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!

Examples

  • 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. It's easy to define concepts like "random variable", but it takes much more effort to show why we're defining it that way, to give examples of it, to explain why the sample space is allowed to disappear, etc. Maybe another way to say it is that the formal definition is the compressed/distilled/abstract version of the idea, but there was also a whole bunch of computation (by human brains) going on that led up to that compressed version.
  • https://twitter.com/ProfJayDaigle/status/948957356984946690 http://jadagul.tumblr.com/post/159881292413/nostalgebraist-i-took-linear-algebra-as-an

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.
  • https://news.ycombinator.com/item?id=18582013