Thursday, October 29, 2009

Proud to be a Monomath

A girl told her father that history was much easier when he was in school. "Why?" he asked. She responded "Because there was so much less of it."

An old joke, but one that ran through my mind as I read the article The Last Days of the Polymath in the Autumn issue of Intelligent Life. That's polymath, not in the Tim Gower's sense of a massive collaborative effort to solve math problems, but the more traditional sense of people who know a lot about a lot, people like Leonardo da Vinci and Ben Franklin. But the article reminisces about an earlier time, when one could learn "a lot" about an area, such as physics, without needing to know all that much, basically what's covered in a first-year college sequence today. As Bill pointed out, we don't even have many polymaths in sense of knows a lot about a lot of math either. 

Advances in knowledge have made it impossible to know a lot about a lot. Advances in communication and travel have made polymaths less important since we can now better pool our talents. I might only know a lot about a little, but there isn't much I can't find a lot about quickly.

Richard Posner's quote in the article caught my eye.
“Even in relatively soft fields, specialists tend to develop a specialized vocabulary which creates barriers to entry,” Posner says with his economic hat pulled down over his head. “Specialists want to fend off the generalists. They may also want to convince themselves that what they are doing is really very difficult and challenging. One of the ways they do that is to develop what they regard a rigorous methodology—often mathematical.

“The specialist will always be able to nail the generalists by pointing out that they don’t use the vocabulary quite right and they make mistakes that an insider would never make. It’s a defense mechanism. They don’t like people invading their turf, especially outsiders criticising insiders. So if I make mistakes about this economic situation, it doesn’t really bother me tremendously. It’s not my field. I can make mistakes. On the other hand for me to be criticizing someone whose whole career is committed to a particular outlook and method and so on, that is very painful.” [Spelling Americanized]
We monomaths develop specialized vocabularies and mathematical tools and models because it helps use deeply understand an area. But much of what Posner says rings true. For example I've seen these "defense mechanisms" kick in quite often from both computer scientists and economists towards those trying to cross into each other's fields.


  1. (A side note - why Americanize the spelling? What's next, sound-processing to Americanize the accent in UK songs and movies?)

    About the defense mechanism, I must say that that in general (there are exceptions) the "softer" the field is and less prone to objective judgment, the more defense mechanisms they have. You should see how it's like in philosophy. Theoretical CS still has relatively high standards of interoperability with other people's minds.

    (sorry for the righteous outburst in the beginning, I'm what might be called a preservationist)

  2. "Specialists want to fend off the generalists. They may also want to convince themselves that what they are doing is really very difficult and challenging. One of the ways they do that is to develop what they regard a rigorous methodology—often mathematical."

    Sorry Mr Posner, but that's wrong, at least in maths/CS. The reason for using specialist mathematical language to because it makes communication much more terse and efficient.

    In principle all mathematics could be done in ZFC set theory, and hence in human language. Doing so would lead to a huge blow-up in size, making mathematics infeasible. For similar reasons we program in high-level languages like Java, not in assembly code.

    It's psychologically interesting that Posner assumes bad intent in the use of mathematics.

  3. This post brought to mind something I just read yesterday, in a review of the Princeton Companion to Mathematics that appeared in Notices of the AMS. Gil Kalai, one of the 5(!) reviewers, wrote:

    "[Reading] a large encyclopedia-type book like The Princeton Companion to Mathematics (PCM) can be discouraging, recognizing how little your little corner of the woods in this huge forest is. Something to take comfort from is the fractal nature of science and of mathematics. A little discovery in a small corner, a concept or a theorem, can make a big difference for the large picture."

  4. I believe that whether one is a monomath or a polymath is an entirely personal decision, and the ability to make that decision on one's own terms is the privilege accorded to an academic.

    However, I find the stated reasons in this post to be misdirected.

    Whether or not one is a polymath is a matter of attitude. At a fairly mundane level, I know academics who are world class researchers, good squash players, good painters, good musicians, ... Nobody is claiming they should win the highest accolade in each area (although some, such as Herb Simon, even managed that - Nobel prize, Turing award and highest honors in psychology). The main point is one of having an inherent curiosity and sufficient agility outside one's comfort zone to be able to maintain diversity.

    To give a completely nonacademic example, T.S. Eliot was a world class poet and a highly reputed publisher/businessman (he genuinely enjoyed and excelled at the latter even while blazing trails in the former). Similarly, close to home, some people have managed good TCS work and ventures into Wall Street...

    So, I think it is dangerous to perpetrate the myth that the world is too vast to have a diverse personality. In fact, that is a sure shot way to keep many otherwise smart students out of CS altogether! But, as I said at the outset, what one does with one's own life is a personal choice that I do not presume to judge. All I am saying is that, given a suitable attitude and temperament, it is still possible and even worthwhile to be a polymath (albeit, more skilled at some aspects than others).


  5. A girl told her father that history was much easier when he was in school. "Why?" he asked. She responded "Because there was so much less of it."

    This anecdote is meant to be funny, but the girl actually makes a totally reasonable point when you consider population growth.

    To make some rough estimates, about 73 billion people lived their lives between 4000 BC (which we might consider the beginning of "history") and 1950. Another 10 billion or so have lived since 1950 (some of whom are still living).

    So if each person is equally historical, the girl might reasonably claim that there is on the order of 14% more history now than when her father was in school.


  6. One of my favorite quotes on this topic is from Bill Thurston's essay, On Proof and Progress in Mathematics:

    "Basic concepts used every day within one subfield are often foreign to another subfield. Mathematicians give up on trying to understand the basic concepts even from neighboring subfields, unless they were clued in as graduate students."

    Also, if you are living in a community of monomaths, that "personal" choice may no longer be available to you—or only at a substantial cost to your time and patience. For instance, a polymath among monomaths must learn all the new jargon, PLUS the translations between different sets of jargon. Monomath specialization also has the unfortunate side effect of obscuring the history of ideas. As an example, I just watched someone give an intro graduate programming languages lecture who was unable to explain why the pipe symbol in ML data types is called a sum type—beyond the handwaving explanation: "it's from logic".

  7. Thanks for linking to the article.

    My favourite is the last bit,
    "Polymaths possess something that monomaths do not. Time and again, innovations come from a fresh eye or from another discipline. [...] But breakthroughs—the sort of idea that opens up whole sets of new problems—often come from other fields. "

    Which lead me to remember I've internalized and strong believe this bit from the designer Bruce Mau's Incomplete Manifesto,
    "40. Avoid fields.
    Jump fences. Disciplinary boundaries and regulatory regimes are attempts to control the wilding of creative life. They are often understandable efforts to order what are manifold, complex, evolutionary processes. Our job is to jump the fences and cross the fields."