al-Ghazālī on memory and knowledge

It is a cliché to note that the modern telecommunications revolution has utterly transformed the relationship between human beings and facts, or “information”, more broadly speaking. It is also a cliché to note that the last such transformation took place with the invention of the printing press that resulted in easy and rapid dissemination of books. I need hardly elaborate on these developments; they’re clichés, after all! But this easy access to vast oceans of information should not be confused with actual knowledge and understanding of this material.

In his book Ghazālī and the Poetics of Imagination, Ebrahim Moosa recounts a fascinating anecdote from the life of Imām Abū Ḥāmid al-Ghazālī. The young genius had just completed an advanced dissertation on Islamic law and was returning to his hometown of Ṭūs (near Mashhad in northeastern Iran) when he was waylaid by highwaymen. Moosa provides a translation of an account by Tāj al-Dīn al-Subkī, Imām Ghazālī’s biographer, in which Imām Ghazālī is said to have said to the chief thief:

I plead with you in the name of Him who keeps you safe to only return to me my dissertation [ta‘līqa]. It will be of little value to you. The leader of the brigands asked me: “What is a ta‘līqa?” I replied: “Books in my bag. I traveled to listen and write it and to have knowledge of it.” He derisively laughed at me and said: “How can you claim to have knowledge, when I have taken it and stripped you of it? You are now without any learning!” After a while, he ordered his men to return my bag. Ghazālī said: “The leader of the brigands turned into an oracle [mustanṭaq] whom God made to speak in order to guide me.” (p. 94, Moosa)

Moosa adds that following this incident, Imām Ghazālī “memorized his prized dissertation” and “came to view memory as a treasure, something that was always available and present to him, while writing was susceptible to ‘theft’” (p. 94, Moosa).

This pre-printing press incident, from perhaps nine hundred years ago, is nevertheless deeply instructive to us, particularly in an era where memory and memorization have come to be seen as quaint, fallible storage media for information. I think the modern view is flawed for at least two reasons.

• Firstly, there is no reason why orally memorized knowledge should not be stored and transmitted faithfully from generation to generation. The paradigmatic examples are, of course, the Vedic chains of transmission that have preserved the Vedas without so much as a change of pitch accent, and the “protection” (ḥifẓ), or memorization, of the Qur’ān by Muslims. Cultures throughout human history have evolved sophisticated techniques for memorizing, accurately recalling, and flawlessly transmitting complex texts. [This is not an argument for abandoning what we possess today, of course; it is an argument for supplementing modern storage media by utilizing the astonishing human capacity to memorize.]
• Secondly, as Imām Ghazālī rightly points out, there is a world of a difference between the knowledge of a fact itself and knowledge of a fact’s location. (To put it in terms of C, this is the difference between the value of a variable a and its address &a.) Knowing that Wikipedia can answer your burning questions about the continuum hypothesis or about the length of the coastline of Britain is certainly far better than not knowing how to answer these questions at all, but knowing that Wikipedia knows is entirely different from actually knowing the facts themselves. (Having an array of pointers is not the same as having an array of values; we need *p at some stage and not just the pointer p.) It is subsequent to actual knowledge of the facts themselves that we can structure them, manipulate them, and create new knowledge.

A quote from Niebuhr

A remarkably powerful quote from Reinhold Niebuhr, one of President Obama’s favorite thinkers and theologians, on the necessarily incomplete and imperfect nature of all human action:

“Nothing that is worth doing can be achieved in our lifetime; therefore we must be saved by hope. Nothing which is true or beautiful or good makes complete sense in any immediate context of history; therefore we must be saved by faith. Nothing we do, however virtuous, can be accomplished alone; therefore we are saved by love. No virtuous act is quite as virtuous from the standpoint of our friend or foe as it is from our standpoint. Therefore we must be saved by the final form of love which is forgiveness.”

A long article from the New York Review of Books on Niebuhr’s work in international relations and on the dangers of nationalistic hubris can be read here.

Auxiliary Verbs, auf französisch

Anyone who has read Enid Blyton’s Malory Towers and St. Clare’s series knows of the legendary terror French verbs struck into the hearts of British schoolchildren of a certain age. No Maiwand or Isandhlwana could have stopped the might of the British Empire, but the slightest whiff of l’imparfait subjonctif would reduce the stoutest-hearted Viceroy into a whimpering schoolboy. At least this was the impression I had of the power of French verbs over the British psyche while growing up.

The French verbal system is no doubt more complicated than the English system, but this hardly means it’s utterly chaotic. With just a little bit of memorization and a little bit of thought (and some occasional hand-waving and some rather more frequent hand-wringing) it is possible to tame the system of conjugation. One of the reasons for the infamous difficulty of the French system is that it preserves many more synthetic (single-word) forms of its verbs than English does.

Auxiliary Verbs, en anglais

I can already see people running away from this post, terrified by its title. (This, of course, presumes that there are people coming to this site in the first place!) Grammar, perhaps because it is so awfully taught, if at all taught these days in school, tends to frighten people like no other subject save perhaps mathematics. This is a tragedy, for grammar is in fact quite beautiful and often very systematic, quite like mathematics. I'm going to focus on what may seem like a rather strange topic for now, but I will try to be clear (and will also use colors!) in order to convey a little bit of the ultimately systematic nature of this topic.

What is an auxiliary verb?

The word “auxiliary” usually means something supplemental, something additional, possibly accompanied by a sense of superfluousness. But let that not mislead us, for auxiliary verbs are by no means superfluous; indeed, their proper use is essential not merely to ensure that a verb is well-formed, but also to convey a whole host of additional meanings that are not present in the verb itself. (In this sense, the auxiliary verbs are much like the auxiliary corps of the Roman army: the Numidian light cavalry, Balearic slingers, Thracian archers and the like, whose specialized combat skills were absolutely essential to the success of the Roman legions.)

Since it’s hanami season …

and since I just mentioned A.E. Housman, and since all the lawns here are currently covered in a beautiful fine dusting of white April snow, I am compelled to put up his famous ode to the cherry, which first appeared in A Shropshire Lad towards the end of the nineteenth century:

Loveliest of trees, the cherry now

Is hung with bloom along the bough,

And stands about the woodland ride

Wearing white for Eastertide.

Now, of my threescore years and ten,

Twenty will not come again,

And take from seventy springs a score,

It only leaves me fifty more.

And since to look at things in bloom

Fifty springs are little room,

About the woodlands I will go

To see the cherry hung with snow.

From Bartleby’s collection of verse.

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).

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 i ” 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?