Paul Halmos Quotes

The best way to learn is to do; the worst way to teach is to talk.

In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some and here it is; read it, absorb it, and forget it.

Applied mathematics will always need pure mathematics just as anteaters will always need ants.

Many teachers are concerned about the amount of material they must cover in a course. One cynic suggested a formula: since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick.

The heart of mathematics is its problems.

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 generality is, in essence, the same as a small and concrete special case.

Mathematics - this may surprise or shock some - is never deductive in creation.

If the NSF had never existed, if the government had never funded American mathematics, we would have half as many mathematicians as we now have, and I don't see anything wrong with that.

Feller was an ebullient man, who would rather be wrong than undecided.

A smooth lecture ... may be pleasant; a good teacher challenges, asks, irritates and maintains high standards - all that is generally not pleasant.

To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing-one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept,and when I want to learn something new, I make it my first job to build one.

I remember one occasion when I tried to add a little seasoning to a review, but I wasn't allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory. The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes. My proposed first sentence was: "The author discusses valueless measures in pointless spaces."

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? Where does the proof use the hypothesis?

The only way to learn mathematics is to do mathematics.

A clever graduate student could teach Fourier something new, but
surely no one claims that he could teach Archimedes to reason
better.

The computer is important, but not to mathematics.