Astute readers may have noticed the lack of a calculus status report last week. Alas, my "real" work intervened. But I'm now up to Lecture 7 ("Derivatives the Easy Way!", slated for next week), so progress has been made. However, I'm finding there's quite a bit of repetition in this little endeavor of mine, which makes writing incredibly long detailed posts about the week's lessons a bit too redundant; I like a bit more variety on my blog. So instead, from now on I'll just mention in passing how things are going unless a particular lecture sparks an interesting train of thought. In this case, I got to thinking about the literary concept of *mimesis*, and how it applies to learning in general, and learning about math and physics in particular. (I __am__ a former English major, after all, and even had a year or so of grad school before opting out of academia altogether.)

First, a bit of history. Back in ancient Greece, *mimesis* primarily referred to the artistic representation of nature, although two philosophers in particular differed dramatically in their interpretations of it. In one corner, I give you Plato, of cave allegory fame, who believed in a divine realm of Ideal Forms; all creation, including Nature, was imitation in his eyes, and artistic imitation was by definition twice-removed from the Ideal -- ergo, all art, or other created fictions, is inferior to the "real" world, which is in turn inferior to the Realm of Ideal Forms. (Incidentally, one of Jen-Luc Piquant's favorite obscure books is *The Place of the Lion*, by Charles Williams, a former crony of C.S. Lewis. Williams was a bit weird, and his novels are even weirder: *TPotL*'s plot revolves around the real world being invaded by the Platonic Ideals, with catastrophic results.)

In the other corner we have Aristotle, who took some time off from insisting we see by shooting rays of light out of our eyes that reflect off nearby objects, to write his famed treatise, *Poetics*. Aristotle was a bit more forgiving of mimetic "make-believe," believing human beings have an inherent need to create artistic fictions as a form of catharsis, although he valued tragedy over comedy, via a rather convoluted process of reasoning. (He was wrong about how human vision works, too.) But our modern aesthetic still owes something to Plato and Aristotle, both of whom distinguished between *diegesis* -- the act of telling, such as indirect narration of action, or, say, lecturing to students about calculus -- and *mimesis*: the act of showing a character's internal thoughts and emotions via his/her external actions. In fact, it's almost a dictum of today's theater, film, and literature: show, don't tell.

Anyone who's taken Philosophy 101 could tell you that much. But in 1946, a literary scholar named Erich Auerbach appropriated the notion of *mimesis* for what his Wikipedia entry claims is "one of the most ambitious works of literary theory undertaken." Indeed, *Mimesis: The Representation of Reality in Western Literature*, is pretty much required reading for serious students of art and literature; it had a profound effect on my undergraduate self, and a copy still graces my bookshelves. Auerbach analyzes literary conventions throughout the history of Western Europe and how they create "a lifelike illusion of some 'real' world outside the text."

Perhaps a bit too lifelike at times: Australian cyber-psychologist Andrew Campbell of the University of Sydney has noted a distinct rise in the number of what he terms "TV show obsessives," even singling out *Buffy* fans as an illustrative example. Apparently we can "develop fantasy-prone personality disorder," to such an extent that we start to focus over much on the fictional "reality," creating fan sites, chat rooms, organizing conventions, writing fan-fic, and the like. Some people have even been known to write entire books analyzing such fictive universes. (Bibliophile Jen-Luc recommends Jasper Fforde's hugely entertaining Thursday Next series of novels, which plays on this innate human tendency by presuming the existence of an entirely separate "reality" for literary plots and characters.) My college English professor described the mimetic moment as the point at which an individual makes the critical connection between his/her experiences and the artistic work and realizes, "Aha! __This__ is __that__!" This kind of emotional and intellectual resonance on the part of the audience is what makes the creative arts so powerful.

