Academia & Ideas & Personal Posted by christian h., 11 Jul 2007 05:32 am

What does a mathematician do??

Very good question. No worries, I am not going to try and explain the end product of what I do. But how is it done? What does a mathematician do all day, or what do mathematicians do all day when they come together?

Well, first of all, all day is a strictly relative term. A famous mathematician once explained that he couldn’t do mathematics for more than six hours a day, and most of the time, this is true for most of us, at least while working alone. The rest of the time we while away reading, or running, or making music, or watching TV, or blogging - while somehow part of the brain keeps working, which can lead to a certain absentmindedness. Only when I am hot on a trail (sadly, a rare occurrence) will I completely concentrate on work for long periods of time - or when the work is strictly routine, like preparing classes, grading papers (an event all too common in my life), refereeing articles for academic journals or any of the other administrative and community duties that make up a large part of any academic’s day for at least nine months of the year.

Well, but what about the times I do work? Mostly, I sit at my desk for some time. I stare at a blank piece of paper. When I get tired of that, I might stare out of the window, or at the wall if in a basement office. After some time - maybe half an hour - I can’t stand it anymore. Now I have several choices: distract a friend and colleague from work and drag them to get some coffee; take a stroll to get the circulation going; or write something on that piece of paper. What I write on that paper will usually be only a slight variation of whatever I wrote on a different piece the day before - a piece now safely filed away in the “circular file”, or else buried somewhere in my office, or at home. The hope is that sooner or later this slight variation will lead me to an insight.

Then, some day, out of nowhere, I may have an idea. This could work! It’s a very good feeling, an idea. Can be an extraordinary high, in fact. Better not to think about it again for a day to preserve the good feeling (I know from experience that the idea is likely to be wrong, or useless). After a suitable time of enjoying the feeling of my idea, I’ll go back and check it out. Does it even make sense? Care is needed - questions may appear easier than they are (especially when under the influence - even a little alcohol makes everything seem clear). What does it imply? More often than not, it turns out the idea is correct, but only led me in a circle - it looks long and interesting travelled one way, but in the end I’m back where I started.

If the idea holds up for a couple days and seems worthwhile, I’ll write to a collaborator (I much prefer joint projects - which isn’t quite compatible with the requirements for such trivialities as tenure) and explain my idea. In most cases, I’ll hear back soon with a mistake I made (it’s too easy to get carried away by your own ideas), often dooming the whole train of thought. Only rarely does everything work out, and then it’s hard concentration until the possibilities are (for the time being) exhausted.

Once a problem is understood its solution has to be brought into a form other mathematicians will want to read, and be able to understand. Solving the biggest problem in mathematics won’t be of any import if you can’t explain the solution to your colleagues (although once you have a considerable track record of being right - something I’m a long way from - those colleagues will make more of an effort, and will also be more prepared to simply trust you on the details). Writing about mathematics is something that’s difficult to do well. Large numbers of published works are error-ridden, (though rarely are the errors central to the argument), and include convoluted, overly-technical, and notationally insane language: “Let P be point Q, which we’ll call R” is an old but accurate joke. It isn’t only literature professors whose writing makes you go “huh??”. Many mathematicians also don’t like spending time writing down things they, after all, have already understood. This problem seems to have been compounded by the advent of the world wide web making instant dissemination of unfinished manuscripts possible.

Discussing mathematics with others is what I find most enjoyable. I am not very discerning - I’ll happily spend hours explaining calculus to my undergraduates - but certainly discussing research with a colleague, or graduate student, is most satisfying. Thinking along with each other, seeing your vague intuition concretized and formalized - or taken apart - by another person, is an exhilarating experience. When working together ideas can be thrown out and discarded at high speed. Because no two mathematicians think alike, I have never come away from a meeting with a colleague without having learned something - even if no new result was produced.

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Responses to “What does a mathematician do??”

  1. on 11 Jul 2007 at 6:31 am 1. JP Stormcrow said …

    Hah! You can’t fool me. From the evidence of my own eyes I can see that mathematicians, like many others…, read, write and comment on blogs all day. (See this flowchart at Crooked Timber.)

  2. on 11 Jul 2007 at 10:03 am 2. James Killus said …

    Ah, but the interesting (to me, anyway) is whether you consider the process to be one of discovery or invention.

    A friend of mine, who is the best mathematician I know, contends that mathematics is 100% a human invention, and that the intent of a proof is solely to convince other mathematicians of its truth (though, of course, he admits that mathematics can have implications in the material world via the scientific modeling process).

    I have heard others claim that mathematics is part of reality, even that it is a superior reality to the mundane world of materiality.

    Of course, I’m guessing that such questions don’t often come up in the course of your day.

