My post on the 300th anniversary of the death of Gottfreid Leibniz just went live at the Wolfram Blog. Check it out.
I realized with some sheepishness, after writing about Bruce Sterling's story "The Black Swan," that I had only read secondhand accounts of Nassim Nicholas Taleb's Black Swan idea, but not his actual book of that title. So I ordered it, and now have read the book itself. A "black swan" as defined by Taleb, is an event of "rarity, extreme impact, and retrospective (though not prospective) predictability." Taleb is an exquant turned iconoclastic mathematician. This book was a bestseller, and like many bestsellers on arcane topics, it is chatty, opinionated, and a bit on the loose and sloppy side.
As I expected, I enjoyed the book a lot. His subject matter appeals both to my taste in mathematics and my aesthetic of fiction. (My favorite section of David Hartwell's The Dark Descent is The Fabulous Formless Darkenss, which is the section for what he and I call "nature of reality" horror  stories in which the parameters of what one thought was possible usually change for the worse.)
Other than having it confirm my belief system (always a pleasure), there were two basic things I got out of reading this.
As I told my husband (David Hartwell) while reading the book, for me this is partly a book about how David's way of doing things is right. The way David publishes books as an editor, the way he runs our small press, the way he collects books, the way he runs the family finances, etc. all tend to limit downside risks while at the same time leaving him exposed to positive Black Swans  serendipity. (Aren't I a good wife for blogging about how I just read a book that tells me how right my husband is?)
In the second half of the book Taleb spends a lot of wordage and emotion on condemning the "intellectual fraud" of the Bell Curve. I don't really need to be able to judge whether he is correct in terms of all of his examples. But it set me thinking about standardized testing.
Last week and this week, the school district is doing extensive testing of my son, which they do every three years. (This is separate from the group standardized testing.) When my husband I meet with the district in June, we will be presented with a sheaf of paper in which, page after page, our son will be raked in terms of percentiles and numbers of standard deviations from the norm. And we will try to make sense of all this data. Again.
Years ago, I read Stephen J. Gould's The Mismeasure of Man (another book which confirmed my belief systems) and I have a deep distrust of standardized testing anyway. And yet these tests mean something, and it is of crucial importance that I understand what. In past cycles of this, I've wasted a lot of time reading up on what is meant by "processing speed" and how to understand things like the results on subtests of IQ tests.
My son tends to produce an anomalous pattern of results on such tests, and so far no one has yet to be able to explain what this actually means, though  who knows  this year's tests may be different. (We expect to be having our independent expert read the results for us this time.)
So I'm reading along through Taleb's attack on the uses of the Bell Curve, and I realize that of course all of these tests are normed to Bell Curves. They wouldn't give them if they weren't. But all around me I see human attributes that do not fit that model: the ability to draw a recognizable portrait, the ability to play the piano, gender, eye color, etc. Many human differences are not meaningfully normalized to the Bell Curve.
And so I realized there was one further aspect of what these tests measure that I need to keep in mind: that the imposition of percentiles and standard deviations on what attributes are measured may be no more realistic than contemplating a spherical cow. We somehow need to try to consider whatever we're told each of these these tests are measuring is distributed BellCurvelike over the population, or whether it is a different kind of thing. I find this a bit daunting, though probably worth trying.
Just in time for Valentines Day, Wolfram Research releases Mathematica Home Edition for $295 (chocolates not included), a small fraction of the full professional edition price: Get it now your own and make your valentines with Mathematica! Find the true mathematical expressions of your love! (Or test out that new sexual geometry without injuring yourself  just kidding.)
Read all about it at MacWorld, MacNN, and Business Week.
My son Peter is really thrilled that decorating Easter eggs with Mathematica, something that he thought up, is used on the announcement page as an example.
(See also my post Who Among You are Geek Enough to Decorate Your Easter Eggs in Mathematica?)
Jeff Hamrick of the Wolfram Research Special Projects Group has done a blog post giving instructions for using Mathematica to analyze the US Presidential Election. He shows how to pull polling data into Mathematica and how to use some of the Mathematica 6 data libraries to create your own Red State/BlueState maps.
