Mathematical musings: On mathematical research; institutional level

• The Cult of the Paper has many virtues but encourages narrowness of focus.

  Commentary. Though one can't seriously quantify such things, I would not disagree with the assertion that the typical research paper in mathematics is more of an intellectual achievement than the typical paper in any other academic subject. Mathematics requires more novelty beyond routine skill -- to pass the "could any expert have done it?" test -- and is refereed more seriously. But this focus on the research paper as the basic unit of research output -- to be provocative let me call this the Cult of the Paper -- has disadvantages, and in particular encourages a narrowness of focus.

• Theorem-proof mathematics is the only subject that rewards virtual research.

  Commentary. The thought arose from the following true story, an overheard conversation between Professors X (from mathematics) and Y (from electrical engineering) about possible candidates for a possible joint appointment. X suggested Z (on the math side) and described his work. Y wasn't so keen on its importance to EE, and suggested A, B, C (on the EE side) and described their work. In each case X responded ``but Z could have done it better". I forget the actual phrasings of the subsequent exchange, but the gist was that X was firmly convinced that ``could have done it better" trumps ``did do it" while Y was firmly convinced of the opposite.

• A certain competitiveness is intrinsic to research, but institutional structures that encourage competitiveness should be viewed with caution.

  Commentary. Scientific research is supposed to be a collaborative endeavor.

• That mathematical work should be judged by its marginal contribution -- a paper is valuable to the extent that the body of mathematics would be diminished if it were not written -- is surely self-evident. Yet few mathematicians acknowledge the logical consequence that proving an important theorem just before someone else does should be assigned little value.

  Commentary. cf. previous comment.

• An economist regarding mathematical research as a product and mathematicians as producers would note it has a distinctive feature: one's customers are basically the same as one's competitors.

  Commentary. Is there an economic theory for such products?

• Just because there are a thousand papers on a topic, that doesn't mean the topic is important or realistic.

  Commentary. Maybe it's just easy to write papers on that topic.