Paul Halmos is quoted as stating “The heart of mathematics consists of concrete examples and concrete problems.”

Google shows me many people quoting him as stating this, but no authoritative reference or citation as to where he actually said it. Does anyone know the source, context, or a way to confirm this?


1 Answer 1


Here is the full quote.

The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.

It is an excerpt from his autobiography, I want to be a Mathematician, page 324. Halmos'autobiography is a very interesting read. Here is another excerpt, a few paragraphs below the previous one, which is also relevant.

For Dieudonne the important result is, I think, the powerful general theorem, from which it is easy to infer all the special cases you want; for me the greatest kind of step forward is the illuminating central example from which it is easy to get insight into all the surrounding sweeping generalities.

  • $\begingroup$ Interesting. The context you provide changes the meaning substantially. What are examples? The Cantor set comes to mind. $\endgroup$ Mar 19 at 2:19
  • $\begingroup$ Examples in fact deserve their own question. I've added math.stackexchange.com/questions/4661941/… . $\endgroup$ Mar 19 at 2:30

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