I think this is very interesting stuff because, for those of you with Mathematica out there, you don't have to rely on how other people choose to analyze and map this data. If you have questions of your own you can introduce your own data and draw your own maps. I will be very curious to see what people come up with.
Yifan Hu at Wolfram Research has come up with a computer model of the I35 bridge that shows how the bridge could have collapsed with the failure of only 3 pieces. He explains:
The picture below shows the computed stresses in a simple 2D model of the I35W bridge, with red meaning more stress. (I got the geometry from news pictures.) There are definitely aspects of the model that are not realistic. For example, the weight of the trusses themselves isn't included. And, of course, it's in 2D.
So what happens if one of the trusses weakens?
It's easy to include this in the computation by adding an upper bound on the stress in that truss. That just adds another inequalitywhich FindMinimum has no problem with.
One can actually compute all this in real time inside Manipulate. Here's an animation of the result:
One sees that when the truss with maximal stress weakens (shown in yellow), the stress spreads out to other parts of the bridge. If one weakens the next truss, then the stress propagates further. And when one weakens yet another truss, then the constraints can't be satisfied at all any moreso there is no static equilibrium for the bridge, and the bridge cannot stay standing.
See it HERE.
In a short essay "The Space of All Possible Bridge Shapes," composed in response to the Minneapolis bridge collapse, Stephen Wolfram suggests design principles that could lead to stronger bridges:
. . . it's been known for a while that the best networks don't have that kind of simple structure. In fact, they almost seem in some ways quite random.
Well, what about bridges? I strongly suspect that there are much better truss structures for bridges than the classic ones from the 1800sbut they won't look so simple.
I suspect one can do quite well by using simple rules to generate the structure. But as we know from NKS, just because the rules to generate something are simple, it doesn't mean the thing itself will look simple at all.
Two students at our NKS Summer School (Rafal Kicinger and Tom Speller) have investigated creating practical truss structures this wayand the results seem very promising.
So what should the bridges of the future look like? Probably a lot less regular than today. Because I suspect the most robust structures will end up being ones with quite a lot of apparent randomness.
Did you know that there exists a "Conservapedia," a conservative reaction to Wikipedia? I discovered this fascinating fact via the Liberal Avenger, which was making fun of their entry on the Moon. My favorite page on Conservapedia is their entry entitled Examples of Bias in Wikipedia. Here are a few choice bits:
 Wikipedia allows the use of B.C.E. instead of B.C. and C.E. instead of A.D. The dates are based on the birth of Jesus, so why pretend otherwise? Conservapedia is Christianfriendly and exposes the CE deception. . . .
 Wikipedia often uses foreign spelling of words, even though most English speaking users are American. Look up "Most Favored Nation" on Wikipedia and it automatically converts the spelling to the British spelling "Most Favoured Nation", even there there are far more American than British users. Look up "Division of labor" on Wikipedia and it automatically converts to the British spelling "Division of labour," then insists on the British spelling for "specialization" also. Enter "Hapsburg" (the European ruling family) and Wikipedia automatically changes the spelling to Habsburg, even though the American spelling has always been "Hapsburg". Within entries British spellings appear in the silliest of places, even when the topic is American. Conservapedia favors American spellings of words. . . .
 Wikipedia has many entries on mathematical concepts, but lacks any entry on the basic concept of an elementary proof. Elementary proofs require a rigor lacking in many mathematical claims promoted on Wikipedia. . . .
 The Wikipedia entry for the Piltdown Man omits many key facts, such as how it was taught in schools for an entire generation and how the dating methodology used by evolutionists is fraudulent. . . .
Oh, goodness. I wonder which mathematical claims were felt to be unchristian or subject to liberal bias.
Conservapedia is aparently a project of Andy Schafly, son of Phyllis.
(Notably absent from the Conservapedia entry on algebra is a discussion of the Arabic origin of the word.)

I've posted a bunch of photos from the Wolfram Technology Conference 2006. Enjoy!
