Why Math? Beauty

Perhaps the most famous proponent of mathematical beauty, G.H. Hardy has argued for the intrinsic value of mathematics by drawing analogies to (other?) art-forms:

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

Putting aside the issue of whether (à la Oscar Wilde) he might have gone too far, his point is well-taken—math is beautiful. However, what does beauty mean in this context? The truth is, the concept of mathematical beauty is multifarious and subtle and perhaps beautiful in its own right (leading us to recursive beauty with mathematics at its root). In this post, I would like to concentrate on two of the most common conceptions of mathematical beauty which are interestingly at odds with each other.

Intricate Patterns
Math is inextricably linked to patterns; in fact, patterns often loom large in many attempted definitions of mathematics. It has been consistently argued for some time now, that the human brain has evolved specifically to identify and notice patterns. In fact, sometimes it does it too well, finding patterns when none are there. It is not surprising, therefore, that patterns play a significant role in our aesthetic appreciation. Surveying the history of art, one cannot help but notice how patterns and regularity are favoured. From the simple majesty of the Parthenon, to the extravagant elaborations of Gothic architecture. Or how about the Alhambra, its intricate arabesques tapestries are a systematic architectural investigation of symmetry. Architectural marvels indeed, but let us not forget to mention the central role geometry and perspective has played in the visual arts; entire books are still being written on these subjects.

Arabesque from Alhambra. [Image credit: Wikipedia.]

Wallpaper from Alhambra. [Image credit: Wikipedia.]

Pythagoreansfirst peekedfamously arguedthis short video

Elegance
“Refinement,” “dignified grace,” “restraint,” “exactness and precision.” These words, taken from the definition, try to capture the flavour and the texture of elegance. At the beginning of the post, I have mentioned that the concept of mathematical-beauty is versatile. The aesthetics of elegance is quite different from the beauty of patterns. Here’s Bertrand Russell:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Notice especially the contrast Russell is drawing between math and painting or music. “[B]eauty cold and austere,” “stern perfections” — this is the language of elegance. In poetry, which Russell favourably compares to math, elegance is achieved through economy of language. In mathematics, elegance is achieved through economy of thought. To wit, most of the theorems in the Elements were already well-known in the ancient world, but Euclid was the first to deduce all of them from 10 axioms. By compressing 13 books and hundreds of deep and powerful theorems into a mere 10 axioms, Euclid has turned himself immortal – the patron saint of modern mathematics. It is for this amazing compression, this hitherto unparalleled economy of thought that we say Euclid Alone Has Looked on Beauty Bare (in the words of Edna St. Vincent Millay).

The Twain Shall Meet
Mathematical beauty is truly awesome—in the original sense of awe-inspiring (though the other sense applies as well). Feelings of awe stem, I think, from size comparison. Seeing something immense is awe-inspiring because we compare it to ourselves. Seeing something  immensely complex is awe-inspiring because we compare it to our puny powers of understanding. The recurring motif is our limited ability to grasp, our inability to behold the entire thing with our own eyes, or our inability to hold it all in our mind, to comprehend. Thus, the stupendously intricate patterns of mathematics are awe-inspiring.

While gargantuan and gargantuanly complicated things are almost always awe-inspiring, it is precisely when we are able to comprehend them that they become truly beautiful. Thus, it is not ridiculously complicated patterns per se which we’re after; what we truly crave are ridiculously complicated patterns arising from laughably simple rules. The beautiful patterns are easy enough to find—mathematics is rife with them. These pique the curiosity, but are not capable of sustaining it. The driving force behind mathematical research are not the patterns, but the promise that behind them we can find a fantastically elegant theory—the simple rules that give rise to these patterns (this is closely related to the idea of mathematical insight).

Far from being two contrasting conceptions of mathematical beauty, intricate patterns and elegance are important parts of a bigger whole. Uniting them together, Lady Mathematics adorns herself in magnificence and splendour.

Granada Alhambra. [Image credit: Wikipedia.]

[Imported from my now defunct blog 10,000 Hours of Mathematics.]

