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  • Joseph Nebus 3:00 pm on Saturday, 31 October, 2015 Permalink | Reply
    Tags: assumptions, , , , , , ,   

    Reading the Comics, October 29, 2015: Spherical Squirrel Edition 


    John Zakour and Scott Roberts’s Maria’s Day is going to Sunday-only publication. A shame, but I understand Zakour and Roberts choosing to focus their energies on better-paying venues. That those venues are “writing science fiction novels” says terrifying things about the economic logic of web comics.

    This installment, from the 23rd, is a variation on the joke about the lawyer, or accountant, or consultant, or economist, who carefully asks “what do you want the answer to be?” before giving it. Sports are a rich mine of numbers, though. Mostly they’re statistics, and we might wonder: why does anyone care about sports statistics? Once the score of a game is done counted, what else matters? A sociologist and a sports historian are probably needed to give true, credible answers. My suspicion is that it amounts to money, as it ever does. If one wants to gamble on the outcomes of sporting events, one has to have a good understanding of what is likely to happen, and how likely it is to happen. And I suppose if one wants to manage a sporting event, one wants to spend money and time and other resources to best effect. That requires data, and that we see in numbers. And there are so many things that can be counted in any athletic event, aren’t there? All those numbers carry with them a hypnotic pull.

    In Darrin Bell’s Candorville for the 24th of October, Lemont mourns how he’s forgotten how to do long division. It’s an easy thing to forget. For one, we have calculators, as Clyde points out. For another, long division ultimately requires we guess at and then try to improve an answer. It can’t be reduced to an operation that will never require back-tracking and trying some part of it again. That back-tracking — say, trying to put 28 into the number seven times, and finding it actually goes at least eight times — feels like a mistake. It feels like the sort of thing a real mathematician would never do.

    And that’s completely wrong. Trying an answer, and finding it’s not quite right, and improving on it is perfectly sound mathematics. Arguably it’s the whole field of numerical mathematics. Perhaps students would find long division less haunting if they were assured that it is fine to get a wrong-but-close answer as long as you make it better.

    John Graziano’s Ripley’s Believe It or Not for the 25th of October talks about the Rubik’s Cube, and all the ways it can be configured. I grant it sounds like 43,252,003,274,489,856,000 is a bit high a count of possible combinations. But it is about what I hear from proper mathematics texts, the ones that talk about group theory, so let’s let it pass.

    The Rubik’s Cube gets talked about in group theory, the study of things that work kind of like arithmetic. In this case, turning one of the faces — well, one of the thirds of a face — clockwise or counterclockwise by 90 degrees, so the whole thing stays a cube, works like adding or subtracting one, modulo 4. That is, we pretend the only numbers are 0, 1, 2, and 3, and the numbers wrap around. 3 plus 1 is 0; 3 plus 2 is 1. 1 minus 2 is 3; 1 minus 3 is 2. There are several separate rotations that can be done, each turning a third of each face of the cube. That each face of the cube starts a different color means it’s easy to see how these different rotations interact and create different color patterns. And rotations look easy to understand. We can at least imagine rotating most anything. In the Rubik’s Cube we can look at a lot of abstract mathematics in a handheld and friendly-looking package. It’s a neat thing.

    Scott Hilburn’s The Argyle Sweater for the 26th of October is really a physics joke. But it uses (gibberish) mathematics as the signifier of “a fully thought-out theory” and that’s good enough for me. Also the talk of a “big boing” made me giggle and I hope it does you too.

    Izzy Ehnes’s The Best Medicine Cartoon makes, I believe, its debut for Reading the Comics posts with its entry for the 26th. It’s also the anthropomorphic-numerals joke for the week.

    Frank Page’s Bob the Squirrel is struggling under his winter fur this week. On the 27th Bob tries to work out the Newtonian forces involved in rolling about in his condition. And this gives me the chance to share a traditional mathematicians joke and a cliche punchline.

    The story goes that a dairy farmer knew he could be milking his cows better. He could surely get more milk, and faster, if only the operations of his farm were arranged better. So he hired a mathematician, to find the optimal way to configure everything. The mathematician toured every part of the pastures, the milking barn, the cows, everything relevant. And then the mathematician set to work devising a plan for the most efficient possible cow-milking operation. The mathematician declared, “First, assume a spherical cow.”

    The punch line has become a traditional joke in the mathematics and science fields. As a joke it comments on the folkloric disconnection between mathematicians and practicality. It also comments on the absurd assumptions that mathematicians and scientists will make for the sake of producing a model, and for getting an answer.

