Manuel Blum has some helpful advice for graduate students starting a PhD. This is a sequel to the previous entry.
Blum refers to the joke that ‘a PhD knows more and more about less and less
until he knows everything about nothing.’ He explains that although, it may seem that one is working on a minute area, if one masters that minute area, it helps one see the broader universe in a new light.
There are different kinds of research in mathematics. One could try to prove something true or expected to be true. One could also search for what is true or disprove what you expect to be true. Blum’s suggestion is that while trying to prove something true, it can be helpful to try disproving it. This can give you a useful insight. The conclusion is that one should keep one’s mind open and be prepared for the unexpected.
For thesis writing, Blum writes that it is better to get the thesis checked and proofread by one’s peers. Not every supervisor is good at or keen on proofreading. The advice article also contains the following quotation:
“First have something to say,
Second say it,
Third stop when you have said it,
Finally give it an accurate title.”
JOHN SHAW BILLINGS [1838-1913]
Blum’s last words are the most inspiring:
One does not have to be brilliant, a genius, to be special.
To do something better than anyone/everyone else. To be
One has only to choose an END
And then DO IT.