My wife said to me (I paraphrase):
The trouble with mathematicians is that you first take something that is obviously true and try to show it is not-so-obviously true by defining a starting point and the rules of inference, so that the veracity of the original statement becomes doubtful, and then you turn around to do the work to show that it is (trivially) true after all.
This happened when I tried to explain the concept of parallel transport to my wife, who is not a mathematician, and who is nevertheless an extremely intelligent and well-educated scientist. It was then that I realized that it is really, really hard to educate, in the root sense of the word: to totally lead a student out and along your line of reasoning until the “Ah-ha!” moment.
Like so many other things in life, mathematical discoveries are built upon a mix of perspiration and inspiration. It is often the brilliant insight after a long period of fruitless toil that heralds the success of a mathematician. Yet many also argue that the time spent mulling over existing ideas and methods is akin to the tilling of a field; only after it can the seed of invention be planted. So often it is through the skeptical re-examination of previous ideas that a new discovery is born; progress arises from asking ourselves why a logical statement is true and how we can make it false. However, I believe that
A good mathematician thinks uncommonly about common things, but a great mathematician explains uncommon things commonly.
A mathematician should not be simple a “logical inference machine.” Our job does not end at the discovery. To elucidate complex arguments and to impart knowledge and techniques are also very large parts of our obligation. It is with this in mind that I write in this blog.
Here, I practice how to explain.
About the author
My name is Willie Wong, and I am a mathematician. I received both my BA (2005) and PhD (2009) in mathematics from Princeton University, under the tutelage of Professor Sergiu Klainerman. My specialty is evolutionary partial differential equations and geometric analysis, with a focus on wave and dispersive equations arising from physics, but I am also keen on harmonic analysis. Most recently my attention has been on mathematical problems arising from general relativity, such as issues relating to the cosmic censorship conjectures, and well-posedness of other Lagrangian field theories.
About the banner
In case you are wondering: the doodle on the upper right is a three-dimensional visualization of a quaternionic fractal. It was produced using Quat and The GIMP.
(Not) about the title
I finally came up with a decent title for this blog, months after its inception. No, I’m not going to explain what it means (first one to guess it gets a cookie!), but it has something to do with nonlinear wave equations, if you want a hint.
