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!
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.
- another related thing is, mathematicians have an aesthetic where you want to remove any non-essential thing. the concrete examples that motivated a definition or theorem will necessarily not be fully general. the search for the "right generalization" leads to a final definition/theorem that looks cleaner/appeals more to the mathematician's aesthetic. but the concrete example is what gave them the intuitions in the first place, so taking that away from the reader results in either confusion or lots of duplicated work for the reader. Henkin's "I do not believe that my crimes against historical discovery are isolated. It is my impression that many mathematical papers are written in a fashion that tends to obscure the process of discovery. Tendencies to find and exhibit neat proofs often result in the suppression of first proofs, and the thirst for general results can squeeze out special cases which may have led the way to discovery."
- 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