The Two Cultures of Mathematics (2000) [pdf]

I wrote a blog post a few years ago[1] making a similar claim about computer science. I said there’s 3 main camps of programmers: - People who enjoy programming because it’s mathematically beautiful (eg Haskell programmers) - People who enjoy programming because they like reasoning about machines, and like mechanical sympathy (eg C programmers) - And people who like programming because it can solve real problems for their users. I got pushback here and elsewhere that lots of people fit…

One technique Prof Gower's mentions (the phenomenon of the concentration of measure) has many applications in machine learning/statistics (which may be of more interest to HN readers than some of the probabilistic combinatorics mentioned in the paper). A few (non-rigourous) examples in ML/statistics are: - The use of the LASSO estimator [0] (minimizer of |Y - β X|₂ + λ |β|₁) for variable selection is justified by Candes & Tao's 2005 results [1, 2] that show, under fairly general conditions…

This observed pattern can be described by personality typing. I like to think of it as 2 classifications of how our brain explores - depth-based vs breadth-based search. Problem solving vs random walk. Directed vs discovered. Convergent vs divergent, etc. Essentially, one type enjoys digger deeper (defined by getting closer towards a goal) into a topic, while another enjoys expanding outward (exploring a topic, no goal required). The majority of people enjoy doing one type of thinking and hate doing…

There are two "cultures" of mathematics in my particular area of interest, namely computational mathematics. I spent an intense year of study reading several hundred papers crossing both cultures. One culture (my "original home culture") is generally referred to as "computer algebra". Here algorithms are developed and favored because they "mostly work". The second culture generally falls under "proof assistance". Here various logic theories are developed and then implemented to support developing proofs as…