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 Halmos More Quotes by Paul Halmos More Quotes From Paul Halmos The only way to learn mathematics is to do mathematics. Paul Halmos learning math way The best way to learn is to do; the worst way to teach is to talk. Paul Halmos mathematics worst way The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method. Paul Halmos education math knowledge Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? ... Where does the proof use the hypothesis? Paul Halmos fighting example doe ...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. Paul Halmos essence special science A smooth lecture... may be pleasant; a good teacher challenges, asks, irritates and maintains high standards - all that is generally not pleasant. Paul Halmos teaching challenges teacher 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. Paul Halmos education math jobs The beginner should not be discouraged if he finds he does not have the prerequisites for reading the prerequisites. Paul Halmos beginners reading doe 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 Halmos special heart profound Applied mathematics will always need pure mathematics just as anteaters will always need ants. Paul Halmos mathematics ants needs [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. Paul Halmos analysis science knowledge 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 Halmos teaching teacher knowledge The heart of mathematics is its problems. Paul Halmos mathematics problem heart Feller was an ebullient man, who would rather be wrong than undecided. Paul Halmos undecided men What's the best part of being a mathematician? I'm not a religious man, but it's almost like being in touch with God when you're thinking about mathematics. God is keeping secrets from us, and it's fun to try to learn some of the secrets. Paul Halmos religious god fun 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." Paul Halmos math class numbers 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 Halmos luck taste math You are allowed to lie a little, but you must never mislead. Paul Halmos mislead littles lying When a student comes and asks, "Should I become a mathematician?" the answer should be no. If you have to ask, you shouldn't even ask. Paul Halmos students answers math 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. Paul Halmos education math teacher
