My dad, a mathematical physicist, a man with an... - Deepstash

My dad, a mathematical physicist, a man with an active mind, but one of few words, was a gentle guide, letting me think through things on my own. We took our time walking, and we also took our time thinking and talking about the basic properties of numbers. In my head, it was easy to cut the pie in half, and then in half again, and again: two, four, or eight pieces. But, somewhere near Main Street, I got stuck on how to reliably create three, five, or six pieces. I started thinking about making bigger numbers out of smaller numbers. 

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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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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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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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