10 ms·

Reasons to use Haskell as a Mathematician (2006)

93 points by cottonseed 12y ago

gone35 12y ago
Shamelessly off-topic, but I can tell the author is a legit pure mathematician: (1) Haskell is Functional (2) Haskell Insulates You (3) Haskell Performs Well [...] (3) Haskell is Declarative Not being ironic, by the way. From experience, often I can tell pure vs applied mathematicians by how hilariously bad/absent-minded they can be at basic arithmetic/algebraic manipulations. Same for theoretical- vs systems-oriented CS people, I've found --almost as if being a horrible programmer somehow gives you academic hipster cred. (EDIT - To clarify: I mean CS academics in grad school, not CS professionals in industry. Also it's just a silly anecdote only marginally correlated with reality: no offense intended to anyone.)
- moab 12y ago
  Inheriting off-topicness, while that stereotype does apply to most mathematicians, a large number of the CS theory people I've worked/TA'd with were actually pretty good programmers, and a few had even taught multiple years of our intro systems course. Of course this is just one school, but I think computer science as a field is still young enough that someone can be an expert in one area and still be reasonably fluent in other areas. Don't think the last point is particularly true either: being a good programmer is valuable even for theoretical CS. Point in case, recently saw the work leading to a paper on SDD solvers: there was a ton of programming and exploration (albeit in Mathematica/Matlab) that led to the final paper. Also, people working on parallel stuff these days (even the theory guys) usually publish CILK/MPI along with the paper. It's great when you can do both :)
  - gone35 12y ago
    Yes: That's definitely the way it should be! I agree 100% that CS is still young and cohesive enough for dedicated people to achieve both breadth and depth, which perhaps means "grand" single-person revolutionary breakthroughs can still be achieved: Karp made a similar point during one of the Turing centenary talks (*An Algorithmic View of the Universe [1,2]), positing that the field is not yet specialized enough to prevent an Alan Turing-like figure today to have a comprehensive view of the state-of-art of the discipline (!). [1] http://dl.acm.org/citation.cfm?id=2322176.2322189 http://dl.acm.org/citation.cfm?id=2322176.2322189 [2] http://www.youtube.com/watch?v=f6df3s3x3zo#t=56m25s http://www.youtube.com/watch?v=f6df3s3x3zo#t=56m25s
- StandardFuture 12y ago
  It is technically impossible to be a good 'CS People' and simultaneously be a 'horrible programmer'. It seems that what you are referring to are people who know basic CS concepts and even know the basic features of a language or two but do not have a 'Hacker' mindset. This does not make them a 'horrible programmer' -- just as a hacker who comes up with 'tricks', without actually understanding that the things he 'created' were already discovered in some 70s CS paper, is not a 'horrible Computer Scientist'. It's not a sufficient method to describe a person's strengths and weaknesses.
  - elliptic 12y ago
    So, basically, you think that there are exactly 0 CS professors who are horrible programmers?
    - StandardFuture 12y ago
      How can your comment completely disregard the entirety of my actual conclusion? >It's not a sufficient method to describe a person's strengths and weaknesses. So, there should be if your classifications are correct. You are assuming that 'CS Professor' is equatable to 'good CS People'. But, my classification may not be so rigid. EDIT: You also still have not defined what it means to be a 'Good Programmer' vs. a 'Horrible' one.
  - gone35 12y ago
