Tagged: philosophy Toggle Comment Threads | Keyboard Shortcuts

  • Joseph Nebus 4:00 pm on Sunday, 23 July, 2017 Permalink | Reply
    Tags: , , , devilbunnies, ecology, , , Human Cull, , philosophy, ,   

    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.

    • goldenoj 8:05 pm on Sunday, 23 July, 2017 Permalink | Reply

      Human Cull may be the most disturbing idea for a comic ever.


      • Joseph Nebus 3:37 am on Tuesday, 25 July, 2017 Permalink | Reply

        It’s a strip that has really, deeply bothered me. I know the cartoonist is talking about “culling” in a comically great overreaction to people who might be a bit annoying in your daily life. But that started out feeling terribly nasty when we’re talking about, like, people who put their jackets over empty seats next to them at the movie theater. And given the turn the world’s taken for the nasty the past couple years it hasn’t improved the strip’s tone any.

        It’s not like “here’s an annoying thing people do” is an inherently bad idea for a comic strip. It drove They’ll Do It Every Time for much of its run, and it underlay comic strips like The Dinette Set or (in its implications) Pluggers. But I think there needs to be a bit more clearly expressed empathy and grace for the approach to really work.


    • The Chaos Realm 3:42 pm on Monday, 24 July, 2017 Permalink | Reply

      Now they have a machine that counts out the tills at shift changes or for money drops–I just saw it in action at a local grocery store. The “good old days” (aka massive headache) of counting it out by hand or trying to make the tills balance out are over, it seems. Whew! LOL


      • Joseph Nebus 3:47 am on Tuesday, 25 July, 2017 Permalink | Reply

        Those have turned up, where I buy stuff, sporadically over the last couple decades. I haven’t worked out what the rhyme or reason for a particular shop having one is, though. It’s not all the shops in a chain and it doesn’t seem to be particularly tied to how urban or rural the place is or how new the shop is. Maybe it’s tied to how often the management thinks cashiers are pocketing change and that’s too idiosyncratic for a mere customer to know.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Wednesday, 31 August, 2016 Permalink | Reply
    Tags: , , hot hands, Julia Sets, , philosophy, , thinking   

    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.

    • sheldonk2014 1:26 am on Wednesday, 28 September, 2016 Permalink | Reply

      I just thought of one Joseph
      How many stiches in an average size shirt


      • Joseph Nebus 10:27 pm on Friday, 30 September, 2016 Permalink | Reply

        That’s … a tough one. I’m not sure for example how the number of stitches is counted for a panel of fabric like makes up the front of a shirt.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Saturday, 9 July, 2016 Permalink | Reply
    Tags: celestial mechanics, , philosophy   

    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.

  • Joseph Nebus 4:00 pm on Saturday, 28 November, 2015 Permalink | Reply
    Tags: artwork, , brute force, , , philosophy, , rectangles, secrets   

    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.

    • BunKaryudo 12:20 pm on Sunday, 29 November, 2015 Permalink | Reply

      I’m feeling pretty pleased with myself after reading this post since I’d actually heard of the Fermi Paradox before. I know it basically just boils down to, “Many scientists estimate that intelligent life should be common in the universe, so where is everyone?” Nevertheless, I’m puffing my chest out and strutting around like a mathematical genius this week.


      • Joseph Nebus 9:38 pm on Thursday, 3 December, 2015 Permalink | Reply

        Oh, do. It’s always fun to run across something and recognize it. And the Fermi Paradox does after all relate to one of those Fermi Problems: if we make some reasonable guesses about how likely aliens are to exist, we’re forced to look for reasons why they’re much less likely than we imagine. There’s good science to be done figuring out why our estimates are wrong, or why our reasoning is misfiring.

        Liked by 1 person

        • BunKaryudo 1:47 pm on Friday, 4 December, 2015 Permalink | Reply

          My own personal theory is that the cosmos is teeming with intelligent life but it’s all hiding from us (while sniggering tee hee hee, probably), or alternatively, there is no other intelligent life in the entire universe. I’m pretty sure it’s either one of those two — or else something in between. (Alright, I admit it. I have no idea.) :)


          • Joseph Nebus 7:43 pm on Friday, 4 December, 2015 Permalink | Reply

            It’s hard to say, really. The obvious alternatives are despairing in different ways. There’s nobody else in the universe, or there’s no way of contacting them; or they all have a social order so strict that in billions of years there’s no defying a quarantine rule. There’s no rule that the universe has to be constructed so that it’s pleasant, but there’s something at least my mind rebels against in facing those options.

