Just a place to jot down my musings.

Thursday, March 31, 2011

Poetry according to A.E. Housman

In The Name and Nature of Poetry, the great Housman writes of his response to poetry:
“Poetry indeed seems to me more physical than intellectual. A year or two ago, in common with others, I received from America a request that I would define poetry. I replied that I could no more define poetry than a terrier can define a rat, but that I thought we both recognised the object by the symptoms which it provokes in us. One of these symptoms was described in connexion with another object by Eliphaz the Temanite: ‘A spirit passed before my face: the hair of my flesh stood up’. Experience has taught me, when I am shaving of a morning, to keep watch over my thoughts, because, if a line of poetry strays into my memory, my skin bristles so that the razor ceases to act. This particular symptom is accompanied by a shiver down the spine; there is another which consists in a constriction of the throat and a precipitation of water to the eyes; and there is a third which I can only describe by borrowing a phrase from one of Keats’s last letters, where he says, speaking of Fanny Brawne, ‘everything that reminds me of her goes through me like a spear’. The seat of this sensation is the pit of the stomach.”
What a line: ‘everything that reminds me of her goes through me like a spear’! 

And how beautifully Housman describes the involuntary physical reactions that Sanskrit theorists have called the sāttvika-bhāvas: stambha (stupefaction), sveda (perspiration), romāñca (horripilation), svara-bhaṅga (voice-cracking), vepathu (trembling), vaivarṇya (pallor), aśru (tears), and pralaya (loss of consciousness).


Friday, March 18, 2011

Adding up lots of things


One of the most fun things to do in math is to add things up, lots and lots of them, the more the better! It's clear, of course, that if you add up a finite number of things—numbers, vectors, polynomials, matrices, whatever—you'll get a finite answer. Things get more interesting when you add up an infinite number of things. But as I've blogged about at length earlier, there are many different meanings of the word “infinite”. So for now, let's just limit ourselves to talking about a countably infinite list of things.

First, a few terms. A sequence is nothing more than a list, usually indexed so you can refer to every element unambiguously. It's usually written \( \{ a_0, a_1, a_2, \dots, a_n, \dots \} \). A(n infinite) series is what you get when you add up all the terms of a sequence. This will look like \( a_0 + a_1 + \dots + a_n + \dots \), and is sometimes more compactly written in “sigma-notation” as \( \sum_{i=0}^{\infty} a_i \) (read as “sum from i equals 0 to infinity of a sub ” or something similar). A series is said to diverge if its value is infinite, and to converge if it adds up to a finite number.

It's immediately clear, of course, that a series like \( 1 + 1 + 1 + \dots \) is going to “diverge” to infinity. (The proof is trivial.) Furthermore, what this means is that even if you take the tiniest number you can imagine, whether that is one-half or one-quadrillionth or one-googolplexth, if you add it to itself an infinite number of times, then you're going to get infinity. (The proof is trivial: any series of the form \( x + x + x + \dots \) is the same as \( x \cdot (1 + 1 + \dots ) \), and we already know that that series diverges.)

So what this means is that the terms of our list have to get successively smaller and smaller if we are to get a series that converges. So what about the series \( 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} + \dots \)? This famous series is called the harmonic series. We can clearly see that each term of the harmonic series shrinks to zero. What does this series converge to?


Why pearls, and why strung at random?

In his translation of the famous "Turk of Shirazghazal of Hafez into florid English, Sir William Jones, the philologist and Sanskrit scholar and polyglot extraordinaire, transformed the following couplet:

غزل گفتی و در سفتی بیا و خوش بخوان حافظ

که بر نظم تو افشاند فلک عقد ثریا را


into:

Go boldly forth, my simple lay,
Whose accents flow with artless ease,
Like orient pearls at random strung.

The "translation" is terribly inaccurate, but worse, the phrase is a gross misrepresentation of the highly structured organization of Persian poetry. Regardless, I picked it as the name of my blog for a number of reasons: 
1) I don't expect the ordering of my posts to follow any rhyme or reason
2) Since "at random strung" is a rather meaningless phrase, I decided to go with the longer but more pompous "pearls at random strung". I rest assured that my readers are unlikely to deduce from this an effort on my part to arrogate some of Hafez's peerless brilliance!

About Me

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Cambridge, Massachusetts, United States
What is this life if, full of care,
We have no time to stand and stare.
—W.H. Davies, “Leisure”