    Sorry I wasn't clear: by 'CS people' I meant to refer to CS academics in graduate school (professors, post-docs, grad students), not CS professionals in industry --admittedly a sign of insularity, I recognize. And what I meant is that, anecdotally, academics working on CS theory (eg theory of computation, computational learning theory) are more likely than not to unmistakeably be what I think everyone on HN would agree to call point-blank "horrible programmers": no use of version control, no tests, little concern for organization/modularity/maintainability, etc. And in part it makes sense since they don't have to hack nearly as much as systems-type CS academics --in many ways their work is closer to pure mathematics than engineering/applied math. That's all. In all just a silly anecdote, although with a kernel of truth. See for instance Scott Aaronson's joke comments about his programming abilities [1]: On the spectrum of computer science, I'm about as theoretical as you can get. One way to put it is that I got through CS grad school at Berkeley without really learning any programming language other than QBASIC (!). [1] http://www.scottaaronson.com/blog/?p=266 http://www.scottaaronson.com/blog/?p=266
- nostrademons 12y ago
  He was a pure mathematician at the time this was written. Now he works for Google, IIRC on self-driving cars: http://www.knowthecosmos.com/destember/destember-experiment-inside-the-mind-of-google-x-engineer-dan-piponi/ http://www.knowthecosmos.com/destember/destember-experiment-... He also won an Oscar this year, his second total: http://www.imdb.com/name/nm0685004/awards http://www.imdb.com/name/nm0685004/awards
  - JoeAcchino 12y ago
    Nice to know, if an oscar-winning pure mathematician working on science-fiction projects found difficult to learn Haskell, I can save myself the trouble.
    - nostrademons 12y ago
      If that's your goal. There are some people who do things because they are difficult, because whatever activity is most difficult is probably the most mind-expanding.
- Bahamut 12y ago
  I don't think you know enough pure mathematicians.
  - gone35 12y ago
    You are probably right --I haven't been spending much time of late with people from my own department. :)
warfangle 12y ago
I've always wondered why computational mathematicians/physicists end up using MATLAB/FORTRAN instead of haskell or similar.
- jbooth 12y ago
  Because they're trying to do work with computers, rather than do research about computing. If the interestingness of the implementation doesn't matter and you just want results, go with MATLAB/FORTRAN/Numpy or whatever and a fast BLAS implementation. Mutable state everywhere, side effects everywhere, you got your results fast and can go finish your paper or your project or whatever.
  - warfangle 12y ago
    Yeah, but with Haskell the code would look more like the actual equations you're working from and not an implementation of the equations (thinking like differential manifolds etc).
    - sampo 12y ago
      How does matrix indexing look like in Haskell?
      - codygman 12y ago
        I found this by looking at the repa docs: mmMult :: Monad m => Array U DIM2 Double -> Array U DIM2 Double -> m (Array U DIM2 Double) mmMult a b = sumP (Repa.zipWith (*) aRepl bRepl) where t = transpose2D b aRepl = extend (Z :.All :.colsB :.All) a bRepl = extend (Z :.rowsA :.All :.All) t (Z :.colsA :.rowsA) = extent a (Z :.colsB :.rowsB) = extent b http://www.haskell.org/haskellwiki/Numeric_Haskell:_A_Repa_Tutorial#Example:_matrix-matrix_multiplication http://www.haskell.org/haskellwiki/Numeric_Haskell:_A_Repa_T...
        sampo 12y ago
segmondy 12y ago
there are a bunch of languages mathematicians should know, haskell is one of them, so is prolog, lisp, APL, FORTRAN, and C. They all have their advantages, and you just have to know which one will give you an edge in your problem domain.
gizmo686 12y ago
As someone who has used Haskell for math, I would comment that the standard prelude (Haskell's standard library) can be quite obnoxious to work with depending on what you are doing. When I use Haskell for math, I normally switch to the numeric-prelude [0]. [0] https://hackage.haskell.org/package/numeric-prelude https://hackage.haskell.org/package/numeric-prelude
- gamegoblin 12y ago
  At least #2 on their list of issues is currently fixed in GHC 7.8
tdees40 12y ago
I'm currently porting a Numpy project into haskell as a self-improvement project, and it's been 69% great. But then I find some library that I have to implement from scratch when I just had it in python. That's a pretty big problem, and I'm willing to put up with it, but most people aren't.
- PieSquared 12y ago
  Just out of curiosity, what libraries were these?
  - tdees40 12y ago
    Scipy optimization libraries primarily.
- iand675 12y ago