In the comments section, in the context of whether one is more likely to survive the collapse of a building using the duckandcover technique or the triangle of life technique (taking refuge next to furniture, not under it), Jonathan Post tells the following story:
When we had "duck & cover" nuclear drills at my Robert Fulton elementary school (P.S. #8), in the late 1950s, I refused to get under my desk, and got sent to the principal's office. I explained to him that we were directly across the East River from downtown Manhattan, and that Wall Street could be ground zero. I explained that the radius of the fireball varied with the 2/3 power of megatonnage, and that the desk would not give even a microsecond of protection. He agreed, and told me not to tell all this to the other students, as it might frighten them. For that matter, he advised me not to scare the teachers. I did not leave my baby teeth for a "tooth fairy." Rather, I had my Mom snailmail them to someone who was researching Strontium90 levels in teeth, for fallout research.
Pakistan has about 2 percent of the world's population living on less than 0.7 percent of the world's land.
Q: What portion of the world's population lives in areas affected by the earthquake? How can you tell? How many of those are under age 18?
Show your work.
Extra credit: What is the population density in the most severely affected areas?
ALSO, there is a fascinating piece by an Indian seismologist, Arun Bapat, about what is to be learned from this earthquake tragedy, including some risk factors to that population your trying to do math about that might not have occurred to you:
. . . let us examine the fate of conventional structures. Press reports and television coverage indicate that there has been extensive damage in the mountainous areas of this region. The area in the vicinity of earthquake epicentre is situated at an altitude of 2,000 to 3,000 meters. Seismic vibrations have more amplitude at higher elevations. For example, take a 30storeyed building. It will have the least vibrations at the level of the ground floor but, as you go higher, the amplitude of the vibrations increase. The earthquake damage in Baramulla, Uri, Poonch, and so on, which are located at heights of about 1,500 to 2000 metres, and at a distance of about 60 to 90 km from the epicentre, was therefore more severe, as compared to the damage at Islamabad or Haripur, which are at a distance of about 60 to 90 km, but situated at an elevation of about 500 metres or so.
Follow the link to the Indian Express news story, "Is there anybody out there?" It is the first one I've seen to give any account of what I've suspected was going on in the quakeravaged hills.
Even in the fuzzy Digital Globe satellite images from 1999  the best I could get of the region over the internet  it is apparent on my nice large monitor that the mountainsides are terraced with farmlands, and their creases are dotted with small white rectangles suggestive of roofs. There were people down there.
Given confidence at my own skills as a cinematographer by a look through the GoogleVideo, I opened my own account and I uploaded this bad little clip I shot on the 28th of September of my kids in their first encounter with Peter Overmann's Wolfram Tones. It took a while for the busy folks at Google Video to approve my clip, but now it has finally been unleashed upon the public.
Back on the 28th,when I first uploaded it, I wrote:
After dinner this evening, I sat my son Peter, who has just started 3rd grade, down at my computer and let him play with Wolfram Tones for the first time. The first interesting thing that happened was that my daughter Elizabeth, who turns 3 in October, started jamming to the Wolfram Tomes soundtrack on the toy piano in the living room. (I had gotten the video camera out to film Peter, and she started while I was getting set up.) After about 10 minutes of fiddling, Peter came up with something he really liked.
I got out the video camera for a kind of personal notetaking to watch how Peter used the program. What happened while I was getting the camera out and turned on, I find quite remarkable: Elizabeth's jamming along with the music coming from the computer is something she usually only does with live music, implying that somehow the music coming from WolframTones passed the Turning test for her.
I know that lots of other mommies foist their children on their husbands and go to conferences all by themselves, but I haven't done anything like that since my children were born, and Peter turns eight next month. So, yippeee, I'm going to the Wolfram Technology Conference and I won't have to interrupt a conversation even once to say, Get down from there! or Don't eat that! (Or at least, I don't think I will.)
My son found this caterpillar in our yard Friday. I don't know what species it is. (Anyone know?) I took its picture because of the cellular automatastyle patterns on its back. I sent it to the folks at Wolfram, who I'm sure could tell me what pattern that is.
(Regarding the ichor it's sitting in, we didn't know if it was injured or whether this was some other kind of secretion.)
I'm a bit unsure what I shold have said this morning when my son's new third grade teacher told me, in the context of a discussion of the math cirriculum, that in the third grade "we don't teach memorization; we teach concepts." Perhaps she was a little too candid.
