Pearls at Random Strung

Religion and narrative

2011-09-05T14:05:00.002-04:00

This post on The Daily Dish, on religion as theater, has got me thinking: All the talk these days of faith versus reason, of religion versus science, seem to me to be misplaced. A lot of this stems from what I think is a misplaced emphasis on religion as “blind faith”. Perhaps we would do better to think of religion as “shared stories” (or better yet, “shared experiences”), for every religious community has a particular narrative about the human condition.

To the “non-believers”, the “outsiders”, it is the “story” that matters—whether the content is “true” or “false”, “historical” or “mythological”, “revealed” or “constructed”. Hence arguments about whether Genesis can literally be true or whether Rāma actually had a bridge built to Laṅkā.

To the “believers”, the “faithful”, the “insiders”, it is the “shared” part that matters—the fact that these stories resonate not just with one person but with an entire community; the fact that this resonance has held true for this community over time (even if, and possible especially if, it has resonated with different concerns at different times); and the fact that these stories will continue to be shared with the community to come, if the current generation does its job right. In fact, I think that this “shared” aspect is so important that the “stories” themselves gradually change over time, emphasizing certain things and downplaying others—but always in a way that allows them to be shared and accepted by the majority of the community.

To the “Truthseeker”, both aspects matter equally—if it is false, then it is not worth pursuing; if it cannot be shared, in at least some dilute form, then it cannot be a goal towards which one can guide others, around which a community can be built.

“God-obsessed”

2011-08-22T14:27:00.000-04:00

A fascinating essay by Tony Woodlief, called “Dreaming God”:

We are god-obsessed and god-seeking and at least the intellectuals of earlier ages—even if they couldn’t bring themselves to belief—recognized this. So many of today’s intellectuals are so far removed from religion that they don’t know the half of how deeply it’s intertwined in the lives and hearts of the rest of us.

And even more provocative:

We are god-obsessed because we have lost God or we are running from God or we are hopelessly seeking Him, and maybe all of these at once.

An extremely interesting analogy:

We are god-obsessed the way a child snatched from his mother will always have his heart and flesh tuned to her, even after he forgets her face. Cover the earth with orphans and you will find grown men fashioning images of mothers and worshipping strong women and crafting myths about mothers who have left or were taken or whose spirits dwell in the trees.
And at the edges of their tribal fires will stand the anthropologist and the philosopher, reasoning that all this mother-talk is simply proof that men are prone to invent stories about mothers, which is itself proof that no single story about a mother could be true, which is proof that the brain just evolved to work that way.

I’ll comment on this one of these days when I get more time.

LaTeX on Blogspot

2011-08-10T21:34:00.007-04:00

Blogspot is pretty bad at displaying math; \( \LaTeX \) is pretty awesome at displaying math. So how do we get Blogspot and \( \LaTeX \) working with each other? Using MathJax!

All it takes is the addition of one line to the HTML code for each blog, and presto! we can include \( \TeX \) and \( \LaTeX \) commands into our blog. And not only can we do inline math, like \( \int_{0}^{2 \pi} \mathrm{d}x = 2 \pi \), but also display math, like:

\[
\begin{eqnarray}
(a + b)^2 & = & (a + b) \cdot (a + b) \\
& = & a(a+ b) + b(a + b) \\
& = & a^2 + ab + ba + b^2 \\
& = & a^2 + 2ab + b^2
\end{eqnarray}
\]

Truly remarkable! Now all I need to do is update all my old math posts using MathJax and good ol’ \( \LaTeX \).

Counting, uncounting, and dust, part four

2011-08-06T02:14:00.008-04:00

I’d been completely side-tracked from this set of posts, and I only just realized I had forgotten to finish it. Last time we checked, we had finally figured out that there are different levels of infinity. We had, however, ended up with a rather strange idea: the idea that the set of all real numbers between 0 and 1 is somehow as big as the set of all real numbers in its entirely. In effect, we have shown that there is no difference between a mountain and a molehill!

We thus have two different lines of inquiry to pursue:

Are there levels of infinity beyond the infinity of the real numbers?
Are there ways to measure the size of a set that can distinguish mountains from molehills?

Levels of infinity
We begin with the idea of a power set: the power set of a set is defined as the set of all subsets of the set. Huh? An example should make things clearer: if \( A = \{ a, b, c \} \), then the power set of \( A \), sometimes written as \( \mathcal{P}(A) \) and sometimes \( 2^{A} \) (we’ll see why), is the set \( \mathcal{P}(A) = \{ \{ a \}, \{ b \}, \{ c \}, \{ a, b \}, \{ b, c \}, \{ a, c \}, \{ a, b, c \}, \{ \} \} \). All those braces are necessary, because every element of \( \mathcal{P}(A) \) is a subset (and not an element) of \( A \) itself. We note that \( A \) itself is an element of \( \mathcal{P}(A) \), as is the empty set \( \{ \} \).

Now it is fairly easy to show that if \( A \) is a finite set of cardinality \( n \) (written \( | A | = n \) ), then the cardinality of the power set of A is \( 2^{n} \). More picturesquely, \( | 2^{A} | = 2^{| A |} \).

Why? Well, let’s count the number of subsets of size \( k \) for every number less than or equal to \( n \):

For \( k = 0 \): There is exactly one subset of \( A \) with zero elements, and that is the empty set
For \( k = 1 \): There are \( n \) subsets with one element each
For \( k = 2\): There are \( \binom{n}{2} \) subsets with two elements (where I am using one of many different notations to represent combinations. More on them elsewhere)

… and so on.
When we add them all up, we somehow magically (a.k.a. the Binomial Theorem!) get \( 2^n \)! Quite magical (not).

This is all very cute, of course, but what happens when \( A \) is not finite? What happens when we take, say \( \mathbb{N} \), as our set? What, in other words, is \( 2^{\mathbb{N}} \)?

We know that for any finite number \( n \), \( 2^n > n \). One of the neat things about Cantor’s diagonalization theorem is that it tells us this is true for any \( n \)—including infinite \( n \)! How?

We recall that we had earlier defined “counting” as establishing a bijective correspondence between two sets. This is how we proved that \( \mathbb{N} \) and \( 2 \mathbb{N} \) were, in fact, the same “size”. So to prove that \( A \) and \( \mathcal{P}(A) \)are not, in fact, the same size, we must show that no correspondence between them can be bijective. And if we can show that no matter what correspondence we take, there is always some element of \( \mathcal{P}(A) \) which gets left out even while every single element of \( A \) is accounted for, then we can prove that \( \mathcal{P}(A) \)is, in fact, bigger than \( A \).

We begin by noting that \( \mathcal{P}(A) \) is at least as large as \( A \). This is true because there is a bijection between \( A \) and the elements of \( \mathcal{P}(A) \) that are all single-element subsets of \( A \). (In symbols, the collection \( \{ a, b, c \} \) has a bijective correspondence with \( \{ \{ a \}, \{ b \}, \{ c \} \} \).) Now consider any function f that maps an element of \( A \) to an element of \( \mathcal{P}(A) \). We need to show that there is at least one element of \( \mathcal{P}(A) \) that cannot be in \( A \) (and let us constantly remind ourselves that an element of \( \mathcal{P}(A) \) is a subset of \( A \) ). How can we find such an element of \( \mathcal{P}(A) \) that does not depend on the particular definition of a correspondence \( f \)?

This is the step in the diagonalization process analogous to picking one digit from every number on the list and modifying it in order to create a new number. But how do we do that with sets? We note that the correspondence \( f \) connects every element \( x \in A\) with an element \( f(x) \in \mathcal{P}(A) \) ; i.e., with a subset of \( A \). Depending on how we define our correspondence, it may be the case that \( x \) is actually an element of the subset that it maps to; or it may be the case that \( x \) is not an element of the subset it maps to. Clever devils that we are, we pick our element of \( \mathcal{P}(A) \) to be the set \( B \) of all those elements of \( A \) which are not elements of the subset they correspond to. (This is like changing the digit we pick so we know our new number differs from every existing number in at least one place.) In symbols, \( B = \{ x \in A | x \notin f(x) \} \).

How do we know that there is no element of \( A \) that can be mapped to \( B \) by \( f \) Why, by the very definition of \( B \) itself! Suppose, for the sake of contradiction, that there does exist a \( y \in A \) such that \( f(y) = B \). By the definition of \( B \), this must mean that \( y \notin B \). But then by the definition of \( B \), we have \( y \in B \)! This is a clear contradiction, which shows that there simply cannot be any such \( y \). In other words, \( \mathcal{P}(A) \) is always larger than \( A \)—regardless of whether \( A \) has finitely many elements or not!

