Wednesday, January 15, 2020

Math is hard, let's go shopping

I find mathematics, with all it's weird terminology and abstraction and equations, hard to understand. That's kind of funny coming from someone like me who makes a living from a branch of mathematics. I find computers and programming to be rather easy to understand — probably because I've had a lot of practice. But computer science is just applied logic and programming is arguably just the study of the computable functions, so you'd think math would come naturally. It doesn't.

One problem I've found is that as much as mathematicians pride themselves on rigor, they tend to be a bit sloppy and leave out important details. Computer scientists don't leave out important details because then the programs won't run. It's true that too much detail can clutter things up, but leaving out the detail and relying on “context” just increases the intellectual burden on the reader.

I will give mathematician's credit for thinking about edge cases perhaps more than a computer scientist would. It can be easy to be a bit complacent with edge cases because the computer will likely do something even if you don't think too hard about what it ought to do. But a good computer scientist tries to reduce the number of edge cases or at least make them coherent with the non-edge cases.*

Mathematicians seem to take perverse pleasure in being obscure. Computer scientists strive to be as obvious as possible because like as not, they are the ones that have to revisit the code they wrote and don't want to have to remember what they were thinking at the time. It's just easier to spell things out explicitly and obviously so that you can get back up to speed quickly when you have to debug your own stupid code. Every time I pick up some literature on category theory, I get hit with a “Wall of Terminology” denser than the “Wall of Sound” on a Phil Spector recording. It's fundamentally simple stuff, but it is dressed up in pants so fancy one has a hard time extracting the plain meaning. What seems to be universal in category theory is my difficulty in getting past page 4.

I once read a mathematical paper that talked about an algorithm with three tuning parameters: α, β, and another α. No decent computer programmer would give the same name to two different variables. Which α was which was supposed to be “obvious” from the context. The brainpower needed to keep track of the different αs was absurd and a complete waste of effort when calling the variable something else, like γ would have done the trick.

And don't ask a mathematician to write computer code. That's the one time they'll leave out all the abstraction. Instead of a nice piece of abstract, functional code, you'll get a mess of imperative code that smashes and bashes its way to a solution with no explanation of how it got there. It's a lot easier to take some abstract, functional code and figure out a more optimal way, probably imperative way to do it than it is to take a more optimal imperative piece of code and figure out the abstract, functional meaning of it.

I've found it to be extremely helpful when a computer paper includes one or two concrete examples of what it is talking about. That way, if I try to go implement code that does what the paper suggests, there's some indication that I'm on the right track. I'm more confident that I understand the paper if I have working code that produces the exact same values the paper's authors got. It's harder to find concrete examples in a math paper, and it is easier to think you know what it says but be far off base if there aren't any examples.

Maybe I shouldn't blame mathematicians so much and look a little closer to home. Perhaps I should study harder instead of demanding to be spoon fed difficult concepts. But then I read Feynman, S&ICP, S&ICM, and Jaynes and discover that maybe I just need a simple explanation that makes sense to me.

Sturgeon's Revelation is “90% of everything is crap”. This is true of both mathematical papers and computer science papers.

*An old joke illustrates the importance of thinking of edge cases: A programmer implements a bar. The test engineer goes in and orders a beer, orders zero beers, orders 999999999 beers, orders -1 beers, orders a lizard, and declares the bar ready for release. The first customer comes in and asks to use the restroom. The bar catches fire and burns down.


John Cowan said...

Quine's article "Mathematosis" from his "intermittently philosophical" dictionary Quiddities is quite informative on this feature of mathematical style.

Otto said...

Also look into Vladmir Arnold's essay On Teaching Mathematics and David Hestenes' essay Mathematical Viruses.