  3. on 11 Jul 2007 at 10:25 am 3. christian h. said …

    Certainly an answer to the question James raises isn’t necessary to do mathematics, but it is discussed quite regularly. I’m not an idealist, so I agree with your friend that mathematics is a human invention.

    Here’s a quite amusing article from the Notices of the AMS (the AMS is our professional society) discussing this issue, as well as mathematical writing, intended for a general audience.

  4. on 11 Jul 2007 at 11:13 am 4. Oaktown Girl said …

    A friend of mine, who is the best mathematician I know

    James is speaking about mathematicians that he knows personally, but it calls to mind a question I have as a total math outsider: is there such a thing as “The Best” mathematician? Or are there several categories of mathematics and calling someone “The Best” would be about as ridiculous as calling someone “The Best” musician in the world?

  5. on 11 Jul 2007 at 11:34 am 5. christian h. said …

    The latter. Like most disciplines nowadays, mathematics is splintered into many sub-disciplines (it’s not as bad as in the sciences, though, I think). Even within one area, saying someone “is the best” makes no sense - there is usually a handful or so (depending on the size of the area) of top people in an area who produce the truly daring new results and breakthroughs - and the rest of us who live off the fact that those handful of top people don’t have enough time to follow all the ramifications of their work.

  6. on 11 Jul 2007 at 11:52 am 6. Oaktown Girl said …

    I saw the math-themed movie called “Pie” (math symbol for 3.14…) but it didn’t stick with me, so I don’t remember much about it (plus, I saw it years ago at a film festival when it first came out).

    I also have seen “Good Will Hunting” of course, and within the past year, I rented the movie “Proof”. I always wonder what real mathematicians think about those movies. If you’ve seen them, can you tell me what specific categories of mathematics they are dealing with? What do you “real” mathematicians think about those movies?

  7. on 11 Jul 2007 at 12:03 pm 7. christian h. said …

    Sorry, I haven’t seen a single one of those movies - mostly on suspicion that I would just cringe all through them. I’ve heard good things about Proof - the play anyway; bad things about Pi; and “yeah right, now everybody will think again that you don’t have to work hard to learn and do math” about Good Will Hunting. My impression was that these movies are more about the mathematician as a person than about any specific mathematics.

  8. on 11 Jul 2007 at 12:37 pm 8. Oaktown Girl said …

    My impression was that these movies are more about the mathematician as a person than about any specific mathematics.

    Well, of course because it wouldn’t have much broad appeal otherwise. Good Will Hunting seemed to be about someone who was an out-of-this-world math genius: a primitive with no formal training blowing the minds of professional mathematicians. I was curious what you thought about that, but obviously you can’t say if you haven’t seen the movie. Maybe James or Bill (or a math lurker coming out of lurking) can share an opinion.

    I enjoyed Proof just fine. I was thinking I may have enjoyed it more if I better understood the math theory they were talking about. But maybe not. There were some math specifics in both movies that you would have been able to identify, but of course I couldn’t tell you what those were.

  9. on 11 Jul 2007 at 12:38 pm 9. JP Stormcrow said …

    Okay, is there where I confess to having a B.A in Mathematics? Nah, too embarassing and too many explanations required, so I won’t. (or maybe I already have … 4 months of blog time is like, gee I dunno, a whole big number of human years.)

    1) Actually, I found my first “real” job via an ad looking for “Math Majors”. I was puzzled after the interview that they would so advertise what was barely an entry-level clerical/technician job. The answer from my cagey new boss was basically that he had found that they generally got smart people who were willing to take the job (i.e. had no other job prospects…) [I know that this is not generally true, but like any “traditional” Liberal Arts degree, if you didn’t catch one of the law/med/grad school or internship trains you were a bit adrift.]

    2)Large numbers of published works are error-ridden

    Somewhere I have a great monograph by Donald Knuth entitled On Theory and Practice (the online stuff of his by that name is related but quite different.) In it he briefly describes the development of ΤeΧ and notes that after the pseudo-code specification was done, it was about as complex as a long “proof” that might be published in a Math journal. He then says that if it were a proof and was so published that it would be reviewed and read by a few practitioners in the area and that a some errors would be found and corrected. Instead since it actually needed to be compiled on a computer and executed, much more effort was expended in actually getting it “right”. In his view the ability to actually test “theories’ in practice was a real feature of computer “science”.

  10. on 11 Jul 2007 at 12:40 pm 10. Zeus said …

    Ahh, the sand mandala of mathematics: To build an intricate design with painstaking attention and energy only to have it swept away. I hadn’t thought of mathematics as Buddhist practice (relinguishing attachment, etc.), but now I can see how it might be.