I've spoken to many a scientist who was inclined to agree with Plato in de-valuing fiction, which is a shame, because I would argue that created fictions present a uniquely effective teaching tool, a way to supplement rather dry college lectures (*diegesis*) with a dose of creativity (*mimesis*) to spark students' excitement and interest in a given subject. It's a critical component of acquiring true knowledge -- actual learning, as opposed to memorizing facts by rote (dubbed "plug and chug" by some educators). Every teacher I know is heartened whenever they see that light bulb of genuine comprehension turn on in a student's brain: "Oh! __This__ is __that__!" In the same way that our favorite works of art, literature, music or theater tend to be those with elements we recognize and can respond to emotionally, we tend to respond more to books, lectures, or classroom curricula that enable us to make similar connections between, say, the abstract concepts of math and physics, and our real-world experiences. If our emotions are engaged as well, even better: that excitement and enthusiasm serve to fuel students' desire to persevere past the inevitable frustrating roadblocks in the quest for knowledge. It's all about making those crucial connections.

What prompted this sudden lapse into deep reflective thought, you may be wondering? Chalk it up to my initial failure to make a crucial connection in my introduction to the fundamental theorem of calculus: the notion of the function. Basically, a function is a set of ordered pairs -- say, *x* and *y* -- where, for every value of *x*, there is only one corresponding value for *y*. This is the sort of jargon that makes my eyes glaze over in record time; it's much easier to wrap my brain around the notion that the function provides a critical link between the derivative and the integral in calculus. The two are essentially two different ways of looking at the same situation, in the given case, a car moving down a straight road. (For a more detailed explanation of those two concepts, see this prior post.) Using the derivative, we can find our speed from our position (a process of subtraction and division); using the integral, we can figure out how far we've traveled based on our speed (a process of multiplication and addition). But the second method is a heck of a lot more labor intensive.

Fortunately, there's an alternative method thanks to the aforementioned fundamental theorem. Normally, if someone asked us to figure out how far we've traveled knowing just the velocity of the car at each point in time along the road -- the so-called velocity function -- we'd use that tediously time-consuming integral method. But it turns out that we'd get the same answer if we just subtracted our beginning position from our end position (the position function). Of course, we wouldn't know our exact ending position offhand, but we would if we could just figure out which position function generates the known velocity function. Apparently, __every__ position function generates a velocity function, and you simply need to trace your steps backward, so to speak. Then you can use the far easier derivative method to get your final answer.

Sounds straightforward enough, right? Except one crucial step was missing: the introductory lecture on the function fails to explain exactly how one finds that position function. Apparently we're going to cover that in Lecture 7. Until then, I'll just have to take the assertion on faith. This is a minor quibble I have with my DVD instructor: insanely slow pacing in his presentation of the material. Enough with the niggardly portioning out of tiny snippets of information, as if they were clues to a mathematical mystery, the solution to which one isn't quite ready to give away just yet. It's one thing to carefully build one's case for calculus, step by step, so as not to lose students along the way. It's quite another to hoard information that could be crucial to clear understanding. Students like me need to have that critical mimetic moment!

Apart from its usefulness as a pedagogical tool, this same interpretation of mimesis ("Aha! __This__ is __that__!") could easily be applied to some of the most revolutionary ideas in physics. Albert Einstein once credited his development of special relativity to a critical insight gleaned years before, as he sat on a train moving away from the station platform -- namely that he would measure time differently from within the moving train than someone standing on the platform. The same goes for Isaac Newton, co-founder, with Gottfried von Leibniz, of calculus. Legend holds that it was watching an apple falling from a tree that gave Newton his critical insight into gravity and his laws of motion.

Newton also provides a useful example of the failure to evoke a critical connection or mimetic response in an intended audience, via a 1676 letter to Leibniz in which he attempted to stake his claim to the "invention" of modern calculus. He employed a Latin sentence that served as an anagram: rearrange all the letters in alphabetical order, and voila! Proof that he had prior knowledge of the key concepts. Except who has time to go to all that trouble? Not me. And not Leibniz, either. Even decoded, it wasn't entirely clear what Newton was getting at. So Newton was too clever for his own good, and the result was an especially long and acrimonious controversy. (Jen-Luc Piquant was delighted last week to find a post by Dave Bacon -- a.k.a., the Quantum Pontiff, often referred to as "His Holiness" by physics bloggers -- detailing, with photos, a re-enactment of the legendary rivalry between Newton and Leibniz, with the help of Shtel-Optimized's Scott Aaronson. That's an example of humorous mimesis: the audience laughs because it makes the connection between historical "fact" and the skittish "fiction" presented on the stage.)

