The case for using computer algebra
Many academics and researchers get annoyed when students use computer algebra programs such as Mathematica to evaluate simple integrals that they maintain should be done by hand. The question I ask is “At what point do you expect your students to switch over to using a computer?”. Most mathematical examples are artificial in that closed form expressions exist. However, in nearly any real problem, this is not the case.
I learnt mathematics using slide rules and tables, then calculators, then computers, then symbolic algebra. To me, this is a valid progression — but not one that everyone should have to go through. I feel that the only way true progress can be made is if we don’t have to learn a whole set of rules. If we had to do calculus using Newton’s geometrical constructs then progress would be very slow. The real question is what are the essential tools and lessons. To me, knowing what a derivative and integral mean “physically” is far more important than knowing how to compute a specific integral.
Many people feel that reliance on computer algebra means that students can’t do calculus by hand and hence the really don’t understand what’s going on, just how to get the answer by computer. Calculus concepts are subtle. However, just knowing the mechanics of computing an integral or derivative does not imply understanding. I believe that it is possible to have true understanding without computation.
From a list of pet peeves of Paul Abbot, physics professor at University of West Alabama.
We learn nothing from those who say: ‘Do as I do.’ Our only teachers are those who tell us to ‘do with me’, and are able to emit signs to be developed in heterogeneity rather than to propose gestures for us to reproduce. In other words, there is no ideo-motivity, only sensory-motivity.
—Gilles Deleuze, Difference and Repetition, page 23
“I know that I am mortal by nature, and ephemeral; but when I trace at my pleasure the windings to and fro of the heavenly bodies I no longer touch the earth with my feet: I stand in the presence of Zeus himself and take my fill of ambrosia”
― Ptolemy, Ptolemy’s Almagest
From the little reading I had done I had observed that the men who were most in life, who were moulding life, who were life itself, ate little, slept little, owned little or nothing. They had no illusions about duty, or the perpetuation of their kith and kin, or the preservation of the State… The phantasmal world is the world which has never been fully conquered over. It is the world of the past, never of the future. To move forward clinging to the past is like dragging a ball and chain.
—Henry Miller, Sexus pages 262, 430; as quoted in Anti-Oedipus, pages 27-8
Flipping through my copy of Carnegie by Peter Krass, which I read a few years ago, I found a torn paper of scribbled quotes from Seneca’s Letters from a Stoic:
Only an absolute fool values a man according to his clothes, or according to his social position, which after all is only something that we wear like clothing. —page 94
What is death? Either a transition or an end. I am not afraid of coming to an end, this being the same as never having begun, nor of transition, for I shall never be in confinement quite so cramped anywhere else as I am here. —page 124
So-called pleasures, when they go beyond a certain limit, are but punishments… —page 144
Only a man of wisdom and experience can really love. —page 230
Clojure is a fascinating language and learning it has presented an intellectual challenge I haven’t experienced in a long time. I guess trying to wrap my head around all the various declensions and forms in Attic Greek was the last thing that challenged me this much.
As I’m learning Clojure, I’m finding all kinds of awesome resources for learning, tools for building and testing, etc. Here’s what I’m finding, conveniently captured in a GitHub Gist that I’ll update as often as necessary:
From the climactic final pages of Catch-22:
Yossarian crossed quickly to the other side of the immense avenue to escape the nauseating sight and found himself walking on human teeth lying on the drenched, glistening pavement near splotches of blood kept sticky by the pelting raindrops poking each one like sharp fingernails. Molars and broken incisors lay scattered everywhere. He circled on tiptoe the grotesque debris and came near a doorway containing a crying soldier holding a saturated handkerchief to his mouth, supported as he sagged by two other soldiers waiting in grave impatience for the military ambulance that finally came clanging up with amber fog lights on and passed them by for an altercation on the next block between a single civilian Italian with books and a slew of civilian policemen with armlocks and clubs. The screaming, struggling civilian was a dark man with a face white as flour from fear. His eyes were pulsating in hectic desperation, flapping like bat’s wings, as the many tall policemen seized him by the arms and legs and lifted him up. His books were spilled on the ground. “Help!” he shrieked shrilly in a voice strangling in its own emotion, as the policemen carried him to the open doors in the rear of the ambulance and threw him inside. “Police! Help! Police!” the doors were shut and bolted, and the ambulance raced away. There was a humorless irony in the ludicrous panic of the man screaming for help to the police while policemen were all around him. Yossarian smiled wryly at the futile and ridiculous cry for aid, then saw with a start that the words were not ambiguous, realized with alarm that they were not, perhaps, intended as a call for police but as a heroic warning from the grave by a doomed friend to everyone who was not a policeman with a club and a gun and a mob of other policemen with clubs and guns to back him up. “Help! Police!” the man cried, and he could have been shouting of danger.
Apocalyptic and ironic depiction of mob-rule and the corrupting-power phenomenon that has become known via the Stanford prison experiment.
Full article here. 8-year-old Morgan speaks the truth.