            Liked by 1 person

            • BunKaryudo 1:08 am on Saturday, 5 December, 2015 Permalink | Reply

              My best guess, and of course it is just a guess, is that life is fairly common, intelligent life less common but around, and technologically advanced civilizations pretty rare.

              Of those technologically advanced civilizations that do exist and that avoid annihilating themselves, perhaps the window during which they are using technology primitive enough for easy detection from afar is fairly brief. If they don’t want to be found or if they have no particular interest in communicating with the local insect life, they might simply not be visible to us.

              I’d hate to think that in a universe of such unbelievable size, we’re the only ones here. Just because I’m not fond of the idea doesn’t mean it’s not the right one, though. Whatever they happen to be, the facts are the facts.


              • Joseph Nebus 7:11 am on Sunday, 6 December, 2015 Permalink | Reply

                The idea that technological civilizations only spend a short time being able or willing to contact an Earth-type civilization is probably the least lonely of the options. It’s still disheartening, though; the universe seems too big to be that effectively empty. But perhaps it is. Probably we’ll only become confident of that if we try to work out all the ways it wouldn’t be effectively empty and see what follows from trying to prove or disprove those.


    • Garfield Hug 1:12 pm on Monday, 30 November, 2015 Permalink | Reply

      Congrats on achieving a new statistic on your blog! Happy 30,000 views! 👍👏


    • davekingsbury 7:46 am on Tuesday, 1 December, 2015 Permalink | Reply

      Your blogs combine education with entertainment in equal proportion … hard to prove but easy to appreciate!


  • Joseph Nebus 3:00 pm on Monday, 5 October, 2015 Permalink | Reply
    Tags: , , philosophy, simulations, zero point energy   

    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.

  • Joseph Nebus 11:16 pm on Tuesday, 3 March, 2015 Permalink | Reply
    Tags: , , , , philosophy, , theology, typesetting   

    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.

    • elkement 11:03 am on Sunday, 8 March, 2015 Permalink | Reply

      I remember that stuff – as one of the most intriguing things I learned in the Linear Algebra class in the first semester.


      • Joseph Nebus 11:51 pm on Monday, 9 March, 2015 Permalink | Reply

        Linear Algebra? I’m intrigued it was put in that course. In my curriculum they were fit into real analysis and mathematical logic instead.

        Liked by 1 person

        • elkement 10:59 am on Tuesday, 10 March, 2015 Permalink | Reply

          It was somewhere in the same chapter / lecture as different types of sets, infinite sets, and Russell’s paradox of the set of all sets and related proof of the inherent contradition …


          • Joseph Nebus 7:59 pm on Thursday, 12 March, 2015 Permalink | Reply

            Ah, I see. I wouldn’t have thought to connect the topics quite that way, although it’s possible I’m just thinking too heavily of how it happened to be done the semesters I took linear algebra, which were pretty heavily biased towards the sorts of matrix and vector space stuff that would be helpful in physics. Maybe I failed to read the chapters the professor chose to skip.

            (I didn’t have much choice: I lost my textbook after the first exam and couldn’t buy or borrow a second copy. Luckily homeworks were assigned by actually writing out the problems, rather than just ‘Chapter 2.3 3-9 odds, 12, 14’, so I could keep up, but it was tougher than it needed to be. I’m not positive the professor wasn’t kind to me with my final grade, or whether having to pay extremely close attention to definitions and proofs in class was better for me than trusting I could check the details in the textbook later on.)

            Liked by 1 person

            • elkement 8:21 pm on Thursday, 12 March, 2015 Permalink | Reply

              Yes, the lecture was mainly matrices, vector spaces, and tensors. The set of sets and Cantor’s diagonal argument etc. were mentioned in one of the first chapters if I recall correctly. Russell’s proof (or some version of it) required mapping elements of a set onto their power set (or something ;-)) so this was introduced right after surjective and injective linear maps.

              I am too lazy now, but I could check – I still do have this textbook, but it is literally falling apart!


              • Joseph Nebus 3:17 am on Saturday, 14 March, 2015 Permalink | Reply

                Ah, OK. Now I see where it’d fit naturally in with the way the instructor was leading the course. It wasn’t something I had expected but I do see how that makes sense.