This seems as good a point as any to segue into discussing a few of the niftier recent applications of mathematical models. Because what are mathematical models, if not visual representations of abstract concepts, with the added advantage of enabling scientists to make useful real-world predictions? Okay, not all mathematical modeling has a practical application. Topologists, for example, are more interested in studying imaginary multidimensional shapes that couldn't exist in our four-dimensional spacetime. When was the last time you bumped into a Calabi-Yau on a leisurely stroll through your local park?

Andrew Lipson is a bit unique among topologists in that respect: he's a computer programmer at an investment bank in England with a PhD in knot theory. In his spare time, he builds improbable mathematical shapes and objects out of Legos; the recreation of Rodin's "The Thinker" (above), a variety of Mobius strips (shown here
with little Lego men walking around the surface), figure-eight knots, a punctured torus, and even a series recreating several famous sketches by M.C. Escher -- the latter created in collaboration with Daniel Shiu. There's definitely a bit of mimesis at play: his creations evoke a spark of recognition in the viewer, assuming the viewer is familiar with Escher in the first place. Or Rodin, or Mobius strips.

But there's no denying that much of the appeal of mathematical modeling for non-scientists lies in how it can help make predictions and therefore, better, more informed decisions -- like deciding where to locate your spiffy new bakery to ensure the greatest likelihood of success. Pablo Jensen, a physicist with the Ecole Normale Superieure in Lyon, France, published a paper last month in *Physical Review E* (Vol. 74, p. 035101) describing a new mathematical model that could help with that. It's based, somewhat improbably, on how various kinds of atoms interact electromagnetically: opposite charges attract, while like charges repel.

Jensen found the same phenomenon among various kinds of retail stores. For instance, a bakery will repel other bakeries -- competition! -- and, for some reason, secondhand stores, but a bakery will attract drug stores and butchers. I especially loved Jensen's quote to CNNMoney.com: "Physicists think they can make a model out of almost everything." In this case, he succeeded: the Chamber of Commerce in Lyon is using his mathematical index to help small business owners determine the best locations for new stories. Jensen next hopes to adapt his model to other cities in Europe and the US. I consider his critical insight -- that the same formulae used to describe atomic interactions could also be used to describe store locations -- to be a mimetic moment: at some point he slapped his forehead and exclaimed, "Eureka! __This__ is __that__!"

I've written before about how physicists mathematically model traffic flow. Now it turns out that researchers at Purdue University have created a mathematical simulation that could help reduce airport congestion -- welcome news to frequent flyers who are routinely plagued by inexplicable delays, on top of lost baggage and a rapid decline in basic in-flight creature comforts. And Joannes Westerink, a mathematician at the University of Notre Dame, is applying mathematics to the task of improving the models used by meteorologists to predict storm surges after a hurricane. That's not an easy feat, since the models must take into account not just the movement of water -- standard fluid dynamics -- but also the effects of winds, atmospheric pressure, tides, waves, and the geometry and topography of the coastal ocean and adjacent floodplain. (Jen-Luc opines that perhaps Lipson could use his Legos to help with the geometry and topography problems.) In each of these cases, the mathematical models are, in a sense, "fictions": creative representations of real-world physical systems and/or phenomena. Connecting the math with the reality gives rise to useful insights.