    The joke within the joke is that it’s actually fine to make absurd assumptions. We do it all the time. All models are simplifications of the real world, tossing away things that may be important to the people involved but that just complicate the work we mean to do. We may assume cows are spherical because that reflects, in a not too complicated way, that while they might choose to get near one another they will also, given the chance, leave one another some space. We may pretend a fluid has no viscosity, because we are interested in cases where the viscosity does not affect the behavior much. We may pretend people are fully aware of the costs, risks, and benefits of any action they wish to take, at least when they are trying to decide which route to take to work today.

    That an assumption is ridiculous does not mean the work built on it is ridiculous. We must defend why we expect those assumptions to make our work practical without introducing too much error. We must test whether the conclusions drawn from the assumption reflect what we wanted to model reasonably well. We can still learn something from a spherical cow. Or a spherical squirrel, if that’s the case.

    Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 28th of October is a binary numbers joke. It’s the other way to tell the joke about there being 10 kinds of people in the world. (I notice that joke made in the comments on Gocomics.com. That was inevitable.)

    Eric the Circle for the 29th of October, this one by “Gilly” again, jokes about mathematics being treated as if quite subject to law. The truth of mathematical facts isn’t subject to law, of course. But the use of mathematics is. It’s obvious, for example, in the setting of educational standards. What things a member of society must know to be a functioning part of it are, western civilization has decided, a subject governments may speak about. Thus what mathematics everyone should know is a subject of legislation, or at least legislation in the attenuated form of regulated standards.

    But mathematics is subject to parliament (or congress, or the diet, or what have you) in subtler ways. Mathematics is how we measure debt, that great force holding society together. And measurement again has been (at least in western civilization) a matter for governments. We accept the principle that a government may establish a fundamental unit of weight or fundamental unit of distance. So too may it decide what is a unit of currency, and into how many pieces the unit may be divided. And from this it can decide how to calculate with that currency: if the “proper” price of a thing would be, say, five-ninths of the smallest available bit of currency, then what should the buyer give the seller?

    Who cares, you might ask, and fairly enough. I can’t get worked up about the risk that I might overpay four-ninths of a penny for something, nor feel bad that I might cheat a merchant out of five-ninths of a penny. But consider: when Arabic numerals first made their way to the west they were viewed with suspicion. Everyone at the market or the moneylenders’ knew how Roman numerals worked, and could follow addition and subtraction with ease. Multiplication was harder, but it could be followed. Division was a diaster and I wouldn’t swear that anyone has ever successfully divided using Roman numerals, but at least everything else was nice and familiar.

    But then suddenly there was this influx of new symbols, only one of them something that had ever been a number before. One of them at least looked like the letter O, but it was supposed to represent a missing quantity. And every calculation on this was some strange gibberish where one unfamiliar symbol plus another unfamiliar symbol turned into yet another unfamiliar symbol or maybe even two symbols. Sure, the merchant or the moneylender said it was easier, once you learned the system. But they were also the only ones who understood the system, and the ones who would profit by making “errors” that could not be detected.

    Thus we see governments, even in worldly, trade-friendly city-states like Venice, prohibiting the use of Arabic numerals. Roman numerals may be inferior by every measure, but they were familiar. They stood at least until enough generations passed that the average person could feel “1 + 1 = 2” contained no trickery.

    If one sees in this parallels to the problem of reforming mathematics education, all I can offer is that people are absurd, and we must love the absurdness of them.

    One last note, so I can get this essay above two thousand words somehow. In the 1910s Alfred North Whitehead and Bertrand Russell published the awesome and menacing Principia Mathematica. This was a project to build arithmetic, and all mathematics, on sound logical grounds utterly divorced from the great but fallible resource of human intuition. They did probably as well as human beings possibly could. They used a bewildering array of symbols and such a high level of abstraction that a needy science fiction movie could put up any random page of the text and pass it off as Ancient High Martian.

    But they were mathematicians and philosophers, and so could not avoid a few wry jokes, and one of them comes in Volume II, around page 86 (it’ll depend on the edition you use). There, in Proposition 110.643, Whitehead and Russell establish “1 + 1 = 2” and remark, “the above proposition is occasionally useful”. They note at least three uses in their text alone. (Of course this took so long because they were building a lot of machinery before getting to mere work like this.)

    Back in my days as a graduate student I thought it would be funny to put up a mock political flyer, demanding people say “NO ON PROP *110.643”. I was wrong. But the joke is strong enough if you don’t go to the trouble of making up the sign. I didn’t make up the sign anyway.

    And to murder my own weak joke: arguably “1 + 1 = 2” is established much earlier, around page 380 of the first volume, in proposition *54.43. The thing is, that proposition warns that “it will follow, when mathematical addition has been defined”, which it hasn’t been at that point. But if you want to say it’s Proposition *54.43 instead go ahead; it will not get you any better laugh.

