I used Arnold’s book for my first ODE class, and it was a blast. I later on learned more about this mathematician and find his philosophy of doing math is really inspiring. Here is a biography of this mathematician I found online. Written in Chinese.
P.S. I’m engrossed by Arnold’s way of doing math (geometry, spatial thinking, and intuition), and agree with the point made by the blog post that there’s also a very algebraic mindset of doing math. When I did my REU with a bunch of talented people, we discussed this question ‘what is math’, and I realize our opinions of this question shapes our math focus as well as way of doing math. So my readers, what do you think? I’m curious to hear your answers.
A supplementary read written by Arnold himself on teaching mathematics, provided by my friend.
After this reading, I realized how WRONG the first article I shared with you. Arnold doesn’t take any stance in the division. Rather, he thinks of math as a part of physics–the cheapest experiment, a simplistic modeling of the science. You can get a glimpse of his ideas by looking at the quotes I selected here.
One quote from the article I really liked:
Jacobi noted, as mathematics’ most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum.
These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity between the east coast of America and the west coast of Africa in geology.
Another quote talking about modeling (or axiom+proof way of doing math):
Complex models are rarely useful (unless for those writing their dissertations).
An alarming quote, a thought that I’ve never encountered before:
I even got the impression that scholastic mathematicians (who have little knowledge of physics) believe in the principal difference of the axiomatic mathematics from modelling which is common in natural science and which always requires the subsequent control of deductions by an experiment.
Another quote from the article, quoting L.Pasteur:
By the way, I shall remind you of a warning of L. Pasteur: there never have been and never will be any “applied sciences”, there are only applications of sciences (quite useful ones!).