t be smaller than zero is easy: the probability of anything is a number greater than or equal to zero, and less than or equal to one. (A few bright people have tried working out ways to treat probabilities that can be negative numbers, or that can be greater than one, but nobody’s come up with a problem that these approaches solve in a compelling way, and it’s really hard to figure out what a negative probability would mean in the observable world, so we leave the whole idea for someone after us to work out.) That was easy enough.

Continue reading “The Big Zero”

Split Lines


My spouse, the professional philosopher, was sharing some of the engagingly wrong student responses. I hope it hasn’t shocked you to learn your instructors do this, but, if you got something wrong in an amusing way, and it was easy to find someone to commiserate with, yes, they said something.

The particular point this time was about Plato’s Analogy of the Divided Line, part of a Socratic dialogue that tries to classify the different kinds of knowledge. I’m not informed enough to describe fairly the point Plato was getting at, but the mathematics is plain enough. It starts with a line segment that gets divided into two unequal parts; each of the two parts is then divided into parts of the same proportion. Why this has to be I’m not sure (my understanding is it’s not clear exactly why Plato thought it important they be unequal parts), although it has got the interesting side effect of making exactly two of the four line segments of equal length.

Continue reading “Split Lines”

Kenneth Appel and Colored Maps


Word’s come through mathematics circles about the death of Kenneth Ira Appel, who along with Wolgang Haken did one of those things every mathematically-inclined person really wishes to do: solve one of the long-running unsolved problems of mathematics. Even better, he solved one of those accessible problems. There are a lot of great unsolved problems that take a couple paragraphs just to set up for the lay audience (who then will wonder what use the problem is, as if that were the measure of interesting); Appel and Haken’s was the Four Color Theorem, which people can understand once they’ve used crayons and coloring books (even if they wonder whether it’s useful for anyone besides Hammond).

It was, by everything I’ve read, a controversial proof at the time, although by the time I was an undergraduate the controversy had faded the way controversial stuff doesn’t seem that exciting decades on. The proximate controversy was that much of the proof was worked out by computer, which is the sort of thing that naturally alarms people whose jobs are to hand-carve proofs using coffee and scraps of lumber. The worry about that seems to have faded as more people get to use computers and find they’re not putting the proof-carvers out of work to any great extent, and as proof-checking software gets up to the task of doing what we would hope.

Still, the proof, right as it probably is, probably offers philosophers of mathematics a great example for figuring out just what is meant by a “proof”. The word implies that a proof is an argument which convinces a person of some proposition. But the Four Color Theorem proof is … well, according to Appel and Haken, 50 pages of text and diagrams, with 85 pages containing an additional 2,500 diagrams, and 400 microfiche pages with additional diagrams of verifications of claims made in the main text. I’ll never read all that, much less understand all that; it’s probably fair to say very few people ever will.

So I couldn’t, honestly, say it was proved to me. But that’s hardly the standard for saying whether something is proved. If it were, then every calculus class would produce the discovery that just about none of calculus has been proved, and that this whole “infinite series” thing sounds like it’s all doubletalk made up on the spot. And yet, we could imagine — at least, I could imagine — a day when none of the people who wrote the proof, or verified it for publication, or have verified it since then, are still alive. At that point, would the theorem still be proved?

(Well, yes: the original proof has been improved a bit, although it’s still a monstrously large one. And Neil Robertson, Daniel P Sanders, Paul Seymour, and Robin Thomas published a proof, similar in spirit but rather smaller, and have been distributing the tools needed to check their work; I can’t imagine there being nobody alive who hasn’t done, or at least has the ability to do, the checking work.)

I’m treading into the philosophy of mathematics, and I realize my naivete about questions like what constitutes a proof are painful to anyone who really studies the field. I apologize for inflicting that pain.

Lose the Change


My Dearly Beloved, a professional philosopher, had explained to me once a fine point in the theory of just what it means to know something. I wouldn’t presume to try explaining that point (though I think I have it), but a core part of it is the thought experiment of remembering having put some change — we used a dime and a nickel — in your pocket, and finding later that you did have that same amount of money although not necessarily the same change — say, that you had three nickels instead.

That spun off a cute little side question that I’ll give to any needy recreational mathematician. It’s easy to imagine this problem where you remember having 15 cents in your pocket, and you do indeed have them, but you have a different number of coins from what you remember: three nickels instead of a dime and a nickel. Or you could remember having two coins, and indeed have two, but you have a different amount from what you remember: two dimes instead of a dime and a nickel.

