I should start this entry with a disclaimer like Wikipedia: “Caution! This article is written like an advertisement.” So it may seem, but what I am advertising can be readily had for a few hundred dollars from software company Mathematica and after reading this you may really want it.
First, I’m sure everyone reading this has had at least one of Martin Gardener’s “Aha” moments: that split second when hours of study have all clicked and the answer suddenly becomes clear. I can remember my first time. It was late at night and the college library was empty and closing. I had to retreat to a vacant classroom (left open all night in a small Ohio college town) to keep trying to figure out what all this epsilon/delta stuff in the calculus book was all about. When it finally hit I felt a brain-rush like I’d never experienced before. It was an intellectual high when your brain goes: “Aha, so that’s how it works!” After that, like a true addict, one can’t stop trying to experience it over and over again and, for some reason, mathematics seems to give more “Aha”s than many other endeavors.
Imagine then if one could have something that could give Aha-moments on demand? In mid-1988 software called “Mathematica” (a name recommended by Steve Jobs, as recently divulged by Stephen Wolfram the developer of Mathematica on the occasion of Steve Jobs passing) blossomed in the fledgling computer world with a program initially written to take advantage of Apple computer graphics capabilities. Suddenly almost every mathematical function known to man could be had for the cost of a few key strokes (albeit sometimes the syntax could be a bit obtuse) for unfailing execution, the results plotted in 3-D if so desired. The software was successful and the product grew and matured until today it stands at version 8.0. The reason why it offers ‘Aha’ moments is for the two-fold thrill of deciphering the commands to calculate your result and the production of the result itself. A quick look at the Mathematica web site will easily convince you of the extreme power and versatility. Today undergrads at MIT solve quantum mechanics problems of undreamed of complexity, in class, with their laptops running Mathematica. And the benefits reach all the way down to grade school. With version 8.0, Mathematica offers a web site of thousands of “Demonstrations” where some phenomena or other, from simple arithmetic to laser physics, is given on a screen with slide buttons: the effects of manually changing variables can be seen instantly. High school teachers love the ability to demonstrate a physics (or chemistry, math, astronomy...anything using math) principle projected on a white board with the ability to write on the board as well to help explain the concept under consideration.
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And now for the odd connection, or not, again I have no answer, just conjecture. For quite a while, I don’t know when he started, but up to the day after the Challenger Space Shuttle disaster on January 28, 1986, Richard Feynman used to teach physics to Hughes employees (like me; I happened to have an office next door to the auditorium he used). Every other Wednesday afternoon he would hold forth in the auditorium on the first floor of building R1 on Imperial Blvd. in El Segundo; at that time it was all Hughes Aircraft. You can verify this astounding fact in his book “Surely You’re Joking, Mr. Feynman?” on p.330 (Actually, in the book, he says he “…used to teach…” And while the book was published in 1985 he was still teaching in 1986; I wonder if this wasn’t one of his famous ploys to keep the celebrity-curious from interfering with his enjoyment of lecturing on physics)?
OK, so I had Feynman as a teacher for 2 hours every other week. So, what didn’t I ask him? Well, I didn’t ask him what he saw in his mind’s eye when someone said “electron.” Wouldn’t that have been keen to hear? I asked him to draw me a copy of the very first Feynman Diagram he ever drew. He couldn’t; he didn’t remember what it was! I also didn’t ask him what his relationship was to Mathematica. But why would I think he had anything to do with Mathematica?
Now for another one of my completely unsubstantiated conjectures. In 1985 physicist and science historian Silvan Schweber published a paper in the “Reviews of Modern Physics,” (vol. 58, p. 452), discussing Feynman’s contributions to quantum mechanics. He mentions that while Feynman was still in high school in 1933 he assigned every symbol provided in the top row of keys on his typewriter a mathematical definition. This allowed him to type complex number equations using a sort of non-standard English language syntax unlike anything in use in mathematics at the time. The illustration provided looks sort of like what Mathematica uses today for entering functions. For instance in Mathematica you can type: “AgeOfUniverse” then a space to signify multiply and “SpeedOfLight.” Hit enter and you get 10 to the 26.1489 power meters; the distance light has traveled since the beginning of the universe. And Stephen Wolfram, the developer of Mathematica worked with Feynman at Caltech for about ten years right before it was released. And there is a famous photo of the two of them huddled together, Feynman writing, Wolfram watching, taken about the time Wolfram was developing Mathematica. And I can almost read Feynman’s lips saying: “Listen Wolfram, I think you need computer software that can be programmed to solve big complex problems by simply writing down the names of functions and giving the range of variables to evaluate!” And, you know what, that’s pretty much what we have!
Oh, yeah, and maybe this isn’t completely unsubstantiated. In 2005 Wolfram wrote: “We [Feynman] talked a lot about how it should work. He was keen to explain his methodologies for solving problems: for doing integrals, for notation, for organizing his work. I even managed to get him a little interested in the problem of language design. Though I don't think there's anything directly from Feynman that has survived in Mathematica.” Yet it’s hard not to think: use Mathematica and get a little help from Feynman!
And, if any further advertising were required to encourage you to try Mathematica, I give you John Forbes Nash (remember the movie “A Beautiful Mind?”): “I spoke on how I had been using MATHEMATICA in my work on the game models…I used the talk materials as prepared for my talk earlier in 2003 in Napoli (which was the latest lecture on the topic of game theory and economic interest). (It happened that progress since then had been slow, because of difficulties in actually finding solutions by computational means.)…were included to illustrate the key topic of the applicability of the…MATHEMATICA software "to complex problems in game theory"”. (July 25, 2003).
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