User:IssaRice/Mental representations in mathematics
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.
- https://twitter.com/ProfJayDaigle/status/948957356984946690 http://jadagul.tumblr.com/post/159881292413/nostalgebraist-i-took-linear-algebra-as-an
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.