Differences of Culture

I attended a few talks last week that highlighted the differences in culture that can exist between different academic disciplines.

In a talk on Monday, a group of students presented to a visiting professor. After listening to the students present, the professor always asked the students to summarize their work in the style they would use to explain their research to someone at a cocktail party. This is a good thing to point out to graduate students: we tend to get overly excited about our own research, and can miss the forest for the trees.

But that was the only question the professor would ask. Repeatedly. To all of the students. He didn't seem to pay much attention to the details of the presentations. He only wanted the big picture.

At a presentation in mathematics / statistics / computer science, there is a certain amount of work expected from the audience. The speaker will explain things, but the audience is expected to 'play along' and attempt to fill in some gaps as the presentation progresses. In my mind, there are two reasons for this. First, there isn't enough time to explain everything and cover enough material to get to the interesting parts. Second, the mental effort spent contemplating what the speaker is presenting keeps the audience engaged. It's the difference between reading a novel (entertaining, but requires little work) and a textbook.

I've noticed this when attending presentations outside of the mathematical sciences. The speaker focuses a lot on the big picture, but does not delve into the details of their methodology. They focus more on their results than how they got their results. Which leaves me uncomfortable: I am supposed to trust the speaker, but I have read too many crap scientific articles to immediately assume everyone has done their methodological due diligence.

I witnessed a different sort of cultural divide later in the week. I attended a seminar with an audience populated by mathematicians, physicists, and meteorologists. The presenter, a mathematician, freely used common mathematical notation like span (of a collection of vectors) and \(C^{2}\) (the space of all twice continuously differentiable functions). The physicists and meteorologists had to ask for clarifications on these topics, not because they did not know them, but because they did not speak of them in such concise and precise terms. In the latter case, for instance, they probably had concepts of a function needing to be 'smooth,' where smooth is defined in some intuitive sense.

This reminds me of an excerpt from a review of textbooks that Richard Feynman undertook later his career. He volunteered to help the public school system of California decide on mathematics textbooks for their curriculum:

It will perhaps surprise most people who have studied these textbooks to discover that the symbol >> or << representing union and intersection of sets and the special use of the brackets and so forth, all the elaborate notation of sets that is given in these books, almost never appear in any writings in theoretical physics, in engineering, in business arithmetic, computer design, or other places where mathematics is being used. I see no need or reason for this all to be explained or taught in school. It is not a useful way to express one's self. It is not a cogent and simple way. It is claimed to be precise, but precise for what purpose?

— from New Textbooks for the 'New' Mathematics by Richard Feynman, excerpted from Perfectly Reasonable Deviations from the Beaten Track

Feynman wrote these words in the middle of the New Math turn in the education of mathematics in public schools. New Math focused on teaching set theory, a building block of much of modern mathematics, starting in elementary school. This wave of mathematics education did not last long in the US. It was certainly gone by the time I traversed the public school system. I didn't learn, formally, about sets until my freshman year course in Discrete Mathematics.

I think most mathematicians would find Feynman's statement that sets are "not a useful way to express one's self" very strange. At least for myself (and I'm the farthest thing from a pure mathematician), I find set theoretic notation extremely useful. But this might be a cultural thing. From the talk attended last week, the span is itself a set (the set of all linear combinations of some collection of vectors), as is \(C^{2}\). And clearly the physicists and meteorologists have gotten along just fine without thinking of these things as sets, or using explicit set notation. Again, probably because they have some fuzzy, intuitive idea in their head for what these things mean.