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  • Joseph Nebus 6:00 pm on Tuesday, 3 January, 2017 Permalink | Reply
    Tags: , , , , mathematicians   

    The End 2016 Mathematics A To Z Roundup 

    As is my tradition for the end of these roundups (see Summer 2015 and then Leap Day 2016) I want to just put up a page listing the whole set of articles. It’s a chance for people who missed a piece to easily see what they missed. And it lets me recover that little bit extra from the experience. Run over the past two months were:

  • Joseph Nebus 3:00 pm on Friday, 4 March, 2016 Permalink | Reply
    Tags: , , , , , mathematicians,   

    A Leap Day 2016 Mathematics A To Z: Conjecture 

    For today’s entry in the Leap Day 2016 Mathematics A To Z I have an actual request from from Elke Stangl. I’d had another ‘c’ request, for ‘continued fractions’. I’ve decided to address that by putting ‘Fractions, continued’ on the roster. If you have other requests, for letters not already committed, please let me know. I’ve got some letters I can use yet.


    An old joke says a mathematician’s job is to turn coffee into theorems. I prefer tea, which may be why I’m not employed as a mathematician. A theorem is a logical argument that starts from something known to be true. Or we might start from something assumed to be true, if we think the setup interesting and plausible. And it uses laws of logical inference to draw a conclusion that’s also true and, hopefully, interesting. If it isn’t interesting, maybe it’s useful. If it isn’t either, maybe at least the argument is clever.

    How does a mathematician know what theorems to try proving? We could assemble any combination of premises as the setup to a possible theorem. And we could imagine all sorts of possible conclusions. Most of them will be syntactically gibberish, the equivalent of our friends the monkeys banging away on keyboards. Of those that aren’t, most will be untrue, or at least impossible to argue. Of the rest, potential theorems that could be argued, many will be too long or too unfocused to follow. Only a tiny few potential combinations of premises and conclusions could form theorems of any value. How does a mathematician get a good idea where to spend her time?

    She gets it from experience. In learning what theorems, what arguments, have been true in the past she develops a feeling for things that would plausibly be true. In playing with mathematical constructs she notices patterns that seem to be true. As she gains expertise she gets a sense for things that feel right. And she gets a feel for what would be a reasonable set of premises to bundle together. And what kinds of conclusions probably follow from an argument that people can follow.

    This potential theorem, this thing that feels like it should be true, a conjecture.

    Properly, we don’t know whether a conjecture is true or false. The most we can say is that we don’t have evidence that it’s false. New information might show that we’re wrong and we would have to give up the conjecture. Finding new examples that it’s true might reinforce our idea that it’s true, but that doesn’t prove it’s true.

    For example, we have the Goldbach Conjecture. According to it every even number greater than two can be written as the sum of exactly two prime numbers. The evidence for it is very good: every even number we’ve tied has worked out, up through at least 4,000,000,000,000,000,000. But it isn’t proven. It’s possible that it’s impossible from the standard rules of arithmetic.

    That’s a famous conjecture. It’s frustrated mathematicians for centuries. It’s easy to understand and nobody’s found a proof. Famous conjectures, the ones that get names, tend to do that. They looked nice and simple and had hidden depths.

    Most conjectures aren’t so storied. They instead appear as notes at the end of a section in a journal article or a book chapter. Or they’re put on slides meant to refresh the audience’s interest where it’s needed. They are needed at the fifteen-minute park of a presentation, just after four slides full of dense equations. They are also needed at the 35-minute mark, in the middle of a field of plots with too many symbols and not enough labels. And one’s needed just before the summary of the talk, so that the audience can try to remember what the presentation was about and why they thought they could understand it. If the deadline were not so tight, if the conference were a month or so later, perhaps the mathematician would find a proof for these conjectures.

    Perhaps. As above, some conjectures turn out to be hard. Fermat’s Last Theorem stood for four centuries as a conjecture. Its first proof turned out to be nothing like anything Fermat could have had in mind. Mathematics popularizers lost an easy hook when that was proven. We used to be able to start an essay on Fermat’s Last Theorem by huffing about how it was properly a conjecture but the wrong term stuck to it because English is a perverse language. Now we have to start by saying how it used to be a conjecture instead.

    But few are like that. Most conjectures are ideas that feel like they ought to be true. They appear because a curious mind will look for new ideas that resemble old ones, or will notice patterns that seem to resemble old patterns.