                Liked by 1 person

            • elkement 8:57 pm on Thursday, 12 March, 2015 Permalink | Reply

              … and as you speak about Real Analysis this is maybe the time to ask a perfectly stupid question, but blame it on differences in our educational systems and my ignorance thereof: When I was a student of physics, I had two math classes in the first year, Linear Algebra and Real Analysis (two semesters each, first and second of my “undergrad” studies, though we had no bachelor degrees back then, only masters – this was just the first year of five).
              So I always thought “Calculus” = “Real Analysis”. But it isn’t, right?
              I know this may sound dumb but I have tacitly made this assumption so often, so I admit my blunder publicly now :-)

              Real analysis was mainly theorems and proofs, “building math from scratch”, series and functions, their properties – continuous, differentiable etc.
              Is “Calculus” more about learning rules how to integrate and differentiate, but without all those detailed proofs? I started thinking about it when I read a book by a science writer (an English major) who tought herself calculus later. It seems it had not been mandatory in her high school. Then I’d understand why colleges would have to teach calculus to make sure everybody has the same background. I can remember I had a few colleagues who came from a high school not at all specialized in science. We have e.g. something like “business highschools”, with accounting classes and the like…. but those students were unlikely to pick a science degree at the university so perhaps nobody cared that they had a really hard time with that rigorous, proof-based math right from day 1.


              • Joseph Nebus 3:24 am on Saturday, 14 March, 2015 Permalink | Reply

                I had to think about this one a bit, but I believe there is a subtle difference between Calculus and Real Analysis. Real Analysis is the study of real-valued functions — how to define them, how to use them, how to manipulate them. But the most interesting stuff to do with real-valued functions that you can teach with the sorts of proofs that new students can follow or reconstruct are generally the things that we get in intro calculus: finding maximums and minimums, finding derivatives, integrating, that sort of thing. So Real Analysis tends to look like “Intro Calculus, only this time you have to do the proofs”.

                Liked by 1 person

  • Joseph Nebus 10:15 pm on Sunday, 22 February, 2015 Permalink | Reply
    Tags: Batman, Bernard Riemann, , , , , integrals, philosophy, power   

    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.

    (More …)

    • ivasallay 8:10 pm on Monday, 23 February, 2015 Permalink | Reply

      I like Robbie and Bobbie, too. I also really like that there were comics that mentioned Georg Cantor and Bernard Riemann. Thank you for your explanations, too!


      • Joseph Nebus 3:20 am on Friday, 27 February, 2015 Permalink | Reply

        Aw, thank you. Georg Cantor’s a really interesting fellow that deserves a proper biography, but I don’t know if I can explain him in one essay fairly.


    • Carrie Rubin 1:15 am on Tuesday, 24 February, 2015 Permalink | Reply

      It’s interesting, because when I tell people a test was negative, there are some who aren’t quite sure what that means. I find it’s better to say, “Your test was normal.” Although that doesn’t always fit the study result (for example, a flu test–it’s either negative or positive), it’s an easier concept for some to grasp.


      • Joseph Nebus 3:22 am on Friday, 27 February, 2015 Permalink | Reply

        I’m not surprised there are people who don’t know what a negative test means. It really is a term of art, at least in this context, and the fact that “negative” is a familiar everyday word only helps it be more confusing. If the result were called, say, “antihomological” (or whatever) it wouldn’t have such connotations and wouldn’t need to be translated into everyday speech.

        Liked by 1 person

  • Joseph Nebus 10:20 pm on Tuesday, 13 January, 2015 Permalink | Reply
    Tags: , Civil War, controversy, , , , , , , philosophy,   

    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.

    • jcckeith 12:30 am on Wednesday, 14 January, 2015 Permalink | Reply

      From what I understood about this post was – aside from the king is god’s annointed and thus is always right and whatever he says is the rule regardless of its veracity – was that all accepted mathematicians of the day stuck with the classical models of mathematics, which in truth, can be difficult to use much less completely comprehend. So Hobbes wanted, like so many people these days, for mathematics to have a much more reasonable, understandable basis? From what i gather from your post, he offered various proofs for his assertions but none were complete or at least none proved his assertions when provided for peer review? And from what else you have said, this Hobbes guy had the deck stacked against him from the beginning because of his background in philosophy?


      • Joseph Nebus 12:08 am on Thursday, 15 January, 2015 Permalink | Reply

        Largely, yes, although I’m hesitant to declare exactly what Hobbes wanted because I haven’t read anything more than excerpts of his work, and he did a lot of work over the course of decades, and it can be hard to tell what the point of original work is until after it’s been thought and reworked, completed, and refined. In many ways mathematical constructs are like inventions, with the first ideas of something a complex and barely functional kludge that requires a couple generations of work to make into an elegant and understandable whole, and Hobbes’s mathematics didn’t get those generations of work.

