Dear Doron,
It's nice to hear from you. It's strange, I thought the essay defended the importance of computation against the conceptual point of view. A referee seemed upset that I had misrepresented the latter. And Michael Harris, who you seem to dislike, also dislikes me, it seems, because I like computers:
[See here]
It's all very complicated... It's hard to tell what side anyone is on.
In all seriousness, as you know, I am wholeheartedly in favor of the use of computers in mathematics. My goal was to describe a common view and explore it. For your information, I am appending the response I sent to Craig Larson last night.
Best wishes,
Jeremy
P.S. Thank you for linking to the original article. I don't mind criticism, as long as others can read what I have said and decide for themselves whether it is objectionable.
*****
Dear Craig,
I think we are mostly in agreement. I like to think of mathematics as serving a dual purpose, like architecture: we want it to be useful and we want it to be beautiful. The buildings we live in should keep us warm and keep us safe from the elements, but they should also give us comfortable, aesthetically pleasing places to live. The same holds of mathematics. There isn't a sharp division between the two constraints: one of the reasons that mathematics is so aesthetically pleasing is that its concepts give us the power to do new things, and the fact that mathematics is so beautiful has encouraged some of the greatest minds to pour enormous energy into its development, leading to useful results. I hope that nothing I wrote suggests otherwise.
The thing is, we don't have to try hard to make a case that computers in mathematics are useful. Numerical and symbolic mathematical methods are used everywhere, in engineering, finance, economic forecasting, policy, and more.
In the essay, I was more interested in exploring the aesthetic. One sometimes finds the attitude that *real* mathematics is the stuff that doesn't involve computation -- it's developing the big ideas that make computation possible, way down the road. Even there, I think we need to come to terms with the role that computers can play.
Best wishes,
Jeremy