Thus, \( 2^{\mathbb{N}} \) must needs be greater than \( \mathbb{N} \) in size. But we also know, through a different diagonalization proof, that \( \mathbb{R} \) is greater than \( \mathbb{N} \) in size. Is it in fact true that \( \mathbb{R} = 2^{\mathbb{N}} \)? As it turns out, this question is better answered by taking \( \mathbb{Q} \) rather than \( \mathbb{N} \) (but as we saw earlier, these two sets are the same size, and so it doesn’t really matter). And the answer, which I will not go into in any detail, is yes. When we construct real numbers by the method of Dedekind cuts, we do in fact map every real number to a subset of \( \mathbb{Q} \). The converse can also be shown to be true.

But regardless, the power set argument shows that there have to be infinities greater than the uncountable set of the real numbers. The power set of the real numbers, \( \mathcal{P}(\mathbb{R}) \), is strictly larger than \( \mathbb{R} \), and its own power set \( \mathcal{P}(\mathcal{P}(\mathbb{R})) = 2^{2^{\mathbb{R}}}\) is larger still, and so on, ad infinitum! (A more concrete example of a set that is of cardinality \( 2^{\mathbb{R}} \) is the set of all functions from \( \mathbb{R} \) to \( \mathbb{R} \).)

Indulge me, por favor

2011-08-05T17:10:00.000-04:00

I don’t normally soar away in flights of fantasy; I enjoy feeling grass underneath my feet and sand between my toes far too much. But just this once, I request you, gentle reader, to forgive my rhapsodizing. If it doesn’t make sense, well, it wasn’t necessarily meant to!

So here goes: Each of us is a link in an infinite chain of being that spans space and time. Each of these links is, of course, comprised of smaller links, ad infinitum; each of this links, of course, participates in a greater link, ad infinitum. Do I exaggerate when I speak of twin infinities? Maybe, and maybe not.

What I’m really trying to say is that everything is doubly emergent.

Now materialist reductionism is the idea that things can be understood entirely by parsing them into their constituent material parts. It is a remarkably powerful, persuasive idea, and the basis of much modern theoretical and practical advancement, but according to at least some thinkers, it cannot explain the phenomenon of emergence. For them, an anthill is more than the sum of its parts; similarly, each of us human beings is more than the sum of our parts. There is something about our complexity that is irreducible to the parts that constitute us. (Note that emergence does not automatically reject materialism; it does, however, reject reductionism.)

But at the same time, we ourselves are also parts of a bigger, emergent reality—society, we call it. We take it for granted and thus forget how much of who we are (of our emergent selves!) is both affected and effected by this layer of abstraction that lies atop organized collections of interacting human beings.

We are different from computers because our operating systems are able to rewire the physical hardware on which they run.

And paradoxically, the more “concrete” and “elementary” our constituents get, the more conceptual and abstract they become! We smash atoms into electrons and protons and neutrons, only to find that these “elementary” particles are probability distributions; we take them apart even further, and are ultimately left with vibrating 26-dimensional strings. And yet somehow causality travels up this chain in powerful, largely well-understood ways!

We have become accustomed to thinking of causality purely in instrumental terms. In that sense, it is of course true that it is the parts that alter the whole. But we forget that the word “cause” used to have a much wider sense. What we think of as the “cause” these days is only the Aristotelian “effective cause”. We have forgotten that other “causes” exist and have real effects. The “formal cause”, for instance, can be seen as the way in which higher layers of abstraction limit and direct lower layers. Again, this does not necessitate a belief in a Platonic realm of Forms. When a carpenter builds a chair, it is obviously true that his tools operating on the wood are the “effective causes” of what is produced. But is it not true that a “formal cause”—an understanding of what it means to be a chair, which is necessarily influenced by his social position—also has a part to play in this? We no longer think of this as causality, but as a result we are unable to fully grasp what’s going on here. Causality goes both upwards and downwards (and maybe sidewards as well!) over the web of existence.

Levels of description matter. “Romeo loved Juliet” is as true as “a certain well-structured collection of organic compounds produced certain levels of serotonin and oxytocin in the presence of a similarly well-structured collection of organic compounds”, but they don’t mean the same thing. Even if you ignore the fact that Romeo and Juliet are literary figures! Levels of description matter, and although the same truth can be expressed at different levels, it is significant in different ways at those levels. This is very similar to Karl Popper’s “Three Worlds”, but I think “Three” is too much and too little: too much, because there is only one world; too little, because that one world exists and interacts at many, many different levels. This is not the same as saying there are “Two Truths”; there aren’t, and there cannot be. But the same truth can be expressed at different levels.

Some who face this tower of concrete-yet-abstract layers dismiss it all as illusion or as emptiness. I think the exact opposite is the case. This is reality: a unified whole, an infinitely diverse, infinitely layered, fractal, organic tower that grows, breathes, becomes self-conscious, tries to comprehend all of itself, and shrinks.

Auxiliary Verbs, auf französisch, part trois

2011-07-19T01:45:00.004-04:00

Now that we’ve got a fairly good sense of the two main auxiliary verbs in French, avoir and être, let’s examine some of the more complicated stuff that inevitably comes along. Earlier posts on French auxiliaries are here and here.

The passive voice

As in English, the auxiliary used to form the passive voice is the form “to be”, which is être, colored dark red here as in an earlier post to distinguish it from both the regular verb être and the auxiliary être. In a way, it’s actually false to distinguish between the passive auxiliary être and the full verb être because both really behave the same way, including the question of requiring gender and number agreement of the adjective. The only reason, perhaps, to make this artificial distinction is to be clear that the passive auxiliary être takes as its adjective a past participle of the main verb.

That’s right: the structure of the passive voice is identical to the structure of the compound tenses using the non-modal auxiliary être. Talk about a recipe for confusion!

Except not really. Because the verbs we’ve seen so far that take être as their auxiliary are all intransitive verbs—they lack a direct object, and so there is simply no way they can form a passive at all. What this means is that it’s only really verbs taking avoir as their auxiliary that will form the passive with être.

And of course, once the passive is formed, it can form its own compound tenses. Since the conjugated verb in the passive voice is être, this means the auxiliary constructions will be formed with avoir + été + past participle (inflected).

So let’s take the sentence “I read a book” and look at every single possible verbal construction. This list will look slightly different from the list above because there I classified the verbs into seven groups based on the TAM in which the conjugated verb was conjugated. Here, though, I will be classifying them into the fourteen possibilities that we get by including the various compound tenses.

présent de l’indicatif

je lis le livre
le livre est lu

présent du subjonctif

(que) je lise le livre
(que) le livre soit lu

passé composé

j’ai lu le livre
le livre a été lu

passé du subjonctif

(que) j’aie lu le livre
(que) le livre ait été lu

imparfait de l’indicatif

je lisais le livre
le livre était lu

imparfait du subjonctif

(que) je lusse le livre
(que) le livre fût lu

plus-que-parfait de l’indicatif

j’avais lu le livre
le livre avait été lu

plus-que-parfait du subjonctif

(que) j’eusse lu le livre
(que) le livre eût été lu

passé simple

je lus le livre
le livre fut lu

passé antérieur

j’eus lu le livre
le livre eut été lu

futur simple

je lirai le livre
le livre sera lu

futur antérieur

j’aurai lu le livre
le livre aura été lu

conditionnel

je lirais le livre
le livre serait lu

passé première forme

j’aurais lu le livre
le livre aurait été lu

Quite lovely, really. And it’s worth noting, although only briefly, that if our sentence had been “I read poetry”, which would give us je lis la poèsie, then all the active forms would remain identical, while all the passive forms would remain identical except for the use of the feminine form of the past participle of the verb lire, which is lue. Poetry is feminine for the French, while a book of poems is masculine.

If we note that the passé première forme is nothing but the compound tense of the conditionnel, which in turn is rather like a conditional or subjunctive-ish (I use the word loosely) version of the futur simple, we can see that we can translate le livre aurait été lu almost word-for-word as “the book would have been read.” The only difference is that English must make use of a modal auxiliary and a non-modal auxiliary to convey what French can convey in a single (non-modal) auxiliary conjugated with the appropriate endings.

Reflexive and reciprocal verbs (les verbes pronominaux)

This gets a little more fun. Reflexive and reciprocal verbs, or the so-called pronominal verbs, are called so because they use the reflexive pronouns. English has lost this pronoun and must use constructions like “myself” or “herself” instead, but other languages preserve it. German, for instance, uses sich; French uses se; and Russian has себя. In all of these cases, the pronoun is used whenever the agent and the patient of the verb are either the same (i.e., reflexive) or doing the same action to each other (i.e., reciprocal).