    Some have called mathematics the “language of God” (which may be closer to the truth than fundamentalists who attempt to take the Bible too literally). My intuition is that mathematics (having only a concentration in math as an undergrad, I don’t know squat about higher theoretical math) is both discovery and invention and even aesthetics, and probably more things besides. There are some aspects of mathematic description that seem to transcend its human origins, pointing to a world being creating beyond our imaginings (I’m thinking primarily of quantum physics, for one), but there are other aspects that are quite practical (and still spectacular) like some of the engineering aspects of math and ways complex algorithms can describe sensory behavior and human traffic patterns, etc. Then there are the elegant three-dimensional designs from integral calculus and paisley/sea-shell fractals from non-linear iterative functions.

    I am inclined to think of mathematics culture, itself, as a kind of interactive cultural substrate that articulates the as-yet unarticulated possible lines and axes of the universe. It is, as Christian describes it, a collective work of art, requiring creativity, analysis, reflection, intuition, patience, often collaboration. Math culture for me, then, is an odd kind of “faith” community, as one never exactly knows when one might be visited by an angel of insight, nor where that insight will lead, nor by whose auspices or which connections one might alight on something useful. The sum totals over time of the so-called failures and successes are a kind of landscape that have produced some fairly spectacular applications and even more spectacular questions.

    (Of course there are also the genius hermits like Grigory Perelman [http://www.nytimes.com/2006/08/22/science/22cnd-math.html?ex=1184299200&en=e88928d5fe503dae&ei=5070], I wonder what your thoughts are on him, Christian?

    Citizen Zeus

  11. on 11 Jul 2007 at 1:59 pm 11. James Killus said …

    The protagonist in Good Will Hunting was not “merely” a math genius, he was an all-round supermind, with an eidetic memory plus nearly complete comprehension of everything he ever heard or read. The authors seemed to have taken every story about every prodigy ever seen and rolled them up into one person.

    He was also good in a bar fight. Quite a fellow indeed.

    And yes, it is a sampling issue to refer to “the best mathematician I know,” in that by “mathematician” I’m referring to the ability to construct formal proofs (and that’s a very general category in itself), as opposed to someone who uses mathematics in some specific, or even practical fashion. For example, although my friend has taught college level courses in statistics, I’m quite certain I know several people who are better statisticians than he is, I know several better numerical methods folks, etc.

    That’s one of the annoying things about the TV show Numb3rs, of course. Charlie is too much the polymath, and spends far too much of his time doing things that engineers, physicists, and computer scientists do, and very little time on “real math.”

  12. on 11 Jul 2007 at 2:38 pm 12. christian h. said …

    Now Numb3rs I’ve actually watched a couple episodes, and enjoyed them. The mathematics seems to be correct - it is applied math, of course, by its very definition, but hey, if it gets us more funding convinces people that math is actually very diverse, and widely applicable it’s all good.

    One thing you clearly can say about Perleman is that he puts his money where his mouth is. He believes that it is the work that should be honored - not the creator, so he doesn’t accept the Fields medal. He has basically no job (of course he seems to largely live off his mother’s small pension as far as I know, and we could discuss if that’s the ethical thing to do when he could easily provide for himself).

    As for discovering errors in published work: a computer program is different in that a tiny error - amounting to a typo - can and will stop it from functioning correctly (at least that’s my experience). That’s because the device “reading” the program, the computer, isn’t very smart. A human reader of an article in mathematics - or any other subject, for that matter - will automatically correct for the smaller mistakes.

    If I read a math paper and actually have to read the details of the proofs, I haven’t really understood what’s happening. Once I do understand, I simply absorb the basic structure of the proof and am confident that I can fill in the details if needed. The human mind develops an intuition for the formalism (I believe we discussed this before when James posted on spatial awareness).

    Plus, there’s a lot that is just routine (this is the reason I don’t know any mathematician who likes Good Will Hunting - like everything else, learning math is repetition, repetition, repetition). The problems I did in my first years at the university (comparable to senior level/ first year grad school stuff in the US) seemed to require an “idea” at the time - now they look no different to me than simple calculus.

  13. on 11 Jul 2007 at 2:43 pm 13. Kiera PSI said …

    I have to admit that I am not a math aficionado. I’m fine with basic math, and work well with the basic theorems of geometry, but take me beyond that and I’m totally, irrevocably, LOST.

    Christian, as a mathematician, have you noticed that at the lower levels, people who do well in geometry generally don’t do so well in algebra and visa versa? Or have I been imagining the phenomenon? I’m making the assumption that those that continue on to “higher math” and become mathematicians and the ilk are those that don’t have a problem with either discipline.

  14. on 11 Jul 2007 at 2:57 pm 14. christian h. said …

    Okay, am just reading the script for Proof. First thing I notice is typical U of C arrogance - putting down Northwestern. Hah!

