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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 forge

by wfo 11y ago

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-edge research paper. Then if you let that knowledge atrophy, you won't be able to understand it without a lot of re-learning when you look at it again. Unfortunately cute tricks like comments and concrete examples won't save you here (if concrete examples even exist -- oftentimes their constructions are so convoluted the abstract theorem is far easier to understand, and often times they are misleading. The theorem covers all cases, but all of the easily understandable concrete examples are completely trivial and don't require the theorem at all.) 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. Do you think perhaps there's a reason why a simple and easy trick like commenting and renaming lemmas has been discovered and solidified as standard practice in programming, but hasn't been adopted in mathematics? Are mathematicians really just SO dumb? The answer is those tricks just aren't good enough. Mathematicians do exposition. They do plenty of explaining. Any textbook and even many research papers spend a huge amount of time explaining what is going on as clearly as is possible. As it turns out the explanation helps, but the material is just plain hard.

akie 11y ago
> Do you think perhaps there's a reason why a simple and easy trick like commenting and renaming lemmas has been discovered and solidified as standard practice in programming, but hasn't been adopted in mathematics? Are mathematicians really just SO dumb? No, they aren't, but the example in question would have been easier to understand if he had tried to explain what was going on, instead of just saying "it is trivial that this or that follows". I think the comparison with "old code syndrome" is pretty much spot on, to be honest.
- conceit 11y ago
  ... especially if code is to be considered in the terms of information theory. Context is important and context can't be encoded without raising the entropy. Also, speech and to a large degree mathematical symbolism are sequential, so multi dimensional problems would have to be broken down into one sequential dimension, raising the entropy exponentially or whatever or loosing information. Can't help it.
timwaagh 11y ago
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.
- madsbuch 11y ago
  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 sit down and learn scrum, read manifests of coding practice etc. The math culture will eventually see itself be forced to go these ways to keep up.
  - gjulianm 11y ago
    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.
    - xiaq 11y ago
      > 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. Have you forgotten the pains that you went through trying to understand the difference between values, pointers and references, lexical and dynamic scoping, static and dynamic typing, and the like? Can you see them with your eyes? Have you ever tried fully explaining any of those to a non-CS major in half an hour? :) Abstractness is pretty subjective. For non-programmers, even the idea of CPU and memory can be abstract. And it doesn't help to open up a computer and point to the hardware; that is like claiming a mathematical paper is not abstract by pointing to the very concrete paper and ink that embodies it.
hinkley 11y ago
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.
- Ntrails 11y ago
  Thing is, I did understand it. Hell, I looked through my notes from maths degree (5 years ago) and guess what - most of it seems like nonsense. The worst bit is that because these were notes to myself - jam packed with comments of "so obviously" followed by a transformation I can make no sense of at all. It makes me pretty sad to think what a waste of time that learning was. Also the flip side of "hehe - I was well smart" is "shit - I'm now a moron"
- singham 11y ago
  Einstein also said "make things simple and not simpler". They will always be things that are irreducible.
- claudius 11y ago
  > "If you can't explain it simply, you don't understand it well enough." is my new favorite Albert Einstein quote. It’s also one of his more moronic quotes. Certainly on a very abstract, dumbed-down level everything can be explained simply; sure, if you cannot give your parents a rough idea what you’re doing, you might want to look into more examples. But there are plenty of things which require a very extensive basis to be understood thoroughly. For example, it is very easy to summarise what a (mathematical) group is and for anyone with a basic understanding of abstract maths, it will be understandable. It’s also very simple to find some examples (integers with addition, ℝ/{0} with multiplication etc.) which might be understandable by laypeople, but you will either confuse the latter or only give examples and not the actually important content. Further, when you have “simply explained” what a group is, can you go on and equally “simply explain“ what a the fundamental representation is and how irreducible representations come about? You just need a certain level of knowledge (e.g., linear algebra) already and not every paper can include a full introduction into representation theory.
xiaq 11y ago
> 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.
- xiaq 11y ago
  To clarify the last sentence: I am not criticizing mathematicians for not doing formal proofs. Often mathematical publications are not final "products" but explorations into new methods. Requiring formal proofs on every publication, with the status quo of computer-assisted proving, will surely impeded the development of mathematics. What I am saying is that I hope one day with the development of computer-assisted proving, the chores involved in doing formal proofs will be reduced to such a degree that mathematicians are more inclined to do them than not.
- Certhas 11y ago
  > writing programs is actually equivalent to doing a mathematical proof Only in the sense that writing in brainfuck is equivalent to writing in Python...
  - xiaq 11y ago
    True, but the point is that in principle you can write very detailed, fully formal proofs, in exactly the same manner programs are written. The rest of my comment then discusses why it is not written that way. Doing formal proofs manually is like writing turing machine instructions. Computer-assisted proving gives you higher level tools, but apparently it is still not powerful enough to be accepted in mainstream. And unlike programming where you get stuck if you haven't invented higher level languages, mathematicians have the widely accepted tool of "it trivially follows". Bourbaki might disagree, though.
efaref 11y ago
Leslie Lamport wrote a paper on this subject over 20 years ago. http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf http://research.microsoft.com/en-us/um/people/lamport/pubs/l... As far as I can tell, nobody has really adopted any of his proposals yet, and most papers continue to be written in pseudo-prose. Remember, mathematics used to be written in full prose (e.g. "the square of the hypotenuse is equal to the sum of the squares of the other two sides") and it took hundreds of years for us to realise that "x^2 + y^2 = z^2" was a better notation. While mathematics may be "genuinely very difficult to understand", so is any complex piece of software. Software engineering techniques might be useful to mathematicians, even if they are all new-fangled and modern.
- cokernel 11y ago
  The structure of Doron Zeilberger's proof of the Alternating Sign Matrix Conjecture is reminiscent of this, though possibly not directly inspired by it: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i2r13 http://www.combinatorics.org/ojs/index.php/eljc/article/view...
  - ezy 11y ago
    "subsubsubsublemma" kind of cracked me up. He seems very dedicated to not numbering the nested schema in this proof. :-)
yummyfajitas 11y ago
While "easy programming tricks" are not all it takes to understand difficult mathematics, writing a math paper like good code makes it drastically easier to understand what's going on. E.g., encapsulation - given a theorem with a highly technical proof, but ideas from the proof are not super important, hide the proof in an appendix. This is directly analogous to private functions which are only used to power a public one. Naming lemmas is also helpful, although not sufficient. Mathematicians aren't dumb. But they do have a set of traditions and a rhetorical style which is often blindly copied. Also, most mathematicians haven't been exposed to good software engineering practices - consider how few use git and instead just email around "paper_v3_chris_edits.tex". Having worked in both the math world and the CS world, math folks can learn quite a bit from software. (And vice versa - e.g., I just wrote a blog post about how some abstract math solved a real software engineering problem for me: http://engineering.wingify.com/posts/Free-objects/ http://engineering.wingify.com/posts/Free-objects/ )
Lawtonfogle 11y ago
A paper cannot teach a layman everything they need to understand the topic, but there are many papers out there I have difficulty understanding because of how they are written, but when reading them with a companion exposition, I can comprehend. It is a balance of getting one's point across while targeting a wide enough audience. It is the same with code. No one should be writing production code at the level where any non-programmer fluent in the language of the comment's could understand. But they should be writing it simplified enough so that performance isn't impacted and that anyone who is maintaining the code should be able to understand it without having to spend extreme amounts of time digesting it. And sometimes a key performance boost will turn into a 'here be dragons'. I do think Mathematicians are optimizing a bit too strongly for similar level peers, but please don't think I'm trying to say the are dumb for doing such.