Why Math? Insight

“Do not all charms fly / At the mere touch of cold philosophy?” Keats famously lamented. Getting his cue from Keats, Poe also keened

Science! true daughter of Old Time thou art!
Who alterest all things with thy peering eyes.
Why preyest thou thus upon the poet’s heart,
Vulture, whose wings are dull realities?

One of the most memorable expressions that have originated from Keats’ poem above is the accusation of Unweaving the Rainbow, which is also the title of a famous book by Dawkins. In the first part of the following short video, Feynman addresses this accusation beautifully.

In most scientists’ frame of mind, understanding only adds to the wonder. On reflection it becomes evident that the same frame of mind is also shared by a huge population of non-scientists. In researching NY-Times most e-mailed articles of the year, challenging cerebral science articles have surpassed expectations. The researchers concluded that readers crave and share articles that inspire awe. Far from robbing the poet of inspiration, as Poe accused, it seems Science indeed only adds to it.

Closer to home, a similar argument can be made by noting the popularity of The Masked Magician and other magic exposés. If these are any indication, most people crave more than what the simple magic show offers. By saying that they want in on the trick, these people are trading the wonder of mystery for the wonder of understanding.

Like with flow, when it comes to providing insights Mathematics is up there at the top of the ladder. Indeed, it can safely be argued that insights are math’s raisons d’être. One of the reasons the Haken and Apple proof of the Four Colour Theorem has been so controversial is precisely because, as Paul Halmos puts it, “we don’t learn anything from the proof.” Halmos was convinced that someday in the future a more elegant solution will be published that will truly explain the theorem.  An interesting historical survey of the 4CT and its philosophical implications briefly quotes Halmos and continues:

The proof gives no indications as to why it is that we only need four colours to colour maps; why not thirteen? What is so special about the number four? As good as an “oracle” gets, it still can’t provide an explanation for that one! Stewart agrees with Halmos and Hersch joins them in their disappointment about there being no “enlightenment” brought out through the proof. Some go as far to claim that mathematics is not about finding answers, but about the methods used to obtain them.

Mathematicians tinker with concepts and every so often realize that many of the problems they’re dealing with share some common features. This is a clear indication that there is an underlying common structure, which they then proceed to abstract, name and define. This process may take a very long time to accomplish. The concept of open sets, so fundamental to topology is such an example. In fact, it took well over a century to put calculus on firm ground (which we now call analysis). Reading some of the history of mathematics, I am often surprised at how recent are the concepts that we happily teach to high-schoolers and undergrads. These concepts hide an astounding level of sophistication and a mountain of historical development.

Once commonalities are delineated and clearly defined, we then regard them as “chunks” and use them as building blocks to go even higher, even deeper. This is the reason that the most advanced research-level mathematics of previous decades is now regularly taught to undergraduates. This is exactly how math progresses.

Mathematics is the language with which we capture the most basic and simple patterns and concepts. We then idealize and expand them. Thus, mathematical insights are insights into the most fundamental patterns we’ve been able to uncover so far. Any progress in mathematical understanding is also a progress in our ability to capture even more fundamental patterns. This explains Wigner’s “unreasonable effectiveness of mathematics.” It also explains why mathematical insights are felt to be so much more penetrating, so much more insightful in comparison to other insights (excepting maybe psychological-philosophical insights, which are of a different kind).

Insight is so fundamental to mathematics that it is inextricably tied to its self-conception. It is a measuring rod against which mathematical achievements are judged. This can be seen in mathematicians’ aesthetic appreciation, an expression of their mathematical value judgments. A theorem is beautiful when it provides insight. A theory is deep when it highlights unexpected connections (think of Galois theory, for example). Euphoria in mathematics is understanding; moments of realization and insight are always awash with joy.

Math enlightens…

[Imported from my now defunct blog 10,000 Hours of Mathematics.]

Why Math? Friends

It is a truism oft bemoaned, but everyone knows that fictional plots portrayed in the popular media tend to perpetuate stereotypes. Mathematicians are cast in the mould of the lone genius, the insane, or both. These romantic images are part of the wider phenomena of the cult of genius and have unfortunately probably been exacerbated by works such as the famous E.T. Bell’s Men of Mathematics.

