User:IssaRice/Mental representations in mathematics

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I think people don't talk enough about mental representations of objects in math, and I think it's horrible!


  • rows of a matrix linearly independent vs rows of a matrix spans domain
  • injective vs surjective and left vs right inverses
  • mental arithmetic -- actually, this is one place where i think there has been a lot of discussion... (example)
  • inf/sup stuff in analysis. I find this so much easier to think about when I draw a line segment and label points on the segment. But many books don't talk about this!
  • sup/inf/liminf/limsup: Tao's piston analogy!
  • computability: many things can be thought of "abstractly" as enumerations and indices, or more "concretely" as programs. Compare Boolos/Burgess/Jeffrey vs Sipser. But there isn't a book that bridges these mental representations!

Why does this happen?

Why don't people talk about this? Some things that could be happening:

  • It happens in-person rather than in writing, so I don't have access to it.
  • Mental representations are too personal. I don't buy this.
  • People have low metacognition.
  • People are elitist or otherwise don't want to share "cheat codes" to make the subject easier for others.
  • I'm wrong about the value of mental representations being important to share. One argument could be like "to really learn math you have to struggle to make your own mental representations".
  • writing math without conveying mental representations is low resistance/the natural thing that happens. Sort of like how writing code without comments/spaghetti code happens by default, unless people are taught how to program.

Related stuff

Other horrible things about mathematics culture:

  • People don't talk enough about their stream of consciousness when thinking about math.
  • Not enough "natural proofs"
  • people don't talk about how they actually find ideas
  • not enough tabular organization
  • reading math is like reading code without a debugger; you can't inspect objects to see "what they look like", what the valid operations are, whether you get an error message when you try to do something invalid, etc.