Social Balance

“The shifting of alliances and rivalries in a social group can be viewed as arising from an energy minimization process. For example, suppose you have two friends who happen to detest each other. The resulting awkwardness often resolves itself in one of two ways: either you drop one of your friends, or they find a way to reconcile. In such scenarios, the overall social stress corresponds to a kind of energy that relaxes over time as relationships switch from hostility to friendship or vice versa.” — from Energy Landscape of Social Balance, Seth A. Marvel, Steven H. Strogatz, and Jon M. Kleinberg, DOI: 10.1103/PhysRevLett.103.198701

Very interesting! More interesting is what they conclude for the future research, viz, the “challenge for the future is to understand its large-scale structure, perhaps even including a characterization of the pathways leading out of the deepest minima–those corresponding to the most entrenched conflict–and toward states of reconciliation.”

The Problem of Mathematics Education in India

This coming from someone who has had his entire education in India and loves the subject may sound a bit alarming. But this is how I find it. Please note that it is just a personal opinion formed based on the random incoherent surveys conducted over the years and discussions within close groups of friends, always–well… almost always–when they were drunk or stoned.

If you ask, majority of the pre-college going students–in their Senior Secondary years, just prior to the college, that is–in India will tell you that they like mathematics. A big chunk of that number belongs to what the Economists call “the great Indian middle class”. Well, that is true from one perspective. They like mathematics because studying pre-college mathematics is a must in India to get into Engineering. Engineering for the Indian middle-class is “a big thing”. You get the picture.

But if you ask me, the other perspective, the way mathematics is taught in most Indian pre-college institutions can never make it a likable subject. Most of you must have seen the Mathematics textbooks in your school days. And then in college days. And then in your kids’ school/pre-college days… I request you to give it another look and then decide if you or your kid can really like it or not. The way the books are written and the way the subject is taught is to program a human-being. You open a book, you’ll see  text that is nothing but a collection of symbols, and algorithms that are to be read and run on a machine. There is hardly any description of the historical importance and the original author’s insight, his fight and vision in reaching the result. The student never feels the sense of discovery the discoverer might have sensed when his/her thought came into the form called “Theorem”.

To be ignorant of the lives of the most celebrated men of  antiquity is to continue in a state of childhood all our days.

The best mathematics books in terms of developing a student’s liking and insight into the subject are never taught when they are needed the most. I have been very fortunate in that regard that when interest in mathematics started waning, I had access to those. But, in India, such books have been inaccessible to most of the students studying mathematics. It is way past the time when those books should have been made the part of curriculum.

The mind is not a vessel to be filled but a fire to be kindled. –Plutarch

But then, a set curriculum ruins the whole purpose of the book for what it is written for, because most teachers at that level never studied those books themselves.

Sometimes, I tend to think that the books at that level, at present, are written by some computer programmers in order to program a child just like a computer is programmed. And we are very good at getting programmed! Due to socio-economic factors mentioned previously, our entire future depends on how well we were programmed. For example, most of the Indian students, at least those who studied a little mathematics at the pre-college level, are good at finding anti-derivatives (which is taught in the name of `Integration’) of some functions. Very few have insight on what really `Integration’ is. They can find surface areas of arcane surfaces and areas under most complex of the curves brooding on the x-y plane using crammed formulae, but they can hardly ever tell you why they needed it or that it is called Riemann Integral. Infinity is another number for them to “produce results” of the questions posed in the exams. They are never taught the profound thought in imaging it in simpler terms. Infinity for them is nothing but some number “out of bounds”–just like a computer–used in simplifying calculations. In short, what we are producing in India in the name of mathematical talent is “human calculating devices.” It is no wonder that India produces so many “skilled” workers in the IT industry.

One of the major forces behind a successful mathematician is his/her teacher who is expected to develop the insight of the subject. The problem with teachers and schools in India, again, is influenced by the changing paradigms in socio-economic scenario. A parent, now-a-days wants the ward to “excel” every aspect of life, be it arts, sports, science, any damn thing. The race of producing sub-standard know-it-all machines has never been so invigorated. The present day teenager is the center of its parents’ hopes. They want it to achieve everything what they could achieve and what they could not, and even more. They want it to get a high-paying position in some multinational, right from the day the kid has entered the school. To heck with the interests or aesthetics! To heck with appreciating ideas. In this rat-race, a teacher is left behind with his values. What should (s)he do? Produce more rats to participate in the race in which no one rat wins the cheese, but several end up biting a little piece and left unsatisfied.

I have observed many examples: kids and teenagers who wanted to study mathematics–or something else for that matter–for their lives, but there will was crushed by the might of the ambition of the great Indian middle-class to reach the top… the economic top.

Probably that’s why we see many brilliant students in India, but not many achieving the heights what Russians, French and Germans do. Probably that’s why a Fields Medal has eluded India despite it being the mother of mathematical thought and its thousands of years’ history of mathematics.