We close with one last mathematical mimetic moment. A physicist at the University of Arizona in Tucson -- improbably named Martin B. Short -- discovered last year that stalactites (those formations hanging from the ceilings of caves) have an underlying shape that can be described by a simple mathematical equation. He followed that up with a paper in the August 2006 *Physics of Fluids*, describing how this same mathematical formula also describes the shape of icicles. (Once again, __This__ is __That__.) A cursory glance at stalactites and icicles would reveal their similarity of shape, so what's the big deal? Well, the physical processes that give rise to these formations are markedly difference: heat diffusion and rising air form icicles, while the diffusion of carbon dioxide gas leads to the growth of stalactites. Yet somehow, these two different processes result in the same underlying shape, described by the same mathematical formula.

In fact, UA's press release on Short's findings described this mathematical shape as being the "Platonic Form" for an icicle (or a stalactite). So we have come full circle back to our pal Plato, by way of making a series of seemingly unlikely connections. And with any luck, we've all been entertained and edified somehow along the way. That's the power of mimesis.

The mathematician Georg Polya wrote a famous book entitled How to Solve It, in which he explored ways to approach mathematical problem solving. In the third section, "A Short Dictionary of Heuristic," he describes problem-solving by analogy, by generalization, by specialization, and by other means that are essentially the realization that this and that are related.

Posted by: David Bindel | October 21, 2006 at 11:11 PM

Let me see if I've got this - you understand the notion of the derivative and integral of a function, but you didn't understand the notion of the function itself until you saw it as the integral of the derivative?

Well, whatever works, I guess. But you also say that the introductory lecture doesn't say how to find the position function. That's because it's assumed that you already knew what it was.

It's one of your inputs. Calculus problems typically go like, "A French lady is thrown off a cliff and her height at any given time is CliffHeight - 16 * Time ^ 2. How fast is she going when she hits the ground?"

They give you the position function, H-16*t^2, and they want you to first, figure out the time it takes to hit the ground (by algebra).

Then they want you to do the calculus to get the velocity function.

Then they want you to take the time from the first step and plug it into the velocity function from the seconds step.

Posted by: andy.s | October 22, 2006 at 07:31 PM

regards differential calculus and the real world: it is wrong! I've serious problems with Ampere's differential concept of electric current as used by Maxwell in his equations when the current is small:

He has something like: electric current I = dQ/dt, where dQ is the amount of charge flowing past any point in a circuit in time interval dt.

Problem: what happens when current is so small only say 1 electron per second passes you? Is I = dQ/dt still true?

The differential equation I = dQ/dt then becomes a statistical average, and a lie. No wonder Maxwell's equations are inconsistent with quantum field theory. Calculus is only strictly valid for continuous, classical fields. You always get discontinuties in the real world, because ultimately QFT is right. This is why QFT gives infinities and requires the renormalization of charge using UV and IR cutoffs to avoid the infinities you get from a continuous field. The continuous differential equations of QFT and their solution are wrong (before renormalization) because they (1) give infinitely strong field effects at distances approaching zero (so the number of particle creation-annihilation loops becomes infinite, and the amount of momenta they carry becomes infinite and unphysical), and (2) give finite vacuum polarization effects down to arbitrarily weak fields out to infinite distances (which again is unphysical, because the entire vacuum of the universe would then polarize until it had cancel every real charge completely!).

Posted by: nigel cook | October 24, 2006 at 04:38 PM

"When was the last time you bumped into a Calabi-Yau on a leisurely stroll through your local park?"

Maybe every time? That's what I understand string theory says, anyway (that the compactification of the extra dimensions is a Calabi-Yau manifold).

Posted by: Charon | October 25, 2006 at 03:35 AM

"Truth, according to Plato, is the domain of philosophy only. Consuming literature was in these days not a solitary act (people did not read books, but listened to 'performances'). Plato's main critique was that theatre was very theatrical, and therefore a rhetorican could persuade his public without necessarily telling the truth."

It is the same today: we like to learn from Professors better than from books, we like to watch performances on stage and film too.

Even writers and especially the best romantic writers are like teachers guiding you thru the plot, leaving the details to your 'imagination'

Oh, and did we forget 'operating theatres'

I just wish doctors, dentists and surgeons could just see what really bad actors they are on the stage. This is Plato's main criticism of the risks of becoming too theatrical.