    And here’s a review (pdf file) of the movie, again from the Notices. Trust me, no mathematical knowledge is required to read it. If I understand correctly, there’s very little actual mathematics in the film.
    The father character apparently worked in game theory (like John Nash, whose life is described in A beautiful mind; is important in economics and related fields - RAND used to be big in it, coming up with theories about winning nuclear wars and stuff), algebraic geometry (that’s my area, by the way - roughly speaking, we study geometric objects that can be described by polynomial equations, that is, equations whose terms can be gotten from the variables using only the operations of multiplication, addition and subtraction - no sines or exponentials) and operator theory (important in quantum mechanics and its further developments).

  15. on 11 Jul 2007 at 3:16 pm 15. Oaktown Girl said …

    The protagonist in Good Will Hunting was not “merely” a math genius, he was an all-round supermind, with an eidetic memory plus nearly complete comprehension of everything he ever heard or read. The authors seemed to have taken every story about every prodigy ever seen and rolled them up into one person.

    Yes, that’s right. There seemed to be absolutely nothing that was beyond that character’s genius grasp. I’m getting irked just thinking about it. Watching that movie, you are led to believe that he could have just as easily been a musical prodigy (any instrument, any style) and/or an Olympic athlete (winter or summer, multiple sports). Plus, of course was charming, grounded, and wise, good-looking, high-functioning socially, and with no outstanding personality quirks.

    Yeah, that’s realism, for ya.

  16. on 11 Jul 2007 at 3:42 pm 16. The Constructivist said …

    Christian, thanks for showing me my life wouldn’t be all that different if I had gone into grad school in math rather than English!

    For me, though, I couldn’t bear to go through the rest of my intellectual life feeling I had jumped out of an airplane not knowing if the chute would deploy or not.

  17. on 11 Jul 2007 at 4:31 pm 17. christian h. said …

    Christian, as a mathematician, have you noticed that at the lower levels, people who do well in geometry generally don’t do so well in algebra and visa versa? Or have I been imagining the phenomenon?

    Good question. I haven’t noticed that, but it’s definitely true that in doing research mathematics one of the important skills is to translate one into the other. Or rather, to view both as the same thing, really. For example the main idea behind modern “algebraic geometry” is to make the following steps:

    1. Start with a geometric object, e.g., a circle.

    2. Notice that the object can be described as set of solutions to some equation - in this case a circle is “the set of all points equidistant from the center”, which suitably expressed in algebraic terms becomes “pairs of real numbers x and y whose square adds up to 1″ (that would be a circle of radius 1 around the point with coordinates (0,0)).

    3. Forget the real numbers, and the geometric object. Instead, just study the algebraic properties of the equation.

    In general I am guessing that geometric intuition and spatial skills are different from those required for algebraic manipulations. Does anyone know more?

  18. on 11 Jul 2007 at 5:45 pm 18. The Constructivist said …

    Not a thing. But I can say that the quality of the teacher makes a big difference. Most of my math profs were inspired teachers, but the one that wasn’t made upper-level calculus a real grind for me, and I couldn’t figure it out or keep up with it on my own.

    As I seem to be in otaku autoethnography mode again (when I’m not reading onechan Uncle Bill Benzon’s gojira tales), despite the bad calculus experience, I did get into non-Euclidean geometry and topology at the end, but I was actually thinking through metaphors, so I was mostly a mathematical fantastist (kind of Platonic idealism meets Guy Gavriel Kay). That is, my “wouldn’t it be lovely?” thought was what if all those fantasy novels where the protagonists leave our reality for some suitably Tolkienesque “one true realm” where everything they do matters for all the rest of the realities could be expressed in mathematical terms? Like a three-dimensional object casts a two-dimensional shadow, so changes in that object cause changes in the shadow, what if they went to some n-dimensional space and everything they did there affected all the other n-1, n-2, n-3 -dimensional spaces….

    You can see why I left math. Figuring out the logic behind existing theorems and writing them up as proofs was hard enough and painful enough (cf. “will the chute open?” analogy above), so I figured actually coming up with new ones would be much harder and much more painful.

  19. on 11 Jul 2007 at 5:50 pm 19. The Constructivist said …

    BTW, I’ve seen courses, Bloggy inquiries, and lists devoted to math and sf, but was wondering if the math and fantasy angle has been discussed as much among actual mathematicians….

  20. on 11 Jul 2007 at 6:01 pm 20. christian h. said …

    There is Flatland, of course. It is about a people living in two dimensions. One day, a (three-dimensional) sphere shows up and intersects their world. Now how could they recognize it? Think about it - how would you recognize that a four-dimensional sphere (still with me?) intersect our threedimensional space? The give-away woyuld be that it suddenly appears out of nowhere as a tiny little dot, becomes a larger and larger balloon, and then shrinks back into nothingness. But I don’t know specifically about any math and fantasy stuff, apart from not-so-serious stories (like, the Klein bottle as some kind of conduit ti the spirit world and such).