    And sometimes conjectures turn out to be false. Something can look like it ought to be true, or maybe would be true, and yet be false. Often we can prove something isn’t true by finding an example, just as you might expect. But that doesn’t mean it’s easy. Here’s a false conjecture, one that was put forth by Goldbach. All odd numbers are either prime, or can be written as the sum of a prime and twice a square number. (He considered 1 to be a prime number.) It’s not true, but it took over a century to show that. If you want to find a counterexample go ahead and have fun trying.

    Still, if a mathematician turns coffee into theorems, it is through the step of finding conjectures, promising little paths in the forest of what is not yet known.

    • elkement (Elke Stangl) 9:38 pm on Friday, 4 March, 2016 Permalink | Reply

      Thanks :-) So you say that experts’ intuition that might look like magic to laymen is actually pattern recognition, correct? (I think I have read about this in pop-sci psychology books) And if an unproven theorem passes the pattern recognition filter it is promoted to conjecture.


      • Joseph Nebus 7:27 am on Wednesday, 9 March, 2016 Permalink | Reply

        I think that there is a large aspect of it that’s pattern recognition, yes. But some of that may be that we look for things that resemble what’s already worked. So, like, if we already have a theorem about how a sequence of real-valued functions converges to a new real-valued function, then it’s natural to think about variants. Can we say something about sequences of complex-valued functions? If the original theorem demanded functions that were continuous and had infinitely many derivatives, can we loosen that to a function that’s continuous and has only finitely many derivatives? Can we lose the requirement that there be derivatives and still say something?

        I realized at one point while taking real analysis in grad school that many of the theorems we were moving into looked a lot like what we already had with one or two variations, and could sometimes write out the next theorem almost by rote. There is certainly a kind of pattern recognition at work here, though sometimes it can feel like playing with the variations on a theme.

        Liked by 1 person

        • elkement (Elke Stangl) 7:37 am on Wednesday, 9 March, 2016 Permalink | Reply

          Yes, I agree – I meant pattern recognition in exactly this way, in a very broad way … searching for a similar pattern in your own experiences, among things you have encountered and that worked. I was thinking in general terms and comparing to other skills and expertise, like what makes you successful in any kind of tech troubleshooting. It seems that you have an intuitive feeling about what may work but actually you draw on related scenarios or aspects of scenarios we had solved.


    • Pen & Shutter 1:09 pm on Saturday, 5 March, 2016 Permalink | Reply

      I understood all that! I definitely deserve a prize … I am no mathematician … And I enjoyed every word! I love your use of English.


    • davekingsbury 3:25 pm on Saturday, 5 March, 2016 Permalink | Reply

      If you’ve nothing for Q, what about Quadratic Equations … though I start twitching whenever I think about them!


      • Joseph Nebus 7:43 am on Wednesday, 9 March, 2016 Permalink | Reply

        I’m sorry to say Q already got claimed, by ‘quaternion’. But P got ‘polynomial’, which should be close enough to quadratic equations that there’s at least some help there.

        Liked by 1 person

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

    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 3:05 pm on Wednesday, 19 March, 2014 Permalink | Reply
    Tags: , , mathematicians, , ,   

    Reading The Comics, March 17, 2014: After The Ides Edition 

    Rather than wait to read today’s comics I’m just going to put in a fresh entry going over mathematical points raised in the funny pages. This one turned out to include a massive diversion into the wonders of the ancient Roman calendar, which is a mathematical topic, really, although there’s no calculations involved in it just here.

    Bill Hinds’s Cleats (March 7, rerun) calls on one of the common cultural references to percentages, the idea of athletes giving 100 percent efforts. (Edith is feeling more like an 80 percent effort, or less than that.) The idea of giving 100 percent in a sport is one that invites the question, 100 percent of what; granting that there is some standard expectable effort made, then, even the sports reporting cliche of giving 110 percent is meaningful.
    Cleats continued on the theme the next day, as Edith was thinking more of giving about 79 percent of 80 percent, and it’s not actually that hard to work out in your head what percent that is, if you know anything about doing arithmetic in your head.

    Jef Mallet’s Frazz (March 14) was not actually the only comic strip among the roster I normally read to make a Pi Day reference, but I think it suffices as the example for the whole breed. I admit that I feel a bit curmudgeonly that I don’t actually care about Pi Day. I suppose that as a chance for people to promote the idea of learning mathematics, and maybe attach it to some of the many interesting things to be said about mathematics using Pi as the introductory note the idea is fine, but just naming a thing isn’t by itself a joke. I’m told that Facebook (I’m not on it) was thick with people posting photographs of pies, which is probably more fun when you think of it than when you notice everybody else thought of it too. Anyway, organized Pi Day events are still pretty new as Internet Pop Holidays go. Perhaps next year’s comics will be sharper.

