Religion and narrative

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”

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

Blogspot is pretty bad at displaying math; \( \mathrm{\LaTeX} \) is pretty awesome at displaying math. So how do we get Blogspot and \( \mathrm{\LaTeX} \) 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 \( \mathrm{\TeX} \)and \( \mathrm{\LaTeX} \) 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:

\[ \begin{eqnarray} (a + b)^2 & = & (a + b) \cdot (a + b) \\ & = & a^2 + ab + ba + b^2 \\ & = & a^2 + 2ab + b^2 \end{eqnarray} \]
Truly remarkable! Now all I need to do is update all my old math posts using MathJax and good ol’ \( \mathrm{\LaTeX} \).

Counting, uncounting, and dust, part four

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:

1. Are there levels of infinity beyond the infinity of the real numbers?
2. 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 \( \{ \} \).

Indulge me, por favor

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

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.

Auxiliary Verbs, auf französisch, part deux

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.

Engineering and science

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”

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

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?

Clouds and geese

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)

A never-ending marvel

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

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.

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

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.

Counting, uncounting, and dust, part two

Extending countability
Having looked at the definition of countability and the natural numbers N, it is but natural (no pun intended) to ask how far this concept can be extended. The natural numbers are pretty awesome, after all, but there are a huge number of things you can't do with natural numbers, including something as basic as subtraction.

Integers
The only operations under which N is closed are addition (that is, you can add two natural numbers and the result will be a natural number) and multiplication, but the latter is pretty limited in the sense that it's basically just repeated addition. So all we can do with natural numbers is "go forth and multiply"; there is no way to reduce them. In other words, there is no "inverse"operation to addition under which N is closed.

Counting, uncounting, and dust, part one

I've wanted to write about this for a while but never really got the chance. One of the coolest things about modern mathematics is infinity: we have found all sorts of ways to study and classify different notions of "size" and "number" that can result in the most counterintuitive results imaginable. I want to look at a few examples, starting today.

Counting
Let's begin with the natural numbers, the set N = {1, 2, 3, ... }. This is "clearly" an "infinite" set, as indicated by the ellipsis. What does "infinite" mean in this case? Now we know that we can add natural numbers together to produce natural numbers; more simply, we can add the number 1 to a natural number n to produce a new natural number n + 1. N is infinite in the sense that there is simply no "end" to it, at least in one direction (because there is such a thing as the smallest natural number, which is 1). To rephrase that, there is simply no way to list every single element of N, whereas for any finite set, no matter how large it may be, you can eventually list out every element.

Before we go any further, let's actually get a handle of what counting means. In real life, when we have to count out a bunch of things, we essentially point at each item once (and only once) and call out the natural numbers in order. We are in effect uniquely indexing every item with one (and only one) of the natural numbers. There is nothing twisted or weird about this; this is the most "natural" (pardon the pun) way of counting things.

What this definition of counting allows us to do is to compare the size of different sets: we can know that a sack of three potatoes is the same numerical size as a sack of three apples because we can index both of them with the set {1, 2, 3}. (It should be clear that it does not matter which potato we're pointing to when we call out a particular number; all that matters that we point to each potato only once and we call out the numbers in order only once.) Now so far this sounds like a ridiculously overdone way of stating the utterly obvious (and I've not even pulled out real mathematical terminology or notation!), but the utility of such abstraction will become more and more evident as we journey along.

Godā Stuti, conclusion and English translation

kavi-tārkika-siṃhāya kalyāṇa-guṇa-śāline |
śrīmate veṅkaṭeśāya vedānta-gurave namaḥ ||

Godā Stuti, 29

iti vikasita-bhakter utthitāṃ veṅkaṭeśāt
bahuguṇa-ramaṇīyāṃ vakti Godā-stutiṃ yaḥ |
sa bhavati bahumānyaḥ śrīmato raṅga-bhartuḥ
caraṇa-kamala-sevāṃ śāśvatīm abhyupaiṣan || 29 ||

Godā Stuti, 28

śatamakha-maṇi-nīlā cāru-kalhāra-hastā
stana-bhara-namitāṅgī sāndra-vātsalya-sindhuḥ |
aḷaka-vinihitābhiḥ sragbhir ākṛṣṭa-nāthā
vilasatu hṛdi Godā Viṣṇucittātmajā naḥ || 28 ||

Resilience and complex systems

Another article in SEED Magazine, by Carl Folk, on the ecological term resilience applied to other complex systems like the global economy. The author defines resilience as "the capacity of a system—be it an individual, a forest, a city, or an economy—to deal with change and continue to develop," in other words, to respond to perturbations and even catastrophes in innovative ways.

So what does this mean? For one,

"systems are understood to be in constant flux, highly unpredictable, and self-organizing with feedbacks across multiple scales in time and space. In the jargon of theorists, they are complex adaptive systems, exhibiting the hallmark features of complexity."

This ability of a complex system to adapt to variations does not necessarily mean that such adaptation is smooth or predictable. Indeed, the non-linearity of feedback loops means that complex systems can behave in what seems erratic (from the perspective of a linear model). This includes wild oscillations between dramatically different positions, such as a lake fluctuating between hyperoxygenation and algal bloom, or for that matter the Earth going from polar regions covered in dense vegetation (as during the Mesozoic era) to iced-over tropical regions (Snowball Earth).

The risks of interconnectedness

This is a tremendously interesting article in SEED Magazine, titled "On Early Warning Signs", by George Sugihara that draws on the author's background in biology to depict some of the limitations of the modern tendency to use linear systems (plus perturbations) to model fundamentally non-linear systems. Among other things, non-linear systems show high levels of interconnectedness that tends to fluctuate (technically heteroskedasticity), which makes predicting their behavior using linear models very difficult.

The author further notes:

"Leading up to the crash, there was a marked increase in homogeneity among institutions, both in their revenue-generating strategies as well as in their risk-management strategies, thus increasing correlation among funds and across countries—an early warning. Indeed, with regard to risk management through diversification, it is ironic that diversification became so extreme that diversification was lost: Everyone owning part of everything creates complete homogeneity. Reducing risk by increasing portfolio diversity makes sense for each individual institution, but if everyone does it, it creates huge group or system-wide risk. Mathematically, such homogeneity leads to increased connectivity in the financial system, and the number and strength of these linkages grow as homogeneity increases. Thus, the consequence of increasing connectivity is to destabilize a generic complex system: Each institution becomes more affected by the balance sheets of neighboring institutions than by its own."

An interesting analogy drawn from nature, this time from plants and pollinators:

"[T]he same hierarchical structure that promotes biodiversity in plant-animal cooperative networks may increase the risk of large-scale systemic failures: Mutualism facilitates greater biodiversity, but it also creates the potential for many contingent species to go extinct, particularly if large, well-connected generalists—certain large banks, for instance—disappear. It becomes an argument for the “too big to fail” policy, in which the size of the company’s Facebook network matters more than the size of its balance sheet."

Very interesting stuff.

Godā Stuti, 27

jātāparādham api mām anukampya Gode
goptrī yadi tvam asi yuktam idaṃ bhavatyāḥ |
vātsalya-nirbharatayā jananī kumāraṃ
stanyena vardhayati daṣṭa-payodharā ’pi || 27 ||