        I think it’s fair to say he wanted a more materialistic mathematics, thinking of the lines you draw with straightedge and pencil and the circles you draw with compass and paper. There’s obvious need for that, especially if you want to do something like turn an abstract concept into an actual machine or building or canal or such. But it isn’t the same kind of work that mainstream mathematics was doing at the time, so in some ways he was working in a different field from other mathematicians.

        Hobbes’s background in philosophy didn’t by itself hurt him; the fields of mathematics and philosophy blur together on many points, and were even more blurry then. Both fields have about equal claims to Descartes as a founder of their modern incarnations, after all, and Berkeley and Pascal and Leibniz as lesser but still noteworthy figures. Philosophers can be surprised to know mathematicians get to put in a claim on Kant as one of their member; I’m sure the reverse happens. It was a more fluid era.

        However, Hobbes’s particular philosophy worked against him, because it was frightfully controversial (then and since) and thus made it harder for people to stand behind him. And it put him politically at odds with the Oxford and Cambridge establishments — there was a fierce battle about how the Universities should be reformed and how free they ought to be — and these were the people who would form the Royal Society and the mainstream of English mathematical thought.

        And, yes, Hobbes didn’t manage to prove the big impressive things he wanted to prove, including some results that carried implication like pi being (if I have it correctly, as I can’t find the page for this right now) equal to 3.2, which everyone by then knew could not be so.

        Liked by 1 person

    • Boxing Pythagoras 1:44 pm on Wednesday, 14 January, 2015 Permalink | Reply

      If you liked Squaring the Circle, you’ll likely enjoy Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander. He also touches on the Hobbes/Wallis feud, but talks about a number of other related discussions from 17th Century mathematics on the philosophical nature of the field.


      • Joseph Nebus 12:09 am on Thursday, 15 January, 2015 Permalink | Reply

        I’ve been looking at that! I’ve enjoyed Alexander’s other books, and mostly been figuring out whether my reading list has gotten short enough to start adding new things to it.

        Liked by 1 person

    • Aquileana 10:16 pm on Sunday, 18 January, 2015 Permalink | Reply

      Excellent post Joseph … I can’t but congratulate you!.

      “Homo homini lupus”, as Hobbes would say!.

      All the best to you. Aquileana :D


      • Joseph Nebus 1:20 am on Monday, 19 January, 2015 Permalink | Reply

        Oh, I’d forgot that Hobbes quote, which is silly since it’s one of his top ones.

        Thanks kindly; I’m glad you enjoyed the post.

        Liked by 1 person

  • Joseph Nebus 11:49 pm on Monday, 8 September, 2014 Permalink | Reply
    Tags: , , , , , philosophy, , tipping,   

    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.

    • howardat58 1:47 am on Tuesday, 9 September, 2014 Permalink | Reply

      If teachers understood that when you bring in negative numbers you don’t just stick them to the left of the numbers you had before, you have actually created a new number system, for a new purpose, and made two copies of the original numbers (the natural numbers), stuck the copies together at zero, with one lot going to the left and one lot going to the right. It is completely arbitrary whether the ones to the right are called the positive numbers or the negative numbers. But of course that is far too mathematical for most people.


      • Joseph Nebus 6:50 pm on Friday, 12 September, 2014 Permalink | Reply

        You are right, that the introduction of negative numbers as this thing sort of slapped onto the left end of the number line sets up for a lot of trouble, particularly in subtraction and probably also in multiplication and even greater-than and less-than comparisons. But I also see why it’s so attractive to introduce it that way; it feels natural, or at least it look that way.

        I wonder if there’s a way to introduce the subject more rigorously but still at a level that elementary school students, who generally aren’t very strong on abstract reasoning, will still find comfortable, and that won’t cause their parents to get upset that they’re being taught something too weirdly different from what they kind of remember from school themselves.


    • Thomas Anderson 2:08 am on Wednesday, 10 September, 2014 Permalink | Reply

      In the case of Moose there, I’m tempted to say that, at least in part, he’s the victim of his education. If I had my way, students at some level would have a math class all about quickly winging it in real life situations. Shortcuts, etc. That would have been very helpful for me. It wasn’t until college physics when I realized you can start fudging numbers until you got “close enough.”


      • Joseph Nebus 6:56 pm on Friday, 12 September, 2014 Permalink | Reply

        He is, yes, certainly a victim there. The training to work out the exact problem specified is a good one, but what’s missed is knowledge that he actually gets to pick the exact problem. That’s a real shame since so much mathematics is a matter of picking the exact problem you want to solve, and how perfectly you have to solve it.

