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.]