*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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MORE IDEAS FROM THEARTICLE

*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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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:

- An initial period of
**concentration**, study, tinkering aka “worrying” about a problem or idea, followed by ..... **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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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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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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