As odd as it might sound, I've never... - Deepstash

As odd as it might sound, I've never been particularly confident of my mathematical abilities. I don't mean the arithmetic part, the part that people usually associate with being a mathematician. ("Hey, let Dan calculate the tip! Ha ha!") It's true that I'm probably better than average at mental math, but that's not really what makes a mathematician a mathematician. My job is to come up with ideas. Sometimes we mathematicians call the things we think about and work with "objects," which doesn't mean triangles, spheres, or other shapes.

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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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