    If you’d like to see either proof rendered as non-head-crushingly as possible, the Metamath Proof Explorer shows the reasoning for Proposition *54.43 as well as that for *110.643. And it contains hyperlinks so that you can try to understand the exact chain of reasoning which comes to that point. Good luck. I come from a mathematical heritage that looks at the Principia Mathematica and steps backward, quickly, before it has the chance to notice us and attack.

     
    • BunKaryudo 5:51 am on Monday, 2 November, 2015 Permalink | Reply

      I must admit, I thought the spherical cow joke was pretty funny. It’s also true, though, that as almost certainly the least mathematically gifted of your readers, it hadn’t occurred to me until I read a bit further that spherical cows might actually be a useful abstraction for certain types of problem. It might also make life easier for farmers since they could roll them back to their byres.

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      • Joseph Nebus 1:13 am on Friday, 6 November, 2015 Permalink | Reply

        I’m sorry to have had your comment hidden a while. WordPress thought it might be spam and I failed to check sooner. I guess it doesn’t understand why spherical cows might be talked about so much.

        Still, yes, in many ways cows could be made easier to work with if they were much more spherical. Even an ellipsoidal cow would offer some advantages.

        Liked by 1 person

        • BunKaryudo 1:09 pm on Friday, 6 November, 2015 Permalink | Reply

          It’s true. Those stubby little cow legs just get in the way.

          Incidentally, don’t worry about my comment going missing for a while. My comments quite often seem to be mistaken for spam by WordPress. Perhaps I should stop wearing the grey trenchcoat, sunglasses and false mustache.

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    • elkement (Elke Stangl) 8:06 am on Wednesday, 18 November, 2015 Permalink | Reply

      My favorite is of course your awesome find of humor in Principia Mathematica :-)

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      • Joseph Nebus 4:08 am on Friday, 20 November, 2015 Permalink | Reply

        I’m so glad you like. I had thought the ‘occasionally useful’ proposition the most famous bit of the Principia Mathematica, but then I suppose ‘most famous’ doesn’t actually mean anyone’s heard of it.

        Liked by 1 person

  • Joseph Nebus 3:04 pm on Monday, 25 May, 2015 Permalink | Reply
    Tags: , assumptions, , , , , higher mathematics,   

    A Summer 2015 Mathematics A To Z: ansatz 


    Sue Archer at the Doorway Between Worlds blog recently completed an A to Z challenge. I decided to follow her model and challenge and intend to do a little tour of some mathematical terms through the alphabet. My intent is to focus on some that are interesting terms of art that I feel non-mathematicians never hear. Or that they never hear clearly. Indeed, my first example is one I’m not sure I ever heard clearly described.

    Ansatz.

    I first encountered this term in grad school. I can’t tell you when. I just realized that every couple sessions in differential equations the professor mentioned the ansatz for this problem. By then it felt too late to ask what it was I’d missed. In hindsight I’m not sure the professor ever made it clear. My research suggests the word is still a dialect rather than part of the universal language of mathematicians, and that it isn’t quite precisely defined.

    What a mathematician means by the “ansatz” is the collection of ideas that go into solving a problem. This may be an assumption of what the solution should look like. This might be the assumptions of physical or mathematical properties a solution has to have. This might be a listing of properties that a valid solution would have to have. It could be the set of things you judge should be included, or ignored, in constructing a mathematical model of something. In short the ansatz is the set of possibly ad hoc assumptions you have to bring to a topic to make it something answerable. It’s different from the axioms of the field or the postulates for a problem. An axiom or postulate is assumed to be true by definition. The ansatz is a bunch of ideas we suppose are true because they seem likely to bring us to a solution.

    An ansatz is good for getting an answer. It doesn’t do anything to verify that the answer means anything, though. The ansatz contains assumptions you the mathematician brought to the problem. You need to argue that the assumptions are reasonable, and reflect the actual problem you’re studying. You also should prove that the answer ultimately derived matches the actual behavior of whatever you were studying. Validating a solution can be the hardest part of mathematics, other than all the other parts of mathematics.

     
    • Lynette d'Arty-Cross 3:28 pm on Monday, 25 May, 2015 Permalink | Reply

      I first ran across this word when I was learning German. One of its meanings is “formation” or “beginning.”
      Interesting post. :)

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      • Joseph Nebus 2:42 am on Tuesday, 26 May, 2015 Permalink | Reply

        I’d thought it had a meaning like that. There are a good number of mathematical terms that are German in origin — well, “eigenvalue”, along with related words like “eigenvector” and “eigenfunction” are evidence of that — though I’m surprised to find one that’s in the process of becoming part of the English mathematician’s vocabulary.

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        • Lynette d'Arty-Cross 4:05 am on Tuesday, 26 May, 2015 Permalink | Reply

          Interesting. Eigen means “own” in the sense that the object is separate, such as
          “my own car.” Roughly 60% of English comes from Latin via Old German, so there’s quite a common history.