  21. on 11 Jul 2007 at 6:07 pm 21. The Constructivist said …

    Hey, maybe I chose the wrong field–here’s a complex take on math and metaphor and here’s a simple one! Both reference Núñez and Lakoff’s Where Mathematics Comes From. Christian, did this get much attention among mathematicians when it came out in 2000?

  22. on 11 Jul 2007 at 6:11 pm 22. The Constructivist said …

    By the way, that John Allen Paulos guy is a good writer. Any other mathematicians writing for general audiences you’d recommend, Christian? Math public intellectuals, so to speak….

  23. on 11 Jul 2007 at 9:21 pm 23. JP Stormcrow said …

    My vote for the most annoying “mathematical” fiction is Ratner’s Star by Don DeLillo. I liked some older DeLillo (and have Underworld and White Noise on my non-shrinking list), but Ratner’s which is centered around a teenage mathematical prodigy and possible signals from outer space, is a mess - but hey! only 448 pages of turgid, frustrating prose to wade through - get right on it christian and report back to us.

    In the tradition of Flatland, which is delightfully subtitled: A Romance of Many Dimensions is AK Dewdney’s Planiverse, which extends the concept to how a lot of different devices “work” in the 2-d world. (Dewdney took the bloom off a bit by devolving into an obstreporous 9/11 crank.)

    G. H. Hardy’s A Mathematician’s Apology is worth a read - he is unabashed in his “competitiveness” and penchant for ranking. And Hardy is of course known for his connection to Srinivasa Ramanujan, the wunderkind Indian mathematician.

    I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

    I had not heard of Grigory Perelman, Ramanujan seems somewhat similar.

    Dr. Free Ride has a post up describing OpenWetWare, an effort to promote the sharing of information, know-how, and wisdom among researchers and groups who are working in biology & biological engineering. I wonder if there are similar efforts in mathematics?

  24. on 11 Jul 2007 at 9:31 pm 24. JP Stormcrow said …

    On a more ridiculous note, I learned a lot today about the true depths of nuttiness of Lyndon LaRouche following links from a Scott McLemee post at Crooked Timber. I knew about the weird politics and conspiracy theories, but did not realize the depth of the wackiness and that it extended into a lot of recent “work” and theories (and conspiracy theories) on scientists and mathematicians. (For instance Kepler, Leibniz and Gauss rule, Galileo, Newton and Cauchy drool!) Check out any issue of DYNAMIS: The Journal Of The LaRouche-Riemann Method Of Physical Economics for some math and geometry with attitude.

    … to treat the effects of such principles in the way in which the satanic Sarpi’s lackey Galileo had attempted sodomic rape on the body of Kepler’s discoveries of the Creator’s universal physical principle [emphasis added]

    and

    Here, Stewart [author of a Calculus textbook] tacitly ignores the discovery of Cusa, that the circle is not a polygon! This might seem a fine point, but it happens to be quite crucial for the future of humanity.[emphasis added again]

  25. on 11 Jul 2007 at 10:17 pm 25. JP Stormcrow said …

    One final quickie (going for the consecutive comment record here … watch out, TC).

    Probably should have a whole thread on the relation of mathematics and reality - it has always fascinated me.

    In particular the “real” numbers have always seemed to have a tenuous connection to the “real” world, other than that they work so freaking well for so much. (and to me this is connected to the dubious status of “limits” — and this is sounding a bit like those LaRouchians.)

    There’s something happening here
    What it is ain’t exactly clear

    This is more Physics than Math and worth a longer discussion (and someone much more knowledgable than I).

    If nothing else I would like to find again a book I read in the ’70s that developed a”discrete” Newtonian physics. One of the author’s points was that in retrospect (and admitting that it was with 20/20 hindsight and the benefit of information theory and the “computing paradigm”) , why would you expect “infinite precision” real numbers to be the ground truth - runs into problems on informational grounds alone. My personal version of this that I developed as a teenager and had fights with Physics types about (and therefor am probably full of shit on it .. for instance see here) is that one resolution of the Achilles paradox is rather than resorting to “limits”, that when the distances get close enough and the time units small enough the notion of “ahead” and “behind” and “now” become imprecise and the whole I am now exactly where you were then construct loses meaning.

    Anyway, JP proving he can “quack” like a duck. But if nothing more I WANT TO FIND THAT BOOK. It probably says nothing like I remember - and the fact that I cannot dredge it up via teh Google probably attests to its importance… but I want it.

  26. on 12 Jul 2007 at 4:59 am 26. JP Stormcrow said …

    I have to check no more than a couple papers a week
    Good so you have plenty of time to read Ratner’s Star and all that great LaRouchian geometry.
    Just wanted to break up JP’s string of comments
    PWNED!