  A nice way to look at it: if you share what code you have had to reimplement, you're making it more than 69% great for the potential Haskellers that follow!
  - tdees40 12y ago
    Would love to! But my employer probably wouldn't be so thrilled about it.
cousin_it 12y ago
I've slowly come to the opinion that laziness is not the best idea, even from a pure mathematical point of view. There are large mathematical benefits to using a strict language like SML. One important benefit to me is that strictness allows you to have an equality operation with a very strong mathematical guarantee, namely that the types for which equality can be automatically defined are precisely the types for which equality is decidable (always terminates). A lazy language throws that out the window. In a strict language you can define both a type of guaranteed-finite lists with decidable equality and a type of potentially-infinite streams, while in a lazy language you only have the latter. I wrote about it in more detail here: http://slepnev.blogspot.ch/2014/06/is-laziness-wrong-for-equality.html http://slepnev.blogspot.ch/2014/06/is-laziness-wrong-for-equ...
- wyager 12y ago
  But mathematics is non-strict. One can define functions that terminate when evaluated lazily but don't terminate when evaluated strictly. These kinds of functions are prevalent in mathematics.
  - cousin_it 12y ago
    Does your belief that "mathematics is non-strict" go as far as saying that the type of natural numbers: data Natural = Zero | Successor Natural must include this member as well: omega :: Natural omega = Successor omega bool1 = (omega == Zero) -- returns False bool2 = (omega == Successor Zero) -- returns False bool3 = (omega == omega) -- whoops, infinite loop The moral is that some parts of mathematics may be lazy, but mathematical induction has gotta be strict. Since many functional programs rely on induction for proofs of termination, correctness and resource usage, that means they should be strict as well.
    - wyager 12y ago
      >Since many functional programs rely on induction for proofs of termination, correctness and resource usage, that means they should be strict as well. Am I incorrect in thinking that any strict program proven to be correct and terminating is also correct as a lazy program?
j2kun 12y ago
I want to hear from people who do math all day every day, and not just programming language theory. I want to hear about algebraic geometers, topolgists, ring theorists using Haskell and saying "Yeah this is really helping me in my work."
- Sniffnoy 12y ago
  As a mathematician I'll certainly endorse a number of the points in the linked post. Because what they mean is that it can be very quick to translate mathematics into code. Maybe it won't run fast or be understandable by anyone else, but I can express complicated ideas (relatively) compactly and quickly and get something working without huge amounts of tedium.
  - j2kun 12y ago
    Can you give me some examples? What do you use Haskell to do? I am also a mathematician so I can handle the jargon.
    - Sniffnoy 12y ago
      Mostly not large things. (Or at least, mostly not things that are large in Haskell.) An example recently -- I wanted to consider, let's say you have two finite rooted trees, S and T, with T having more vertices; and you you have a bijection f from the set of subtrees of S (subtrees have to have the same root as the original) to the set of subtrees of T, with certain restrictions on it; and then to f you associate a system of linear equations; and if the solution set to this system isn't contained a hyperplane aligned with the axes, then (if there exists an f for which this is true) I'll say "S interferes with T". And then I want to check for various T whether there is any smaller S that interferes with them. Basically without something like Haskell, or at least something like Haskell-or-ML (I didn't need typeclasses and I don't know how much I needed laziness), it seems like that would have been a complete mess. Of course, the above example is perhaps not the best as it meant I had to [either look up a library, or, what I actually did] implement row-reduction in Haskell, obviously not a great language for it! But ultimately it wasn't too terrible. And the rest as I've said would have been terrible in another language.

