Reminds me to get around to Piper Harron's thesis. Which was made to be seriously readable.
Math seems to have a culture of systematically bad writing (as Bill Thurston discussed).
I no longer understand my PhD dissertation
181–190 of 281 posts
Re: I no longer understand my PhD dissertation
#182I've been working with Haskell* for a couple of years and it is quite often that I work with code that I don't fully understand. I'll come across a terse bit of code, then carefully take it apart to see what it does (by taking bits and pieces out and giving them names instead of passing in using point-free notation and also adding type annotations). Once I see the whole picture, I make my own change and then carefully re-assemble the original terse bit of code. One could ask the question: wasn't the verbose version better? I'm going to lean on the side of no. If I left this verbose and other bits verbose then it would be hard to see the whole picture.
I think doing maths would be better if it was done interactively with software. If equations were code then you could blow it up and look into fine details and then shrink it to a terse form while software keeps track of the transformations to make sure what you write is equivalent. Maybe it's time to add a laptop to that paper pad?
* not arguing anything language specific here except that Haskell makes use of variety of notations that makes the code shorter and more like mahts. More so than most languages.
Re: I no longer understand my PhD dissertation
#183Earlier quoted context omitted.
I don't know... I'm taking a theory of computability class right now and I'm sure glad that the pumping lemma for regular languages is called "the pumping lemma for regular languages"
But isn't that an attempt at a "descriptive" name? The intuition for the person who discovered this lemma, clearly envisioned the underlying process as a device pumping out new strings belonging to the language.
I like to imagine they envisioned the strings themselves being "pumped" up by inserting new characters in the center.
Re: I no longer understand my PhD dissertation
#184Earlier quoted context omitted.
since when are standard tools of good education 'cute little tricks'? mathematics is hard. mathematicians are some of the smartest people in the world. yet the current tendency to not do these things or not do them enough is a consequence of a certain culture, efficiency constraints and even the near universal use of LaTeX.
I completely agree on the culture point: Programming and math _is_ essentially the same (Curry Howard isomorphism) and the problems are indeed equally complex. The difference is that programming is driven on economical terms, hence agility, flexibility, etc. has been developed. Mathematics is driven in the university sphere, where mostly intrinsic motivation drives. Not many mathematics professors have the urge to si…
I don't think that programming and math are the same in a practical sense. I am studying math and CS and they're fairly different. Programs deal with specific things, types and data that you manipulate and see with your eyes. Maths deal with abstract concepts for which finding examples can be pretty difficult.
Also, while programming, you can design your functions and their interfaces before writing them down. In maths, that's impossible: you're constrained by what can/cannot be done.
Scrum or coding practice doesn't apply here. Maths don't have to "keep up", they're already way ahead of their applications. It also doesn't need to be fast or flexible, just rigorous. And while motivation and intuition helps (specially when learning) some concepts cannot be motivated or given an intuition, and even in that case you still have to learn the small details and formalizations.
Re: I no longer understand my PhD dissertation
#185TLDR: The author independently re-discovered what you may know as Old Code Syndrome. I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. This guy's basically in the position of a fairly new developer who's just been asked to do non-trivial update of his own code for the first time. All those clever one-liners he put into his code made him feel sm…
It would be really nice if all it took to understand difficult mathematics were some easy programming tricks. The problem with looking at old code is you forget what is going on or what the purpose of different components are. The problem with looking at old mathematics is that it is genuinely very difficult to understand. You work very hard to be an expert in a field and get to a level where you can read a cutting-e…
It's also the jargon.
I remember years ago telling a coworker that I was an ACM member and subscribed to the SIGPLAN proceedings. He looked at me and with all sincerity asked, "You can understand those things??"
To which I responded, "About half," but I totally sympathized with his question. Both Math and CS need the reincarnation of Richard Feynman to come and shake things up a bit. There's too much of the 'dazzle them with bullshit' going on. It's no wonder that it takes so long for basic research to see application in real scenarios. You people bury your research under layers of obfuscation about half the time. Does it really help anybody to do that? Why do you do that?
