Just a place to jot down my musings.

Friday, January 31, 2014

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!

Sunday, January 5, 2014

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!





Why pearls, and why strung at random?

In his translation of the famous "Turk of Shirazghazal of Hafez into florid English, Sir William Jones, the philologist and Sanskrit scholar and polyglot extraordinaire, transformed the following couplet:

غزل گفتی و در سفتی بیا و خوش بخوان حافظ

که بر نظم تو افشاند فلک عقد ثریا را


into:

Go boldly forth, my simple lay,
Whose accents flow with artless ease,
Like orient pearls at random strung.

The "translation" is terribly inaccurate, but worse, the phrase is a gross misrepresentation of the highly structured organization of Persian poetry. Regardless, I picked it as the name of my blog for a number of reasons: 
1) I don't expect the ordering of my posts to follow any rhyme or reason
2) Since "at random strung" is a rather meaningless phrase, I decided to go with the longer but more pompous "pearls at random strung". I rest assured that my readers are unlikely to deduce from this an effort on my part to arrogate some of Hafez's peerless brilliance!

About Me

My photo
Cambridge, Massachusetts, United States
What is this life if, full of care,
We have no time to stand and stare.
—W.H. Davies, “Leisure”