93 points by cottonseed 12y ago

Shamelessly off-topic, but I can tell the author is a legit pure mathematician: (1) Haskell is Functional (2) Haskell Insulates You (3) Haskell Performs Well [...] (3) Haskell is Declarative Not being ironic, by the way. From experience, often I can tell pure vs applied mathematicians by how hilariously bad/absent-minded they can be at basic arithmetic/algebraic manipulations. Same for theoretical- vs systems-oriented CS people, I've found --almost as if being a horrible programmer somehow gives you academic hipster cred. (EDIT - To clarify: I mean CS academics in grad school, not CS professionals in industry. Also it's just a silly anecdote only marginally correlated with reality: no offense intended to anyone.)

moab 12y ago
Inheriting off-topicness, while that stereotype does apply to most mathematicians, a large number of the CS theory people I've worked/TA'd with were actually pretty good programmers, and a few had even taught multiple years of our intro systems course. Of course this is just one school, but I think computer science as a field is still young enough that someone can be an expert in one area and still be reasonably fluent in other areas. Don't think the last point is particularly true either: being a good programmer is valuable even for theoretical CS. Point in case, recently saw the work leading to a paper on SDD solvers: there was a ton of programming and exploration (albeit in Mathematica/Matlab) that led to the final paper. Also, people working on parallel stuff these days (even the theory guys) usually publish CILK/MPI along with the paper. It's great when you can do both :)
- gone35 12y ago
  Yes: That's definitely the way it should be! I agree 100% that CS is still young and cohesive enough for dedicated people to achieve both breadth and depth, which perhaps means "grand" single-person revolutionary breakthroughs can still be achieved: Karp made a similar point during one of the Turing centenary talks (*An Algorithmic View of the Universe [1,2]), positing that the field is not yet specialized enough to prevent an Alan Turing-like figure today to have a comprehensive view of the state-of-art of the discipline (!). [1] http://dl.acm.org/citation.cfm?id=2322176.2322189 http://dl.acm.org/citation.cfm?id=2322176.2322189 [2] http://www.youtube.com/watch?v=f6df3s3x3zo#t=56m25s http://www.youtube.com/watch?v=f6df3s3x3zo#t=56m25s
StandardFuture 12y ago
It is technically impossible to be a good 'CS People' and simultaneously be a 'horrible programmer'. It seems that what you are referring to are people who know basic CS concepts and even know the basic features of a language or two but do not have a 'Hacker' mindset. This does not make them a 'horrible programmer' -- just as a hacker who comes up with 'tricks', without actually understanding that the things he 'created' were already discovered in some 70s CS paper, is not a 'horrible Computer Scientist'. It's not a sufficient method to describe a person's strengths and weaknesses.
- elliptic 12y ago
  So, basically, you think that there are exactly 0 CS professors who are horrible programmers?
  - StandardFuture 12y ago
    How can your comment completely disregard the entirety of my actual conclusion? >It's not a sufficient method to describe a person's strengths and weaknesses. So, there should be if your classifications are correct. You are assuming that 'CS Professor' is equatable to 'good CS People'. But, my classification may not be so rigid. EDIT: You also still have not defined what it means to be a 'Good Programmer' vs. a 'Horrible' one.
- gone35 12y ago
  Sorry I wasn't clear: by 'CS people' I meant to refer to CS academics in graduate school (professors, post-docs, grad students), not CS professionals in industry --admittedly a sign of insularity, I recognize. And what I meant is that, anecdotally, academics working on CS theory (eg theory of computation, computational learning theory) are more likely than not to unmistakeably be what I think everyone on HN would agree to call point-blank "horrible programmers": no use of version control, no tests, little concern for organization/modularity/maintainability, etc. And in part it makes sense since they don't have to hack nearly as much as systems-type CS academics --in many ways their work is closer to pure mathematics than engineering/applied math. That's all. In all just a silly anecdote, although with a kernel of truth. See for instance Scott Aaronson's joke comments about his programming abilities [1]: On the spectrum of computer science, I'm about as theoretical as you can get. One way to put it is that I got through CS grad school at Berkeley without really learning any programming language other than QBASIC (!). [1] http://www.scottaaronson.com/blog/?p=266 http://www.scottaaronson.com/blog/?p=266
nostrademons 12y ago
He was a pure mathematician at the time this was written. Now he works for Google, IIRC on self-driving cars: http://www.knowthecosmos.com/destember/destember-experiment-inside-the-mind-of-google-x-engineer-dan-piponi/ http://www.knowthecosmos.com/destember/destember-experiment-... He also won an Oscar this year, his second total: http://www.imdb.com/name/nm0685004/awards http://www.imdb.com/name/nm0685004/awards
- JoeAcchino 12y ago
  Nice to know, if an oscar-winning pure mathematician working on science-fiction projects found difficult to learn Haskell, I can save myself the trouble.
  - nostrademons 12y ago
    If that's your goal. There are some people who do things because they are difficult, because whatever activity is most difficult is probably the most mind-expanding.
Bahamut 12y ago
I don't think you know enough pure mathematicians.
- gone35 12y ago
  You are probably right --I haven't been spending much time of late with people from my own department. :)

warfangle 12y ago

I've always wondered why computational mathematicians/physicists end up using MATLAB/FORTRAN instead of haskell or similar.

jbooth 12y ago
Because they're trying to do work with computers, rather than do research about computing. If the interestingness of the implementation doesn't matter and you just want results, go with MATLAB/FORTRAN/Numpy or whatever and a fast BLAS implementation. Mutable state everywhere, side effects everywhere, you got your results fast and can go finish your paper or your project or whatever.
- warfangle 12y ago
  Yeah, but with Haskell the code would look more like the actual equations you're working from and not an implementation of the equations (thinking like differential manifolds etc).
  - sampo 12y ago
    How does matrix indexing look like in Haskell?
    - codygman 12y ago
      I found this by looking at the repa docs: mmMult :: Monad m => Array U DIM2 Double -> Array U DIM2 Double -> m (Array U DIM2 Double) mmMult a b = sumP (Repa.zipWith (*) aRepl bRepl) where t = transpose2D b aRepl = extend (Z :.All :.colsB :.All) a bRepl = extend (Z :.rowsA :.All :.All) t (Z :.colsA :.rowsA) = extent a (Z :.colsB :.rowsB) = extent b http://www.haskell.org/haskellwiki/Numeric_Haskell:_A_Repa_Tutorial#Example:_matrix-matrix_multiplication http://www.haskell.org/haskellwiki/Numeric_Haskell:_A_Repa_T...
      - sampo 12y ago

segmondy 12y ago

there are a bunch of languages mathematicians should know, haskell is one of them, so is prolog, lisp, APL, FORTRAN, and C. They all have their advantages, and you just have to know which one will give you an edge in your problem domain.

