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Yes. There are some deep insights in this exposition. The irony is (in my opinion) APL is the worst Array/Matrix based programming language. In fairness it was

Yes. There are some deep insights in this exposition. The irony is (in my opinion) APL is the worst Array/Matrix based programming language. In fairness it was also the first, but compared to Matlab or Julia it is not as expressive and feels much harder to use.

solinent 6y ago
There's a certain mathematical elegance to APL, I think. When the language is terse enough it helps you visualize and work with the language as a tool of thought--Matlab attempts to map actual mathematics to ASCII which is not that successful for me at least, since it meets a middle ground where it's too difficult for me to think quickly purely in Matlab and it's too high level for it to be useful as a practical language. Engineers love it for prototyping, though, so maybe I just haven't worked with Matlab enough.
- blululu 6y ago
  Fair point - personally I found APL to be a little too terse to be readable in ASCII. I think that there is a big difference in the affordances of a chalkboard/paper and a monospaced text editor, and to me APL is too close to paper based notation where it is easy to read and write a larger set of symbols. Matlab has some dedicated notation around matrices but uses more text heavy descriptions beyond that which feels better suited to a command line. Julia takes an even more text based approach and supports notations like list comprehensions which feel easier to learn, read and use than a set glyphs.
- Someone 6y ago
  “Engineers love it for prototyping, though” Makes perfect sense. Matlab is for engineers, not for mathematicians. They use computer algebra systems, proof assistants, etc. Difference is that engineers (and physicists) want answers and don’t care about how they are obtained, while its the reverse for mathematicians. I think APL, although it, too, is a language for computing numbers, spiritually is a bit closer to mathematics than Matlab.
  - solinent 6y ago
    I'm a bit of both, so I guess I take the radical approach--straight from mathematics to C++/ASM/FPGA/ASIC. Ultimately programming languages are just an alternate notative system for mathematics--formal language theory actually formalizes and generalizes this, it's what us Computer Scientist's specialize in generally. Since the computer is just a glorified calculator with memory (sorry Apple), we can fit the whole thing into a formal mathematical framework.
patrec 6y ago
If you think APL is less expressive than Matlab, you probably haven't really grasped it, IMO. Having said that, Matlab is optimized for manipulating matrices and replicating the notation of normal linear algebra and has excellent implementations of basically any numerical algorithm that frequently comes up in this . So writing something like chol(X'*X+diag(eig(X))) in an APL will look uglier and quite possibly slower and less accurate and depending on the numerical routines you need, require extra implementation work on your part. But that overhead is constant factor, more or less anything you can express well in matlab can be expressed straightforwardly in APL, too, if you have the right numerical routines. That's not true in the other direction though: there's a lot of stuff in APL you cannot express adequately in matlab at all. For example J (and these days Dyalog as well IIRC) have an operation called under which basically does this: u(f,g) = x => f^-1(g(f(x)). So you can write geometric_mean = u(mean, log). It is completely impossible to implement something like "under" in matlab. Admittedly the J implementation at least of deriving a generalized inverse for an arbitrary function f is a somewhat ill-defined hack, but this is still something that is both conceptually and practically quite powerful. Also, whilst Matlab is really clunky for anything that is not a 2D array and hardcodes matrix multiplication as the one inner-product, APL has more powerful abstractions for manipulating arbitrary rank arrays and a more general concept of inner products. Also, APL has some really dumb but cherished-by-the-community ideas that make the language less expressive and much more awkward to learn, e.g. the idea of replicating the terrible defect of normal mathematical notation where - is overloaded for negation and subtraction to every other function.
- henrikeh 6y ago
  > It is completely impossible to implement something like "under" in matlab. I’m a little curious about this. Does J have a notion of the relationship between certain functions and their inverse? What is it that enables “under” in J which makes it impossible in Matlab?
  - klibertp 6y ago
    > Does J have a notion of the relationship between certain functions and their inverse? Yes. Many built-in words have inverses assigned, and you can assign inverse functions to your own words with :. https://code.jsoftware.com/wiki/Vocabulary/codot https://code.jsoftware.com/wiki/Vocabulary/codot EDIT: and here's a table with predefined inverses: https://code.jsoftware.com/wiki/Vocabulary/Inverses https://code.jsoftware.com/wiki/Vocabulary/Inverses
yiyus 6y ago
How do you define expressiveness to arrive to that conclusion?