Good thing that here in the Future we can all work out multiplication problems from first principles whenever we need to multiply, isn't it?
AND NOW FOR SOMETHING COMPLETELY DIFFERENT: In the midst of all this mess, I got an email from one of the folks at Wolfram telling me about their lovely new Internet widget WolframTones, which is essentially the aesthetics of A New Kind of Science rendered as sound. One of the potentially revolutionary strengths of Mathematica is it's ability to render mathematics as sound, allowing us to gain greater understanding of math using our faculties for appreciating and understanding music. I've been looking into this myself, reading up on the neurology of math. One interesting book on this subject I have in hand is Functional Melodies: Finding Mathematical Relationships in Music by Scott Beall.
But what is special about the Wolfram version, and sets it apart from other attempts to integrate mathematics and music, is that it takes on the gnarly natural mathematics derived from Wolfram's attempt to parse the complexity of the geometry of nature. The piano selections remind me of Philip Glass's "Closing," which I think of as the best Thinking Music I have in my iTunes.
One of my summer projects is to teach Peter his multiplication tables before school starts. Our exceptionally fine school district had an extremely difficult time teaching him his basic addition and subtraction facts in the first and second grade, and I have no reason to believe that they will have more success in the third grade with multiplication. Peter has David's amazing associative memory, and while associative memory is great for learning about, say, red efts, since calling to mind all the information you have about surrounding concepts such as salamanders and newts gives you context and allows you to interpolate information you don't know. But for recalling information about, say, the number seven it is a disaster (as my Google search link handily illustrates: nearly a billion results for the numeral 7; only a hundred and sixteen million for the word seven spelled out).
I have arrived at this formulation: Memory is something I do; memory is something that happens to David and Peter. So in order to get Peter over certain academic hurdles, I need to teach him how to work at memory. Simple recitation does not do it for numbers. The public schools have a slightly more complex technique that boils down to repetition which has failed us utterly, so far.
So I have been looking for alternatives. One of the alternatives, has been assigning multiplication problems to particular places in a classically organized "memory palace" structured around the pool area and grounds of the hotel where the International Conference on the Fantastic in the Arts is held. Each memory place is a place where he found a memorable creature. (The iguana he spotted by the whirlpool is given the spot 7 X 7 and is names Fortunine to invoke 49; the place he found a favorite caterpillar is designated 8 X 8, and the caterpillar is named "Sticky Boy" to invoke 64.) This was succeeding up to a point, but lacked a structure that could be extrapolated upon.
So this afternoon, I hit upon the idea of building a Great Pyramid, complete with a Lego Pharaoh, to illustrate the concept of perfect squares in a way that could be generalized to other multiplication problems, and would also allow us to deduce the existence of prime numbers.
The hardest part was sorting his vast and diverse collection of Legos for the collection of 200odd square Legos with four bumps on them. This allowed us the make the first 8 layers of the pyramid. Starting with one, I had him tell me what the product of each number was when multiplied by itself; then we collected the right number of square Legos,; then we built the next layer of the pyramid. (Because of the tightness of the fit needed for the Legos near the middle, I did the middle parts, and he did the perimeters.) Having verified that the square of each number indeed yielded a square, we moved on to rectangles; and then we demonstrated experimentally that there are some numbers of blocks that can't be made into rectangles (the example we tried was 19). Then I explained about prime numbers.
I am pleased with this Lego activity, but also think that it would not have worked if I had not first helped him memorize the perfect squares using the poolside memory palace. I was taking things he had memorized as arbitrary concepts and giving them a more conceptually based architectural structure which can be extrapolated from.
PS: I must say I'm rather hardpressed to understand the intended mneumonics of the "Fact Triangles" in the Everyday Mathematics curriculum that Peter's school uses.
Through most of my career as a mother, I have made it a point of aligning my interests with my children's interests. This has taken me to many interesting places, taught me many interesting things, and even gotten me published in the science magazine Nature (reprint on Fantastic Metropolis).
I have made an exception for annoying fads, especially the Pokémon thing. (See my May 18th, 2003 post, "Pokémon Infestations and Other Matters.")