by wfo 11y ago

akie 11y ago

> Do you think perhaps there's a reason why a simple and easy trick like commenting and renaming lemmas has been discovered and solidified as standard practice in programming, but hasn't been adopted in mathematics? Are mathematicians really just SO dumb? No, they aren't, but the example in question would have been easier to understand if he had tried to explain what was going on, instead of just saying "it is trivial that this or that follows". I think the comparison with "old code syndrome" is pretty much spot on, to be honest.

conceit 11y ago
... especially if code is to be considered in the terms of information theory. Context is important and context can't be encoded without raising the entropy. Also, speech and to a large degree mathematical symbolism are sequential, so multi dimensional problems would have to be broken down into one sequential dimension, raising the entropy exponentially or whatever or loosing information. Can't help it.

timwaagh 11y ago

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.

madsbuch 11y ago
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 sit down and learn scrum, read manifests of coding practice etc. The math culture will eventually see itself be forced to go these ways to keep up.
- gjulianm 11y ago
  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.
  - xiaq 11y ago
    > 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. Have you forgotten the pains that you went through trying to understand the difference between values, pointers and references, lexical and dynamic scoping, static and dynamic typing, and the like? Can you see them with your eyes? Have you ever tried fully explaining any of those to a non-CS major in half an hour? :) Abstractness is pretty subjective. For non-programmers, even the idea of CPU and memory can be abstract. And it doesn't help to open up a computer and point to the hardware; that is like claiming a mathematical paper is not abstract by pointing to the very concrete paper and ink that embodies it.

hinkley 11y ago

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.