        (Admittedly, Moose, by the definition of his character, would struggle with a problem simpler to calculate too.)


        • Thomas Anderson 7:27 pm on Friday, 12 September, 2014 Permalink | Reply

          That’s a great way of putting it. I feel like a lot of my mathematics woes would have been avoided if I’d learned fact that the “right” answer is the one that’s as accurate as you need it to be.

          Ugh, I think back to college physics and wasting time trying to juggle so many decimals just because I thought the most accurate answer was the most desirable one, when I could have done the question in a quarter the time by saying “okay pi is 3, g is 10, and air resistance is a figment of an overactive imagination.”


          • Joseph Nebus 7:16 pm on Sunday, 14 September, 2014 Permalink | Reply

            Decimals have this strange hypnotic effect on the human psyche. I don’t know what TV Tropes would call it but you see a piece of this in a science fiction show where a character is shown to be very smart by reeling out far more digits than could be meaningful or even useful, like the time Commander Data gave the travel time between galaxies down to fractions of a second.

            It’s kind of a shame that fractions are kind of clumsy to work with, since the description of something as, say, one-quarter can avoid the trap of thinking that you have more precision than 0.25 really entitles you to. (And, yes, you can add a note about your margin for error, but I’m not convinced that people really internalize that, not without a lot of practice.)


    • elkement 8:14 am on Thursday, 11 September, 2014 Permalink | Reply

      The joke involving those perfectly rational perfectly informed beings is my favorite – as it has several ‘levels’…


  • Joseph Nebus 3:08 pm on Monday, 4 August, 2014 Permalink | Reply
    Tags: birthdays, , keytar, philosophy, platypus, ,   

    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.)


    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.


    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.

    View original post

  • Joseph Nebus 7:03 pm on Saturday, 10 May, 2014 Permalink | Reply
    Tags: , , philosophy, sphere, , Zeno   

    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.

    • BunnyHugger 12:32 am on Sunday, 11 May, 2014 Permalink | Reply

      When we were talking about this before, I realized that there’s a paradox or at least a bit of sophistry to be made out of the idea of coloring a point. (I expect I’m not the first person to think of this, though I did hit on it independently.) A point has no area. Something has to have an area, however, to be colored: it has to take up space. So if we choose some arbitrary point on a piece of paper and use a red marker to color in a section of the paper that contains that point, it is still the case that the point itself is not red. This can be repeated with any arbitrary point within the red section. Thus, no point within the red section is in fact red.


      • Joseph Nebus 2:26 am on Tuesday, 13 May, 2014 Permalink | Reply

        You’re right, yes, and it isn’t just a bit of sophistry. I think this idea is tied rather closely to the division between intensive and extensive quantities, and that’s worth some further attention so I might write a sequel post to this.

        The division there is one that I know from statistical mechanics, where there are some physical properties — such as atomic weight — that are inherent to every molecule of a substance, while there are others — such as density — that really aren’t. This demands an answer to the question of where it does come from. This is probably just another version of the heap paradox, I admit, but it’s one that a narrow-minded physics major can’t dismiss as a vapid argument over words.


    • Jim 5:27 pm on Monday, 12 May, 2014 Permalink | Reply

      Answer to question: ideally, at the touchdown point on the runway.

      Oh, wait, wrong kind of plane.

      Yes, I know Earth is an oblate spheroid, not a sphere.


      • Joseph Nebus 2:27 am on Tuesday, 13 May, 2014 Permalink | Reply

        There’d also be the takeoff point too, come to think of it.

        (I believe the contemporary favorite term for the shape of the earth is actually “geoid”, which covers all the ways that the planet is a little bit off being spheroidal, but isn’t nearly so satisfying to say, particularly as it’s too obviously just a way of saying “earth-shaped”.)


    • Tim Erickson 10:49 pm on Monday, 19 May, 2014 Permalink | Reply

      I especially like, “If you tried to point at a tangent point, you’d miss it.”


      • Joseph Nebus 6:59 pm on Wednesday, 21 May, 2014 Permalink | Reply

        My, thank you kindly. It’s so easy accepting marks on paper as things that it’s easy to forget they’re only approximations, and precision is difficult.


  • Joseph Nebus 12:21 am on Wednesday, 19 February, 2014 Permalink | Reply
    Tags: , drills, philosophy, , ,   

    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.

    • Jon Awbrey 12:28 am on Wednesday, 19 February, 2014 Permalink | Reply

      There must be a Jessica Rabbit joke in here somewhere …


    • fluffy 9:46 pm on Sunday, 18 May, 2014 Permalink | Reply

      In this case I think the intuitive approach that everyone uses is much more clear and easily-visualized than the “correct” original design.