Now, of course, the three languages use their reflexive pronouns differently. French and German inflect the pronoun for person, as English does too, sort of. Witness I talk to myself and He talks to himself; in French, je me parle and il se parle; and in German, ich spreche mich and er spricht sich. Russian does not; and so these two would work out to be я себе говорю and он себе говорит. This is also the case in Hindi: मैं अपने आप से बात कर रहा हूँ and वह अपने आप से बात कर रहा है. It is worth noting, however, that in both French and German, it is only in the third person that the reflexive pronoun is distinct from the direct and the indirect pronouns—sich versus ihn or ihm, in the case of German, se versus le/la or lui.

Another interesting difference comes into play with the use of grammatical cases. Russian, as a language that makes heavy use of cases, declines себя in the appropriate case in every context: себя in the accusative and genitive, себе in the dative and prepositional, and собой in the instrumental. This parallels the declension of the (non-reflexive) first- and second-person pronouns меня and тебя. German, which also uses cases fairly extensively, declines the first and second person pronouns for case (thus mich and mir, dich and dir), but not the reflexive pronoun itself (sich is both accusative and dative). French, however, makes no distinction among the direct (i.e., accusative), indirect (i.e., dative), and reflexive pronouns in the first person and second person in the singular and in the plural: me, te, nous, and vous serve all these purposes. Only in the third person is any distinction made:

in the singular, we have:

masculine (il): le, lui, and se
feminine (elle): la, lui, and se

in the plural, we have, for both masculine (ils) and feminine (elles): les, leur, and se

The point of all of this is to build up to some of the trickiness concealed by the use of the reflexive pronoun se in the French reflexive and reciprocal verbs.

The most common categories of reflexive and reciprocal verbs are:

the so-called “inherently reflexive” verbs like se souvenir, meaning “to remember”, which just happen to take the reflexive
reflexive verbs which are formed from non-reflexive verbs, such as se laver (“to wash [oneself]”) from laver (“to wash [someone else]”)
reciprocal verbs, which are typically formed from non-reflexive verbs, and which are conjugated in the plural, such as se parler (“to speak [to each other]”).

In all of these cases, the auxiliary verb is always, without exception, être. Therefore, the default position is that the past participle must agree in gender and number with the subject of the verb. Thus, “I remember” is je me souviens (also the motto of Quebec!), while the passé composé of “she remembered” is elle s’est souvenue. And the passé composé of “I’ve washed (myself)” is je me suis lavé (since I’m male).

But—and there always seems to be one in French!—this past participle agreement falls apart in two further cases:

when a perfectly well-formed reflexive verb is followed by a part of the body that is a direct object (so “she washed (herself)” is elle s’est lavée but “she washed her hands” is elle s’est lavé les mains)
when the reflexive pronoun is not a direct object (so “we [all females] spoke among ourselves” is nous nous sommes parlé, because the structure of the verb parler demands the use of an indirect object with the preposition à, and so the se in se parler is really an indirect reflexive pronoun)

The strange rules of past participle agreement in French can be summarized thus:

If the auxiliary verb, whether avoir or être, is preceded by a direct object (whether reflexive or not), then the past participle must agree with the direct object.
If not, then:

if the auxiliary verb is avoir or if it is être followed by a part of the body, there is no agreement. This is why the compound tenses of the passive voice do not show agreement between the subject and the past participle été.
if the verb is être, the participle agrees with the subject of the verb. This is why the simple tenses of the passive voice do show agreement between the subject and the past participle of the underlying verb.

A few final examples to clear everything up:

With avoir:

“I [male] see the girl”: je vois la fille
“I [male] have seen the girl”: j’ai vu la fille
“I [male] see her”: je la vois
“I [male] have seen her”: je l’ai vue
“I [male] have read the books”: j’ai lu les livres
“I [male] have read them”: je les ai lus
“The books which I’ve read are interesting”: les livres, que j’ai lus, sont intéressants [the direct object of lire precedes it]

With the passive voice:

“The girl is seen”: la fille est vue
“The girl has been seen”: la fille a été vue [note that été is masculine and vue is feminine]

With être:

Intransitive verbs:

“The girl leaves”: la fille part
“The girl (has) left”: la fille est partie
“The boy leaves”: le garçon part
“The boy has left”: le garçon est parti

Reflexive verbs:

“She washes (herself)”: elle se lave
“She has washed (herself)”: elle s’est lavée
“She washes her face”: elle se lave le visage
“She has washed her face”: elle s’est lavé le visage

That should be it, I think, for now, at least on the topic of French auxiliary verbs. Mon dieu!

Auxiliary Verbs, auf französisch, part deux

2011-07-19T00:54:00.002-04:00

Last time, we took a very brief look at the general structure of the French verbal system. Let’s now take a gander at the auxiliaries! (For that was the whole point of this, n’est-ce pas?)

Instead of getting bogged down in hypothetical forms, let’s start by looking at the compound tenses of the verb parler. Paralleling English, the structure of this construction is avoir + past participle (with the word avoir colored dark green to show that it is the auxiliary verb and not the usual verb avoir that means “to have”). We can conjugate the verb avoir in every tense and mood that we can conjugate parler in; these compound tenses have different names. To keep things simple, let’s just look at the first person singular I speak in all its various forms in French.

présent de l’indicatif

simple: je parle
compound (passé composé): j’ai parlé

présent du subjonctif

simple: (que) je parle
compound (passé du subjonctif): (que) j’aie parlé

passé simple

simple: je parlai
compound (passé antérieur): j’eus parlé

imparfait de l’indicatif

simple: je parlais

compound (plus-que-parfait de l’indicatif): j’avais parlé

imparfait du subjonctif

simple: (que) je parlasse
compound (plus-que-parfait du subjonctif): (que) j’eusse parlé

futur

simple: je parlerai
compound (futur antérieur): j’aurai parlé

conditionnel

simple: je parlerais
compound (passé première forme): j’aurais parlé

We can arrange the tenses in the indicative mood in the following temporal sequence:

Past context (with no necessary temporal distinction between the imparfait and passé simple)

j’avais parlé — I had been speaking / j’eus parlé — I had spoken
je parlais — I was speaking / je parlai — I spoke

Present context

j’ai parlé — I have spoken
je parle — I speak / am speaking

Future context

j’aurai parlé — I will have spoken
je parlerai — I will speak

Up to this point, things are not very different from English. However, things are about to get a lot more complicated. This is because French also allows the verb être to function as an auxiliary verb in some specific cases:

when the main verb is an intransitive verb of motion or of change of state

but then sometimes these verbs can end up with avoir if they’re then used transitively!

when the main verb is a reflexive or a reciprocal verb.

This is trickier than English, certainly, but not yet insurmountable. For if we just had a list of verbs that took être as the auxiliary verb, we would be fine. Alas, one more wrinkle: in the construction être + past participle, the past participle must agree with the subject of the verb.

So let’s take the verb partir (“to leave”) and conjugate it in the first-person singular. (Since I’m male, I will use the masculine singular form of the past participle of partir, which is simply parti. If I were female, I would use partie, and if I were instead conjugating the verb in the first-person plural, I would use partis.)

présent de l’indicatif

simple: je pars
compound (passé composé): je suis parti

présent du subjonctif

simple: (que) je parte
compound (passé du subjonctif): (que) je sois parti

passé simple

simple: je partis
compound (passé antérieur): je fus parti

imparfait de l’indicatif

simple: je partais

compound (plus-que-parfait de l’indicatif): j’étais parti

imparfait du subjonctif

simple: (que) je partisse
compound (plus-que-parfait du subjonctif): (que) je fusse parti

futur

simple: je partirai
compound (futur antérieur): je serai parti

conditionnel

simple: je partirais
compound (passé première forme): je serais parti

This is what we get. But we’re not yet done. There are still a few murky pools to wade through before reaching the shores of clarity.

The full verbs être and avoir

As in English, the two non-modal auxiliary verbs used in French are also independent, fully functional verbs. This means that they too can take an auxiliary verb in order to generate a compound tense. The past participle of avoir is eu, and that of être is été. For whatever reason, both these verbs take the auxiliary avoir, so that, for example, the sentence j’ai une voiture (“I have a car”) becomes j’aurai eu une voiture in the futur antérieur and j’eusse eu une voiture in the plus-que-parfait du subjonctif.