Is it possible to remember correctly both the total number of coins you have, and the total value of those coins, while being mistaken about the number of each type? That is, could you remember rightly you have six coins and how much they add up to, but have the count of pennies, nickels, dimes, and quarters wrong? (In the United States there are also 50-cent and dollar coins minted, but they’re novelties and can be pretty much ignored. It’s all 1, 5, 10, and 25-cent pieces.) And can you prove it?

Reblog: Kant & Leibniz on Space and Implications in Geometry


Mathematicians and philosophers are fairly content to share credit for Rene Descartes, possibly because he was able to provide catchy, easy-to-popularize cornerstones for both fields.

Immanuel Kant, these days at least, is almost exclusively known as a philosopher, and that he was also a mathematician and astronomer is buried in the footnotes. If you stick to math and science popularizations you’ll probably pick up (as I did) that Kant was one of the co-founders of the nebular hypothesis, the basic idea behind our present understanding of how solar systems form, and maybe, if the book has room, that Kant had the insight that knowing gravitation falls off by an inverse-square rule implies that we live in a three-dimensional space.

Frank DeVita here writes some about Kant (and Wilhelm Leibniz)’s model of how we understand space and geometry. It’s not technical in the mathematics sense, although I do appreciate the background in Kant’s philosophy which my Dearly Beloved has given me. In the event I’d like to offer it as a way for mathematically-minded people to understand more of an important thinker they may not have realized was in their field.

Frank DeVita

        

Kant’s account of space in the Prolegomena serves as a cornerstone for his thought and comes about in a discussion of the transcendental principles of mathematics that precedes remarks on the possibility of natural science and metaphysics. Kant begins his inquiry concerning the possibility of ‘pure’ mathematics with an appeal to the nature of mathematical knowledge, asserting that it rests upon no empirical basis, and thus is a purely synthetic product of pure reason (§6). He also argues that mathematical knowledge (pure mathematics) has the unique feature of first exhibiting its concepts in a priori intuition which in turn makes judgments in mathematics ‘intuitive’ (§7.281). For Kant, intuition is prior to our sensibility and the activity of reason since the former does not grasp ‘things in themselves,’ but rather only the things that can be perceived by the senses. Thus, what we can perceive is based…

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Where Rap Music and Discrete Mathematics meet.


It’s the weekend; why not spread a bit of mathematics humor, using the basic element of mathematics humor, the Venn diagram?

Interestingly, Venn diagrams are also an overlap between Mathematics Humor and Philosophy Humor.

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Some More Comic Strips


I might turn this into a regular feature. A couple more comic strips, all this week on gocomics.com, ran nice little mathematically-linked themes, and as far as I can tell I’m the only one who reads any of them so I might spread the word some.

Grant Snider’s Incidental Comics returns again with the Triangle Circus, in his strip of the 12th of March. This strip is also noteworthy for making use of “scalene”, which is also known as “that other kind of triangle” which nobody can remember the name for. (He’s had several other math-panel comic strips, and I really enjoy how full he stuffs the panels with drawings and jokes in most strips.)

Dave Blazek’s Loose Parts from the 15th of March puts up a version of the Cretan Paradox that amused me much more than I thought it would at first glance. I kept thinking back about it and grinning. (This blurs the line between mathematics and philosophy, but those lines have always been pretty blurred, particularly in the hotly disputed territory of Logic.)

Bud Fisher’s Mutt and Jeff is in reruns, of course, and shows a random scattering of strips from the 1930s and 1940s and, really, seem to show off how far we’ve advanced in efficiency in setup-and-punchline since the early 20th century. But the rerun from the 17th of March (I can’t make out the publication date, although the figures in the article probably could be used to guess at the year) does demonstrate the sort of estimating-a-value that’s good mental exercise too.

I note that where Mutt divides 150,000,000 into 700,000,000 I would instead have divided the 150 million into 750,000,000, because that’s a much easier problem, and he just wanted an estimate anyway. It would get to the estimate of ten cents a week later in the word balloon more easily that way, too. But making estimates and approximations are in part an art. But I don’t think of anything that gives me 2/3ds of a cent as an intermediate value on the way to what I want as being a good approximation.

There’s nothing fresh from Bill Whitehead’s Free Range, though I’m still reading just in case.