      • Joseph Nebus 6:52 pm on Wednesday, 21 May, 2014 Permalink | Reply

        It might be, if (say) propositions A and C turn out to be logically exclusive so there’s no point there being any overlap. On the other hand, making explicit the concept that there might be a case where A and C both hold, and then consciously ruling that out, could also be useful in organizing one’s thoughts about the problem. I suppose it does depend on what you’re trying to use the diagrams for.


    • snidelytoo 9:31 pm on Wednesday, 6 August, 2014 Permalink | Reply

      I should have pointed out In The Usual Place that we recently had help in celebrating Venn’s Birthday:

      I did not try all the combinations, but I did work through several (all the ones on the left paired with its equal-latitude partner on the right, plus all of the mammal ones, and one or two more.


      • Joseph Nebus 2:22 pm on Thursday, 7 August, 2014 Permalink | Reply

        I admit that’s what I did too. I tried a couple of the variations and then figured I didn’t real need to do all 36 (or which ever it was).

        Back when they did that Valentine’s Day romantic-comedy-scene generator I only got through two or three of them before my love established that I’d never seen, like, Say Anything or The Lost Boys (I forget how that one came up) and was horrified. This was before I admitted the Indiana Jones news.


  • Joseph Nebus 11:42 pm on Monday, 28 October, 2013 Permalink | Reply
    Tags: , philosophy, ,   

    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.

    (More …)

  • Joseph Nebus 11:27 pm on Tuesday, 22 October, 2013 Permalink | Reply
    Tags: , philosophy, , , student responses,   

    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.

    (More …)

    • fluffy 7:08 pm on Wednesday, 23 October, 2013 Permalink | Reply

      See also: Douglas Adams’ argument that the universe has zero population.


    • elkement 12:17 pm on Friday, 25 October, 2013 Permalink | Reply

      You have convinced me once more that probability is among the non-intuitive things in mathematics! Nassim Taleb, irreverent trader-turner-philosopher, states that he often found that even professors in statistics make that type of mistake if statistics problems are not presented to them as text book problems, but phrased in natural language in passing. He gave this example: There are two hospitals in a city, a big one and a small one. You know that in one hospitals 60% of all children born in a certain period are boys. Is this rather the big or the small hospital? Taleb said that most ‘experts’ picked the big one though chances for a deviation of the expection value are larger for the small one, thus the smaller sample size.


      • Joseph Nebus 4:47 am on Monday, 28 October, 2013 Permalink | Reply

        Probability is definitely one of the big non-intuitive things (and I’ve been meaning to get a follow-up to this written; it’s been a busy week).

        I’m really startled by the Taleb example, though. I’m curious how the question was presented in the wording since … well, it does feel to me (and to my spouse) that it should be obvious the bigger deviation from the expected average is more likely to happen in the smaller hospital, but perhaps there’s something in the phrasing that throws people. (It’s probably possible to get any response to any probability question by phrasing it right.)


  • Joseph Nebus 10:57 pm on Friday, 26 April, 2013 Permalink | Reply
    Tags: , , , , obituaries, philosophy,   

    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.

    • elkement 11:46 am on Saturday, 27 April, 2013 Permalink | Reply

      I had once enjoyed Simon Singh’s book on Andrew Wiles’ life-task of developing a proof for Fermat’s Last Theorem … and I also wondered how many people actually really have followed each step.


      • Joseph Nebus 4:52 pm on Wednesday, 1 May, 2013 Permalink | Reply

        I realize on thinking about it I’m not sure whether I’ve read Singh’s book on Andrew Wiles or not. I know I’ve read at least one of his books, but my records on this stuff are all a mess.

        Great mathematics puzzles are often split between those which have great stories behind them and those which are just, “this nagged at mathematicians for a long while, until someone had a breakthrough”. I wonder if the difference is actually whether the problem was something that could be explained to a lay audience briefly, or whether at least one Great Weird Mathematics Personality got involved, or whether there just are some things that have histories that can’t be made interesting no matter how important they are. (Keep in mind, I’m a person who owns multiple pop histories of containerized cargo and am glad to.)


    • elkement 8:27 pm on Tuesday, 30 April, 2013 Permalink | Reply

      Joseph, I enjoy following your blog(s), so I have nominated you for the Liebster blog award. These are the “rules”, but feel free to respond in any way you prefer – I did not really follow the rules last time ;-)


      • Joseph Nebus 4:49 pm on Wednesday, 1 May, 2013 Permalink | Reply

        Well, goodness, thank you. I intend to give a reply in kind, and appreciate your thinking of me.


