The semester’s over and I’ve been cleaning off my desk. I found an old issue of the Notices of the AMS (February 2009) with a bookmark in it. It was Freeman Dyson‘s Einstein Lecture entitled “Birds and Frogs.” Here are some good quotes from it.
He opens with:
Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time.
Throughout the article he give examples of birds (Descartes, Weyl, Manin, etc.) and frogs (Bacon, Besicovitch, Von Neumann, etc.).
Mathematics needs both birds and frogs. Mathematics is rich and beautiful because birds give it broad visions and frogs give it intricate details. Mathematics is both great art and important science, because it combines generality of concepts with depth of structures. It is stupid to claim that birds are better than frogs because they see farther, or that frogs are better than birds because they see deeper. The world of mathematics is both broad and deep, and we need birds and frogs working together to explore it.
Of David Hilbert and the 23 problems he presented at the 1900 ICM Dyson wrote:
Hilbert himself was a bird, flying high over the whole territory of mathematics, but he addressed his problems to the frogs who would solve them one at a time.
I recommend the article; it is very interesting. While reading it I thought that maybe “walls and floors” might be another good metaphor to use in place of “birds and frogs.” Both walls and floors are important in constructing a building.
One last quote I liked was by the physicist Leo Szilard: “Let your acts be directed towards a worthy goal, but do not ask if they can reach it: they are to be models and examples, not means to an end.” Dyson used it in reference to the Riemann hypothsis and other supremely challenging mathematical problems.
This reminds me of a quote that our college president likes to cite at commencement. It was how our college’s founder (and a founding father of the United States) Benjamin Rush wanted to be remembered: “he aimed well.”
Me? I’m definitely not a frog. I am a bird wanna-be. My wings aren’t very strong, but I try to fly around and survey the beautiful lands.
David Richeson: Division by Zero 
I’m a bird wanna-be too, which is why my favorite question on my final exam for my History of Math students this semester was: “Are quaternions numbers? Defend your answer.” It’s an example of a question for which both “yes” and “no” are correct answers.
And it is why my favorite proof that I’ve seen this year is Lindemann’s proof that π is transcendental. It shows some of the same kind of insight that Euler’s solution of the Basel problem shows–an intimacy with numbers.
Every field of research “needs both birds and frogs.” This post reminds me why I hate the poorly contexted Einstein quote, “imagination is more important than knowledge.”
You may be a bird wanna be in Mathematics but you are definitely a high flyer as an educator.
Would like to share with you.
Chen, Too few “birds”—Response to Dyson’s “Birds and Frogs”
Introduce you a great article for all Mathematician and Computer Theoretists.
Click to access R1.pdf
Interesting response. Thank you!