From Hamming:

"I thought hard about where was my field going, where were the opportunities, and what were the important things to do. Let me go there so there is a chance I can do important things."

It seems he is talking about the important to the world aspect. He wants to have a big impact on the world, and be where the action is. The goal is to make a name for yourself, or to at least have a hand in the next big transformations.

But there is also the "important to you" aspect. In Hamming's case, those two notions of importance align. But not so for everyone.

Quoting again:

"I went home one Friday after finishing a problem, and curiously enough I wasn't happy; I was depressed. I could see life being a long sequence of one problem after another after another."

So, he is happiest when working on problems that have big "important" implications for the world. Good for him; I'm glad he discovered that about himself, and followed what made him happy.

So now for my actual point: I'd encourage a person to actually first and foremost focus on what is important to them personally — what makes them happy — rather than what seems "important" from some external perspective.

I think a lot of people will decide, like Hamming did, that they want to be where the action is, that they want to participate in transforming the world, that that is what makes them happy. But to put that choice on a pedestal as though it is the True Goal — to put "important to society" above "important to oneself" is putting the cart before the horse. It's how you get a bunch of unhappy people chasing after other people's dreams.

It's actually somewhat touched upon in TFA, with:

"The following recipe assumes you're very ambitious."

Indeed — like Hamming was. But not everyone is, and not everyone needs to be to be happy. I am just slightly irked by our somehow reserving the word "great" for ambitious people's accomplishments.