However, math, like all branches of knowledge, is an intensely collaborative activity. This is true in Newton’s sense of on the shoulders of giant, but also in the literal sense. Today especially, a lot of mathematics is the result of joint research (sometimes, massively so). But even when a single name appears on the published paper, the updated cliché of “it takes a village to raise a mathematician” still almost always apply. I cannot claim that there are no self-made mathematicians, Ramanujan was one. I also cannot argue that there are no mathematical loners; there are quite a few. However, I would certainly like to propose that “doing” mathematics as a group, as a social activity, is one of the pleasures and, for some, one the motivating reasons of continuing to do it at all.

Pray don’t be alarmed by the apparently sudden change of topics, but whenever the subject of Harry Potter is brought up among my literary friends, they seem to be completely baffled by its success. In the course of our discussions, I think we have reached a plausible hypothesis. It is not uncommon for a community of readers to form and start speculating, constructing elaborate theories, suggesting interpretations, etc. In this respect Harry Potter is not unique; it is a common phenomenon among many other series of books (multivolume fantasy epics are especially prone to this, apparently) as well as TV series. The only difference is that in the case of Harry Potter, possibly due to publishers sudden realization of the existence of a hitherto untapped segment of the market—“young adults”—the community has reached a critical mass (whereby we again reach the safe harbour of mathematics, having landed in the port of Graph Theory and its applications to Social Networks). Many young people have come to simultaneously realize the exquisite pleasures of speculating together, and having serious, sometimes even philosophical, discussions.

Why do such communities spring around books and other media and how do they grow? I suspect that such communities form both because they are inherently enjoyable and because they are meaning-constitutive. In the previous post I have described how Aristotle’s interpreted the meaning of life to be about flourishing and striving to improve. A later philosopher, Epicurus, had a different interpretation. According to modern mavens, Epicurus’ philosophy is largely misunderstood. It is true that he advocated pleasure (and more importantly to him, the absence of suffering) as the purpose of life, but he was not a hedonist. For example, he advocated restrain and moderation, since excessive behaviours would only lead to greater pains down the road. He was one of the first Utilitarian thinkers (though he was trying to maximize a different function than later Utilitarians), and like Mill after him he seemed to recognize the importance of both “higher” and “lower” pleasures.

Epicurus’ school was not school-like at all; it was called “The Garden,” and was simply a community of friends living together. In Epicurus’ philosophy, friends are an essential and indispensable requirement for happiness. Hume famously said “truth springs from argument among friends,” and Epicurus would have wholeheartedly concurred. Passionate intellectual discussions (what Hume calls here “arguments”) among friends are not only one of the “high” pleasures, but may also be inherently meaning-constitutive in myriad ways.

The first and most straightforward way is the affirmation of mutual worth. If I deem something to be important, and my friends do too, this affirmation gives merit and justification for our shared interests. However, the same function could be served when the subject-matter is perceived to have a social value attached to it, some prestige. In both of these cases, mathematics is viewed through the lens of anthropology or sociology as a social construct (see for example, Reuben Hersh’s What Is Mathematics, Really?).

There is another way in which friendly arguments matter a great deal. Aristotle’s philosophical grandfather was Socrates, the namesake of the Socratic Method. We shall probably never know why Socrates developed his famous method, but I suspect that he must have really enjoyed arguing with his friends; in a good way. Socrates is famous for asking his interlocutors to define some common concept—usually a virtue like justice, bravery, wisdom, etc.—and then proceeding to tear their definitions one after the other until they gave up and asked him for his definition. Throughout this whole process Socrates maintains that he does not have the faintest idea of what the proper definition could be, and if he is reputed to be wise it is only because he knows that he doesn’t know anything. Many subsequent writers have been fairly skeptical of Socrates’ protestations of ignorance; and reading the dialogues one does get an uncanny feeling that Socrates has a hidden agenda, that he knows exactly what he’s driving at (though this might well be due to the fact that Plato wrote these dialogues after the fact and, significantly, after Socrates’ demise). However, I do think Socrates is in earnest.