And well when we watch House, Scrubs, ER and a whole plethora of hospital dramas we soon realise it is not about the patient but about the ego, vanity and drama in the doctors', nurses', consultants', specialists' lives ... and they would try and convince you they know what they are doing when they don't, and their authority rests on the inability to challenge hierarchy - in which medicine (and heart surgery) have placed themselves on top of the hierarchical pyramid, since even kings (good night al saud) and presidents need to put their lives literally in a surgeon's hand. Yes even the 'wicked' Dick Chenney has a whole retinue of docyors trying to keep him alive and make him inmortal.

The mark of the beast, may not be bowing to the $US dollar, but becoming enslaved to the medical profession and pharmaceutical industries. But het don't worry about me I'm probably just a mad poet. lol!

Posted by: Quasar9 | October 25, 2006 at 06:48 AM

Maths writes the programs

by trial and error we define which work

We analise the airpor data

and we can 'predit' traffic (and the weather) too

But people shop in supermarkets what is on the shelve, you cannot buy or choose to buy what is not available on the shelve. So people do not choose what they want to buy, but people choose from what someone else chose to put on the shelf. Of course there is a dynamic interaction between the the supplier - the consumer - the supplier.

But if I had a $dollar for every coffe I've been 'served' and did not enjoy, I'd be close to a dollar millionair by now. So my guess is that the number of coffees (or even beers drunk) is not directly significant to the pleasure obtained.

I only brought this in, because on the radio there were talking about how many people get drafted or dragged into plays (theatre) even concerts (gigs) they'd really rather not be at (or pay for) but out of some false sense of loyalty to family and friends - and I'm not just on about the terrible school pantos or plays.

But the number crunching is not aware of these nuances. 5o million visited this country or that country does not really tell us why they did so.

And 50 million use this drug or that drug does not tell us it is any good - any more than the coffeee sales do. Ten million had this treatment or surgery tells us no more about the quality or efficacy of said treatment and surgery, than saying that all children's parents attended the school play, tells us about the quality of the play

And so many 'people' went to see such and such a film or read such and such a book, tells us little about the film or the book - it just helps hollywood accountants budget which films to finance

So Jen, here's wishing you a mighty fine day!

Posted by: Quasar9 | October 25, 2006 at 07:02 AM

Charon makes a good point. And while you walk through that park, just think of the hundred trillion neutrinos zipping through your body each second! If you still have a TV with rabbit ears, try this: tune it to a blank station and turn down the brightness knob. A few percent of the flickering "snow" is actually cosmic microwave background radiation. The fires of creation are still burning, passing through each of us as we laugh and sleep and think.

http://www.astro.ubc.ca/people/scott/faq_basic.html

I'm glad to see you sticking with the calculus lessons. Just wait, in no time at all you'll be defining 1-forms on differentiable manifolds!

Posted by: Blake Stacey | October 25, 2006 at 10:45 AM

Some excellent comments here but I wanted to respond specifically to Andy's disbelief over my ignorance of the function. It's one of those concepts that is so basic to scientists, that they forget it's just jargon to the Common Man. "Let X be a function of Y" conveys no real information to a non-scientist, even though we might be able to blithely rattle it off on a test. True understanding -- e.g., knowing the underlying connections and how it all fits together -- is not the same thing as casual recitation.

Nor am I entirely convinced that Starbird (DVD instructor) assumed his audience already knows what a function is -- the jury's still out on that. It might be that he's deliberately taking a very different approach in tailoring the course to non-scientists. And for the most part, I find his conceptualizations effective; he just moves a bit too slowly for my tastes, and left out one critical bit of information. From here on out, the lectures focus almost exclusively on the different ways to apply the fundamental theorem of calculus. And that serves to make the notion of the function crystal clear -- defining it by all the different things it can do.

Posted by: JenLucPiquant | October 27, 2006 at 03:07 PM