With être as a full verb, things look a little more interesting. We say Jacques est heureux for “Jack is happy” and Marie est heureuse for “Marie is happy”; it is clear that the adjective has to agree in gender and number with the subject. However, because être takes avoir as its auxiliary, the participle été remains unchanged. Thus, in the plus-que-parfait du subjonctif (I pick this only because this strikes me as being perhaps the most recondite of French verbal forms!) these two sentences become Jacques eût été heureux and Marie eût été heureuse, respectively.

Engineering and science

2011-07-17T11:42:00.005-04:00

This is a fascinating quote from Eugene Ferguson’s Engineering and the Mind’s Eye (p. 13):

“The philosopher Carl Mitcham gives design and invention their proper places in the scheme of things by observing that ‘invention causes things to come into existence from ideas, makes world conform to thought; whereas science, by deriving ideas from observation, makes thought conform to existence.’ ”

While a lot can be said about it, including a somewhat artificial distinction, I think, between science and invention, what struck me immediately was the similarity between this verse and the famous verse of Bhavabhūti from the Uttararāmacarita, which I've already written about. I cite the Sanskrit here again:

laukikānāṃ hi sādhūnām arthān vāg anuvartate |
ṛṣīṇāṃ punar ādyānāṃ vācam artho ’nuvartate ||

A loose English translation: “For the good people of this world, speech conforms to reality; for the seers of old, reality conforms to speech.”

“Desiderata”

2011-07-17T11:31:00.002-04:00

It was quite intriguing to learn that Cap’n Jack Sparrow bears upon his back a tattoo of a famous poem, now used as a devotional on occasion: Desiderata, by Max Ehrmann. Here is the poem (now ruled to be out of copyright even though its author only passed away in the 1940s), thanks to Wikipedia:

Go placidly amid the noise and haste,

and remember what peace there may be in silence.

As far as possible without surrender

be on good terms with all persons.

Speak your truth quietly and clearly;

and listen to others,

even the dull and the ignorant;

they too have their story.

Avoid loud and aggressive persons,

they are vexations to the spirit.

If you compare yourself with others,

you may become vain or bitter;

for always there will be greater and lesser persons than yourself.

Enjoy your achievements as well as your plans.

Keep interested in your own career, however humble;

it is a real possession in the changing fortunes of time.

Exercise caution in your business affairs;

for the world is full of trickery.

But let this not blind you to what virtue there is;

many persons strive for high ideals;

and everywhere life is full of heroism.

Be yourself.

Especially, do not feign affection.

Neither be cynical about love;

for in the face of all aridity and disenchantment

it is as perennial as the grass.

Take kindly the counsel of the years,

gracefully surrendering the things of youth.

Nurture strength of spirit to shield you in sudden misfortune.

But do not distress yourself with dark imaginings.

Many fears are born of fatigue and loneliness.

Beyond a wholesome discipline,

be gentle with yourself.

You are a child of the universe,

no less than the trees and the stars;

you have a right to be here.

And whether or not it is clear to you,

no doubt the universe is unfolding as it should.

Therefore be at peace with God,

whatever you conceive Him to be,

and whatever your labors and aspirations,

in the noisy confusion of life keep peace with your soul.

With all its sham, drudgery, and broken dreams,

it is still a beautiful world.

Be cheerful.

Strive to be happy.

al-Ghazālī on memory and knowledge

2011-05-24T14:13:00.005-04:00

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

2011-05-05T03:06:00.001-04:00

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

2011-04-12T18:24:00.001-04:00

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.

So what do we have in French? Three tenses: the present (le présent), the preterite (le passé simple), and the future (le futur). There is one further aspectual distinction drawn in the past tense through the use of a synthetic form, which is the imperfect (l’imparfait). A modal distinction is drawn in the future through another synthetic form, producing the so-called conditional mood (le conditionnel). The présent and the imparfait come in two moods, indicative (l’indicatif) and subjunctive (le subjonctif).

We could, of course, organize these seven forms differently. Instead of grouping them as we have done, we could group them by verbal stem, which would give us

the present stem, from which arise

the présent de l’indicatif and
the présent du subjonctif

and derived from the présent de l’indicatif is the imparfait de l’indicatif;

the future stem, which is almost always just the infinitive itself, from which arise

the futur and
the conditionnel;

the preterite stem, which is often related to the past participle, from which we get

the passé simple

and derived from the passé simple is the imparfait du subjonctif.

In addition to these synthetic forms, we have a number of compound forms that make use of auxiliary verbs. As with English, the auxiliary verb is the one that is conjugated in a particular tam, and the main verb remains in an unconjugated (usually participial) form. Thus we have:

from the auxiliary in the présent indicatif or subjonctif, the present perfects:

the passé composé de l’indicatif and
the passé composé du subjonctif;

from the auxiliary in the imparfait indicatif or subjonctif, the pluperfects:

the plus-que-parfait de l’indicatif and
the plus-que-parfait du subjonctif;

from the auxiliary in the futur, the future perfect (futur antérieur)
from the auxiliary in the conditionnel, the future perfect conditional (passé conditionnel)
from the auxiliary in the passé simple, the past perfect (passé antérieur)

And of course, every one of these 14 different verbs can be either active or passive, generating one more layer of complexity.

As with English, French uses non-modal auxiliary verbs to convey semantic and syntactic nuances. There are many similarities between the English and French auxiliary verb systems.

As with English, French uses two non-modal auxiliary verbs, avoir (“to have”) and être (“to be”).
As with English, both avoir and être are also full-blown verbs in their own right, operating independently of their role as auxiliary verbs.
The structure of the French auxiliary construction involves a conjugated form of the auxiliary verb combined with a past participle of the main verb, just as it does in English (although this is not entirely identical in the two languages).
Both English to be and French être are also used to form all passive voice constructions.

There are, however, a number of differences between the English and French auxiliary verbs.

Whereas English uses the construction to be + gerund to indicate a progressive or continuous aspect to the main verb, French does not do so. In the present tense, this is built into the French verb itself: thus, while in English I eat (habitually) is different from I am eating (currently, continuously, or progressively) in French we would use je mange for both. In the past tense, French uses the separate imparfait verbal form to indicate progression or continuation.
Whereas English possesses modal auxiliary verbs, French does not. This is a major, major structural difference between the two languages, and indeed between the Germanic languages (English, Dutch, and German, among others) on the one hand and the Romance (French, Spanish, Italian, and the like).
Whereas English possesses only one non-modal auxiliary verb (to have) to indicate the perfect aspect, French uses both avoir and être for the same semantic function.

Now, while it was possible to speak of English verbs as possessing principal parts, this is not quite the case in French because French verbs are far more inflected. Instead, it makes more sense to speak of a French verb as possessing a number of different verbal stems in addition to the usual suspects like the past participle and present participle.

To take an example, the exceedingly regular verb parler (“to speak”) has the following parts:

present stem: parl–
preterite stem: parl–
future stem: parler–
present participle ‘stem’: parlant–
past participle ‘stem’: parlé–

As noted above, one can use the three verbal stems to generate the other synthetic verbal forms such as the conditionnel and the imparfait subjonctif. And I will explain the use of the word ‘stem’ in the last two forms presently (no pun intended). But before that, it is extremely important to remember that a lot of verbs, especially the most frequently used, are much more complicated. Verbs not only have different stems for different tenses, but may go so far as to switch stems within the same tense. Thus, looking at the regular verb parler and the trickier verb boire (“to drink”):

présent de l’indicatif

I speak: je parle; I drink: je bois
We speak: nous parlons; we drink: nous buvons
They speak: ils parlent; they drink: ils boivent

présent du subjonctif

[It is necessary that] I speak: (que) je parle; [that] I drink: (que) je boive
[It is necessary that] we speak: (que) nous parlions; [that] we drink: (que) nous boivions
[It is necessary that they speak: (que) ils parlent; that they drink: (qu’) ils boivent

imparfait de l’indicatif

I was speaking: je parlais; I was drinking: je buvais
We were speaking: nous parlions; we were drinking: nous buvions
They were speaking: ils parlaient; they were drinking: ils buvaient

To return to an earlier point of digression, I used the word ‘stem’ with the present and past participles earlier. The reason for this is that French, unlike English, is an intrinsically gendered language, so that every single noun is either masculine or feminine in gender. Furthermore, since adjectives must agree in gender with the nouns they modify, it follows that the participles agree (but only in some cases, not all) with the nouns they modify. In general, the bare form of the participle, the so-called ‘stem’, represents the masculine singular; we add an –e to make it feminine and an –s to make it a plural.

This is by no means a complete look at the complex verbal system of French, but it gives us a sense of what we're up against once we get into the intricacies of the auxiliary verbs.

Auxiliary Verbs, en anglais

2011-04-08T01:59:00.002-04:00

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.)