Any mathematician reading the philosophical dialogues of Socrates cannot but notice their distinctive conjecture\counter-example structure. During the protean processes of problem-solving, mathematicians would conduct a great many number of “experiments”: seeking examples, conjecturing, finding counter-examples, non-examples, testing the boundaries and conditions of theorems and lemmas, etcetera. Throughout such broad experimentation, one starts to get a feeling for the problem, its gestalt, to really “grok” the problem. I’ve found that the conjecture\counter-example (point\counter-point, argument\counter-argument) works particularly well with two minds (or more) – then things really get moving; conjectures, examples, and counter-examples are bandied about like Ping-Pong balls, and new insights are gained. Thus, I suspect that Socrates may well have needed his interlocutors to make definitions that he can shoot down, and through a recursive process of refinement form his own ideas and intuitions about the subjects at hand. This sort of friendly argument is an example of the way in which friendship can act as a conduit for the betterment of the friends.

To conclude, the followings apply to Mathematics viewed as a cognitively demanding collaborative activity. The gifting of one’s full and sustained attention and intellectual efforts is indeed a profound gift. Harking back to the previous post, there is research into the hypothesized phenomena of group flow; which may well be the reason friendly arguments are a “higher pleasure.” Beyond that, the good will (so crucial to philosophical discussions of friendship) of the friends toward each other is manifested by their service as a sounding board for ideas – giving each other fertile ground for the cultivation of insight. Furthermore, engagement in a shared interest for the purpose of achieving a common goal is a strong form of affirmation. Finally, the tribulations involved in a meaningful but difficult cerebral pursuit impart an “historical property” that deepens and enriches the friendship.

Math, friends, and coffee – the good life…

[Imported from my now defunct blog 10,000 Hours of Mathematics.]

Why Math? Flow

One of the greatest mathematicians of the 20th century, Paul Erdȍs, had a favourite response to the age-old question of the meaning of life: “to prove and conjecture.” While this might seem quaint, or naïve, an answer befitting such a unique and colourful character, maybe he wasn’t so far off.

The quest for the meaning of life is closely related to the pursuit of happiness. However, happiness is much more versatile a concept than what our day-to-day use of the word would suggest. One of the many biases that are inherent in our way of seeing the world is called future discounting (though similar concepts go by many other names), which roughly means that we tend to ascribe more importance to our present feelings and state than to our future feelings and state. This is the reason behind procrastination. The ancient Greeks called this concept Akrasia, and Aristotle was one of the firsts to conduct a thorough philosophical analysis of this concept. The human condition is such that even though we all know it would be better for us to exercise regularly, eat healthfully, and perform our tasks promptly, very few of us do any of those things consistently (not to mention all of them). This schism between what we know would be “better for us,” and what we actually do is the reason for separating happiness from pleasure.

Aristotle advocated the eudaimonic life, which roughly translates as “human flourishing.” The purpose of life, the meaning of happiness, is the creation of a good project. A life well lived and full of meaning is a life that has successfully created a good project – one would like to look back at the end of one’s life and judge that it has been a good project, a good life.

Modern thinkers have identified this craving for meaning as a fundamental motivation of human actions. People are willing to disregard pecuniary incentives and choose a lower-paying more meaningful vocation over a higher-paying one that is less meaningful to them. It is also the reason for the taking-up of hobbies, and crucially for our discussion, for the strive to master one’s hobby – to develop expertise and become better.

Many positive psychologists maintain that the drive for mastery and self-actualization is the key to understanding human optimal experience. This optimal experience has been dubbed flow by Mihaly Csikszentmihalyi, a modern pioneer in this subject. In simplistic terms, flow is a state of complete immersion in a task or activity, where the entire world disappears, when even you disappear; there is only the activity – your entire being is completely focused to the exclusion of every other consideration. Many gamers have experienced this or a similar feeling. It is often reported in serious sports activity, and consistently so at the high-levels of strategy games like chess. Some educational research even suggests that it is manifested when you “lose yourself in a good book.”