  27. on 12 Jul 2007 at 5:37 am 27. christian h. said …

    Mathematicians use preprint servers to share knowledge, and conferences. You have to understand there aren’t that many of us. In my specific area (algebraic K-theory) there are maybe 200-300 serious researchers worldwide. Our conferences are kind of like a travelling circus, I always say. The speed with which new research is produced is much slower than in the sciences (on the other hand, nothing ever becomes “outdated”, really - at worst, something can seem old-fashioned). I have to check no more than a couple papers a week (physicist friends of mine claim they have to read dozens of abstracts a day). Anyway, gotta run. Just wanted to break up JP’s string of comments…

  28. on 12 Jul 2007 at 8:54 am 28. nnyhav said …

    New day coming.

  29. on 12 Jul 2007 at 3:11 pm 29. James Killus said …

    In rereading this comments thread there are three separate tangents I’d like to go off on, but I’m just going to briefly note them rather than follow them, because, hey, only so many hours in the day.

    If reverse order, I’m reminded of the old saying, “You need real numbers for rational problems and rational numbers for real problems.” I had a professor at RPI who expressed the opinion that differential equations were only a special limiting case of difference equations and that computing would eventually convince people of this simple truth and difference equations would be taught instead of differential equations, save for some purely mathematical courses.

    Hasn’t happened yet, of course.

    Farther up, in the back-and-forth on math prodigies and prodigies in general, I’m reminded that certain areas of mathematics, like some sorts of music, are ammenable to prodigies because they do not depend upon experience so much as knowledge and thought. Some fields of knowledge and achievement simply demand a certain amount of living-through-it, quantity time as opposed to quality time, to borrow a phrase from another kind of discussion. It’s very hard for me to imagine a 12 year old writing a great novel, for example, or being a brilliant political consultant.

    Finally, there is in certain areas of pop culture creation something that is sometimes called “the Superman Problem,” though I’ve seen it in another context as “the Natty Bumppo problem” Simply put, the tale of the “competent man” tends to head toward making him a paragon, and the saga of Superman winds up with a hero too powerful to have a meaningful dramatic opposition. There was a time in the 1950s, when the Superman comics character could re-ignite dead suns with his heat vision. So the stories tended to wind up being either Superman once more twarting Lois Lane in her quest to prove that Clark Kent was Superman, or Lex Luthor finding one more way to put some Kryptonite into something he hadn’t put it in before. Eventually they ramped down old Supes’ powers a bit, but the Problem remains.

    Ah, and one more tangent: the “super smart guy” thing in science fiction has often wound up giving the protagonist some kind of psychic powers, which feels like an out and out cheat. The real dynamic in “super smart guy” stories can either be on its effects on the guy (which is seldom satisfying because SF writers aren’t that smart) or its effects on the people around the guy. In that sense, one interesting feature about superhero stories isn’t that they are super, but that they are heros, i.e. that they are admired, rather than outcasts. Of course, there are many other stories that take the opposite approach.

  30. on 12 Jul 2007 at 3:58 pm 30. Oaktown Girl said …

    I’m reminded that certain areas of mathematics, like some sorts of music, are ammenable to prodigies because they do not depend upon experience so much as knowledge and thought.
    Image Hosted by ImageShack.us

    Don’t sue! Buy this t-shirt here!
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  31. on 12 Jul 2007 at 5:18 pm 31. Oaktown Girl said …

    While I was able to make a grand slam reply to a fantastic set-up line by James, I hope I have not inadvertently derailed truly thoughtful responses to his insightful and well-thought out comment.

    So whoever may be lurking about, pay no attention to the dulcet tones of Ralph Wiggums’ nose-o-phone and please chime in!

  32. on 12 Jul 2007 at 7:09 pm 32. The Constructivist said …

    Little did Oaktown Girl know that with one pithy comment she proved the eternal value of the WAAGNFNP….

  33. on 12 Jul 2007 at 9:12 pm 33. Oaktown Girl said …

    Well,thanks, TC. But the image (in #30) got vaporized from my ImageShack account. Sometimes that just happens - an image will disappear and then reappear a little later, just a temporary glitch in the system. But I’m not going to take any chances, so I’ve edited the comment to link to where I found the image, and even spelled out the website name. I would have credited the image in the first place except that I was at work when I did it and that’s always a combo-plate of rush-job and sneaking around the boss.

    Anyway, maybe the image won’t disappear again now. We love you, Ralph Wiggum!

  34. on 12 Jul 2007 at 9:16 pm 34. James Killus said …

    Hey, I’m still stuck trying to get my heat vision to ignite this match…

    I think I’ll light this one with my good looks. –The Congress of Wonders

  35. on 13 Jul 2007 at 7:39 am 35. christian h. said …

    Quick report by the Tribunus Laticlavius from an undisclosed Cafe somewhere in a large Midwestern city. James is right that there are areas of mathematics where not much formal education is needed - but only a few, and even in those usually heavy-duty methods have a much better chance of success (yeah, I admit it, I hate combinatorics! Stop nagging!).