Before looking at particular languages, let's dive a little more deeply into this “whole host of additional meanings”. Every verb has a semantic core, a meaning that is somehow present in every single form of its use (except for idiomatic usages and the like, of course). But in addition to this core meaning of the verb, the most basic action that it can convey, there are a number of additional nuances that can be conveyed: the time when this action takes place; the status of the action, i.e., whether it’s only just begun or in progress or about to end; the likelihood of the action happening, and so on. I picked these three particular sets of nuances not because they are exhaustive, but because they have their own names in grammar: tense, aspect, and mood respectively. Every language on earth has some way or the other of conveying these three things, and they are usually intertwined in strange ways that seem entirely non-strange because they are utterly internalized by native speakers. For this reason, they're sometimes referred to collectively as tam.

One more idea remains, that of grammatical voice, but I will get to it later.

For now, though, let's look at the tam structure of English. This list here may seem strange, even wrong; if it strikes you so, I request you, gentle reader, to hold back your criticism just a little bit longer and to accept what I say at face value for now. (This is not to say that my descriptions are perfect; far from it!)

t: English has two tenses, the non-past and the past.
a: English can represent a wide range of aspects, but in addition to an unmarked aspect whose meaning depends on tense, there are two further aspects that are intimately connected to the verb, the perfect and the continuous/progressive.
m: English has at least two moods, the indicative and the subjunctive. Some include a third, the imperative. English can express all sorts of complex combinations of moods using additional modal verbs, which are also of interest to us here; what is normally considered the future tense in English is strictly speaking a modal verb.

The English verb system
Enough of this, let's see this system in operation! Let's take a verb, and see what forms we can generate. To celebrate the coming of spring, let's take a verb describing an activity that can finally be performed outdoors: swimming. All grammatical forms that convey this core semantic value can be constructed from one of three so-called principal parts: the infinitive (in this case, to swim), the preterite (here, swam), and the past participle (here, swum). Not all English verbs have three distinct principal parts; I picked this because it does. A fourth part, which is not principal because it can be produced fairly regularly from the infinitive, but which is important because of its frequency of occurrence, is the gerund (here, swimming).

[A minor note: the gerund is identical in form to the verbal noun, but they serve dramatically different purposes and must never be confused. Consider the difference between the gerund swimming in the sentence I am swimming and the verbal noun swimming in the sentence Swimming is fun. Here's a simple rule of thumb to differentiate between them: if you can substitute the infinitive in the place of the word without destroying the original meaning, it is the verbal noun. The first use of swimming is a gerund because I am swimming means something very different from I am to swim, whereas the second use of swimming has to be a verbal noun because both Swimming is fun and To swim is fun essentially mean the same thing.]

On its own, using just the principal parts, we can produce the following forms.

Non-past tense, indicative mood:

I swim.
You swim.
He swims.
We swim.
They swim.

Non-past tense, subjunctive mood:

(It is necessary that) … I swim.
… you swim.
… he swim.
… we swim.
… they swim.

Past tense, indicative mood:

I swam.
You swam.
He swam.
We swam.
They swam.

A few things are already clear.

First, English verbs show almost no conjugations at all. In the non-past indicative, the only change is the introduction of the -s in the third person singular.
Second, the non-past indicative and subjunctive are almost identical, with the only difference being that the subjunctive does not add the -s.
And finally, it is abundantly clear that the bare English verb is sadly incapable of conveying any further meaning whatsoever. It desperately needs to be assisted, to be supplemented, to be augmented by something else in order to produce any sort of nuanced meaning at all.

Did someone just call for assistance? In march the auxiliaries!

The English auxiliaries

Loosely speaking, we can classify all English auxiliary verbs into two classes: modal and non-modal.

There are three major non-modal auxiliary verbs in English: to be, to have, and to do. Of these, to do is mostly used to deal with negation. Both to be and to have act as aspectual auxiliaries, allowing us to convey additional nuances connected to time and to status of completion.
The main non-modal auxiliaries, which convey additional nuances of meaning connected to likelihood and certainty, are shall / should, will / would, can / could, may / might, and must.

The major difference between the modal auxiliaries on the one hand and the non-modal auxiliaries is that the latter also serve as full verbs in the language: to be means to exist, to have means to possess, and to do means to enact or to perform. The modal auxiliaries, on the other hand, are deficient; they have neither infinitives nor past participles. Another major difference between the modal and non-modal auxiliaries will soon become apparent.

In order to distinguish between these different uses, I will represent all auxiliary verbs in dark green and all modal verbs in dark blue hereon.

Non-modal auxiliaries in English
So what is the structure of an auxiliary construction in English? It is always the auxiliary verb that is conjugated, while the main verb that is being augmented takes some sort of indeclinable form.

With have, this form is the past participle, so that the form is to have + past participle
With be, this is the gerund, so that the form is to be + gerund

I will represent this indeclinable part of the verb in bold.

Thus, in the plain sentence I swim, the verb to swim can give us the following auxiliary usages:

With have (this is the so-called perfect aspect, since it communicates the completion of an action):

I have swum.
I had swum.

With be (this is the continuous-progressive aspect):

I am swimming.
I was swimming.

Our understanding of the structure of the auxiliary allows to generate yet another form, the so-called perfect progressive. This involves taking the verbal structure to be + gerund and treating its conjugating part (the verb to be) as the basis for generating the indeclinable portion of the verbal structure to have + past participle. This gives us

to have + past participle of to be + gerund,

or more simply

to have been + gerund,

where been is green to show that it came from the auxiliary be and bold to show that it is indeclinable. Thus, two forms of this construction:

I have been swimming.
I had been swimming.

Modal auxiliaries in English
The modal verbs in English are strange beings. They are highly deficient, lacking even the most basic accoutrements of regular verbs like the infinitive, the past participle, and the gerund. This, when combined with our knowledge of the non-modals have and be, tells us something:

English does not permit non-modal auxiliaries to be followed by a modal auxiliary.

So what is the structure of a modal auxiliary construction? As with the non-modal construction, this also involves one part that can conjugate (at least to the extent that English modal verbs conjugate) and one indeclinable part. The modal auxiliary construction is

modal verb + bare infinitive,

where what I mean by the “bare infinitive” is simply the infinitive minus the ‘to’ at the beginning.
This structure also immediately tells us another rule:

English does not permit modal auxiliaries to be followed by another modal auxiliary.

This is simply because modal verbs lack infinitives, so there is no way we can arrange them into this kind of structure. It’s also worth noting that this is not the case for non-modal auxiliaries; as we saw, we can indeed have the sequence to have been + gerund.

Anyway, let’s look at some examples. In properly prescribed English (not that this is followed any longer), the future tense is expressed in the first person using the modal shall and in the second and third persons using the modal will. So the two sentences I swim and He swims can be put into the future thus:

I shall swim.
He will swim.

If that’s all there were to modal auxiliaries, there wouldn’t be much to write home about. Things get interesting, though, because modal auxiliaries can be followed by non-modal auxiliaries. This is, of course, because the non-modal auxiliaries have and be do have bare infinitives. Thus, we can add considerable temporal nuance to our proclamations for the future thus:

I shall swim.
I shall be swimming.
I shall have swum.
I shall have been swimming.

The Passive Voice
The auxiliaries we saw so far help expressed semantic nuances: nuances of meaning to do with time and likelihood. There is one more construction in English that uses an auxiliary verb, but this is largely for syntactic purposes: the passive voice. (This is not to say that the English passive voice is semantically identical to the English active voice; it is not.)

Slightly confusingly, the auxiliary verb for this is also to be; I will distinguish it from the non-modal auxiliary be by coloring this one dark red. The passive voice of a verb is formed thus:

to be + past participle,

and hence this must not be confused with the auxiliary construction using to have.

Examples, as always. The sentence Jack eats the fruit is in the active voice, and can be transformed into the passive voice as The fruit is eaten (by Jack). As before, the verb is here is a full-fledged verb, and can be put into all of the various combinations of modal and non-modal auxiliaries illustrated above as follows. It is worth noting, however, that some of the combinations do sound strange to the ear, even if they are technically permissible.

The fruit is eaten.
The fruit was eaten.
The fruit is being eaten.
The fruit was being eaten.
The fruit has been eaten.
The fruit had been eaten.
The fruit has been being eaten. [This is a very weird construction, and I cannot recall seeing it anywhere. It would only make sense in a very limited semantic range, such as The fruit has been being eaten by Jack since 10 am this morning, but it has still not been eaten completely! And even here we would probably put the first clause in the active voice and say Jack has been eating the fruit since 10 am this morning, but it has still not been eaten completely!]
The fruit had been being eaten. [Same caveat as above.]
The fruit will be eaten.
The fruit will be being eaten. [This too sounds a little weird to the ear.]
The fruit will have been eaten.
The fruit will have been being eaten. [This is perhaps even weirder than its non-modal counterpart, if that is even possible!]

