Paul R Halmos - Collected Quotes

"Many people think of mathematics itself as a static art - a body of eternal truth that was discovered by a few ancient, shadowy figures, and upon which engineers and scientists can draw as needed." (Paul R Halmos, "Innovation in Mathematics", Scientific American Vol. 199 (3) , 1958)

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. [...] The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. 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 [...] the language and notation are those of ordinary informal mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study." (Paul R Halmos, "Naive Set Theory", 1960)

"It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts." (Paul R Halmos, "Selecta: Expository writing", 1983)

"The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases." (Paul R Halmos, "Selecta: Expository writing", 1983)

"A teacher who is not always thinking about solving problems - ones he does not know the answer to - is psychologically simply not prepared to teach problem solving to his students." (Paul R Halmos, "I Want to Be A Mathematician", 1985)

"Mathematics is not a deductive science - that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork." (Paul R Halmos, "I Want to Be A Mathematician", 1985)

"The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me - both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension." (Paul R Halmos, "I Want to Be a Mathematician", 1985)

"To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess." (Paul R Halmos, "I Want to be a Mathematician", 1985)

"A good stock 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." (Paul R Halmos)

"Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs." (Paul R Halmos)

"Pure mathematics can be practically useful and applied mathematics can be artistically elegant." (Paul R Halmos)