    Oaktown Girl’s picture is priceless. Reminds of something my father used to say: a child prodigy is someone who’s as smart at age five as (s)he is at age 50.

    As far as the Superman problem: I’ll just say I don’t like Q. Never have. All-powerful characters stink.

    Finally, as for the world being discrete (well, more like “foam” if you take quantum field theory into account): true enough, but that doesn’t undermine the usefulness of limits at all. That’s like saying “who needs statistics, we could just observe every single event” or something…

  36. on 13 Jul 2007 at 9:44 am 36. JP Stormcrow said …

    “who needs statistics, we could just observe every single event” or something…

    Well, we could. That is what Patch-22 is for.

    In part how I am led to probably overvalue the role of the discrete is wrapped up in another longer topic that I will just touch on. How the powerful computing/information metaphor informs so much of my (and others) worldviews and to some extent “traps” us in that narrative. Similar to how Victorians adopted “clockwork/mechanical” narratives of the world - Twain has some interesting writings in that vein.

    That is not to say that the computing/information paradigm is wrong, it just isn’t as right as we often think that it is. But it has aided progress on many problems that we were stuck on before. (I view much of the evolution/genetics/protein story as being part of this whole comprehensive metaphor.)

    As I said - topic for another day, but the real trick is to figure out the next big “mindset” now. Then you can play Babbage and Ada, and maybe get a crappy programming language named after yourself. (Actually you’ll get whatever is the equivalent of a crappy programming language in the new “paradigm” named after you.)

    .. my bet - it will involve, really internalizing the parallel vs. serial views of processing and processes - you can see the seeds already. But we may be individually too dumb to ever really conceptualize it.

    Oh, but I still ♥ limits… and always will.

  37. on 13 Jul 2007 at 10:48 am 37. Oaktown Girl said …

    I’m pretty pressed at work here today, no time to google or wiki anything. Can anyone give a quick primer on limits and discrete and such? Is there such thing as a quick primer on those?

    Christian - yeah, that illustration of Ralph Wiggum is fantastic. I really appreciate illustrations that convey a lot of emotion or story line and do it so simply and seemingly effortlessly. I guess that’s what you would call “deceptively simple”.

    The slight upturn of the nose where he’s shoved the clarinet in, the suit and tie denoting this is a formal performance or recital, and the perfectly blissful expression on his face - so happy and yet so completely unaware (which is maybe why he’s so happy).

    Come to think of it, you could almost just as easily switch Pooh with Ralph and have The Tao of Ralph Wiggum.

  38. on 13 Jul 2007 at 12:08 pm 38. JP Stormcrow said …

    Pushing along another one of the divergent threads which have emerged here.**

    I too get very tired of superhero’s powers - because I never really understand the limits. Is Superman having a good fight or a bad fight? Who the hell knows? Interesting that they ramped his powers back at some point. Usually in the interest of fresh sensation they keep pushing it higher to the point of absurdity. For instance in the Matrix series. (And I assume that it has been noted that Jedi skills seemed to deteriorate between the time of Episode 3 and 4.) It is one of the things that detracts a bit from Harry Potter for me, and leaves me very cold towards things like Crouching Tiger.

    ** Back at MB’s place I once posted my variant of the Cauchy convergence test for Internet discussions.

    A discussion is convergent, if and only if, for any amount of attention a>0 there is a reply N such that any subsequent subsequence of replies generates less total attention than a.

  39. on 13 Jul 2007 at 12:33 pm 39. JP Stormcrow said …

    Oaktown,

    I saw at Wikiversity that they have a module on Introduction to Limits, but it presumes familiarity with the concepts in College Algebra - which they also have. I have never gone through them, but from a cursory glance they look to be reasonable.

    Here is my Bush Impeachment Paradox to illustrate limits.

    If Bush is a “distance” x from being impeached and by every subsequent action he cuts that distance in half, will he ever reach impeachment. (Assume each subsequent action occurs in half the time of the previous action.)

    Answer as if you are:
    a) Nancy Pelosi
    b) David Broder
    c) Glenn Reynolds
    d) Sparky the Wonderdog
    e) none of the above.

  40. on 13 Jul 2007 at 1:30 pm 40. Kiera PSI said …

    Answering as “Sparky the Wonderdog”:

    Nope, nope, nope, so I’ll just save us all a lot of grief and bite ‘em in the ass now!