gizmo686 12y ago

As someone who has used Haskell for math, I would comment that the standard prelude (Haskell's standard library) can be quite obnoxious to work with depending on what you are doing. When I use Haskell for math, I normally switch to the numeric-prelude [0]. [0] https://hackage.haskell.org/package/numeric-prelude https://hackage.haskell.org/package/numeric-prelude

gamegoblin 12y ago
At least #2 on their list of issues is currently fixed in GHC 7.8

tdees40 12y ago

I'm currently porting a Numpy project into haskell as a self-improvement project, and it's been 69% great. But then I find some library that I have to implement from scratch when I just had it in python. That's a pretty big problem, and I'm willing to put up with it, but most people aren't.

PieSquared 12y ago
Just out of curiosity, what libraries were these?
- tdees40 12y ago
  Scipy optimization libraries primarily.
iand675 12y ago
A nice way to look at it: if you share what code you have had to reimplement, you're making it more than 69% great for the potential Haskellers that follow!
- tdees40 12y ago
  Would love to! But my employer probably wouldn't be so thrilled about it.

cousin_it 12y ago

I've slowly come to the opinion that laziness is not the best idea, even from a pure mathematical point of view. There are large mathematical benefits to using a strict language like SML. One important benefit to me is that strictness allows you to have an equality operation with a very strong mathematical guarantee, namely that the types for which equality can be automatically defined are precisely the types for which equality is decidable (always terminates). A lazy language throws that out the window. In a strict language you can define both a type of guaranteed-finite lists with decidable equality and a type of potentially-infinite streams, while in a lazy language you only have the latter. I wrote about it in more detail here: http://slepnev.blogspot.ch/2014/06/is-laziness-wrong-for-equality.html http://slepnev.blogspot.ch/2014/06/is-laziness-wrong-for-equ...

wyager 12y ago
But mathematics is non-strict. One can define functions that terminate when evaluated lazily but don't terminate when evaluated strictly. These kinds of functions are prevalent in mathematics.
- cousin_it 12y ago
  Does your belief that "mathematics is non-strict" go as far as saying that the type of natural numbers: data Natural = Zero | Successor Natural must include this member as well: omega :: Natural omega = Successor omega bool1 = (omega == Zero) -- returns False bool2 = (omega == Successor Zero) -- returns False bool3 = (omega == omega) -- whoops, infinite loop The moral is that some parts of mathematics may be lazy, but mathematical induction has gotta be strict. Since many functional programs rely on induction for proofs of termination, correctness and resource usage, that means they should be strict as well.
  - wyager 12y ago
    >Since many functional programs rely on induction for proofs of termination, correctness and resource usage, that means they should be strict as well. Am I incorrect in thinking that any strict program proven to be correct and terminating is also correct as a lazy program?

j2kun 12y ago

I want to hear from people who do math all day every day, and not just programming language theory. I want to hear about algebraic geometers, topolgists, ring theorists using Haskell and saying "Yeah this is really helping me in my work."

Sniffnoy 12y ago
As a mathematician I'll certainly endorse a number of the points in the linked post. Because what they mean is that it can be very quick to translate mathematics into code. Maybe it won't run fast or be understandable by anyone else, but I can express complicated ideas (relatively) compactly and quickly and get something working without huge amounts of tedium.
- j2kun 12y ago
  Can you give me some examples? What do you use Haskell to do? I am also a mathematician so I can handle the jargon.
  - Sniffnoy 12y ago
    Mostly not large things. (Or at least, mostly not things that are large in Haskell.) An example recently -- I wanted to consider, let's say you have two finite rooted trees, S and T, with T having more vertices; and you you have a bijection f from the set of subtrees of S (subtrees have to have the same root as the original) to the set of subtrees of T, with certain restrictions on it; and then to f you associate a system of linear equations; and if the solution set to this system isn't contained a hyperplane aligned with the axes, then (if there exists an f for which this is true) I'll say "S interferes with T". And then I want to check for various T whether there is any smaller S that interferes with them. Basically without something like Haskell, or at least something like Haskell-or-ML (I didn't need typeclasses and I don't know how much I needed laziness), it seems like that would have been a complete mess. Of course, the above example is perhaps not the best as it meant I had to [either look up a library, or, what I actually did] implement row-reduction in Haskell, obviously not a great language for it! But ultimately it wasn't too terrible. And the rest as I've said would have been terrible in another language.