solinent 6y ago

There's a certain mathematical elegance to APL, I think. When the language is terse enough it helps you visualize and work with the language as a tool of thought--Matlab attempts to map actual mathematics to ASCII which is not that successful for me at least, since it meets a middle ground where it's too difficult for me to think quickly purely in Matlab and it's too high level for it to be useful as a practical language. Engineers love it for prototyping, though, so maybe I just haven't worked with Matlab enough.

blululu 6y ago
Fair point - personally I found APL to be a little too terse to be readable in ASCII. I think that there is a big difference in the affordances of a chalkboard/paper and a monospaced text editor, and to me APL is too close to paper based notation where it is easy to read and write a larger set of symbols. Matlab has some dedicated notation around matrices but uses more text heavy descriptions beyond that which feels better suited to a command line. Julia takes an even more text based approach and supports notations like list comprehensions which feel easier to learn, read and use than a set glyphs.
Someone 6y ago
“Engineers love it for prototyping, though” Makes perfect sense. Matlab is for engineers, not for mathematicians. They use computer algebra systems, proof assistants, etc. Difference is that engineers (and physicists) want answers and don’t care about how they are obtained, while its the reverse for mathematicians. I think APL, although it, too, is a language for computing numbers, spiritually is a bit closer to mathematics than Matlab.
- solinent 6y ago
  I'm a bit of both, so I guess I take the radical approach--straight from mathematics to C++/ASM/FPGA/ASIC. Ultimately programming languages are just an alternate notative system for mathematics--formal language theory actually formalizes and generalizes this, it's what us Computer Scientist's specialize in generally. Since the computer is just a glorified calculator with memory (sorry Apple), we can fit the whole thing into a formal mathematical framework.

patrec 6y ago

If you think APL is less expressive than Matlab, you probably haven't really grasped it, IMO. Having said that, Matlab is optimized for manipulating matrices and replicating the notation of normal linear algebra and has excellent implementations of basically any numerical algorithm that frequently comes up in this . So writing something like chol(X'*X+diag(eig(X))) in an APL will look uglier and quite possibly slower and less accurate and depending on the numerical routines you need, require extra implementation work on your part. But that overhead is constant factor, more or less anything you can express well in matlab can be expressed straightforwardly in APL, too, if you have the right numerical routines. That's not true in the other direction though: there's a lot of stuff in APL you cannot express adequately in matlab at all. For example J (and these days Dyalog as well IIRC) have an operation called under which basically does this: u(f,g) = x => f^-1(g(f(x)). So you can write geometric_mean = u(mean, log). It is completely impossible to implement something like "under" in matlab. Admittedly the J implementation at least of deriving a generalized inverse for an arbitrary function f is a somewhat ill-defined hack, but this is still something that is both conceptually and practically quite powerful. Also, whilst Matlab is really clunky for anything that is not a 2D array and hardcodes matrix multiplication as the one inner-product, APL has more powerful abstractions for manipulating arbitrary rank arrays and a more general concept of inner products. Also, APL has some really dumb but cherished-by-the-community ideas that make the language less expressive and much more awkward to learn, e.g. the idea of replicating the terrible defect of normal mathematical notation where - is overloaded for negation and subtraction to every other function.

henrikeh 6y ago
> It is completely impossible to implement something like "under" in matlab. I’m a little curious about this. Does J have a notion of the relationship between certain functions and their inverse? What is it that enables “under” in J which makes it impossible in Matlab?
- klibertp 6y ago
  > Does J have a notion of the relationship between certain functions and their inverse? Yes. Many built-in words have inverses assigned, and you can assign inverse functions to your own words with :. https://code.jsoftware.com/wiki/Vocabulary/codot https://code.jsoftware.com/wiki/Vocabulary/codot EDIT: and here's a table with predefined inverses: https://code.jsoftware.com/wiki/Vocabulary/Inverses https://code.jsoftware.com/wiki/Vocabulary/Inverses

yiyus 6y ago

How do you define expressiveness to arrive to that conclusion?