God Plays Dice

From Steele this morning, a metaphor for mathematics I hadn't heard before: mathematics is like an oil painting. Basically, people doing oil paintings start by making a very rough sketch of the painting and then progressively build up the details of the figures. (I've never painted in oil, so correct me if I'm wrong.)

Mathematics is similar. In research one only starts out with a vague idea of the result and then progressively refines it; in teaching one first gives a sketch of an argument and then comes back and fills in the details. Teaching was the context here; often in classes which depend on measure-theoretic probability, which this is, we first give a semi-formal proof of a result and only later come back and fill in the σ-fields, justify the magic words like "dominated convergence", and so on.

Compare perhaps Hackers and Painters by Paul Graham, which compares the two title groups.

J. Michael Steele, a professor of statistics here at Penn, has a page of humorous quotes, some of which refer to probability and statistics. He also has some "rants", which include the non-ranty Advice for Graduate Students in Statistics. I can't vouch for how good this advice is -- it's hard for me to judge anything that purports to advise someone going through the process that I am currently a part of -- but it at least sounds good, and I've found myself thinking of little pieces of this page from time to time over the last few months. (I took a class from Steele last semester, and probably discovered this back in September; looking for information that advises me on what to do as a graduate student is one of the less damaging ways in which I procrastinate. It helps that these pages rarely say "don't procrastinate!", although they do say things like "work consistently", which is similar. I characterize this as "less damaging" because I suspect that I may have internalized some of the advice I've seen, and will remember it when I am seriously immersing myself in research. That date is not all that far in the future.)

I'm currently reading The Cauchy-Schwarz Master Class (link goes to MAA review, which pretty well explains the book's raison d'etre and contents) by J. Michael Steele. (How did I find this book? I'm taking Steele's course "Probability Inequalities and Machine Learning"; he alludes to this book every so often.)
It is full of interesting inequalities, and I may have something to say about them later, but I just wanted to share the following, where Steele is talking about George Polya and his book How to Solve It:

Some of the richest of Polya's suggestions may be repackaged as the modestly paradoxical question: "What is the simplest problem you cannot solve?".... Perhaps no other discipline can contribute more to one's effectiveness as a solver of mathematical problems.

I think this is worth remembering.