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

Goalless endeavors

My friend Carl and I recently had a short conversation about zen meditation. He brought up some points that all basically referred to the end goal of meditation: health and mindfulness benefits, etc. To which I replied that meditation is necessarily a goalless endeavor. Yes, meditating will make you think clearer and harmonize your body with your mind and all that, but if you go into meditation trying to do these things, then you will not achieve these things. That’s the paradox of meditation. And here’s a quick psychoanalytical explanation of why I think that is from a paper:

The “self” and the “ego” must not be confused to be the same. Instead we must understand the ego to be an element within the self, a device for the self’s cognition. The self creates a cognitive framework in which the ego may recognize (re-cognize) self from non-self, in which the ego may create its home, thereby completing the individual’s identity in what Levinas calls the “I.” The ego moves outward toward the world in understanding, adequating the world around it and reducing other to same: thus the ego is the totalizing element within the self. In order to totalize, the ego must be self-referential, it must return to itself in its process of adequation; thus the ego assumes a teleological structure.

This teleological structure is exactly the same as any goal-oriented task. You say “I want to accomplish X, and I will do Y to get it.” After doing Y, you have X. This posits the end, X, before acting out Y, thus it is teleological. This structure is absolutely related to the totalizing movement of the ego, so it could be called an egoism, for better or for worse.

Our society largely follows this general teleological trend, which may have been why Carl wasn’t used to the idea of a goalless endeavor (which he admitted to). Now, I’ve been doing zazen meditation for 4 years, and I’ve studied the Japanese Kyoto School of philosophy, so I should have a good understanding of this idea. Today I meditated for the first time in about 6 or 8 weeks, and I noticed that my mind was racing, all over the place. You’re supposed to let go of your thoughts, but I couldn’t – they just kept racing, it was terrible. Reflecting on that later, I realized what happened: I had been so entrenched in goal-oriented thinking (eg. programming, problem solving, generally trying to figure out life) over the past few weeks that my mind had conditioned itself to work teleologically: I was constraining my thinking to a known end goal. As soon as I removed that end goal by meditating, my mind unleashed it’s pent-up creative chaos and went every which way.

This release is a great thing! Since this morning, I’ve had a few good ideas relating to a paper I’m writing, and I’ve been super productive. But I have two important points to make:

  1. Even if we have an intellectual understanding of this goal-oriened framework and the downsides that come with it, the natural tendency of our ego is to revert back to it. The ego is a powerful structure in the psyche, it’s quite overbearing at times, and it takes continual practice to let it go. This is something that meditation accomplishes. Regression is possible: if you stop meditating, you may go back to egoism. And, because it is so powerful, it tends to manifest itself in multiple layers of our society, from the way we structure businesses (greed and power) to the way we approach relationships (why are you really friends with/dating this person? for the things you get out of it?).
  2. The mind is decidedly not consumed by the ego. It can release the ego when necessary, such as during meditation, or during the pop-psychology state of “flow.” And, when the mind is not consumed by the ego, it is allowed to freely produce. It can be creative to its fullest capacity. The immense ability for creativity is one of humanity’s greatest assets and something that should be sought after. Ay, this egoless, creative state, achievable and practicable via meditation, certainly should be valued.

I’m not saying meditation is the only way to go about egolessness, but it is a quite effective one, with little barrier to entry. Martial arts, painting and drawing, practicing mathematics, reading literature, playing sports; these can all be goalless endeavors.

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 Resources for Learning, Building, Developing, Experimenting

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.