Mathematical aptitude as Gift

I recently came across an article by Jo Boaler which argued against the usage of mathematics as a “gatekeeping” tool for children, and against thinking of children as having an “innate gift” for mathematics for fear that this would contribute to greater inequity among the populations who dive into mathematics as a profession. This article, especially when read in the context of Boaler’s work in reshaping school education in California, has upset a number of people, including parents of children who are seen as “gifted”.

As someone who was a math major, who did not pursue formal graduate study of mathematics, whose dissertation drew upon a number of abstract mathematical concepts, who continues to study higher mathematics for pure pleasure, and at least one of whose kids is obsessed with numbers, I have a lot to say on this subject! In case you don’t want to read this long, rambling post, here are a number of positions that I think can all be held simultaneously without fear of self-contradiction:

Unsubstantiated conclusions

• There is genuine variation in innate potential for (certain kinds of) mathematical thinking. 
• Consequently, there are indeed certain people who possess this potential to a much greater extent than others.
• Practice and perseverance will allow people to maximize their innate mathematical potential (which may be higher than they think).
• Conversely, some people will not fulfill their innate mathematical potential if they don’t put in this work. This could be due to a whole range of factors, including personal choice, disabilities, or environmental factors.
• Mathematical education at the (American) school level is narrowly focused on a small part of the spectrum of mathematical thinking, so that performance at the school level is not a strong predictor for genuine mathematical potential. 
• In fact, I would assert even more strongly that doing well in school math is neither necessary nor sufficient to do well in college-level math or higher.
• Most people have no real idea of what higher mathematics is actually like, and are quite shocked when they realize that many mathematicians might not actually be great at mental math or other flashy demonstrations of school-level computational performance.
• Mathematical education at the (American) school level is indeed serving a “gatekeeping” function that has deleterious consequences for students.

As a way to make sense of this complex bundle of thoughts, and in order to connect it to my previous musings on this blog on mathematics, meditation and Indian philosophy (specifically the school of Mīmāṃsā), I will posit the following: 

We must distinguish between mathematical study as an end in itself (kim), mathematical study as a means to some other end (kena), and mathematical study as a process or procedure (katham) to be followed to produce some other end.

A biographical interlude

In my undergraduate days, I studied mathematics at one of the most demanding places in the US. In that context, I would rate myself as middle-of-the-road: I did decently, but there were also a number of students who were far ahead. (This was a a humbling lesson for me, and one I very much needed to learn!) I realized that these students could be conveniently classified into two types: 

1. Some students had studied a lot more mathematics than others: their high school math programs had been very intense (often French or Eastern European), or they had had the chance to take a lot of college math while in high school itself. These folks were ahead because they had put in the time and the effort into studying mathematics, and were already familiar with content that was otherwise brand-new to me. To use a commonly-misused term, they were more “mathematically mature”. However, I also noted that some of these students were not necessarily much faster at grasping totally new ideas, or at extrapolating to unusual domains. 
2. And there were others who were just … different. They too had studied a lot more mathematics than others, but you could sense that this was just natural to them. Mathematical facts were obvious to them that were inscrutable to us after hours of bashing our heads against a page of symbols. Where we struggled in the foothills of abstraction, they frolicked freely in the heights, like Sherpas who grew up amidst Himalayan peaks. 

Seeing people with a real mathematical gift was humbling, liberating, but also (in retrospect) constricting for myself. For someone who had always done well in mathematics in school to realize there were people who effectively operated on a different (and yes, non-intersecting) mathematical plane than mine was not easy initially. Once I absorbed that lesson, though, I felt liberated: I didn't need to compete with them! I could do math at my own level, at my own pace, and I could derive pleasure from it. This sense of liberation has allowed me to continue to study math for my own edification, long after departing the hallowed halls of academia. This allows me to enjoy the mysterium tremendum of mathematics: a near-religious sense of awe, marvel, and humility in the face of mathematical structures of immense beauty—which would likely not have been possible had I been trying to grok them to crunch a problem set under time pressure. 

At the same time, in retrospect, I can also see that my recognition of the gift of mathematics kept me from pursuing mathematical study in graduate school. I thought to myself that graduate education in mathematics must surely be the preserve only of those who were truly gifted, with no place for ordinary folks like myself. This, however, was an act of self-limitation, and not necessarily true. 

On an innate Gift for mathematics

It is transparent to me that there is natural variation in people’s affinity for mathematics, and that this often manifests in its extreme forms at very young ages. Consider, as examples of outliers, the extraordinary feats of Carl Friedrich Gauss and Terence Tao in their childhood. Closer to home and a lot more down to earth, I see my son playing day and night with numbers, delighting in their combinations and patterns. (This may perhaps be the mysterium fascinans aspect of mathematics!) 

I want to zero in on this sense of play, for I think it is critical to understanding how real mathematics often operates: It is an autonomous domain, with its own objects and rules, where the only goal of the game is to continue playing the game. (In describing mathematics this way, I do not intend to support either a Platonic or a formalist philosophy of mathematics; I'm merely trying to capture the phenomenology of doing mathematics in a state of flow.) This, as I have said above, has a strongly aesthetic flavor to it: it is enchantingly beautiful to a few humans, baffling to most, and repulsive to some. 

