I’m pretty confused that this is as necessary as it is, particularly with writing that involves a lot of math and math notation. I don’t understand how people get the insight and motivation necessary to write that kind of thing without explaining what the point of it is or giving examples of how to apply it as part of their expositions.
Your (johnswentworth’s) posts don’t seem to suffer nearly as much from this, at least from a quick skim of the ones you said on another thread you’d like distilled, but e.g. Infra-Bayesianism seems maybe important (it’s about something I was thinking about anyway), but I’ve “bounced off” the sequence, as apparently have enough others for there to be a job offering distilling it. The same thing happens sometimes when I’m reading academic papers.
I have nothing against rigor and I actually enjoy math, but when writing about models or systems one wants to apply to something that isn’t purely mathematical, these are the basic parts I generally see and how I react to them:
Motivation. This is great! Tell me what problem you’re trying to solve and sketch your solution. People tend to do this only at the beginning of a piece, if they do it at all, which in my opinion is a mistake. I want to know why you care about proving that theorem, and if you don’t say I can’t always guess.
Examples. These are great! If you introduce a construct, show me one. Make it just non-trivial enough that there’s a point to using the abstraction at all. Ideally make its components things I can already recognize and understand without the abstraction. Again, there often are some but not enough; any significant point made that isn’t illustrated with an example forces me to come up with one on my own or else I just won’t understand it.
Text description of math. This varies a lot. If it’s showing the correspondence between the real problem being solved and the abstractions and notation used, that’s fine. Other statements relating mathematical objects to one another are okay if I already have a good grasp of what those objects are (less so if some or all of them were only introduced in the current piece and are non-obvious in terms of the underlying constructs). Proofs written in text tend to frustrate me. They require extreme amounts of thought per sentence because they’re written very elliptically (stating “X has property Z” while assuming I remember “X has property Y” and “property Y implies property Z”, often when neither of these latter things is stated in the piece at all), and when I do figure out what they’re saying, 90% of it is extremely obvious and uninteresting, but the 10% that makes the proof work isn’t highlighted in any way. I largely end up skipping them, and since it seems like they’re probably one of the hardest parts to write, that feels like a big waste.
Symbolic math. There’s no other way to be so precise so concisely, but I find myself having to refer back to definitions constantly to figure out what they’re saying, and long blocks of nothing but symbols are even more impenetrable than prose proofs; I end up skipping such blocks if there’s any way I can figure out what’s being said without decoding them. In programming we all know single letter variable and function names make code hard to understand (with a few common exceptions like naming a loop index “i”); I think symbolic math would be much easier to read if it looked more like a computer program.
Now, I do also see writing that’s almost all motivation and examples, and that doesn’t seem right either. It doesn’t create sufficient technical understanding that I could implement the ideas in a computer program, or adapt them to very different cases than the ones presented. But to summarize, some skills you say are important in distilling are:
Ability to independently generate intuitive examples when reading mathematical arguments, or having a mathematical discussion
Ability to extract the core intuitive story from a mathematical argument
And, well, I honestly don’t understand how anyone can even think about math, and particularly not how they can come up with useful math, without having examples and an intuitive story in mind already.
I’m pretty confused that this is as necessary as it is, particularly with writing that involves a lot of math and math notation. I don’t understand how people get the insight and motivation necessary to write that kind of thing without explaining what the point of it is or giving examples of how to apply it as part of their expositions.
Your (johnswentworth’s) posts don’t seem to suffer nearly as much from this, at least from a quick skim of the ones you said on another thread you’d like distilled, but e.g. Infra-Bayesianism seems maybe important (it’s about something I was thinking about anyway), but I’ve “bounced off” the sequence, as apparently have enough others for there to be a job offering distilling it. The same thing happens sometimes when I’m reading academic papers.
I have nothing against rigor and I actually enjoy math, but when writing about models or systems one wants to apply to something that isn’t purely mathematical, these are the basic parts I generally see and how I react to them:
Motivation. This is great! Tell me what problem you’re trying to solve and sketch your solution. People tend to do this only at the beginning of a piece, if they do it at all, which in my opinion is a mistake. I want to know why you care about proving that theorem, and if you don’t say I can’t always guess.
Examples. These are great! If you introduce a construct, show me one. Make it just non-trivial enough that there’s a point to using the abstraction at all. Ideally make its components things I can already recognize and understand without the abstraction. Again, there often are some but not enough; any significant point made that isn’t illustrated with an example forces me to come up with one on my own or else I just won’t understand it.
Text description of math. This varies a lot. If it’s showing the correspondence between the real problem being solved and the abstractions and notation used, that’s fine. Other statements relating mathematical objects to one another are okay if I already have a good grasp of what those objects are (less so if some or all of them were only introduced in the current piece and are non-obvious in terms of the underlying constructs). Proofs written in text tend to frustrate me. They require extreme amounts of thought per sentence because they’re written very elliptically (stating “X has property Z” while assuming I remember “X has property Y” and “property Y implies property Z”, often when neither of these latter things is stated in the piece at all), and when I do figure out what they’re saying, 90% of it is extremely obvious and uninteresting, but the 10% that makes the proof work isn’t highlighted in any way. I largely end up skipping them, and since it seems like they’re probably one of the hardest parts to write, that feels like a big waste.
Symbolic math. There’s no other way to be so precise so concisely, but I find myself having to refer back to definitions constantly to figure out what they’re saying, and long blocks of nothing but symbols are even more impenetrable than prose proofs; I end up skipping such blocks if there’s any way I can figure out what’s being said without decoding them. In programming we all know single letter variable and function names make code hard to understand (with a few common exceptions like naming a loop index “i”); I think symbolic math would be much easier to read if it looked more like a computer program.
Now, I do also see writing that’s almost all motivation and examples, and that doesn’t seem right either. It doesn’t create sufficient technical understanding that I could implement the ideas in a computer program, or adapt them to very different cases than the ones presented. But to summarize, some skills you say are important in distilling are:
And, well, I honestly don’t understand how anyone can even think about math, and particularly not how they can come up with useful math, without having examples and an intuitive story in mind already.