Flow is an intoxicating and euphoric state. Many professionals and amateurs, in all fields of activity, constantly try to bring it about.  However, Flow is hard to achieve, something to strive for. Only by continually trying to push yourself, go beyond your current limitations and attain new heights, could you hope to experience such a state. Csikszentmihalyi has a famous diagram that illustrates his theory of flow as being a perfect balance between the skill-level of the performer and the challenge-level of the task at hand. Therefore, this intensely pleasurable state could theoretically be experienced by engaging in virtually any challenging activity over extended periods of time, while trying to adaptively develop the appropriate skills to meet the challenge.

Flow is a perfect balance between skill-level and challenge-level. [Image credit: Wikipedia.]

I do not know any other activity that is as flow-conducive as mathematics. Ask practically any mathematician what first drew them into math, and they would probably describe a flow experience – usually in the form of a difficult problem successfully solved (this is often the turning point). Mathematics is an Ali-baba cave of puzzles, teasers, riddles, conundrums, mysteries, paradoxes, and enigmas – there is something for everyone, at all levels. This richness means that it is always possible to find a challenge that would perfectly balance your skill-level, inducing a state of flow.

While flow is supremely enjoyable and may be inherently valuable, it is also extremely significant for its role in improvement. The desire to attain flow is a contributor to the intrinsic motivation to improve. Furthermore, flow is also a sign both that improvement has occurred, and that further improvement is about to take place. Thus, flow is intimately related, on many levels, to our self-actualization mastery-driven projects. Csikszentmihalyi has famously written that “[o]nly through freely chosen discipline can life be enjoyed and still kept within the bounds of reason.”

This freely chosen discipline was also Aristotle’s answer to the Akrasia problem. Only through consistent work (slow and steady) can we overcome our natural future-discounting bias, and work toward building a meaningful life. A meaningful life, therefore, is a life of striving toward a worthwhile goal, thereby constantly improving ourselves.

In many mastery-driven activities, the attainable skill-level is socially bounded. In competitive sports, for example, it is very clear what an ultimate level of achievement is. In the rare event that such a level is attained, it is unclear where to go from there – it would seem that the project is finished, the objective achieved. This may be the reason for the famous post-competition depression among elite athletes. Mathematics is different in this respect. Solving an interesting and difficult problem simply leads to many other interesting and difficult problems. One can always strive higher, always dive deeper, always develop one’s skills further – math is unbounded, an infinite frontier.

In conclusion, the optimal-experience of flow is one of the strong motivations for taking-up mathematics. The drive for mastery and self-actualization then pushes forward and propels one to scale new heights. Eventually comes the realization that mathematics is an unbounded terra incognita, full of treasures. Thereafter, this land of mysteries forever beckons, promising the opportunity to forget oneself in the flow of a new adventure. By that time, one has come to include math as part of one’s life-project and engaging in it enriches and provides more meaning to one’s life. Some are so stricken with it that they even come to make it the central project of their lives, thereby concluding that “the meaning of life is to prove and conjecture.”

Math beckons…

[Imported from my now defunct blog 10,000 Hours of Mathematics.]

Why math?

Tell most people you’d like to study math, and more likely than not their eyes will glaze over. Mathematics, for most people, is the epitome of dullness. Belittling of one’s math abilities is a point of pride: “I was never any good at math…” though the same people would be ashamed to say “I was never any good at reading.” This sad state of affairs has been decried for many years. Writers much more talented than me have investigated the possible causes and tried to set the record straight; showing how creative and exciting math truly is. C.P Snow in his Two Cultures and Lockhart in his A Mathematician’s Lament, have covered this ground more eloquently than I ever could.

The very fact that you are reading this blog suggests that you do not need convincing, so I will proffer no argument. Instead, I would like to explore the reasons and motivations that would drive a person to dedicate a large portion of their time (or life) to mathematics. The intent is to explain and show, to provide a glimpse into the mind.

The world is thy oyster and before you are thousands paths, myriad possibilities, a cacophony of pleasant and important activities – so why study math?

[Imported from my now defunct blog 10,000 Hours of Mathematics.]