"If you can't explain it simply, you don't understand it well enough." is my new favorite Albert Einstein quote.
Re: I no longer understand my PhD dissertation
#186TLDR: The author independently re-discovered what you may know as Old Code Syndrome. I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. This guy's basically in the position of a fairly new developer who's just been asked to do non-trivial update of his own code for the first time. All those clever one-liners he put into his code made him feel sm…
> I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. If it only happened on mathematical papers... It is all over the place, in books, classes, etc. Few professors tell you what are they doing and why; they just throw formulas, theorems and symbols at you. Intuition is not given its necessary importance, first because it's hard to grasp some con…
+1 on the final point. It may be difficult to appreciate how terse terse is here.
Re: I no longer understand my PhD dissertation
#187Let us give some credit to the professors who read the dissertation and understood what was going on.
Re: I no longer understand my PhD dissertation
#188Earlier quoted context omitted.
since when are standard tools of good education 'cute little tricks'? mathematics is hard. mathematicians are some of the smartest people in the world. yet the current tendency to not do these things or not do them enough is a consequence of a certain culture, efficiency constraints and even the near universal use of LaTeX.
I completely agree on the culture point: Programming and math _is_ essentially the same (Curry Howard isomorphism) and the problems are indeed equally complex. The difference is that programming is driven on economical terms, hence agility, flexibility, etc. has been developed. Mathematics is driven in the university sphere, where mostly intrinsic motivation drives. Not many mathematics professors have the urge to si…
[deleted]
Re: I no longer understand my PhD dissertation
#189Earlier quoted context omitted.
since when are standard tools of good education 'cute little tricks'? mathematics is hard. mathematicians are some of the smartest people in the world. yet the current tendency to not do these things or not do them enough is a consequence of a certain culture, efficiency constraints and even the near universal use of LaTeX.
I completely agree on the culture point: Programming and math _is_ essentially the same (Curry Howard isomorphism) and the problems are indeed equally complex. The difference is that programming is driven on economical terms, hence agility, flexibility, etc. has been developed. Mathematics is driven in the university sphere, where mostly intrinsic motivation drives. Not many mathematics professors have the urge to si…
The differences you note are what I consider cultural differences.
Math _indeed_ have the idea about interfaces and implementations. They are called something different (exitensisal types from a PL point of view).
I will recommend you to look into type theory, in particular the book "types and programming languages" provides a very good introduction.
In CS we have two areas that make it completely clear that math and CS are coupled, that is complexity theory and programming languages. After having taken a course in each, and pondering a bit, it should indeed be possible to see the that they are the same.
Re: I no longer understand my PhD dissertation
#190TLDR: The author independently re-discovered what you may know as Old Code Syndrome. I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. This guy's basically in the position of a fairly new developer who's just been asked to do non-trivial update of his own code for the first time. All those clever one-liners he put into his code made him feel sm…
It would be really nice if all it took to understand difficult mathematics were some easy programming tricks. The problem with looking at old code is you forget what is going on or what the purpose of different components are. The problem with looking at old mathematics is that it is genuinely very difficult to understand. You work very hard to be an expert in a field and get to a level where you can read a cutting-e…
> Programming has existed for, say, 50-100 years. We have recorded mathematical history going back thousands, with contributions from most of the most brilliant human beings to ever exist.
Mathematics with a solid logic foundation has also existed only for the past century, and writing programs is actually equivalent to doing a mathematical proof as some have already pointed out.
The actual problem is that when programming you are talking to computers, so you have to lay out each step, or the computer will not know what to do. Sure you can use libraries to do very complex things all at once, but then that's because those libraries have already laid out the steps.
But when doing mathematics you are talking to humans and you often skip steps in a very liberal way. There is indeed a library of theorems that is written down, but there is also an unwritten library of "it trivially follows", which, if a mathematician is asked to actually write it down, might feel humiliated.
When you think that there is something so trivial that computers must be able to do, you still have to find or invent a library for that. When you think there is something in mathematics so trivial that it must be true, you just need to convince your audience.
One day all mathematical papers will come with a formal proof, but that day has not yet arrived.