Incidentally, this autonomy of mathematics also poses a danger to those who most love it, in that it might entirely devour a person’s passion, energy and even sanity, if they do not retain a strong tie to the world of concreta. This is why I have called it a “Gift” with a capital ‘G’: to evoke the German word Gift which means “poison”, although it is a cognate of the English gift as well.

Mathematics as kim, kena and katham

The experience of the truly mathematically gifted thus clearly involves treating mathematics as an end in itself, as a source of joy pursued for its own sake. In the language of Mīmāṃsā, this is a kim, an end-goal in itself. Note that this might actually be a real challenge in following the standard school curriculum, where joy is not typically listed as a learning objective! For such students, self-paced self-study may in fact be the right answer.

But there are many who, talented though they may be in mathematics, are interested not in mathematics as an end but in mathematics as a means, as an instrument for some other goal. Scientists, engineers, data scientists, financial analysts, accountants: all of these professionals acquire fluency in some domains of mathematics and apply it to some other problems, creating value for the world and revenue for their bosses and a home with a view for themselves. (Even professional mathematicians who may be doing math as a way to pay the bills would fall into this category.) In the language of Mīmāṃsā, this is the use of mathematics as a kena or as a sādhana, an instrument for realizing a different end-goal. For many (though not all) such students, the structure of math education in school actually works, with its emphasis on procedure and algorithm and its de-emphasis of seemingly unnecessary tools like proofs. This divide often persists through college as well, where courses like calculus, linear algebra and differential equations might be taught both in mathematics departments as well as in engineering departments in wildly different ways.

Finally, we find yet another (mis)use of mathematics, this time as a process and not an outcome, with some other goal in mind (Mīmāṃsā would call this katham, a “how”). Thus, the education system seeks some kind of restriction on student outcomes, whether by design or by accident, and must accomplish this goal through some means of gatekeeping. The process may end up being a sequence of math instructions that feel more like a robotic algorithm, or perhaps like an obstacle course, where students must jump through the hoop of Algebra and across the moat of Geometry and scale the wall of Pre-calculus … all of this just to get into a good college where they in fact plan to study Old Norse sagas. This sort of gatekeeping can and no doubt does damage some students’ self-confidence and motivation to learn. For those who have the potential to utilize mathematics as a tool, such self-limitation and external gatekeeping may well keep them out of potentially rewarding occupations. For others, who may privately enjoy the free play of beautiful mathematics, it may poison the well and damage their associations with this source of great joy.

To me, the use of the Mīmāṃsā framework helps narrow down the problem quite specifically: It is the fetishization of mathematical performance and its misuse as a gatekeeping tool that is the real issue, not the existence of mathematical gifts in some fraction of the population. But it may not be clear why mathematics as gatekeeping is problematic, because modern society has come to privilege performance in mathematics so much.

An analogy to classical music

When I read the article, my first thought was immediately Paul Lockhart’s “A Mathematician’s Lament”, in which he describes the current state of K12 math education by constructing a parallel universe in which writing down sheet music is similarly treated as a gatekeeping tool, without regard to the aesthetic experience of actually listening to, performing, or composing music. I think this analogy is actually a very powerful one and can serve to illuminate the issue at hand.

Because music does not act as a gatekeeping tool in our society, we are much more comfortable with acknowledging that musical genius exists, whether it be as a composer or as a performer. Nevertheless, it is also true that most people can learn to perform some level of music, if taught properly, even if they will never grace Carnegie Hall or feed their families with this skill. (We forget that, for most of human history, when recording and playback devices did not exist, people would have had to sing or hum for themselves if they wanted to hear something musical while walking down the street!)

The analogy to music opens up yet another dimension which is underemphasized in mathematical pedagogy: cultivating taste and developing an aesthetic sense. We cannot all be stage performers; we might not even want to perform any kind of music ourselves; but we may learn to appreciate music and derive deep satisfaction from it. It might not be the same delight that a practitioner enjoys, and it certainly will not be a source of income, but it can be a source of joy. As someone who is neither a professional mathematician nor a professional musician, but who derives comfort and delight and, yes, a sense of proximity to the Divine Infinitude from the consumption of mathematical ideas and musical performances alike, I wish we encouraged all our children and all humans to take the idea of play more seriously and to cultivate a deeper sense of aesthetics. As the Taittirīya Upaniṣad says:

|| raso vai saḥ || 

The Divine truly is aesthetic savoring.

 

 

Information and Sāṅkhya

Alfred Borgmann’s Holding On To Reality opens with the provocative words:

Information can illuminate, transform, or displace reality. When failing health or a power failure deprives you of information, the world closes in on you; it becomes dark and oppressive. Without information about reality, without reports and records, the reach of experience quickly trails off into the shadows of ignorance and forgetfulness.

This is a very interesting and provocative passage, but I picked it up because of its uncanny resemblance to a famous verse from Īśvarakṛṣṇa’s Sāṃkhyakārikās:

prīty-aprīti-viṣādâtmakāḥ prakāśa-pravṛtti-niyamârthāḥ |
anyônyâ-’bhibhavâ-’’śraya-janana-mithuna-vṛttayaś ca guṇāḥ ||

“The three guṇas [which constitute all non-sentient reality] have the respective natures of joy, non-joy, and sorrow; they act to illuminate, transform, and restrain, respectively; and they have the capacity to ground, produce, combine, and suppress one another.”

I need to bone up on my Sāṃkhya and my Borgmann further before I stretch this analogy, but at least at first glance, it appears as if there is something potentially illuminating lurking in the shadows here. (One thing that strikes me: the three guṇas of Sāṅkhya are regarded as entirely distinct from the individual persons, puruṣas, who alone are conscious observers and actors. This suggests that information itself isn’t enough: it presupposes the existence of conscious observers who can recognize something as being information.)

Open-source software vs open-source religion

It struck me today that there is a duality between the way in which open-source programmers work and the way in which philosophy and theology was done in classical India. (This might seem like a strange duality to bring up, but I do it because people frequently make misleading analogies between computer programming and Sanskrit grammar and other Indian disciplines. Also, because it’s my blog, so there!)

The point of open-source programming is that everyone can edit the source code. You fork a Github repository, make your changes, push them, and perhaps they might get accepted into the main code repository itself. Of course, it’s important that you (broadly) use the same sort of compiler/interpreter that the project is intended to work with, otherwise your code may not quite work the same way on others’ machines. 

In short, for open-source software, the rule is:

change the source, keep the interpreter ( / compiler / whathaveyou).

The structure of classical Indian thought is the precise opposite of this. Most of the oldest philosophical texts were written in laconic, even ambiguous, sūtras. Later philosophers then came to write commentaries on these texts, while their disciples wrote super-commentaries on the commentaries, while their disciples wrote super-super-commentaries on the super-commentaries on the commentaries on the sūtras, and so on.

(Lest you think I am joking, here are just two examples:

1. The foundational text of the Nyāya tradition is the Nyāya-sūtra of Gautama. This gathered the following tower of commentaries:
• Vātsyāyana’s Nyāya-sūtra-bhāṣya
• Uddyotakara’s Nyāya-sūtra-bhāṣya-vārttika
• Vācaspati Miśra’s Nyāya-sūtra-bhāṣya-vārttika-tātparya-ṭīkā
• Udayana’s Nyāya-sūtra-bhāṣya-vārttika-tātparya-ṭīkā-pariśuddhi
2. The foundational text of all Vedāntic schools is the Brahma-sūtra or Vedānta-sūtra of Bādarāyaṇa, sometimes also called Vyāsa. If we look just at one branch of commentaries on this foundational text, the Advaita branch, we get the following picture:
• Śaṅkara’s Advaita Brahma-sūtra-bhāṣya
• Vācaspati Miśra’s Bhāmatī
• Amalānanda’s Vedānta-kalpa-taru
• Appayya Dīkṣita’s Vedānta-kalpa-taru-parimala
• Lakṣmīnṛsiṃha’s Ābhoga
• Akhaṇḍānanda’s Ṛju-prakāśikā
• Padmapada’s Pañcapādikā
• Prakāśātman’s Pañcapādikā-vivaraṇa
• Citsukha’s Pañcapādikā-vivaraṇa-tātparya-dīpikā
• Nṛsiṃhāśrama’s Pañcapādikā-vivaraṇa-prakāśikā
• Nṛsiṃhāśrama’s Vedānta-ratnakośa)

The thing to keep in mind is that each of these commentators took the exact wording of his predecessors extremely seriously. And yet we obtain these complex forking interpretations of these texts, often at loggerheads with each other. The purpose of a commentary was ostensibly to elucidate the meaning of the base-text it commented on—but more often than not, it was to show that the base-text could be read to yield the conclusion you knew it had to yield.

In short, for classical Indian intellectuals, the rule is:

keep the source, change the interpreter.

Reflections on the Revolution in Isla Nublar, epilogue

This is the epilogue to a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

I realize now that I did not explain at all why I picked the title for this series of posts on Jurassic World. It is of course meant to call to mind Edmund Burke’s classic critique of the French Revolution, which has the much more elaborate title Reflections on the Revolution in France, And on the Proceedings in Certain Societies in London Relative to that Event. In a Letter Intended to Have Been Sent to a Gentleman in Paris.

Burke is sometimes caricatured as being an arch-conservative and stick-in-the-mud, but it is important to realize that in the context of his society, he was actually rather on the progressive wing. Burke was a supporter of the American freedom struggle, a champion of the rights of the Irish (he was of Irish descent himself), and vehemently opposed to the East India Company’s depredations in India. And yet, he was fundamentally opposed to the French Revolution, because he saw in it the release of forces deeper and more dangerous than those unleashed by any other revolution: rationalism, utterly convinced of its own correctness and of the total mistakenness of all other positions, willing to overthrow everything that came before it in the name of a new world order founded on pure reason unencumbered by primitive inherited idiocies.

What happens on Isla Nublar is a revolution in that sense: a full overturning of the existing order (a re-volvo) and the institution of a new system based on different principles.

Reflections on the Revolution in Isla Nublar, part three

This is the third of a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

In the previous installations of this series, I first wrote about the hugely significant role clothing plays in understanding the transformations of individuals in Jurassic World. I then expanded on the idea of clothing to enclosures in general, which led naturally to relations of power and dominance, which were then contrasted with relations of trust and understanding (familial relations) and with relations of unification for survival (conjugal relations). It is now time to naturally move from conjugal relations to the theme of evolution, which of course underpins the entire Jurassic Park franchise.

Reflections on the Revolution in Isla Nublar, part two

This is the second of a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

Last time, we took a very close look at some of the significant ways in which clothing works in Jurassic World. I will avoid diving into such detail this time around, as I have a vast range of topics to cover. (Need I say that this will be massively full of spoilers if you haven’t seen the movie?)

2. Enclosures

I’m thinking of enclosures fairly broadly here as any sort of covering that affords protection from the elements. In this sense, clothing is of course a kind of enclosure protecting human skin: the shedding of clothing is the discarding of a layer of insulation. But there are a number of other such enclosures in the film: the gyroball, for instance, literally encloses the two boys in a protective shell as they travel through the park—

Reflections on the Revolution in Isla Nublar, part one

This is the first of a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

I haven’t blogged for a regrettably long time, so I figured I would mark my return to the scene by reflecting on a movie that hasn’t done too badly for itself this summer: Jurassic World. Given my obsession with dinosaurs (par for the course for anyone who saw Spielberg’s original miracle), this experience was particularly fun for me.

I suppose it needs no spoiler warning to say that Jurassic World is about (yet another) dinosaur going crazy on a theme-park island, with lots of humans dying gruesomely while the protagonists make it through with all body parts intact and with an increase in their wisdom. But what struck me even while watching the movie was that it very artfully sets up a number of structural polarities, only to dissolve some of them—but not all!—into an even more complex jumble. Postmodern neopaleontology? Bring it on!

What I will do now is look at a few motifs that are repeated throughout the film, creating striking parallels and generating resonances.

A meditation on mathematics as meditation

Following the previous posts on bhāvanā, bringing-into-being, and its role in meditation, literature, and human creative activity in general, I couldn’t help but relate some of those ideas to mathematics.

Many (pure) mathematicians are content to take the mathematical structures they explore as givens, which they can figure out or manipulate in interesting and sometimes profoundly beautiful ways. This is a naïve version of Platonic realism, which grants to mathematical structures an objective existential status that lies beyond human beings. (This also means that sentient alien species should have exactly the same mathematical ideas as we do, that the Vulcans would accept that Euclid’s axioms generate the same results as us, and so on.)

Philosophers of mathematics who are formalists of various stripes hold instead that mathematics comes down to playing games with symbols: pushing arrows and boxes and Greek and Hebrew characters around based on well-defined rules. (Cue Wittgenstein.) They reject the idea that mathematicians “discover” mathematical structures; rather, they formulate new rules and new symbols and manipulate them.

While it may certainly seem from the outside that this is all mathematicians do, and while Western models of logic separate formal syntax and formal semantics in a way that seems to encourage this line of thought, it does not gel with my personal experience of actually doing proofs. Seldom can a real proof be hit upon simply by pushing symbols around on a piece of paper. At least for me, thinking about a hard mathematical problem involved trying very hard to “see” what was going on behind the symbols: symbols barely came into it. Strangely enough, the harder the proof, the more I thought I was seeing something that was already there! And even in those cases when one does merely shuffle things around, there is a crucial psychological difference between staring at a bunch of symbols on a page and hitting the Eureka moment. The proof is complete, I would argue, only with the latter. (This is not a “proof” that formalism is wrong, but merely an observation that it doesn’t fit with the phenomenology of at least some mathematicians.)

So where does bhāvanā come into the picture here? I would like to suggest (without proof, hehe) that what makes a proof a proof is precisely the fact that when(ever) it is understood correctly, it reliably and unfailingly brings into being in our minds a mathematical truth, in a manner that is at least intersubjectively valid, if not objectively. A proof is the means by which a particular mathematical end (a fact, a theorem, a lemma, or whatever) is attained. 

I have been vaguely inclined towards this manner of thinking ever since I read one of the greatest math books written in recent times in my opinion, Tristan Needham’s Visual Complex Analysis. Needham takes perhaps the most aesthetically remarkable branch of modern mathematics and offers a fabulous tour of its key features and structures in a manner that emphasizes visual and geometric thinking over the algebraic. (That is, he encourages you to prove things not by pushing symbols on paper but by visualizing, rotating, and dilating mathematical structures.) Given my prior bias towards visualizing mathematical structures, this book has been particularly enjoyable to read. (Perhaps my favorite exercise in visualization is the one in which I had to “see” the complex logarithm multifunction twisting and lifting the complex plane into an infinite helix.)

This process of visualizing a mathematical object is both deeply personal and yet objectively available. Two people who visualize a mathematical object will both agree on its key characteristics and its relevant properties, and may yet visualize it in ways that differ quite dramatically (and yet inexpressibly) from each other. To me, this situation bears a thought-provoking resemblance to Hindu/Buddhist meditative exercises in which devotees are asked to bring-into-being a particular deity in their minds, and are usually given elaborate visual descriptions of the deity’s characteristics to aid them in the process. Two different devotees may thus both bring-into-being very different versions of the same deity in their own minds, while yet agreeing fully on all of the key features possessed by this deity. The former half allows them to “take ownership” of the deity, in a sense; the latter half lets them participate in a shared conversation with others about the deity. Of course, by comparing meditative exercises with mathematical proofs, I intend to make neither religious claims about mathematical entities nor mathematical claims about religious entities!

On goals, systems, and bhāvanā

An article by James Clear called “Forget About Setting Goals. Focus on This Instead” [Oh Upworthy, how I truly hate thee!] has been doing the rounds recently. It has received a lot of attention, but when I finally sat to read it a couple of weeks ago, I found myself deeply bothered by something I couldn’t quite get a handle on. It is only just now that I’ve realized what the problem was, and the answer came to me from Mīmāṃsā.

What on earth does a nearly 3,000-year-old Hindu tradition of ritual hermeneutics have to do with any of this? As it turns out, a lot! Mīmāṃsā’s primary intellectual concern has been with the Vedic sacrifice: how it works, how its descriptions in various ritual texts cohere, how it is organized, and so on. To do so, it has developed a formidable arsenal of techniques and frameworks. One of these, the concept of bhāvanā, was widely used and taken up in disciplines far outside Vedic ritual exegesis, including literary theory and imagination / meditation. (See my prior post on imagination for some other uses of the concept of bhāvanā in South Asia.) The time has now come to apply bhāvanā to yet another problem: motivating people to stay on track with difficult projects!

What is bhāvanā?

To massively oversimplify things, and with apologies to Andrew Ollet’s excellent article, a bhāvanā, a bringing-into-being, is a particular action (or a set of actions) designed to create something, undertaken by an agent. Every such bhāvanā has three essential components to it:

1. The desired end which the agent is trying to bring into being through this operation
2. The instrument using which the agent is carrying out the operation
3. The procedure which the agent is following with the instrument to bring about the desired end

These three are respectively called the kim (the “what”), the kena (the “by what”), and the katham (the “how”). These three things are very different from each other. Confusing them can be fatal to understanding how things are actually supposed to work.

The standard introductory Mīmāṃsā handbooks usually explain bhāvanā with an example from Vedic sacrifice. Here’s a rather different, much more quotidian scenario where the three components are nevertheless clearly distinguishable.

After a long, grueling day at work, you come home utterly famished. You don’t want to go out to get dinner, so you decide to make yourself a quick dinner. You have a microwaveable mac & cheese sitting in the freezer, so you take it out, read the instructions on the packet (you don’t really ever cook), stick it in the microwave, and a few minutes later, satiate your hunger with some piping hot coagulated carbs and fats.

In this scenario, it’s pretty clear that something new was created: the state of the world, and more importantly your own state, was transformed in this scenario. In not-so-technical Mīmāṃsā non-jargon, some sort of bhāvanā thingie occurred here. So what were the components of this bhāvanā? What was created?

It is tempting to think of the mac & cheese dinner as being what is created: after all, before you cooked it, it was just a frozen lump of carbs and dairy and preservatives, and it was your cooking it that transformed it into an (arguably) edible mush. However, this would be a major mistake, according to Mīmāṃsā: the mac & cheese was not the desired end of your actions. It wasn’t why you undertook all these steps. Instead:

1. the real end, the kim, must be the resolution of your hunger. 
2. the mac & cheese is the means, the kena, by which your hunger is resolved. 
3. the way you resolve your hunger, the katham, is by following the procedure outlined on the packet to cook and serve the mac & cheese.

Goals and Systems

If my quick overview of bhāvanā didn’t foreshadow it clearly enough, it should be clear what my beef with Clear’s piece is: he conflates kims and kenas when talking about goals, and therefore overemphasizes the importance of systems (which are not quite kathams). To see how he does this, let’s look at the section where he distinguishes between goals and systems:

What’s the difference between goals and systems?

• If you’re a coach, your goal is to win a championship. Your system is what your team does at practice each day.
• If you’re a writer, your goal is to write a book. Your system is the writing schedule that you follow each week.
• If you’re a runner, your goal is to run a marathon. Your system is your training schedule for the month.
• If you’re an entrepreneur, your goal is to build a million dollar business. Your system is your sales and marketing process.

Look carefully at the four things he describes as goals: winning a championship, writing a book, running a marathon, and building a million-dollar business. And look carefully at the four types of people he describes as having these goals: the coach, the writer, the runner, and the entrepreneur. Now, many of us would agree with these things as being described as “goals” (which to me only reinforces the fact that we live in a deeply instrumentalist society). But are they really goals? Are they more like the mac & cheese or like satiating the hunger for a hungry person?

• For a coach who is hungry for a win, winning a championship will certainly satiate his hunger. But it is not obvious to me that people become coaches in order to win championships. It can be argued that the real purpose of being a coach is to, well, coach a bunch of players to the best of their abilities, so that they can perform superlatively on the field. If the team can do that consistently, then they may very well end up winning a championship. It seems to me that winning the championship is really just a means (a kena) to the real end (kim): the joy that comes from watching people do their best on the field. (It is possible to win a championship and yet be dissatisfied, because perhaps your opponent defaulted; it is possible to lose a championship and yet be pleased, because you did your absolute best and fulfilled your “duties”, so to speak.)
• Writers don’t write in order to create books; they write books in order to do something else: tell a story, persuade their readers to act, create emotional states in their readers, convey some valuable information, or even simply feed their families. The book is clearly just a kena. The kim is whatever motivates the writer to write.
• As with the example of the coach, it may well be that running a marathon is a real kim for some people. However, it is again quite likely that there are other satisfactions here: enjoying the endorphin rush, raising money for a valuable cause, staying in shape, or whathaveyou. In all of those cases, the marathon is just a kena that is subordinated to the more significant kim.
• Again, it certainly is the case that some entrepreneurs are just in it for the money. In that simplest of cases, the business is the kena to their real kim: making a boatload of money. But as Guy Kawasaki has said many a time: “make meaning, not money” is the heart of entrepreneurship. Whether your business is worth a billion dollars or a hundred, the real purpose should be to do something that creates meaning for you and for the people you engage with. In such a view, the mere instrumentality of the business is even more strongly pronounced.

None of the four examples of goals here is clearly and precisely a kim: an end that people strive for and desire. Some of these could possibly be treated as kims, but Mīmāṃsā argues (and Clear would agree with this, as he himself writes) that to do so would be to fundamentally misunderstand the nature of each of these sets of actions.

Clear has correctly identified one problem: the so-called “actionable goals” he describes are notoriously bad at actually getting achieved. The real reason for that is because they aren’t in fact goals: they are means to other ends. These other ends are the real goals we should keep in mind. In the absence of those real goals, people wouldn’t act at all!

The problem isn’t with goals at all; it is with the fact that Clear is using two terms (“goal” and “system”) to describe three distinct pieces (the [real] goal, the means to the goal, and the procedure; to use a different metaphor, the destination, the means of transport, and the particular path you navigate). Because of this conceptual blurring, he emphasizes “systems” more than they can bear: what he calls systems are just glorified procedures. 

The real system is the the whole triplet that Mīmāṃsā describes, and this systems needs all three pieces to succeed: a real goal to motivate us to act, a means by which this goal can be achieved, and a procedure that can be followed (with all the useful tips that Clear provides).  

ADDENDUM

One of the subheadings in Clear’s piece gives the game away to the reader who is keyed into Mīmāṃsā: He describes one of the faults of the goal-based approach as being the fact that “Goals reduce your current happiness.” This is exactly what sets kim apart from kena in Mīmāṃsā! Take the typical injunction that Mīmāṃsā analyzes: yajeta svarga-kāmaḥ, “The heaven-seeker should perform a sacrifice.” Lots of Hindus have desired to perform sacrifices, and still do to this day. But Mīmāṃsā argues that the real end here, the real kim, here has to be heaven. The sacrifice itself is only the means by which this end is brought about. And one of the arguments is this: a sacrifice is a difficult, expensive, resource-intensive, and physically taxing undertaking. No rational (pleasure-maximizing, pain-minimizing) human would perform a sacrifice for its own sake. Therefore, a sacrifice must be the means to some other end: the promised result. To follow Clear’s line of reasoning, you would focus on the individual actions of a sacrifice (its procedure) but ignore the coherence of the sacrifice itself as an instrument, and altogether forget about the real goal, the heaven that is the result of the sacrifice!

On imagination, meditation, and bringing-into-being

In this fascinating interview, Tanya Luhrmann addresses the tremendous importance of imagination in religious traditions such as American evangelism. The idea that religion is “belief”, the affirmation of the truth-value of some proposition, is a particularly Western, Protestant, understanding of religion, and is profoundly different from the religious experiences of people from, say, the dharmic traditions. (Or for that matter, from the experiences of Orthodox Christians.) Luhrmann says about kataphatic prayer:

It makes what is imagined in the mind more real. In kataphatic prayer you are saying that certain of your mental images are significant, and you are making these images more sensorially rich, you are allowing yourself to imagine them more vividly. The demand of religion is to teach you that the world as you know it is not the world as it is—and to teach you the capacity to see the world as it is, as something good. So you’ve got to make what is imagined real, and you’ve got to make it good.

The obvious response of the outsider to something like this is to describe it as clearly false, or “merely” imagined. And in a certain sense, the outsider is right: it is the believer who has imagined a particular religious experience into being, for which there is most likely no objective correlate. But Luhrmann argues that this attitude misses the heart of the experience as the insider experiences it: as something real, indeed as something more than real—because they create a new reality for the insider. It makes the insider more likely to feel loved, and thus to become more loving. Luhrmann thinks that something like this may even help reverse the erosion of social ties that people complain about today.

A number of Luhrmann's ideas squarely fit in with late medieval South Indian Hindu thought as is described in More than Real: A History of the Imagination in South India by David D. Shulman.

Shulman focuses on the importance given in medieval South India to the force of imagination: to the fact that human being are at their core imaginative creatures, who shape reality by imagining it together. 

• Sometimes this imagination is internal to the person: Shulman tells the story of an impoverished devotee of Śiva who constructs in his mind a temple so beautiful that Śiva prefers to dwell there instead of in the vast granite temple that a king has built for him (said to be the Kailāsanātha temple of Kanchipuram). 
• At other times, this imagination is intersubjective: Shulman describes in great detail a performance from the Kūḍiyāṭṭam dance-drama tradition of Kerala, in which a solitary skilled dancer transforms an empty, prop-less stage into a story-universe through the combination of his gestures and through the shared imaginations of the entire audience. 

Worship is imagined in the same way—by imagining our iṣṭa-devatās in our minds and by that very act bringing them into being. The word used for bringing-into-being is bhāvanā, a word borrowed from Mīmāṃsā ritual hermeneutics that refers to the power of a sacrifice to bring into being its fruit.[*]

Shulman points out that the philosophical and theological systems in which these systems developed in South India were staunch defenders of ontological realism, but of course not of physicalist, materialist reductionism. (Paradoxically, it was Advaita Vedānta that was fairly skeptical of the positive power of imagination.) He writes, comparing 16th-century India and Europe:

In Europe, the ancient dichotomy of mind and matter hardened into a fully desubjectified theory, or evolving set of theories, about the status of objects within an external, natural world. In India, the dichotomy is itself questionable, and the metaphysics of inner and outer took another course. Broadly speaking, in one conceptual system the imagination became increasingly associated with pathology, while in the other it tended to be understood as therapeutic. (p. 278)

Suspension of disbelief is the wrong way to think about what's going on here (at least in the Indian context; it may well hold for Luhrmann's evangelicals). We don’t lie to ourselves about something being there when it isn’t; we construct it with our mental acts—and by doing so, we make it real.

And towards the end of the book, Shulman also touches very briefly upon Ibn ‘Arabī, in whose vast work khayāl, “imagination”, is profoundly related to the structure of the universe and to the relationship between man and God.


[*] This explains the use of words like bhāvayāmi in much devotional Carnatic music. The singer-devotee is trying to actualize the deity in the minds of all those present at the performance. 

“Practical” Universities

David Brooks has a new column out in the New York Times in which he argues that online education will force a transformation of universities. Drawing on Michael Oakeshott, he argues (or really, just states) that universities today offer two kinds of knowledge: technical (the what) and practical (the how). Brooks claims that, because technical knowledge can easily be transmitted online, we will see people gravitating towards MOOCs where they pick up “just the facts, ma’am” from star online teachers. However, since practical knowledge can only be picked up from experience, he thinks that universities will shift increasingly towards offering this sort of irreplaceable knowledge.

Leaving aside the merits and demerits of Brooks’s piece, I am quite intrigued that he ignores another, crucial, kind of knowledge that universities offer: the why. Now sometimes this knowledge seems like anti-knowledge from the outside, because it is about limits, about ends, and about asking the right kinds of questions. But these are critical issues to think about—admittedly, not for everybody, but for society as a whole. A city full of carpenters, or of philosophers, is not a city but an unnatural monoculture.

This is all the more surprising because a threefold distinction of knowledge was known to Aristotle, who called them epistemē, technē, and phronesis. (Of these three, phronesis directly overlaps with Brooks’s practical knowledge; while technē seems to largely make up, but not exactly correspond to, technical knowledge.) A polis needs all three to flourish. I am curious to know where Brooks thinks epistemē will be found in his post-MOOC world.

The moral lessons of The Hobbit

After having defended Peter Jackson’s Hobbit from some of its most common critiques, I shall now turn to some of the things I enjoyed the most about the movie. Most of these come from Tolkien’s Hobbit, but again, just because a story sounds good in one mode does not automatically mean that it will “click” when told in a different mode.

The great challenge in a visual depiction of the Quest of Erebor is this: this is an event on a much smaller scale than the War of the Ring. For viewers who are looking for epic set-piece battles (or as a little birdie put it to me, for those “who don't necessarily share the love for the books and just walked in to see Orlando Bloom buckle his swash”), there really isn’t anything in the Quest of Erebor that provides this, except for the climactic Battle of the Five Armies. Even the One Ring, which is mildly important for the War of the Ring, plays a minor role here. Its importance wasn’t even recognized by most people at the time of the Quest!

But as Tolkien repeatedly tells us, small does not have to mean insignificant, and this applies to everything, from the hobbits to the Quest itself. 

Boredom

When asked to define “boredom” recently, I came up with this ‘masterpiece’:

Boredom: an external symptom of the deeper inability to be at peace with oneself, characterized by restlessness and by discontent at the inability of external stimuli to permanently fill a soul-shaped void in one’s life. 

(Yes, I did tag this wisdom. Tags don’t always mean what they mean; their vācyârtha can be overriden by a lakṣyârtha, and sometimes they suggest a vyaṅgyârtha over and above and/or alongside that.)

“Transcendence and Self-Transcendence”

Michael Polanyi: chemist and philosopher, brother to Karl Polanyi, historian and sociologist. One of Michael Polanyi’s important shorter pieces is this one, called “Transcendence and Self-Transcendence”, published in 1970.

Some of the interesting bits that emerge (no pun intended) from this work:

I introduce the concept of hierarchical levels. A machine, for example, cannot be explained in terms of physics and chemistry. Machines can go wrong and break down—something that does not happen to laws of physics and chemistry. 

In fact, a machine can be smashed and the laws of physics and chemistry will go on operating unfailingly in the parts remaining after the machine ceases to exist. Engineering principles create the structure of the machine which harnesses the laws of physics and chemistry for the purposes the machine is designed to serve. Physics and chemistry cannot reveal the practical principles of design or co-ordination which are the structure of the machine.

And:

The more intangible the matter in the range of these hierarchies, the more meaningful it is. This is my criticism of all redactionist, mechanistic programs founded on the Laplacean ideal which identifies ultimate knowledge with an atomic topography, the lowest level of the universe.

Most provocative:

I have elaborated in schematic fashion a multiple hierarchy which leads on to ever more meaningful levels. Each higher level is more intangible than the one below it and also enriched in subtlety. And as these more intangible levels are understood a steadily deeper understanding of life and man is gained. These understandings constitute transcendence in the world. 

Unbridled detailing, the ideal advocated by Laplace and his modern followers, not only destroys our knowledge of things we most want to know; it clouds our understanding of elementary perception—our first contact with the world of inanimate matter and of living beings and our initial act of self-transcendence.

I must confess that these fragments of his essay, pulled out of context, do not convey its full meaning. The whole thing is worth reading.

The Arabic nominal system, intro

Arabic has a somewhat tricky system of case-inflections for its nouns. I’ve decided to note down some patterns I’ve seen in order to act as a دلالة الحائرين, a Guide for the Perplexed (in whose company I fall).

In general, Arabic nouns vary along four dimensions:

• number: singular (mufrad, مفرد), dual (muthannā, مثنى), plural (jam‘, جمع)
• gender: masculine (mudhakkar, مذكر) and feminine (mu’annath, مؤنث)
• state: definite, indefinite, and construct (iḍāfa,  إضافة)
• case: “nominative” (marfū‘, مرفوع), “accusative” (manṣūb, منصوب), and “genitive” (majrūr, مجرور)

The Latinate case terms are pretty horrendous when it comes to describing the possible meanings of the Arabic cases, and so I will use only the Arabic terms for them. For the rest, I shall use the English. All of the other terms should be fairly clear, except for the “construct state”, which is generally absent in Indo-European languages. (I’m not going to go into the semantics of the construct state here.)

A fifth dimension, animacy, is implicit and shows up in the formation of plurals. Briefly: when an animate object modified by an adjective takes the plural, then both noun and adjective must be in the plural. But when an inanimate object modified by an adjective takes the plural, then the adjective must be in the feminine singular—regardless of the gender of the object. Consider the four nouns: walad (p. awlād) “boy”, bint (p. bināt) “girl”, qalam (masc., p. aqlām) “pen”, sayyāra (fem., p. sayyārāt), “car”.  Modified by the (regular) participial adjective mukhtalif-  “writing”, we see (in the marfū‘ case):

• awlād mukhtalifūn(a)
• bināt mukhtalifāt(un)
• aqlām mukhtalifa(tun)
• sayyārāt mukhtalifa(tun)

A few random notes:

• Why “regular” adjectives? It’s because Arabic has two different kinds of plurals: the regular or “sound” (sālim, سالم) plural, which is formed regularly for (almost) all feminine nouns and adjectives and for many masculine nouns and adjectives (including all participial adjectives), and the “broken” (maksūr, مكسور) plural, which is formed for some masculine nouns and some masculine adjectives. The most common adjectives and nouns are often the ones that form the craziest broken plurals. Precisely how different broken plurals are formed doesn’t concern us here; all we care about are the ways in which they inflect.

• The regular feminine noun ends in a t, the so-called tā’ marbūṭa(t), but this is not pronounced in pausa. I’ve therefore written it in brackets (which has the added benefit of vaguely resembling the way it’s written in Arabic, as a ة). When case-endings are added, though, the tā’ marbūṭa(a) is pronounced, along with the endings.

• Incidentally, the Arabic word for inflections, i‘rāb (إعراب), is the verbal noun of the verb a‘raba (أعرب) formed from the root ‘-r-b (ع ر ب), which is also the source of the words “Arab” and “Arabic”. In fact, that particular form of the verb literally means “to make something Arab(ic)” or “to Arab-ize”. [I have no basis for this hypothesis, but if the claims made by Bohas, Guillaume, and Kouloughli in The Arabic Linguistic Tradition about the motivation behind Sībawayhi’s nomenclature are correct—i.e., that Sībawayhi picked as grammatical terms action nouns (maṣdar, مصدر) that describe the speaker’s intention—then it would seem that pronouncing the case-endings would stem from the speaker’s desire to make his speech as Arab as possible.]

Actual patterns to follow.

Strings, branes, and all that

 I came across a few articles today on Plus, an online mathematics magazine, which presented some of the basics of theoretical physics—string theory and M-branes in particular—in a simple, accessible way. If you’re interested, a good way to do this would be to start “From Newton to Einstein and beyond” and to try “Tying it all up” before you “Meet the mother theory”. 


Of course, the hard math underlying all these theories really is essential if you want to grok this stuff. I would recommend Sir Roger Penrose’s The Road to Reality, about two-thirds of which is spent in actually explaining the math behind such things as general relativity. Stephen Hawking wrote in A Brief History of Time that his editor warned him that every equation he put in would scare away half of his readers. By this metric, Sir Roger’s editor was probably content to have an audience of five. Including himself. (I’ve worked through less than a third of the book so far!) The Road to Reality tries to cover, quite literally, everything. Reading it will do to your mind what diving into a P90X training program will do to an obese Indian bureaucrat’s body.

If you're stuck on a problem

This is a profound statement by Robert Pirsig, from Zen and the Art of Motorcycle Maintenance:

An egoless acceptance of stuckness is a key to an understanding of all Quality, in mechanical work as in other endeavors.

How does knowledge grow?

The Name of the Rose is a staggeringly marvelous book that has just about everything imaginable in it; insofar as this is true, it is a veritable bestiary of literary and theological ideas about pre-Renaissance Europe. Like the Sanskrit mahākāvyas, it has everything; but it is nevertheless novel in that it questions the pre-existing order in ways that older, more conservative works don’t.
For now, I wish to point out a fascinating passage from the book, addressed to the reader by the narrator Adso:

Learning is not like a coin, which remains physically whole even through the most infamous transactions; it is, rather, like a very handsome dress, which is worn out through use and ostentation. Is not a book like that, in fact? Its pages crumble, its ink and gold turn dull, if too many hands touch it.

This passage would have caught my eye on most days, given that it strikes me as (understandable but) entirely wrongheaded, but it struck me all the more forcefully since I had earlier been translating some Sanskrit subhāṣitas. Specifically, this one:

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.