  • Joseph Nebus 11:18 pm on Sunday, 16 December, 2012 Permalink | Reply
    Tags: , , dollar coins, knowledge, philosophy, two coins, two dimes   

    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?

    • Geoffrey Brent 11:51 pm on Sunday, 16 December, 2012 Permalink | Reply

      If you have n coins, of four possible denominations, then there are (n^3-n^2)/6 possible choices of denominations (n+1 choose 4-1). However, the maximum value is at most 25n and hence there are no more than 25n possible values.

      For n> 12, this means there are more possible choices than there are possible values, hence there must be two different choices with the same value.

      Constructive proof: 6x5c vs 25c + 5x1c.


      • Geoffrey Brent 1:32 am on Monday, 17 December, 2012 Permalink | Reply

        …oops, that n^2 should just be n. But it’s the n^3 part that matters.


        • Geoffrey Brent (@GeoffreyBrent) 11:00 pm on Saturday, 22 December, 2012 Permalink | Reply

          …in fact, now I look again, (n+1 choose 3) is wrong. I didn’t bother to give a full explanation of how I came to that number, and if I had, I’d have realised my error at the time.

          But since the correct number of combinations is slightly MORE than (n+1 choose 3), the proof still holds.

          The argument works like this: make n+1 pencil marks in a line. Between those, you have spaces for n coins:


          now, pick any 3 of those pencil marks, and use those to identify a combination of coins:

          – All spaces up to the first pencil mark (possibly none) are filled by 1c
          – All spaces between first and second are filled by 5c
          – All spaces between second and third are filled by 10c
          – All spaces after third (possibly none) are filled by 25c

          It’s easy to see that each possible choice of 3 pencil marks produces a different combination of coins, so there are at least (n+1 C 3) possibilities. What I missed was that this method doesn’t allow choosing zero nickels or dimes, so I undercounted the total possibilities.

          The correct answer here SHOULD have been (n+3 choose 3), i.e. (n+3)(n+2)(n+1)/6. (I’ll leave proof to the readers, but I’m pretty sure I’ve got it right this time.)

          Also, since the minimum value with n coins is n, and the maximum is 25n, there are actually only (24n+1) possible values from n coins.

          Put those together, and it turns out that n=9 is large enough to guarantee existence of non-unique combinations.

          The same sort of proof will show existence of non-unique combinations for sufficiently large n, as long as you have at least 3 denominations in integer values (or just rational values is good enough). OTOH, if your denominations are 1, sqrt(2), and pi, then no two combinations can ever have the same sum.


    • Chiaroscuro 2:05 am on Monday, 17 December, 2012 Permalink | Reply

      Yup! Four dimes, or, three nickels and a quarter. Both are forty cents.

      I use less coins than Mr. Brent but take more money to do it. The follow-up question: Can either less money than 30 cents or less coins than four be improved upon?



      • Geoffrey Brent 2:54 am on Monday, 17 December, 2012 Permalink | Reply

        I’ve satisfied myself that both of those are minima. Full proof is too cumbersome to enter via margin^Wphone, but I’ll post it when I get home.


    • Geoffrey Brent (@GeoffreyBrent) 11:43 am on Monday, 17 December, 2012 Permalink | Reply

      Okay, terminology: a “matched pair” is a set of two different combinations of coins, both of which have the same number of coins and the same total value, but different coins.

      We can describe these with two quads of numbers: a1, a5, a10, a25 are the number of pennies/nickels/dimes/quarters in the first such combination, and b1…b25 ditto for the second. Being matched means that a1+5a5+10a10+25a25=b1+…+25b25 and a1+…+a25=b1+…+b25.

      Obviously if ax and bx are both nonzero, we can get another matched pair with fewer coins and lower value by subtracting the same denominations from both combinations. So we know that with a minimal pair (in either total value or total coins), at least one of a1 and b1 is zero, and similarly for the other values.

      Some useful lemmas:

      (i) A matched pair must contain at least three different denominations altogether. (If you only have two denominations in use, then the number and value of coins uniquely defines the combination, so the combinations can’t be different.) Among other things, this means that either 1s or 25s must be included in one of the combinations.

      (ii) For a minimal pair, a1 and b1 must be multiples of 5, for obvious reasons.

      (iii) If all the denominations in one combination are higher (lower) than all the denominations in the other, they can’t be a matched pair: since each combination must have the same total number of coins, we can pair them up, and each coin in combination B will be strictly higher (lower) than its mate in A, so B must have a higher (lower) value than A.