For the future, auxiliary verbs in other languages.

Since it’s hanami season …

2011-04-01T12:35:00.004-04:00

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

2011-03-31T23:59:00.001-04:00

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

2011-03-18T14:38:00.003-04:00

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 \( \{ a_0, a_1, a_2, \ldots, a_n, \ldots \} \). A(n infinite) series is what you get when you add up all the terms of a sequence. This will look like \( a_0 + a_1 + \ldots + a_n + \ldots \), 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 \( 1 + 1 + 1 + \ldots \) 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 \( x + x + x + \ldots \) is the same as \( x \cdot (1 + 1 + \ldots ) \), 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 \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} + \ldots \)? 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?

As it turns out, the harmonic series does not converge at all! There are a number of proofs for this, but the simplest one I remember is this:

Take the series \( 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \ldots \), which will clearly diverge. Now rewrite it as \( 1 + \frac{1}{2} + \left( \frac{1}{4} + \frac{1}{4} \right) + \left( \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} \right) + \ldots \)

Comparing this term-by-term and bracket-by-bracket to the harmonic series, we find that
1 = 1
1/2 = 1/2
(1/3 + 1/4) > (1/4 + 1/4)
(1/5 + 1/6 + 1/7 + 1/8) > (1/8 + 1/8 + 1/8 + 1/8)
and so on. But if the series on the right-hand side already diverges, then the harmonic series, which is clearly bigger from its third term onwards, has to diverge too!

I'm not going to get into the various tests that we can perform to check if our favorite series converges or diverges. However, just in case you think at this point that it can never be possible to add up an infinite number of things to get a finite number (à la Zeno), consider this series:
1/2 + 1/4 + 1/8 + …
Or, in our mathematical notation for series, Σ _i=1^∞ (1/2)ⁱ.

Each term is half the size of the previous term; this sort of series is called a geometric series. Without getting into all the mathematical details of proving how some geometric series can converge, it can still be instructive to compare the ratio of two successive terms of our series with two successive terms of the harmonic series.

The ratio of two successive terms a_i+1 / a_i for the geometric series is (1/2)ⁱ⁺¹ / (1/2)ⁱ , which simplifies to a constant 1/2, as is clear to us from the way we constructed the series in the first place. For the harmonic series, this works out to (1 / (n + 1) ) / (1 / n), or, less clumsily,
n / (n + 1).

As n gets larger and larger, the terms of the geometric series get smaller and smaller at the same rate. But in the case of the harmonic series, the ratio gets closer and closer to 1 as n gets larger and larger; in other words, even though the terms of the harmonic series are getting smaller, they're nevertheless not getting smaller “fast enough”, whatever that means.

Anyway, if we take as a given that our geometric series converges, how do we find out what this value in fact is? Well, we could simply add up the terms one by one and see what we get, but this can often be a slow process, and furthermore, this process can only give us an approximation to the final answer. Is there some nifty way to find out the exact value to which our geometric series converges?

As it turns out, there is a fun way to do this. Since we “know” that our series converges to a finite value, let's call this value, say, S. That is, S = 1/2 + 1/4 + 1/8 + …

So what is (1/2) S then? (1/2) S = 1/4 + 1/8 + …

But wait! This is almost the same series as our initial series! Let's subtract:

S - (1/2) S = (1/2 + 1/4 + 1/8 + …) - (1/4 + 1/8 + …)

or (1/2) S = 1/2

which means that S = 1. In other words, 1/2 + 1/4 + 1/8 + … = 1. Cool!

What's more, we can tell that this easily generalizes to any geometric series where the ratio of successive terms is less than one (obviously, since if the ratio is greater than 1, the terms are growing larger and the series necessarily has to diverge). So if we write this ratio in general as x, we can write our geometric series as a₀ + a₀x + a₀x² + a₀x³ + … , or as Σ _i=0^∞ a₀xⁱ. Now, it's clear that the term a₀ is common to all of them and can simply be extracted from the entire sum, so that really the only geometric series we need to evaluate is Σ _i=0^∞ xⁱ, which is the same as 1 + x + x² + x³ + ….

So how do we evaluate this? We can do the same thing as we did earlier:

Let S = 1 + x + x² + x³ + ….

Now clearly xS = x + x² + x³ + …, so subtracting this from the first, we get

(1 - x) S = (1 + x + x² + x³ + …) - (x + x² + x³ + …)

so that S = 1 / (1 - x). Beautiful!

Let's take the series 1 + x + x² + x³ + … again. So far we've just been interested in the x-side of things. But there is another way to look at them: as the infinite sequence (1, 1, 1, 1, …); in other words, we can shift our focus to the coefficients of the series. When we move away from the x to the coefficients, we must temporarily abandon our earlier restraints on x as being less than 1 and all that; we are considering this as a formal power series, where the xs are simply “a clothesline on which we hang up a sequence of numbers for display”, in the words of Herbert Wilf in his remarkable book generatingfunctionology. This immediately tells us that the geometric series is nothing more than one particular kind of formal power series (if this wasn't clear to us earlier), and suggests to us that there may be many more formal power series out there that we can try to sum up! From the perspective of formal power series, writing

1 + x + x² + x³ + … = 1 / (1 - x)

means that the series (1 + x + x² + x³ + …) and the series (1 - x) are formally inverses of each other. If we then decide to take the x-side of things into account (i.e., if we consider 1 + x + x² + x³ + … no longer as a formal power series but as an actual series where the value of x matters), then for all values of x where the series converges, we can use its inverse series to quickly compute its value.

Take another formal power series, this time based on the sequence (1, 2, 3, 4, …). We would write this formal power series as 1 + 2x + 3x² + 4x³ + …. How do we add this up? Here, things get quite interesting, since there are at least two ways to do this:

We can rewrite the formal power series 1 + 2x + 3x² + 4x³ + … as

(1 + (x + x) + (x² + x² + x²) + (x³ + x³ + x³ + x³) + …), which can be rewritten as
(1 + x + x² + x³ + …) + x (1 + x + x² + x³ + …) + x²(1 + x + x² + x³ + …) + …, which, awesomely enough, can be rewritten as
(1 + x + x² + x³ + …)(1 + x + x² + x³ + …), which, of course, is
1 / (1 - x)².

Alternatively, if we know some calculus, we will know that the first derivative of xⁿ is nx^n-1. Since I haven't yet figured out how to get LaTeX output on blogspot, I'll improvise notation and write the first derivative as D. In other words, I'll write Dxⁿ = nx^n-1. Now, it should be clear from this formula that the formal power series 1 + 2x + 3x² + 4x³ + … can be written as

Dx + Dx² + Dx³ + Dx⁴ …, which, thanks to the linearity of differentiation, can be written as
D(x + x² + x³ + x⁴ + …), and where the internal formal power series can be summed to get
D(x / (1 - x) ), which, if we remember our rules of differentiating quotients, can be simplified to produce
1 / (1 - x)², as with the first method!

Clouds and geese

2011-02-16T14:23:00.001-05:00

Yigal Bronner and David Shulman's " 'A Cloud Turned Goose': Sanskrit in the Vernacular Millennium" is an absolutely fascinating journey through a new phase of Indian intellectual and literary production in Sanskrit and in the vernaculars in the last thousand years. Their argument is important and complex, and I will not do it injustice by reproducing it here. All I wish to do is to present one short portion of their analysis of Vedānta Deśika's Haṃsasandeśa, a messenger-poem (dūta-kāvya) that consciously takes as its model Kālidāsa's famous Meghadūta or Meghasandeśa while simultaneously surpassing it in multiple ways.

As background, Kālidāsa's original depicts a yakṣa, a semi-divine being, who is separated from his beloved and thus recruits a cloud as his messenger to take his love to her, just as Rāma sent Hanumān as a messenger to Sītā when they were separated. Vedānta Deśika applies this idea to Rāma himself, who recruits a majestic haṃsa, a goose (translated as "swan" in earlier English versions because the goose is an unfairly maligned bird in English culture!), to fly south to Laṅkā and convey his love to Sītā after Hanumān has returned with news of her. This sets the stage, as Bronner and Shulman beautifully show, for a virtuoso performance of spatial, temporal, and intertextual layering.