Ntrails 11y ago
Thing is, I did understand it. Hell, I looked through my notes from maths degree (5 years ago) and guess what - most of it seems like nonsense. The worst bit is that because these were notes to myself - jam packed with comments of "so obviously" followed by a transformation I can make no sense of at all. It makes me pretty sad to think what a waste of time that learning was. Also the flip side of "hehe - I was well smart" is "shit - I'm now a moron"
singham 11y ago
Einstein also said "make things simple and not simpler". They will always be things that are irreducible.
claudius 11y ago
> "If you can't explain it simply, you don't understand it well enough." is my new favorite Albert Einstein quote. It’s also one of his more moronic quotes. Certainly on a very abstract, dumbed-down level everything can be explained simply; sure, if you cannot give your parents a rough idea what you’re doing, you might want to look into more examples. But there are plenty of things which require a very extensive basis to be understood thoroughly. For example, it is very easy to summarise what a (mathematical) group is and for anyone with a basic understanding of abstract maths, it will be understandable. It’s also very simple to find some examples (integers with addition, ℝ/{0} with multiplication etc.) which might be understandable by laypeople, but you will either confuse the latter or only give examples and not the actually important content. Further, when you have “simply explained” what a group is, can you go on and equally “simply explain“ what a the fundamental representation is and how irreducible representations come about? You just need a certain level of knowledge (e.g., linear algebra) already and not every paper can include a full introduction into representation theory.

xiaq 11y ago

> 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.

xiaq 11y ago
To clarify the last sentence: I am not criticizing mathematicians for not doing formal proofs. Often mathematical publications are not final "products" but explorations into new methods. Requiring formal proofs on every publication, with the status quo of computer-assisted proving, will surely impeded the development of mathematics. What I am saying is that I hope one day with the development of computer-assisted proving, the chores involved in doing formal proofs will be reduced to such a degree that mathematicians are more inclined to do them than not.
Certhas 11y ago
> writing programs is actually equivalent to doing a mathematical proof Only in the sense that writing in brainfuck is equivalent to writing in Python...
- xiaq 11y ago
  True, but the point is that in principle you can write very detailed, fully formal proofs, in exactly the same manner programs are written. The rest of my comment then discusses why it is not written that way. Doing formal proofs manually is like writing turing machine instructions. Computer-assisted proving gives you higher level tools, but apparently it is still not powerful enough to be accepted in mainstream. And unlike programming where you get stuck if you haven't invented higher level languages, mathematicians have the widely accepted tool of "it trivially follows". Bourbaki might disagree, though.

efaref 11y ago

Leslie Lamport wrote a paper on this subject over 20 years ago. http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf http://research.microsoft.com/en-us/um/people/lamport/pubs/l... As far as I can tell, nobody has really adopted any of his proposals yet, and most papers continue to be written in pseudo-prose. Remember, mathematics used to be written in full prose (e.g. "the square of the hypotenuse is equal to the sum of the squares of the other two sides") and it took hundreds of years for us to realise that "x^2 + y^2 = z^2" was a better notation. While mathematics may be "genuinely very difficult to understand", so is any complex piece of software. Software engineering techniques might be useful to mathematicians, even if they are all new-fangled and modern.

cokernel 11y ago
The structure of Doron Zeilberger's proof of the Alternating Sign Matrix Conjecture is reminiscent of this, though possibly not directly inspired by it: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i2r13 http://www.combinatorics.org/ojs/index.php/eljc/article/view...
- ezy 11y ago
  "subsubsubsublemma" kind of cracked me up. He seems very dedicated to not numbering the nested schema in this proof. :-)

yummyfajitas 11y ago

While "easy programming tricks" are not all it takes to understand difficult mathematics, writing a math paper like good code makes it drastically easier to understand what's going on. E.g., encapsulation - given a theorem with a highly technical proof, but ideas from the proof are not super important, hide the proof in an appendix. This is directly analogous to private functions which are only used to power a public one. Naming lemmas is also helpful, although not sufficient. Mathematicians aren't dumb. But they do have a set of traditions and a rhetorical style which is often blindly copied. Also, most mathematicians haven't been exposed to good software engineering practices - consider how few use git and instead just email around "paper_v3_chris_edits.tex". Having worked in both the math world and the CS world, math folks can learn quite a bit from software. (And vice versa - e.g., I just wrote a blog post about how some abstract math solved a real software engineering problem for me: http://engineering.wingify.com/posts/Free-objects/ http://engineering.wingify.com/posts/Free-objects/ )

Lawtonfogle 11y ago

A paper cannot teach a layman everything they need to understand the topic, but there are many papers out there I have difficulty understanding because of how they are written, but when reading them with a companion exposition, I can comprehend. It is a balance of getting one's point across while targeting a wide enough audience. It is the same with code. No one should be writing production code at the level where any non-programmer fluent in the language of the comment's could understand. But they should be writing it simplified enough so that performance isn't impacted and that anyone who is maintaining the code should be able to understand it without having to spend extreme amounts of time digesting it. And sometimes a key performance boost will turn into a 'here be dragons'. I do think Mathematicians are optimizing a bit too strongly for similar level peers, but please don't think I'm trying to say the are dumb for doing such.