    Jenny Campbell’s Flo and Friends (March 15) comes back to useful mental arithmetic work, in this case in working out a reasonable tip. A twenty-percent tip is, mercifully, pretty easy to remember just as what’s-her-name specifies. (I can’t think of the kid’s name and there’s no meet-our-cast page on the web site. None of the commenters mention her name either, although they do make room to insult health care reform and letting students use calculators to do arithmetic, so, I’m sorry I read that far down too.) But as ever you need to make sure the process is explained clearly and understood, and Tina needed to run a sanity check on the result. Sanity checks, as suggested, won’t show that your answer is right, but they will rule out some of the wrong ones. (A fifteen percent tip is a bit annoying to calculate exactly, but dividing the original amount by six will give you a sixteen-and-two-thirds percent tip, which is surely close enough, especially if you round off to a quarter-dollar.)

    Steve Breen and Mike Thompson’s Grand Avenue (March 15) has the kids wonder what are the ides of March; besides that they’re the 15th of the month and they’re used for some memorable writing about Julius Caesar it’s a fair thing not to know. They derive from calendar-keeping, one of the oldest useful applications of mathematics and astronomy. The ancient Roman scheme set three special dates in the month: the kalends, which seem to have started as the day of the new moon as observed in Rome; the nones, when the moon was at its first quarter; and the ides, when the moon was full.

    But by the time of Numa Pompilius, the second (traditional) King of Rome, who reformed the calendar around 713 BC, the lunar link was snapped, partly so that the calendar year could more nearly fit the length of the time it takes to go from one spring to another. (Among other things the pre-Numa calendar had only ten months, with the days between December and March not belonging to any month; since Romans were rather agricultural at the time and there wasn’t much happening in winter, this wasn’t really absurd, even if I find it hard to imagine living by this sort of standard. After Numa there were only about eleven days of the year unaccounted for, with the time made up, when it needed to be, by inserting an extra month, Mercedonius, in the middle of February.) Months then had, February excepted, either 29 or 31 days, with the ides being on the fifteenth day of the 31-day months (March, May, July, and October) and the thirteenth day of the 29-day months.

    For reasons that surely made sense if you were an ancient Roman the day was specified as the number of days until the next kalend, none, or ide; so, for example, while the 13th of March would be the 2nd day before the ides of March, II Id Mar, the 19th of March would be recorded as the the the 14th day before the kalend of April, or, XIV Kal Apr. I admit I could probably warm up to counting down to the next month event, but the idea of having half the month of March written down on the calendar as a date with “April” in it leaves me deeply unsettled. And that’s before we even get into how an extra month might get slipped into the middle of February (between the 23rd and the 24th of the month, the trace of which can still be observed in the dominical letters of February in leap years, on Roman Catholic and Anglican calendars, and in the obscure term “bissextile year” for leap year). But now that you see that, you know why (a) the ancient Romans had so much trouble getting their database software to do dates correctly and (b) you get to be all smugly superior to anyone who tries making a crack about the United States Federal Income Tax deadline being on the Ides of April, since they never are.

    (Warning: absolutely no one ever will be impressed by your knowledge of the Ides of April and their inapplicability to discussions of the United States Federal Income Tax. However, you might use this as a way to appear like you’re making friendly small talk while actually encouraging people to leave you alone.)

    Tom Horacek’s Foolish Mortals (March 17), an erratically-published panel strip, calls on the legend of how mathematicians “usually” peak in their twenties. It’s certainly said of mathematicians that they do their most important work while young — note that the Fields Medal is explicitly given to mathematicians for work done when they were under forty years old — although I’m not aware of anyone who’s actually studied this, and the number of great mathematicians who insist on doing brilliant work into their old age is pretty impressive.

    Certainly, for example, Newton began work on calculus (and optics and gravitation) when he was about 23, but he didn’t publish until he was about fifty. (Leibniz, meanwhile, started publishing calculus his way at about age 38.) It’s probably impossible to say what Leonhard Euler’s most important work was, but (for example) his equations describing inviscid fluids — which would be the masterpiece for anybody not Euler — he published when he was fifty. Carl Friedrich Gauss didn’t start serious work in electromagnetism until he was about 55 years old, too. The law of electric flux which Gauss worked out for that — which, again, would have been the career achievement if Gauss weren’t overflowing with them — he published when he was 58.

    I guess that I’m saying is that great minds, at least, don’t necessarily peak in their twenties, or at least they have some impressive peaks afterwards too.

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