      (iv) Therefore, if a minimal matched pair only has three denominations involved, one combination MUST be the middle denomination (only) and the other MUST have the high and low denominations. (e.g. if we have 1s, 5s, and 25s, one will combination will be all 5s and the other will be 1s and 25s).

      Now, with those lemmas established, let’s let’s look at the minimal number of coins:

      If we have a minimal pair that uses pennies at all, then by (ii) one of the two combinations must have at least 5 pennies. We already know there’s a solution with 4, so this is not minimal.

      If our minimal pair doesn’t use pennies, then it must use 5s, 10s, and 25s, and from (iv) above that means one combination (call it B) is all 10s and the other (A) is 5s and 25s. At least one 25 and one 5 in A requires at least 3 10s in B to match the value, but then we don’t have the right number of coins. So we need at least 3 coins in A, which means a total value of at least 25+5+5=35, which then requires at least 4 coins in B, which gets us to the solution Chiaroscure gave above.

      Next, minimal value:

      If our minimal pair uses 1s, then one combination (call it A) must have at least 5 1s, and from (iii) it must have at least one other coin (10c or 25c), for a total of 6 coins. This means combination B must have at least 6 coins, all of value at least 5, so its value must be at least 30c. It can only be *exactly* 30c if B has exactly 6 5s, which then leaves 5 1s and 1 25 as the only possible solution for A.

      If our minimal pair does not use 1s, then we know A uses 5s and 25s, B uses only 10s. We’ve already established that the smallest solution to this requires at least 4 10s, so total value at least 40, which is worse than the solution I gave above.



  • Joseph Nebus 3:33 am on Thursday, 9 August, 2012 Permalink | Reply
    Tags: , , , philosophy, ,   

    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…

    View original post 700 more words

    • BunnyHugger 7:42 pm on Thursday, 9 August, 2012 Permalink | Reply

      As Joseph knows, I have a framed portrait of Kant in my dining room (I have to stop calling things that; now it’s our dining room) that was taken from a set of prints from the Moscow Observatory — I believe from the 1940s — celebrating people who had made contributions to astronomy. I like that I have a souvenir recognizing Kant as an astronomer.

      It used to bother me that people call the nebular hypothesis “the Laplace theory” when Kant’s work on it was earlier. (I have also heard it called “the Kant-Laplace theory,” but, I think, usually by philosophers.) However, then Wikipedia told me that Kant himself may have gotten the rudiments of it from Swedenborg, and no one ever calls it “the Swedenborg-Kant-Laplace” theory, as far as I’ve heard.

      I hope that DeVita’s article called back ideas for you (regarding both Leibniz and Kant) that I tried to explain to you in our cabin on the Amsterdam-Newcastle ferry while fighting off seasickness.


  • Joseph Nebus 6:31 am on Saturday, 19 May, 2012 Permalink | Reply
    Tags: , philosophy,   

    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.


    View original post

    • BunnyHugger 6:44 am on Saturday, 19 May, 2012 Permalink | Reply

      I spent three hours drawing Venn diagrams on the board today.


      • Joseph Nebus 12:16 am on Sunday, 20 May, 2012 Permalink | Reply

        I probably should’ve spent more time drawing Venn diagrams while I was trying to get my students to follow probability problems. I’m still learning how many examples the lower-level and non-major students need.


    • donna 6:01 pm on Saturday, 19 May, 2012 Permalink | Reply

      But in Discrete, shouldn’t it be integers??? Or at least Rationals??? :)


  • Joseph Nebus 9:50 pm on Sunday, 18 March, 2012 Permalink | Reply
    Tags: , , Cretan Paradox, , , , , philosophy, scalene,   

    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.

    • BunnyHugger 9:54 pm on Sunday, 18 March, 2012 Permalink | Reply

      I liked the Loose Parts one better before I scrolled down and saw the superfluous caption.


      • Joseph Nebus 1:03 am on Tuesday, 20 March, 2012 Permalink | Reply

        Oh? I thought the caption was a fine magnifier. The joke was fully-formed without it, but adding that the guy was there for hours put a frosting of absurd extremity on the proceedings.


        • BunnyHugger 8:50 pm on Tuesday, 20 March, 2012 Permalink | Reply

          Hm. It felt to me like the cartoonist felt he had to explain the joke to me.


Compose new post
Next post/Next comment
Previous post/Previous comment
Show/Hide comments
Go to top
Go to login
Show/Hide help
shift + esc
%d bloggers like this: