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.

 

 

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.

“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.

Verbal knowledge (śābda-bodha) in early modern India

(This is unlikely to be of interest to anybody but me, but then again most things on this blog fall into that category!)

In all things associated with language—grammar, semantics, exegesis, hermeneutics, literary theory, you name it—the Sanskritic intellectual tradition developed perhaps the most sophisticated toolkit available to pre-modern humans. One of the most substantial advances on this front was made by the Navya Naiyāyikas (the “New Logicians”), who came up with a formal terminology designed to make communication unambiguous (or at least as unambiguous as can be possible in a natural language). Their terminology was adopted by intellectuals with wide-ranging commitments and projects, such as Mīmāṃsā (hermeneutics), Vyākaraṇa (grammar), and Vedānta (philosophical theology, for lack of a better description).

Jan Houben’s article “‘Semantics’ in the Sanskrit tradition ‘on the eve of colonialism’” shows how Vaiyākaraṇas (grammarians), Naiyāyikas (logicians), and Mīmāṃsakas (exegetes) all used terminology borrowed from the Navya Naiyāyikas to precisely and unambiguously describe the kind of “verbal knowledge” (śābda-bodha) produced in a speaker of Sanskrit upon hearing the sentence:

rāmo ’nnaṃ pacati 

“Rāma cooks rice”

The question for all three groups of intellectuals is the same: what is the main meaning-bearing element of a sentence, and how does it relate to all the other elements? All three agree on what the sentence conveys to a speaker of Sanskrit; where they differ is on the “keystone” of the sentence.

Definitions of negation

The Sanskrit grammatical tradition loves to name and define its terms, often in verses. For example, the privative prefix, which manifests itself as a- or an- (and is cognate with the Greek a- in words like atom), is called naÑ in the Pāṇinian tradition. A concise verse (from the voluminous Śabdakalpadruma) summarizes the various meanings that this prefix can express:

tat-sādṛśyaṃ virodhaś ca tad-anyatvaṃ tad-alpatā |

aprāśastyam abhāvaś ca nañarthāḥ ṣaṭ prakīrtitāḥ ||

The tradition enumerates six different meanings that can be expressed by naÑ:

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:

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.

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.

On freedom, choice(s), and democracy

Snowed in, I spent all of today reading a very interesting book by Prof. Loren J. Samons II of Boston University, with a very provocative thesis. The book's not-so-subtle title, What's Wrong with Democracy? From Athenian Practice to American Worship, belies its careful argument. Although the bulk of the book involves a close study of historical sources in order to examine the actual practices of the government of the Athenian polis, the author's overarching motivation is not the reassessment of contemporary perceptions of a long-extinct society merely for the sake of historical understanding. His point, rather, is to 

"present and foster criticism of modern democracy … [that is] aimed at the philosophical foundations of democracy, the popular conception of democracy, the practice of representative government through democratic elections, and the social and intellectual environment generated by democratic thought and practice in contemporary America" (p. 1). 

I hesitate to offer a summary of the book for fear of oversimplifying its complex, historically sensitive argument. Very crudely put, Prof. Samons: 

True wisdom

This pearl, from pg. 2 of Raymond Smullyan's The Tao is Silent, is particularly relevant to me right now as I procrastinate, neither working (as I could be) nor sleeping (as I ought to be).

Had I been Laotse, I would have added the following maxim—which I think is the quintessence of Taoist philosophy:

The Sage falls asleep not 

because he ought to 

Nor even because he wants to

But because he is sleepy.

True words of wisdom indeed.

Smell and memory

I've never read Marcel Proust's À la recherche du temps perdu, but I've certainly heard about the famous madeleine episode in which the narrator's experience of seeing, smelling, and tasting a madeleine leads to a flood of past memories (and a voluminous novel!). This is perhaps the best-known contemporary account of the incredible power of the sense of smell, and its profound connection to our deepest emotions. The smell of a favorite dish, of a lover's cologne, of petrichor after summer rains, of a flower: all of these smells can immediately and instantaneously transport us to a different time and place and emotional state.

It is perhaps for this reason that smell plays an extremely important role in Indic culture. David Shulman has written a fascinating article called "The Scent of Memory in Hindu South India" (Res: Anthropology and Aesthetics 13 (1987): 123–133) in which he explores some of the connections among smells, memories, emotions, and aesthetics. 

Bhavabhūti on language

The Sanskrit dramatist Bhavabhūti is reckoned second only to Kālidāsa in his command over the Sanskrit language, his exquisite depiction of the emotions, and his mastery over the art and the science of literary composition. His Uttararāmacarita is a fascinating and powerful retelling of the seventh chapter of the Rāmāyaṇa, the Uttara Kāṇḍa. This powerful, even tragic, chapter is in fact not included in many versions of the Rāmāyaṇa, in particular Kampan's Irāmāvatāram (also known as the Kamba-ramayanam).

One of the things that makes Bhavabhūti so fascinating is his self-conscious reflection on the power of language and art. While Sanskrit literature in general abounds in such reflection—indeed, there is a Ṛgvedic verse on speech uttered in the voice of Speech personified, which promises to bestow speech upon those Speech favors—Bhavabhūti seems to take it to a whole new level in his work. One verse in particular struck me, because of its self-contained meaning, its particular signification in the immediate context of the play, and its more general meaning in the context of literary works as a whole.

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

On Pūrva Mīmāṃsā

Prof. Francis Clooney's article "Why the Veda has No Author" [1] is an excellent survey of the early work of the Hindu school of thought known as (Pūrva) Mīmāṃsā. The word pūrva also means "early" or "prior", but Prof. Clooney's reference here to the "early work" does not mean the distinction between (Pūrva) Mīmāṃsā and Uttara Mīmāṃsā, better known as Vedānta, but to the historically earliest work done by scholars who identified themselves as Mīmāṃsakas: Jaimini, the 2nd century BC author of the Pūrva-mīmāṃsā-sūtra, and Śabara, the great commentator who came four hundred years later. This is important because the school of Mīmāṃsā later splits into two traditions, the Bhāṭṭa, deriving authority from the formidable Kumārila Bhaṭṭa, and the Prābhākara, from Prabhākara Miśra. (The initial long 'ā' in the names of the schools, as opposed to the short 'a' in the names of their founders, is not a typo but a well-established Sanskrit grammatical phenomenon known as vṛddhi. In this particular case, vṛddhi is effected by the use of the so-called aṆ suffix, which has the sense of "descendant" or "offspring".)

Prof. Clooney notes that Mīmāṃsā is

"a worldview and mode of inquiry that developed in a context of reflection on the meaning of the action of sacrifice. In Mīmāṃsā, notions such as "God," the "sacred text," the "author" and the "anthropocentric ordering of reality" were already subjected to a radical critique more than two thousand years ago, and the primary vehicle of this critique was an uncompromising commitment to a reworking of religious discourse on a ritual basis." (p. 660)

The earliest textual evidence for this particular Weltanschauung is Jaimini's Pūrva-mīmāṃsā-sūtra, which, as Prof. Clooney notes, argues that

"[1] religion includes meanings and values appropriate to human beings, but the sum of its meaning necessarily exceeds the human perspective;

[2] the sacred Sanskrit-language Scripture known as the Veda is not a 'book' to be read, nor a source of information about a world exterior to itself; and

[3] the Veda has no author, no meaning beyond the words and the sacrificial intentions themselves; one cannot appeal to a pre-verbal intention to get beyond the words." (p. 660)

The Mīmāṃsakas operated in a religious world dominated by "two interconnected systems", that of the "traditional sacrifices, which had been practiced and described even a thousand years earlier," and that of the "Veda, the totality of the texts relevant to the sacrifices" (p. 661). This dynamic duo of text and ritual, each highly structured on its own and also deeply linked to the other, was what the Mīmāṃsakas investigated and defended against all attacks, both from within the "Hindu" system, such as the Naiyāyika natural theologians, and from without, such as the Buddhists and Jains. A large part of the Mīmāṃsā enterprise thus boils down to a sophisticated process of exegesis whose initial motivation was to understand instructions for the performance of ritual, but whose eventual scope broadened to pretty much any situation where texts were read and actions were performed.

Turning again to Prof. Clooney, "each generation of Mīmāṃsakas sought to elaborate more and more sweeping rules by which to govern the preceding set of meta-rules, although the search for generality was constantly subverted by the precise demands posed by the concrete ritual situations to which the interpreter eventually would have to return" (p. 662). It wasn't that the Mīmāṃsakas formulated ritual handbooks (those had been composed and circulated possibly a thousand years before Jaimini); what they were concerned with was a process closer to what we might consider legal reasoning. (Don Davis's new book, The Spirit of Hindu Law [2], constructs a useful analogy between Mīmāṃsaka thinking and the complexities of navigating the US tax law system.) In the process, they also defended particular positions within the philosophy of language—but always with an eye to preserve the integrity of the sacrifice.

More interestingly, from our perspective outside the dynamic dyad, the Mīmāṃsakas also asked and answered (to their satisfaction, at least) questions regarding the underlying assumptions of their worldview, about "the overall purposes of sacrifices, or the basis on which one can say that the Vedic scriptures are absolutely true, or the manner of verifying the efficacy of sacrifices" (p. 662). This was all the more important because the great rivals of the Brahminical orthodoxy were the Buddhists and Jains, who questioned (as we might today) the whole purpose and efficacy of the Vedic sacrificial system.

Prof. Clooney points out insightfully that

"Mīmāṃsā's response to the whole range of criticisms was to rethink its world without reliance on any single viewpoint, effectively undercutting the possibility of a single perspective. It sought a justification for sacrifice that needed no external validation, either from active gods or satisfied humans, and that required the positing neither of any supernatural realities nor a reliable world order beyond that of good Sanskrit texts, well-performed sacrifices, and a set of rules for integrating the two.

"Jaimini and his commentator Śabara primarily were concerned with achieving a right understanding of the rules of sacrificial action and sacrificial text so as to ensure that what one saw and heard at a sacifice would cohere—be intelligible—regardless of what anyone might say about it from some particular perspective. In discovering these rules they sought to repplace the 'laws of the cosmos' with the 'laws of language and ritual,' and reliance on gods and humans (as norms for meaning) with an appreciation for the harmony of text and action (and everything accompanying them, even in orthodox society as a whole) that underlay the well-wrought sacrifice." (p. 663)

Read the rest of the awesome article to find out more about the Mīmāṃsā worldview, and what it may mean for us today when we no longer follow the Vedic system of sacrifice.

<UPDATE>

I simply had to include this line from the article that compares the Mīmāṃsā exegetical enterprise to Foucault:

"The Mīmāṃsā theory of authorlessness [apauruṣeyatva; emphasis in the original], with its underlying concern to 'liberate' the sacred texts from its author, approaches Michel Foucault's influential interpretation of the common (though not universal) way in which the author-function is used as a societal tool to restrain language, assign responsibility, and confine the meaning of texts. Foucault shows how this author-function … actually restricts texts by ordering them to a designated author, whose intentions determine what the texts are allowed to mean … [Mīmāṃsā] decided that reference to the author as the privileged source of meaning limited the Veda, cut it off from its ritual context, and made it liable to the mistakes and limitations of the author."

</UPDATE>

References
[1] Clooney, Francis X. "Why the Veda Has No Author: Language as Ritual in Early Mīmāṃsā and Post-Modern Theology," Journal of the American Academy of Religion, Vol. 55, No. 4 (Winter), pp. 659–684. Available here from JSTOR.

[2] Davis, Jr., Donald R. The Spirit of Hindu Law. Cambridge University Press, 2010. Available here from Amazon.

"Inception"

As I mentioned to a friend recently, Inception is likely to be the That Movie of this fall: the movie that sparks off intense, passionate 2 am conversations in college dorms. I'm still trying to digest the movie, but for now, let me just ask the big question: Is the whole movie a dream?

Possible responses:

• Yes
• No
• It doesn't matter
• insert-some-other-response-here

For what it's worth, I thought this graphic was incredibly helpful in organizing the different levels of the movie. (It need hardly be said that you should only open this link if you've already seen the movie. If you haven't seen it yet, go see it!)

Reconciling God and Evolution

This is a short and interesting article by Prof. Sarah Coakley at Harvard Divinity School. She argues that "the three most profound problems for Christian theism since the advent of Darwinism, so profound as to cause many to see Darwinism as a "defeater" of Christian belief" are:

1. "[T]here is the issue of how we should understand the relation of God's providence to prehuman dimensions of creation and their development."
2. "[T]here is the issue of how God's providence can relate to the specific arena of human freedom and creativity."
3. "[T]here is the problem of evil, the question of why what happens in the first two realms manifests so much destructiveness, suffering, and outright evil, if God is indeed omnipotent, omniscient, and omni-benevolent."

Her response, particularly to (1), essentially boils down to this: "God is that-without-which-there-would-be-no-evolution-at-all; God is the atemporal undergirder and sustainer of the whole process of apparent contingency or "randomness," yet—we can say in the spirit of Augustine—simultaneously closer to its inner workings than it is to itself." * This allows her to conceive of God as ever-intervening (indeed, without whose intervention at every stage the world would simply cease to be), but whose interventions are masked by the order of the universe (which in itself stems from God).

From what little I know, this view is fairly close to the classical theologians of Islam, particularly Imâm Ghazâlî (without the trinitarian interpretations, of course!). Imâm Ghazâlî in no way denies that the world is orderly and is governed by "natural" laws; what he says, though, is that this order is a contingent one that is sustained by God at every moment. I've always thought that this is a pretty neat way to fully and healthily accommodate scientific investigation of the spatiotemporal universe while not losing sight of religious / spiritual insight. (It's certainly a far more nuanced position than either "it's obvious that a tall old man in a white beard made all this" or "it's obvious that we're nothing more than a handy way to help a few large molecules self-replicate"!)

I'm not convinced by her later argument, that recent discoveries showing the importance of cooperation in evolution help build the case for getting evolutionary biologists and theologians talk to each other. I suspect that given the nature of the scientific method, no dialogue between a scientific discipline and a non-scientific discipline can be well founded on the basis of particular scientific explanations. (All it takes is one paper that shows that what we think is cooperation is really just selfish genes, and poof! vanishes your bridgehead.) No, I think true dialogue must depend on two things:

1. The fact that science and theology are both committed to using reason reasonably to discover something about the world—with the caveat that "world" likely means different things to scientists (probably depending on their institutional and disciplinary affiliation!) and theologians.
2. The fact that it is not particular scientific theories or hypotheses that bind scientists together as a community, but the scientific method per se. In other words, if you want to show that sensible science and intelligent religion are not incompatible, then find a way to accommodate the scientific method into your theology (or vice versa, if you prefer).

In case you think faith and reason are irreconcilable and separated by an unbridgeable gap—welcome to theology! Thinkers affiliated to all the major religious traditions I can think of have invested great efforts into thinking rationally about God and the universe and all that jazz, and I think the modern debate is deeply impoverished by our inability and unwillingness to incorporate their ideas into our discourse. For what it's worth, I think there are at least two solid reasons for encouraging a fruitful dialogue of this sort:

1. Like it or not, the scientific method truly has transformed the depth and breadth of our knowledge of the physical world. It is the single best method we have come up with that explains how the world is (or, more precisely, how the world is not). It would be utter folly for serious modern thinkers to ignore these insights when trying to understand our place in the cosmos—particularly when it is the case that many, if not most, earlier thinkers were willing to do the same.
2. Like it or not, the scientific method alone cannot really offer guidance of any sort—emotional, social, moral, spiritual. And while this does not mean that the only alternative has to be a religious worldview, it does mean that our scientific insights must be contextualized within a broader intellectual framework. What these frameworks can be is up to us—whether we pick mechanistic deterministic nihilism or evanescent illusionary idealism or infinitely fruitful pan(en)theism or whatever else floats our boat—but we need something more.

* This is not a rigorous statement as it stands. The article does a better job of explaining why this may be the case, but also acknowledges that this is but one starting point for a deeper philosophical and theological investigation of a problem that, as is currently being debated, really could do with more sophisticated, more inclusive arguments from both sides.

New Swadharma post

New post up on Swadharma entitled "The many paths to felicity".

"The pale blue dot"

"[A] mote of dust, suspended in a sunbeam."

Such an amazing video. Carl Sagan was a wise man. A demonstration of our insignificance in time and space could scare the weak into passivity and fatalism and despair at the pointlessness of it all; it takes courage and wisdom to generate a moral vision of compassion and connectedness from that insight.

(h/t: sszz)

Another lesson in perspective

This time restricted to our own planet.

If you've ever wondered how deep the oceans are, if you've ever wanted to really know, really feel in your bones how far below sea level the oceanic depth plunges, wonder no more.