Paul Halmos divided progress in math into three categories: concepts, explosions, and developments. This was in his 1990 article “Has progress in mathematics slowed down?”. (His conclusion was no.) This three-part classification not limited to math and could be useful in other areas.
Concepts are organizational ideas, frameworks, new vocabulary. Some of his examples were category theory and distributions (generalized functions).
Explosions solve old problems and generate a lot of attention among mathematicians and in the popular press. As Halmos puts is, “hot news not only for the Transactions, but also for the Times for a day, for Time for a week, and for student mathematics clubs for many months.” He cites the solution to the Four Color Theorem as an example. He no doubt would have cited Fermat’s Last Theorem had he written his article five years later.
Developments are “deep and in some cases even breathtaking developments (but not explosions) of the kind that might not make the Times, but could possibly get Fields medals for their discoverers.” One example he gives is the Atiyah-Singer index theorem.
The popular impression of math and science is that progress is all about explosions though it’s more about concepts and developments.
Related: Birds and Frogs by Freeman Dyson [pdf]
2 thoughts on “Concepts, explosions, and developments”
In Thomas Kuhn’s ‘The Structure of Scientific Revolutions’, concepts are termed paradigms (standard examples of paradigms are Newtonian mechanics or quantum mechanics), and developments are examples of what is termed ‘normal science’ – the incremental advance of knowledge by those working in the framework of a paradigm. I don’t recall if he discussed the type of advance which Halmos termed explosions, but surely they are also a type of normal science.
There’s an analogy between mathematical concepts and scientific paradigms. One way that they differ is that scientific paradigms are about what we believe to be true, but mathematical concepts are about organizing what is known to be true.
For example, a lot of classical mathematical results have been reformulated in terms of topology, category theory, etc. These aren’t new results but new ways of looking at old results. But there is some similarity to Kuhn’s paradigms.