4 ms·

Consider a graph G, with vertices (a,b,...,z). Say, we want to find the shortest path between vertices a and m. A mathematician might write an algorithm called

Consider a graph G, with vertices (a,b,...,z). Say, we want to find the shortest path between vertices a and m. A mathematician might write an algorithm called shortest_path(G, a, m) which might call the algorithm breadth_first_search(a) etc. Functions are called on objects to determine their properties. Or to modify them into other objects. This is what you see in math or theoretical CS papers. In python, in popular libraries, the above will be attained by something like G.shortest_path(a,b) which seems to imply as per OO that graph G has a property shortest_path. In mathematics we don't think like this. Because it prevents mathematical abstraction. An abstract algorithm, like say Eigenvalues can be applied to a matrix M or to a graph G. In a functional language, the user would just do Eigenvalue(M) or Eigenvalue(G) as needed. In python, these would be M.eigenvalues() and G.eigenvalues(), which makes them distinct.

klyrs 4y ago
What you're saying is not true of the most popular graph package in Python (networkx). Graphs there are objects with attributes (edges, nodes, etc) and methods (add_edge, etc), but algorithms such as shortest paths are functions as you have called most natural: shortest_path(G, a, b)
- abdullahkhalids 4y ago
  Well, I am glad that at least this library is being functional, rather than object oriented. I wish all the others did as well.
  - klyrs 4y ago
    It's still object-oriented, though. The graphs themselves are objects, which helps with ergonomics. There are also functions that operate on graphs, because the alternative would be to stuff perhaps thousands of methods into the graph class -- which would be extremely annoying to maintain as well as abysmally slow.
osigurdson 4y ago
Since you capitalized it, it looks like a static method on the “G” class. There is nothing OO about static methods - this is purely and organizational construct.
cft 4y ago
I think it's the superclass A of both G and M that defines the eigenvalue() function. Then when both G(A) and M(A) inherit from A, it captures the mathematical relationship. The subclasses indeed should implement the eigenvalues computations differently, like in real mathematics. Funcional programming is used in Coq because it captures logical chains, for theorem PROOFS, but not for specific COMPUTATIONS, such as eigenvalues.
- abdullahkhalids 4y ago
  I understand that this is possible. But this is not how mathematicians think. If you got together a bunch of mathematicians and asked to invent a programming language to do mathematics, they would never ever invent the concept of a class. Mathematics (outside of stuff like category theory etc) is simply sets and mappings between sets. Anything more complicated is forcing the user to think in a way that mathematics does not.
  - klyrs 4y ago
    > But this is not how mathematicians think. If you got together a bunch of mathematicians and asked to invent a programming language to do mathematics, they would never ever invent the concept of a class. Hi, mathematician here. Please read the source of SageMath: written by mathematicians, for mathematicians. Therein, you will find thousands of classes. Almost all computer algebra systems, written by mathematicians for mathematicians, have a notion of classes. > Mathematics (outside of stuff like category theory etc) is simply sets and mappings between sets. This is an extremely narrow-minded view of mathematics. Mathematicians thrive on abstraction, (category theory is literally mathematics; that's a very strange exception for you to carve out), and if we were to boil everything down to "simply sets and mappings between sets" then we'd be bogged down in utter tedium and nothing would ever get done. Object-oriented programming, specifically class hierarchies and inheritance, are extremely valuable for doing all sorts of math at high levels of abstraction. Hell. If I were to be especially pedantic, I'd point out that category theory itself is "simply sets and mappings between sets" except that sets aren't quite large enough so it's actually "categories and mapping between categories".
CuriouslyC 4y ago
Since the eigen decomposition is specifically a property of the matrix representation of the graph, but other things like shortest path distance between two nodes are naturally enough computed on graphs formed by objects with pointers, perhaps the correct interface is G.adjacency_matrix().eigenvalues().

klyrs 4y ago

What you're saying is not true of the most popular graph package in Python (networkx). Graphs there are objects with attributes (edges, nodes, etc) and methods (add_edge, etc), but algorithms such as shortest paths are functions as you have called most natural: shortest_path(G, a, b)

abdullahkhalids 4y ago
Well, I am glad that at least this library is being functional, rather than object oriented. I wish all the others did as well.
- klyrs 4y ago
  It's still object-oriented, though. The graphs themselves are objects, which helps with ergonomics. There are also functions that operate on graphs, because the alternative would be to stuff perhaps thousands of methods into the graph class -- which would be extremely annoying to maintain as well as abysmally slow.

osigurdson 4y ago

Since you capitalized it, it looks like a static method on the “G” class. There is nothing OO about static methods - this is purely and organizational construct.

cft 4y ago

I think it's the superclass A of both G and M that defines the eigenvalue() function. Then when both G(A) and M(A) inherit from A, it captures the mathematical relationship. The subclasses indeed should implement the eigenvalues computations differently, like in real mathematics. Funcional programming is used in Coq because it captures logical chains, for theorem PROOFS, but not for specific COMPUTATIONS, such as eigenvalues.

abdullahkhalids 4y ago
I understand that this is possible. But this is not how mathematicians think. If you got together a bunch of mathematicians and asked to invent a programming language to do mathematics, they would never ever invent the concept of a class. Mathematics (outside of stuff like category theory etc) is simply sets and mappings between sets. Anything more complicated is forcing the user to think in a way that mathematics does not.
- klyrs 4y ago
  > But this is not how mathematicians think. If you got together a bunch of mathematicians and asked to invent a programming language to do mathematics, they would never ever invent the concept of a class. Hi, mathematician here. Please read the source of SageMath: written by mathematicians, for mathematicians. Therein, you will find thousands of classes. Almost all computer algebra systems, written by mathematicians for mathematicians, have a notion of classes. > Mathematics (outside of stuff like category theory etc) is simply sets and mappings between sets. This is an extremely narrow-minded view of mathematics. Mathematicians thrive on abstraction, (category theory is literally mathematics; that's a very strange exception for you to carve out), and if we were to boil everything down to "simply sets and mappings between sets" then we'd be bogged down in utter tedium and nothing would ever get done. Object-oriented programming, specifically class hierarchies and inheritance, are extremely valuable for doing all sorts of math at high levels of abstraction. Hell. If I were to be especially pedantic, I'd point out that category theory itself is "simply sets and mappings between sets" except that sets aren't quite large enough so it's actually "categories and mapping between categories".

CuriouslyC 4y ago

Since the eigen decomposition is specifically a property of the matrix representation of the graph, but other things like shortest path distance between two nodes are naturally enough computed on graphs formed by objects with pointers, perhaps the correct interface is G.adjacency_matrix().eigenvalues().