  41. on 13 Jul 2007 at 1:41 pm 41. Oaktown Girl said …

    Answering (#39) as Nancy Pelosi -

    Impeachment for Bush is forever “off the table” - the corporate media and right wing punditocracy might say mean things about us. However, I’m ready to bring Articles of Impeachment up against Sparky the Wonderdog right now for biting President Bush in the backside. I’m sure the Republicans will support me on that, and their approval is the only thing I care about, even though no matter how much I cave in to their agenda I’ll never get it. (Just like Bill Clinton).

  42. on 13 Jul 2007 at 1:56 pm 42. The Constructivist said …

    On superhero powers, the whole Dragonball franchise shows the only way to go is self-parody. Wait, so does The Authority. That’s why Ambush Bug was the best Superman character ever! (Miracleman may be the exception that proves the rule; I say may b/c I never read it till the end)

    Same problem with magic in fantasy and technology in sf, eh?

  43. on 14 Jul 2007 at 3:49 am 43. Bill Benzon said …

    On behalf of all past, present, and future band members, I think it’s time we bring on the GNF, starting with the benighted West Coast of the USofA.

    In particular, on behalf of the late Rahsaan Roland Kirk, who really could blow it out his nose — I’ve seen and heard him do it — I protest. Ralph Wiggum IS NOT, I repeat, IS NOT, a member in good standing of the International Guild of Nose Instrumentalists (IGNI — which is also the start of “ignition” as in “bring on the GNF”)..

    Finally, and speaking from experience, if you want to get a giggle out of 5 year olds, while at the same time making adults a bit queasy, take that plastic recorder or — even better — a finely crafted wooden one from Germany, stick it in which ever nostril is most open at the time (it alternates), and give it a good blow. Works every time.

    * * * * * *

    On the subject of mathematics, I’ve had no training in math beyond a rather so-so high school education that stopped with trigonometry — though I did make it through a rather rigorous couse on symbolic logic in college that went as far as defining numbers in terms of set theory. But, I’ve got decent mathematical intuitions and have spent a lot of time with people who know and use real math — though none of them were/are trained as mathematicians. I sometimes think of myself as being in the business of taking soft mushy stuff like poetry and music and developing ways of thinking about them that will require real math to take the ideas further.

  44. on 14 Jul 2007 at 7:52 am 44. christian h. said …

    Oaktown Girl, very quickly and vaguely about limits, and “continuity”: let’s say you take the function that assigns to each point on the earth’s surface the surface temperature. Now take some point on earth, say the intersection of State Street and Congress Parkway in Chicago (that’s where I am right now). If you start somewhere else and walk towards that point, then the temperature will “approach” the temperature at that intersection. In other words, if you take two points very close to each other, then the temperatures won’t be very different; in fact, if you get close enough you can make the temperature difference as small as you wish (it can change quickly - think of a wall between a heated room and Januray Minnesota - but it won’t “jump”).

    At least, that’s the way the world used to be understood. But it presupposes that there is such a thing as an arbitrarily small increment in temperature, and arbitrarily small distances - which, it turns out, isn’t quite right.

    “Discrete” means that there is some minimum distance, or temperature difference you can have. The real world seems to be even more complicated than that, though.

  45. on 14 Jul 2007 at 11:23 am 45. Oaktown Girl said …

    Ralph Wiggum IS NOT, I repeat, IS NOT, a member in good standing of the International Guild of Nose Instrumentalists (IGNI — which is also the start of “ignition” as in “bring on the GNF”)..

    What?? That’s an outrage! He should at least be made an honorary member, or official mascot!

    if you want to get a giggle out of 5 year olds…

    OK, well one of Uncle Bill’s duties at the WAAGNFNP World Gathering ‘08 will be entertaining the kids…or the ones that haven’t been eaten by Gojira or Lord Astaroth, or “accidentally” put in The Trunk by the Minister of Justice.

    Just kidding. I like kids. I really do. Except bratty ones. The brats (young or old) at the ‘08 World Gathering really are going in The Trunk.

  46. on 14 Jul 2007 at 2:29 pm 46. Kiera PSI said …

    Who have you been stuffing in my Trunk now? Sheesh. No kids in the trunk, they make too much of a mess…they turn it into a “fort”, or a “cave”, or something else “fun”. Only adult miscreants treat it with the proper respect and fear it as the extreme punishment it is.

  47. on 14 Jul 2007 at 3:55 pm 47. Oaktown Girl said …

    OK, OK. No kids in The Trunk. We can’t have The Trunk getting messed up by a bunch of snotty brats.

    Maybe for the kids we’ll just go with cages instead. Cages are easier to hose down.

  48. on 14 Jul 2007 at 5:06 pm 48. Bill Benzon said …

    The kids can come over to Mostly Harmless where I’m telling stories about Gojira and her friend, Sparkychan, not to mention onechan and imoto and whomever else wanders through cyber-Kansas.