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          • Joseph Nebus 4:23 am on Tuesday, 26 May, 2015 Permalink | Reply

            If the eigenvalue, or other eigen-thing, has to be rendered in English only it’s usually turned into “characteristic value” or “characteristic vector” or so on. And that’s fair enough.

            The eigenvalues (or other things) can be seen as kind of the spectroscopic analysis of a mathematical object. (This is a very loose metaphor.) If you’ve got a mathematical object describing a system, then the eigenvalues (eigenvectors, eigenfunctions, et cetera) can be simpler ways to describe how the system behaves.

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    • scifihammy 4:00 pm on Monday, 25 May, 2015 Permalink | Reply

      Your A to Z is a good idea. Interesting reading :)

      Liked by 1 person

      • Joseph Nebus 2:41 am on Tuesday, 26 May, 2015 Permalink | Reply

        Thank you. It’s not my idea, but I can at least recognize a fruitful one when it’s presented to me.

        Liked by 1 person

    • Barb Knowles 10:54 pm on Tuesday, 26 May, 2015 Permalink | Reply

      I just think it’s a very cool word.

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    • sheldonk2014 8:48 pm on Thursday, 11 June, 2015 Permalink | Reply

      I like the idea that you have to validate a solution,it sounds very definitive

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      • Joseph Nebus 4:13 am on Saturday, 13 June, 2015 Permalink | Reply

        It’s, sadly, the boring part of learning something. You know the part of mathematics class where you get, say, the root of a polynomial and then you’re supposed to go back and put it in to the polynomial and see if it really is zero? That’s validation. But for a simple problem it’s dull because if you did the work right there’s nothing being revealed.

        Where it’s important is when you try modeling something new and interesting because you don’t know that you made all the right choices in your model. (Also you might not be sure you calculated things based on the model right.) But that’s not something most mathematics classes reach, not below the upper levels of undergraduate life anyway.

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  • Joseph Nebus 8:24 pm on Sunday, 12 February, 2012 Permalink | Reply
    Tags: , assumption, assumptions, , , , , penalty, , , ,   

    Figuring Out The Penalty Of Going First 


    Let’s accept the conclusion that the small number of clean sweeps of Contestants Row is statistically significant, that all six winning contestants on a single episode of The Price Is Right come from the same seat less often than we would expect from chance alone, and that the reason for this is that whichever seat won the last item up for bids is less likely to win the next. It seems natural to suppose the seat which won last time — and which is therefore bidding first this next time — is at a disadvantage. The irresistible question, to me anyway, is: how big is that disadvantage? If no seats had any advantage, the first, second, third, and fourth bidders would be expected to have a probability of 1/4 of winning any particular item. How much less a chance does the first bidder need to have to get the one clean sweep in 6,000 episodes reported?

    Chiaroscuro came to an estimate that the first bidder had a probability of about 17.6 percent of winning the item up for bids, and I agree with that, at least if we make a couple of assumptions which I’m confident we are making together. But it’s worth saying what those assumptions are because if the assumptions do not hold, the answers come out different.

    The first assumption was made explicitly in the first paragraph here: that the low number of clean sweeps is because the chance of a clean sweep is less than the 1 in 1000 (or to be exact, 1 in 1024) chance which supposes every seat has an equal probability of winning. After all, the probability that we saw so few clean sweeps for chance alone was only a bit under two percent; that’s unlikely but hardly unthinkable. We’re supposing there is something to explain.

    (More …)

     
    • Chiaroscuro 5:40 am on Monday, 13 February, 2012 Permalink | Reply

      A much nicer explanation of the sort of thing I just did with a fair amount of the [1/x] button in the windows XP calculator and some messing around. Indeed, it’s some very rough assumptions made; but we’ve got to start somewhere, and this is a good place to start.

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      • nebusresearch 4:42 am on Wednesday, 15 February, 2012 Permalink | Reply

        Oh, ew, you worked it out from trying out different percentages until you found one that matched?

        Actually, that’s a respectable numerical-solution technique, called “regula falsi”, that I should probably explain since it’s powerful, simple, and works. I’ll make a note of that.

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        • Chiaroscuro 4:22 am on Thursday, 16 February, 2012 Permalink | Reply

          Oh goodness no. I *estimated* a few times to get to the proper neighborhood, then figured a way to reverse what I was doing. “So 23% yields… and how about 20%.. hmmm. lower. How about 16.6%.. too low. okay, then we do this in reverse and start with 1/6000…”

          It is a wonderful method for ‘ballpark figures’, quite true.

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          • Joseph Nebus 7:33 am on Monday, 20 February, 2012 Permalink | Reply

            Ah, OK, I follow now. I think I can tie this in to something I’d wanted to talk about anyway, too, so I appreciate the hook.

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