This verse, from which their paper takes its name, is as follows in the original:

lakṣmī-vidyul-lalita-vapuṣaṃ tatra kāruṇya-pūrṇam
mā bhaiṣīs tvaṃ marakata-śilā-mecakaṃ vīkṣya megham |
śuddhair nityaṃ paricita-padas tvādṛśair deva-haṃsair
haṃsī-bhūtaḥ sa khalu bhavatām anvavāyâgra-janmā || (Haṃsasandeśa 1.33)

Lakṣmī, a streak of lightning, graces a body
full of compassion and dark as emerald.
Don't be afraid when you see
a cloud, at whose feet
great seers, birds of your feather,
cluster in worship.
It's a cloud turned goose—
the firstborn in your line.

If it sounds strange to you, so it should. The cloud in question, at least on one level, is the dark icon of Varadarāja-svāmi / Viṣṇu in Kāñcīpuram, where our travelling goose arrives as a pilgrim at Rāma's suggestion. On the breast of this image we find Lakṣmī, who is thus appropriately likened to a flash of lightning within the dark monsoon cloud. At the feet of this emerald-coloured Viṣṇu lie the most accomplished Śrīvaiṣṇava devotees, the nitya-sūris, who have been granted the honorific title of deva-haṃsa, 'heavenly geese' (haṃsa or paramahaṃsa are common titles for advanced ascetics or seers). What is more, the cloud-cum-god himself has become yet another goose in the sense that he is identical to the supreme reality, refereed to already in the Upaniṣads by this same word, haṃsa. As such, he must be the First Goose, and hence the founder of the entire species of which our messenger is the latest representative. So, as the modern commentator S. Narayana Iyengar remarks, there are three conspicuous meanings for the word haṃsa in this one verse: 'a swan [sic], a pure ascetic, [and] the Supreme Spirit.' But there is a fourth one as well: for clearly Kālidāsa's cloud is once again invoked in the most direct metapoetic manner conceivable. The cloud-messenger has truly become a goose, after serving as the firstborn in the line or genre of messenger-poems … It is also very striking that a description of the current iconic image of the god in Kāñcīpuram is given to us as a future projection from out of the past which is the poem's present.

I want to briefly sort out the complex chronological references that Vedānta Deśika is playing with in his poem. There are three conceptually distinct chronologies here, which are all being merged into one through a variety of strategies.

The first is the historical sequence in which the major texts were composed (i.e., in our world): first comes Vālmīki's composition of the Rāmāyaṇa; next comes Kālidāsa's Meghasandeśa; subsequently come the Tamil devotional compositions of the Āḻvārs and Kampaṉ's version of Rāma's tale, the Irāmāvatāram; and finally, nearly a thousand years after Kālidāsa, comes Vedānta Deśika's Haṃsasandeśa.

The second is the internal sequence of time within Kālidāsa's work. His yakṣa hero is aware of the actions depicted in the Rāmāyaṇa, and as Bronner and Shulman show, depicts a geography in which mythic and historical past all blend in.

The third is the internal sequence of time within Vedānta Deśika's work. Hanumān has just returned from his reconnaissance mission to Laṅkā and has just told the whole story to Rāma, who now pines even more for his beloved and thus recruits a goose as yet another messenger to fly south.

These are all tied together by both literary conceit and narrative skill. In the Sanskrit universe, Vālmīki is not just the composer of the Rāmāyaṇa but an actual character in the story, who composes the work well after Rāma's coronation. Consequently, as Vedānta Deśika sees it, the point of time at which his work is set is actually earlier than the act of the composition of the Rāmāyaṇa itself. His work thus acknowledges its debt to Kālidāsa while also inverting the chronological sequence between the two, thus giving rise to "future projection[s] from out of the past which is the poem's present."

What is the message that Rāma wants to give Sītā? An incredibly beautiful, poignant message of love, whose power is heightened immeasurably by the recognition of the fact that for Vedānta Deśika, this God speaking of God's love for God, giving the devotee a glimpse of the ever-present nature of Divine love.

deha-sparśaṃ malaya-pavane dṛṣṭi-sambhedam indau
dhāmaîkatvaṃ jagati bhuvi câbhinna-paryaṅka-yogam |
tārā-citre viyati vitatiṃ śrīvitānasya paśyan
dūrī-bhūtāṃ sutanu vidhinā tvām ahaṃ nirviśāmi || (Haṃsasandeśa 2.40)

I will not provide Bronner and Shulman's beautiful analysis of this verse, and will leave you only with their translation. (p. 26)

Our bodies touch
in the southern wind.
Our eyes meet
in the moon.
We live together in a single home—
the world, and the earth
is the one bed we share.
The sky scattered with stars
is a canopy stretched above us.
Think of this, my lean beauty:
However far you may be,
I still find my way
into you.

Reference

2006. Yigal Bronner and David Shulman. "A Cloud Turned Goose: Sanskrit in the Vernacular Millennium." The Indian Economic and Social History Review, 43.1.

A never-ending marvel

2011-01-30T21:55:00.001-05:00

An always-evolving, never-ending, always-fresh fractal landscape from Vimeo:

Surface detail from subBlue on Vimeo.

The structures that evolve out of this are incredible. This is both jamāl and jalāl, both mysterium fascinans and mysterium tremendum.

Counting, uncounting, and dust, part three

2011-01-23T12:12:00.001-05:00

Sequences and convergence
We saw earlier that the rational numbers Q are much more versatile than the natural numbers N for doing a wide variety of mathematical operations. However, there is still one thing the rational numbers cannot do adequately: it is possible to construct sequences of rational numbers that do not converge to a rational number. In formal terms, the rational numbers are not complete.

Why does this seemingly obscure mathematical quirk matter? Because this is the foundation of calculus. The ideas of sequences and limits are fundamental to calculus, and if such limits don't exist meaningfully, then neither does calculus. But there is another, intuitive reason for why completeness matters.

Take the sequence {1, 1/2, 1/3, 1/4, …}. It should be intuitively obvious that this set of numbers is getting smaller and smaller, and that although it does not contain the value 0, it is inexorably heading towards it. But if you take a sequence like {1, 1.4, 1.41, 1.414, 1.4142, …}, then we "know" intuitively that this sequence converges to the number √2, which we have proven to be irrational. This suggests that there are lots of "holes" in the set of rational numbers, making them incomplete and even discontinuous, in an intuitive sense. (In terms of topology, this intuition is expressed by the idea that the rational numbers are totally disconnected, which roughly means that between every two rational numbers there is at least one irrational number, so that if we tried to walk along the number of rational numbers, we would be forced to jump constantly from one rational number to another.)

When we faced similar problems with N and with Z, we extended them by adding all sorts of missing numbers. We can do the same thing now with Q, extending it to the set of real numbers R, although the mathematical details are not very pretty. The usual non-rigorous method is say that rational numbers have decimal expansions that are either terminating (like 0.3158) or at least repeating (like 0.333... for 1/3), whereas irrational numbers have non-terminating, non-repeating decimal expansions (like 1.4142… for √2). (For those who don't have queasy stomachs, there are at least three different ways of constructing R that are often taught: the easy way by constructing equivalence classes of Cauchy sequences of rational numbers, the standard way by constructing Dedekind cuts, and the coolest way—at least in my book—by continued fractions.)

These details don't really matter; the twofold point is that we can construct the set of real numbers R so that it is complete (i.e., doesn't have any "holes" the way Q does) and that we have a nice decimal representation for every real number. (In formal terms, completeness can be described in a few different ways, one way being that every sequence of real numbers whose terms get closer and closer to each other—every Cauchy sequence, to be precise—will eventually converge to a real number.) To return now to our initial question: how large is R?

Cantor's Diagonalization Proof
By now, all our intuitions about comparing the sizes of sets have been utterly trashed. We no longer really have a good intuition for how large R must be, and it would be entirely forgivable to suppose that R is also countably infinite, just like N. We would be utterly wrong.

The mathematician Georg Cantor offered a beautiful proof of the uncountability of R using the idea of diagonalization. The full-blown version of the proof is a bit tricky, as always, but here I will sketch out a proof of the fact that there are in fact far more irrational numbers between just 0 and 1 than there are natural numbers in their entirety. (Clearly, if this is true, then all of R must also be strictly larger than all of N.)

As with our proof of the irrationality of √2, we will achieve this by contradiction. So let us suppose that the irrational numbers between 0 and 1 are countable; that is, we can enumerate them in a list indexed by the natural numbers N. Now, since we know that irrational numbers have non-terminating, non-repeating decimal expansions, we simply go ahead and write down all the irrational numbers in a list, which would look something like this:
1—0.548574930…
2—0.353495364…
3—0.579473594…
4—0.123534967…
5—0.432874239…
6—0.549412390…
and so on. This is possible because we have supposed that the irrational numbers are countable.

