If all you have is a hammer, everything looks like a nail.

The most important idea I've blogged about so far is Taking Ideas Seriously, which is itself a generalization of Zvi's More Dakka. This post is an elaboration of how to fully integrate a new idea.

I draw a dichotomy between Hammers and Nails:

A Hammer is someone who picks one strategy and uses it to solve as many problems as possible.

A Nail is someone who picks one problem and tries all the strategies until it gets solved.

Human beings are generally Nails, fixating on one specific problem at a time and throwing their entire toolkit at it. A Nail gets good at solving important problems slowly and laboriously but can fail to recognize the power and generality of his tools.

Sometimes it's better to be a Hammer. Great advice is always a hammer: an organizing principle that works across many domains. To get the most mileage out of a single hammer, don't stop at using it to tackle your current pet problem. Use it everywhere. Ideas don't get worn down from use.

Regardless of which you are at a given moment, be systematic because Choices are Bad.

Only a Few Tricks

I am reminded of a classic speech of the mathematician Gian-Carlo Rota. His fifth point is to be a Hammer (emphasis mine):

A long time ago an older and well known number theorist made some disparaging remarks about Paul Erdos' work. You admire contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos' work could be reduced to a few tricks which Erdos repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert's collected papers contains Hilbert's papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert's beautiful results have been completely forgotten. But on reading the proofs of Hilbert's striking and deep theorems in invariant theory, it was surprising to verify that Hilbert's proofs relied on the same few tricks. Even Hilbert had only a few tricks!

The greatest mathematicians of all time created vast swathes of their work by applying a single precious technique to every proble... ,,,,,,,,,