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. 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."
- when i work on problems and i want to write my solution, i even notice a sense of embarrassment or tediousness when i think about writing about all the mistakes i made. my most recent example is https://machinelearning.subwiki.org/wiki/User:IssaRice/Moral_public_goods_example That page displays a very cleaned up version of what i actually played around with when trying to find the general rule. I first tried to reproduce the 90% number for the specific example given in the post. then i started to generalize the number of peasants, and finally i decided to generalize all the parameters. i didn't realize the x_n cancels after solving the quadratic so i had an extra parameter. i coded up the example in python, so that i could plug in a bunch of numbers and get some intuition for the space of possible tax rates. after canceling x_n, i had two parameters: alpha and the fraction N_p/N_n. i generated a heatmap in python to visualize what that looked like (and lots of fumbling around here as i looked up how to make a heatmap and so forth). it was only at the very end where i realized i could use just one parameter, the % of wealth owned by the nobles. anyway, all this to show that when i ended up writing up my solution, i didn't want to bother writing up any of the mistakes (because it would take so much time! because it's ugly! because it's so embarrassing to think that i need to graph shit on a computer to find a solution.)
- https://twitter.com/ProfJayDaigle/status/948957356984946690 http://jadagul.tumblr.com/post/159881292413/nostalgebraist-i-took-linear-algebra-as-an
- babble/prune: another phrasing is to say that by default, writers naturally apply a harsh pruning strategy, so that even though they've "babbled out" lots of wrong ideas or naturally-occurring ideas, the vast majority of these get pruned out before being committed in writing (or they get written down but edited out later). when this happens, the actual causal history of the writer's thoughts get distorted, and appears mysterious to the reader ("how could anyone have come up with this? he must be a genius").
- another phrasing: once you figure out the answer to something, your mind twists in a certain way, and then you just want to straightforwardly say the answer, because it seems so natural to you now. but this writing algorithm ("understand the answer and then just say it naturally") is very different from modeling your reader's naive state of mind, then incrementally updating its state so that the final state matches what's in your mind.
- similar idea: https://twitter.com/johncarlosbaez/status/1249055680033460225 'Most mathematicians are not writing for people. They're writing for God the Mathematician. And they're hoping God will give them a pat on the back and say "yes, that's exactly how I think about it".'
- something i've noticed as i've been writing https://taoanalysis.wordpress.com/ is that for the meatier/longer exercises, i'm often exhausted by the time i've written up the solution in a clean/rigorous way that i'm too tired actually explain what's really going on/why i'm doing some manipulation. This seems bad, but idk how to fix it other than going even more slowly that i already am.
- http://cognitivemedium.com/tat/index.html -- see discussion of hidden representations.
- see michael nielsen quote about "minimal canonical examples" http://cognitivemedium.com/tat/index.html (is this the same as the above bullet?)
Other horrible things about mathematics culture:
- People don't talk enough about their stream of consciousness when thinking about math.
- one reason i've updated towards making videos instead of writing text is that with text, i find it pretty annoying to read stream-of-consciousness style writing (esp. in math, since you can't even see at what speed someone is writing, or where their attention is pointed toward), but with videos you get a much better sense of how someone is thinking in the moment.
- 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.