But that is changing. Or, at least I hope that is changing…

EDIT (Friday, May 29 2009): More discussions reveal more problems. Another major problem is the language. You know what I mean!

Mathematical vs. Verbal Reasoning

George Bernard Shaw (1856 — 1950) once said that as a boy he (1) let someone assume that a=b, (2) permitted several steps of algebra, and (3) found that he had accepted a proof that 1=2. This incident had a deep impact on Shaw’s thought process and forever after, he distrusted assumptions and algebra. The conclusions of a mathematical theory can be retranslated into words, but rarely can they be found by verbal reasoning.

George Bernard Shaw
George Bernard Shaw

In reply to Shaw’s criticism of the formal mathematical reasoning, the economist Philip H. Wicksteed (1844 — 1927) nicely puts:

“Mr Shaw arrived at the sapient conclusion that there “was a screw loose somewhere”– not in his own reasoning powers, but — “in the algebraic art”; and thenceforth renounced mathematical reasoning in favour of the literary method which enables a clever man to follow equally fallacious arguments to equally absurd conclusions without seeing that they are absurd. This is the exact difference between the mathematical and literary treatment of the pure theory of political economy.”

Philip H. Wicksteed
Philip H. Wicksteed

In other words, a mathematical idea, if correctly put and checked, is better than the one that is put in words, for words can never possibly explain in thousands what an equation can. That si where the importance of mathematical shows.

Inside a regular n-gon

I found this neat little problem. Solution in the comment on next Wednesday, unless someone comments with the solution before I do.

A point is chosen randomly inside a regular n-gon. What is the probability that it is closer to the center of the n-gon than it is to the periphery?

LaTeX woes

I am using seminar class for my slide presentation due in a week’s time. I always use the awesome natbib package to manage the citations in my documents. When I ask \LaTeX to keep the landscape orientation of the slides, natbib’s ‘\citet’, ‘\citet*’, ‘\citep’ etc. don’t work the way I expect them to work. Only the numerical entries for the citations show up. But when I keep the orientation of the slides to be portrait, the commands work. Here is the relevant preamble (when the commands work; they don’t work when I remove the ‘portrait‘ option from the ‘documentclass‘ declaration):

\documentclass[a4,portrait]{seminar}
\input{seminar.bug}
\input{seminar.bg2}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{type1cm}
\usepackage{graphicx}
\usepackage{fancybox}
\usepackage{fancyhdr}
\usepackage[authoryear,super,sort]{natbib}
\usepackage[colorlinks]{hyperref}
\bibpunct{(}{)}{;}{a}{,}{,}

Googling did not help me much. 😦 Any help?

‘New Math’ and other Mathematics Videos

‘New Math’ – Tom Lehrer

‘Lose Yourself in the Digits’

e^{i \pi}

‘What you know about math?’

3.141592653589793238462643383279502884197169399375105820974944592307816…

(or Mathematical Pi to the tune of ‘American Pie’)

by Ken Ferrier and Antoni Chan

Pi, pi, mathematical pi

Lyrics:
A long, long time ago
Long before the Super Bowl and things like lemonade
The Hellenic Republic was full of smarts
And a question resting on the Grecian hearts
Was “What is the circumference of a circle?”
But they were set on rational numbers
And it ranks among their biggest blunders
They worked on it for years
And confirmed one of their biggest fears
I can’t be certain if they cried when irrationality was realized
But something deep within them died
the day they discovered pi.
They were thinking

CHORUS:
Pi, pi, mathematical pi
3 point 14 15 92
65 35 89 7
932384 62
6433832 7 (not rounded)

Well this kind of pie is different than most
It hasn’t got berries, ain’t spread on toast
And that’s how it’s always been
We keep extending its decimal places
Pushing our computers through their paces
But we’ll never reach the end

So why the fascination with
A number whose end is just a myth
Whence the adulation
For mental masturbation
It might have something to do with the stars
To calculate distances from afar
But that’s just a guess ’bout the way things are
Regarding the precision of pi
I am pondering

Now I feel that I should mention
Pi is applicable in any dimension
At least as far as I know
If there were no Pi we’d be missing things
Like marbles and mugs and balls of string
And sports such as soccer and curling
The orbs in their celestial paths
Navigate along elliptical graphs
Ellipses have pi in them too
Just one side of them has grew
You can see pi in most everything
It’s in Cornell’s Electron Storage Ring
And also in slinkies and other springs
And that’s why it’s important to know pi
You should memorize

Once one night I had a dream
That pi was gone and I had to scream
Cause all pi things had disappeared (pause)
Can you imagine a world like that
Circles aren’t round and spheres are flat
It’s the culmination of everything we’ve feared
‘Twas a nightmare of epic proportions
One that gave me brain contortions
Oh wait! I mean contusions
They put me in some institutions
But then I escaped and now I’m free
To sing of the virtue of pi