Here is where the diagonalization part comes in. We now essentially have a table of numbers that comprises of all the digits of all the irrational numbers between 0 and 1. So now, let's take a separate, infinitely long piece of paper, and let's walk down the diagonal of the table, writing down the first digit of the first number, the second digit of the second number, …, the i^th digit of the i^th number, … until infinity. (We're assuming we have an infinite amount of time, so this doesn't matter! Gotta love math.) So based on the table we have up here, we will have the number (let's call it x) 0.559572…

We then take a second infinitely long piece of paper and copy down the digits of x, with one key difference: we add 1 to every digit after the decimal point. (As a convention, we can suppose that if x contains a 9 somewhere, then we write down 0 instead, but this is just really a minor thing.) Let's call this new number y; in our case this is going to be the number 0.660681…

So what's so great about what we've done so far? Here is the kicker, the awesome punchline, the crazy part: y is an irrational number that is not part of our list at all!

y is irrational because it is non-terminating and non-repeating. This may not be obvious right at the start, because it is of course possible to imagine that we somehow randomly happen to pick an infinite number of numbers which set up an infinite diagonal of 9s, which would then make the decimal expansion of y contain an infinite number of 0s and thus terminate. However, if this happens, then we can quite easily reorder our original list so that our diagonal is no longer comprised of 9s. (Remember that when we count, it does not matter which particular element gets indexed to which natural number; the only thing that matters is that it can be done. So such a reordering is totally legitimate.)
y is not on our list because, well, it differs from every number on our list at one position, at the very least! This is the neat part: we have consciously constructed y so that the i^th digit of y is different from the i^th digit of the i^th number on our list.

What this means, more simply, is that our initial supposition—that we can enumerate the irrational numbers between 0 and 1—is false. No matter how many irrational numbers we count off against the natural numbers, there will always be more. The set of irrational numbers between 0 and 1 must therefore be strictly larger than the whole set of natural numbers N or integers Z or rational numbers Q (which are all the same size, as we have already seen). Just imagine that: between just two numbers—the smallest gap there is in the integers!—lie more numbers than are encompassed by all of the rational numbers! This fact should blow your mind.

Uncountable Sets
But wait! There is more. Let us now define one more mathematical term: the open interval (a, b), which is sometimes written as ]a, b[ to avoid ambiguity, is the set of all real numbers x that lie between a and b but are not equal to them. In formal mathematical notation, (a, b) = {x ∈ R | a < x < b}. So, using this notation, we can say that we just proved that the size of the set of irrational numbers in the open interval (0, 1) is strictly larger than the size of the set of natural numbers. (Just so we are clear, the open interval (0, 1) contains all the real numbers between 0 and 1, which means it contains all the irrational numbers and all the rational numbers between 0 and 1.)

Just as we looked first at the natural numbers N and then compared them to the even numbers 2N, we can now look at a subinterval of the open interval (0, 1) and see how it stacks up in size with the bigger interval. Let's take the open interval (0, 1/2). This is clearly half as "long" as the open interval (0, 1)—but does it contain "half" as many elements?

Now that we are talking about sets strictly larger than N, we can no longer establish size by naïvely counting all the elements. However, we can certainly use the formalized intuition of counting that we have developed, which is to say, the idea of the bijection. In other words, we can still compare the size of two sets by checking to see if it possible to construct a way to uniquely map every element of one set to exactly one unique element of the other, and vice versa.

There is a very simple bijection between (0, 1/2) and (0, 1):

We can map any x in the interval (0, 1/2) to the number 2x, which is clearly in the interval (0, 1),
We can map any y in the interval (0, 1) to the number y / 2, which is clearly in the interval (0, 1/2).

In other words, the open interval (0, 1/2) is "just as large" as the open interval (0, 1). And furthermore, our intuition should now suggest to us that there was nothing special about picking the interval (0, 1/2); in fact, every open interval (a, b) is "just as large" as (0, 1):

Given any x in the interval (0, 1), we can map it to the number a + (b - a)x, which must lie precisely inside the interval (a, b).
Given any y in the interval (a, b), we can map it to the number (y - a) / (b - a), which must lie inside the interval (0, 1).

Another way to think about this is that the whole real line is really nothing more than the open interval (0, 1) suitably stretched out (or compressed), and hence is just the same size. This should give us a sense of how truly large the size of the real numbers is: we can take as tiny an interval of real numbers as we like, and we will still have more real numbers inside that interval than there are natural numbers. Yet another way to think about it is that every rational number is separated from every other rational number by as many real numbers as there are on the entire real line as a whole. If this does not blow your mind, nothing can.

One proof that the square root of 2 is an irrational number

2011-01-23T12:00:00.002-05:00

Here is an elementary proof of the fact that \( \sqrt{2} \) is irrational. But what does that mean? An irrational number is, well, a number that is not rational! So let's begin with a definition of the set of rational numbers \( \mathbb{Q} = \{ \frac{p}{q} | p, q \in \mathbb{Z}, q \neq 0 \}\). As we saw earlier, this is to be read: \( \mathbb{Q} \) is the set of all fractions \( \frac{p}{q} \) such that both \( p \) and \( q \) are integers, \( q \) not being \( 0 \), and with \( p \) and \( q \) in lowest terms. That simply means that they have no prime factor in common. (Let us recall that we can always write every natural number uniquely as a product of a number of prime factors.) What this definition means, at least for us right now, is that we're not going to be looking at numbers like \( \frac{12}{10} \); we're going to assume that all numbers have been reduced to their lowest form, like \( \frac{6}{5} \).

Another minor note: by our definition, a number like \( \frac{1.5}{3} \) would not count as a rational number. This is not true, of course, because we know we can simply rewrite \( 1.5 \) as \( \frac{3}{2} \), and then our number becomes \( \frac{\frac{3}{2}}{3} \), which is the same as \( \frac{1}{2} \), which is clearly a rational number.

So when we say a number is irrational, we're saying it's crazy in some really fundamental way: there really is no way to write it as the ratio of integers. (If you're wondering why we write \( \mathbb{Q} \) for the rational numbers, it's from the word "quotient".)

The proof of \( \sqrt{2} \) being irrational is a proof by contradiction, that is to say that we shall suppose that it is in fact rational, and show that this necessarily leads to a logical impossibility.

So let us indeed suppose, for the sake of contradiction, that \( \sqrt{2} \) is rational. In other words, suppose we can write \( \sqrt{2} = \frac{p}{q} \), where \( p, q \) are non-zero integers that have no prime factors in common. Through elementary algebra, we can rearrange this to \( p^2 = 2q^2 \). Evidently \( p^2 \) is an even number.

Now if \( p^2 \) is an even number, it must be the case that \( p \) is also even. This is because if \( p \) were odd, then \( p^2 \) would also clearly be odd. (This may seem like a strange reason, but it’s actually a basic law of logic: the statement “A implies B” is the same as “not-B implies not-A”. Note that this is completely different from saying “the statement ‘A implies B’ is the same as ‘not-A implies not-B’.” This is not true in general!) Since \( p \) is even, we can write it as \( p = 2n \) for some integer \( n \). In that case, \( p^2 = 4 n^2 \).

But we already have \( p^2 = 2q^2 \). This must mean that \( 4 n^2 = 2q^2 \), or more simply, \( q^2 = 2n^2 \). But this is impossible! If \( q^2 \) were even, then \( q \) would also be even, which would mean that you could write \( q = 2m \) for some integer \( m \). But then \( p, q \) would have a prime factor in common, which violates our initial supposition! It must therefore be the case that our initial supposition was itself false, which means that \( \sqrt{2} \) has to be irrational.

Now you may not be entirely convinced by this sort of proof by contradiction. Suspicion is a good thing when trying to work through math proofs because it is so easy to inadvertently overlook something critical. In particular, it may not be entirely clear why it is so problematic that \( p \) and \( q \) both end up even. (Everything else flows quite naturally and simply, or so I hope!) The answer to that is simple. Suppose \( p, q \) are in fact even. Then, as we saw above, we can write \( p = 2n, q = 2m \). This means that \( \sqrt{2} = \frac{p}{q} = \frac{2n}{2m} = \frac{n}{m} \). But then we are back where we started! We can square both sides and rearrange to discover that \( n^2 = 2m^2 \), but we know by now that this will end up showing that both \( n \) and \( m \) are even. In other words, we would end up looping through this set of equations until kingdom come. This shows us, then, why it simply has to be the case that \( \sqrt{2} \) irrational.