My earliest memory of anything like a mathematical idea comes from a childhood walk with my dad. We left the house and made our way toward downtown Metuchen, the tiny town in central Jersey where I grew up, to a little luncheonette called the Corner Confectionery. I can still picture it: the rack of newspapers, magazines, and comic books; the ice-cream treats in the back corner; the long counter with stools, where I used to sit and spin until I was told to stop. It was about a mile-long walk, reserved for special occasions.

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The Myth and Magic of Generating New Ideas

newyorker.com

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A Big Question

Where do ideas come from? That’s a big question. Here’s a smaller one: Where do mathematical ideas come from? I’ve wondered about this from the time I first contemplated being a mathematician until long after I officially became one.

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The recipes for new ideas

How do Eureka moments happen? They do feel contingent & magical, arising in consciousness on their own. But looking at the history of breakthrough ideas there seem to be a rough 2-phase process: 

  1. An initial period of concentration, study, tinkering aka  “worrying” about a problem or idea, followed by .....
  2. Unconscious processing. This may mean walking, running, meditating etc... 

It's that feeling of being untethered that gives the mind free rein to recombine the information from step 1 into something new. 

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The Myth and Magic of Generating New Ideas

newyorker.com

Where ideas come from

Ideas come to us when we have a specific problem but we do not focus on solving the problem directly.

It’s not at all obvious how to go about thinking up some new twist on these things. A new idea can feel like a remarkable discovery

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The Myth and Magic of Generating New Ideas

newyorker.com

Learning rate and limits to learning

Some people take longer to develop the relevant building-block insights to progress in deep subjects like math. The ability to acquire understanding may vary from person to person.

It is highly unlikely that less intelligent people have a limit to deep subjects. The obstacle to learning advanced mathematics is that it may take you longer than a very smart person.

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The Competence Trap (and Why it Keeps You From Trying New Things)

scotthyoung.com