I realized in the middle of the night, night before last, that there was something big I had been missing about the whole phenomenon. Here is an outtake from what I wrote about it:
One puzzling phenomenon I've observed watching 2nd graders is how kids, who are only just getting basic addition and subtraction of multidigit numbers by the tail, can spend literally hours trading Pokemon cards (by which I mean 2 or 3 hours at a time). The decisions of whether or not to trade are based on multiple factors, some of which are linear functions like how many hit points does a given card have (or is the sum of the hit points of the two cards you are offering me equal to or greater than the hit points of the card of mine you want), and some of which are binary (is it a "shiny", i.e. a holographic card).
. . .
I spotchecked Peter's sense of the relative value of cards back in February. I had him show me what he thought of as his three best cards. I priced them on Cardorder.com. The cheapest of them came in at $47.00. I then had him show me three of his cards that he thought of as "notsogood." Cardorder.com priced those between 75 cents and $3.00.Given what I know of the scholastically measurable of the math skills of the kids in question, there has to be some kind of preverbal calculation going on. They seem to me to be carrying out complex calculations involving multiple variables of different types, and arriving at basically correct conclusions via some kind of folkmath.
. . .
One other implication of this phenomenon, it seems to me, is that the equals sign, as a piece of mathematical notation, is highly socially embedded. I remember something about a second grade playground bead market at Ravenna during recess that spontaneously emerged and then spread until teachers banned it after a few weeks. It may be that there is a developmental phase around 7 or 8 in which the social embedding of trade is explored.
I would be interested in your anecdotes about young kids and card trading. I've decided to investigate further.
I should also say that this realization was inspired partly by Munir Fasheh's essay "Can We Eradicate Illiteracy Without Eradicating Illiterates?", an expansion on a paper given at a UNESCO meeting in Paris, on 910 September, 2002, to celebrate the International Literacy Day. The meeting was entitled "Literacy as Freedom."
In it, he dscribes his realization of his illiterate mother's mathematical sophistication:
My 'discovery' of my illiterate mother's mathematics, and how my mathematics and knowledge could neither detect nor comprehend her mathematics and knowledge, mark the biggest turning point in my life, and have had the greatest impact on my perception of knowledge, language, and their relationship to reality. Later, I realized that the invisibility of my mother's mathematics was not an isolated matter but a reflection of a wide phenomenon related to the dominant Western worldview. In this sense, the challenge facing communities everywhere, is to reclaim and revalue the diverse ways of learning, teaching, knowing, relating, doing, and expressing. This reclaiming has been the pivotal theme of my thinking and work for the last two decades.
My concern is not about statistical measures  for example, how many learn the alphabet  but about our perception of the learner and what happens to her/him in the process of learning the alphabet. My concern is to make sure that the learner does not lose what s/he already has; that literacy does not replace other forms of learning, knowing, and expressing; that literacy is not considered superior to other forms; and that the learner uses the alphabet rather than be used by it. My concern is to make sure that in the process of eradicating illiteracy, we do not crush illiterates.
In the 1970s, while I was working in schools and universities in the West Bank region in Palestine and trying to make sense out of mathematics, science and knowledge, I discovered that what I was looking for has been next to me, in my own home: my mother's mathematics and knowledge. She was a seamstress. Women would bring to her rectangular pieces of cloth in the morning; she would take few measures with colored chalk; by noon each rectangular piece is cut into 30 small pieces; and by the evening these scattered pieces are connected to form a new and beautiful whole. If this is not mathematics, I do not know what mathematics is. The fact that I could not see it for 35 years made me realize the power of language in what we see and what we do not see.
Her knowledge was embedded in life, like salt in food, in a way that made it invisible to me as an educated and literate person. I was trained to see things through official language and professional categories. In a very true sense, I discovered that my mother was illiterate in relation to my type of knowledge, but I was illiterate in terms of her type of understanding and knowledge. Thus, to describe her as illiterate and me as literate, in some absolute sense, reflects a narrow and distorted view of the real world and of reality. A division, which I find more significant than literate and illiterate, would be between people whose words are rooted in the culturalsocial soil in which they live  like real flowers  and people who use words that may look bright and shiny but without roots  just like plastic flowers.
(It's a neat essay. Read the whole thing.)