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> But it’s mathematical capabilities in some cases, and certainly it’s mathematical potential, outscales Mathematica. ... > Mathematica’s abilities, for insta

> But it’s mathematical capabilities in some cases, and certainly it’s mathematical potential, outscales Mathematica. ... > Mathematica’s abilities, for instance, in abstract algebra are seriously lacking. (In this instance, GAP (open source) exceeds Mathematica pretty seriously.) For someone who is not familiar with FriCAS, can you give an example or two (assuming a math background)? I'm interested in specific cases where open source tooling outperforms Mathematica.

thechao 8y ago
I only have experience with OpenAxiom & Axiom; however, the core definition language (Spad) is a dependently typed module-centric modern language... from 1981. It’s like an alternative “what if” language designed by mathematicians to write CASes. It has the oldest implementation of list- and set- comprehensions I know of. (Disclosure: I wrote a paper about adding JET-based forward AD using single-definition generics to the language.) The packages contain a complete implementation of the Risch algorithm; I’m not sure Mathematica has managed that.
- qubex 8y ago
  According to [0], ”For indefinite integrals, an extended version of the Risch algorithm is used whenever both the integrand and integral can be expressed in terms of elementary functions, exponential integral functions, polylogarithms, and other related functions.” [0] http://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.html#23196 http://reference.wolfram.com/language/tutorial/SomeNotesOnIn...
  - reikonomusha 8y ago
    Most CASes use some form of Risch, but there are various levels of implementation. Axiom is the only known CAS that implements all of the mathematics for every case of Risch, the so-called “algebraic” and “transcendental” cases. Wolfram’s page uses vague language (“an extended form of”) and doesn’t actually confirm with precision what it is able to do.
fdej 8y ago
There are more specific cases where open source tooling outperforms Mathematica than where the opposite holds. This especially true when you move away from basic calculus and towards research math. For example, many number theorists don't care about Mathematica at all because it has virtually no functionality for modular forms and L-functions which Pari/GP and Sage on the other hand support very well. However, nothing open source currently comes close to Mathematica for symbolic and symbolic-numerical calculus (expression simplification, symbolic integration, solving PDEs, etc.). Also, one of Mathematica's main strengths is providing a lot of functionality in a neat package where diverse components are very easy for the user to glue together, with automagical interfaces where the user doesn't need to know what happens behind the scenes (or at least that's how Wolfram PR spins Mathematica being closed source). Meanwhile, Sage combines a lot of functionality but you can't go very far before you run into visible duct tape, API inconsistencies and things that don't quite work together automatically when you think they ought to. Its documentation is also not as good (Sage has very detailed documentation, but often not well structured). Many of the same issues that exist in open source software in other fields, really.

thechao 8y ago

I only have experience with OpenAxiom & Axiom; however, the core definition language (Spad) is a dependently typed module-centric modern language... from 1981. It’s like an alternative “what if” language designed by mathematicians to write CASes. It has the oldest implementation of list- and set- comprehensions I know of. (Disclosure: I wrote a paper about adding JET-based forward AD using single-definition generics to the language.) The packages contain a complete implementation of the Risch algorithm; I’m not sure Mathematica has managed that.

qubex 8y ago
According to [0], ”For indefinite integrals, an extended version of the Risch algorithm is used whenever both the integrand and integral can be expressed in terms of elementary functions, exponential integral functions, polylogarithms, and other related functions.” [0] http://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.html#23196 http://reference.wolfram.com/language/tutorial/SomeNotesOnIn...
- reikonomusha 8y ago
  Most CASes use some form of Risch, but there are various levels of implementation. Axiom is the only known CAS that implements all of the mathematics for every case of Risch, the so-called “algebraic” and “transcendental” cases. Wolfram’s page uses vague language (“an extended form of”) and doesn’t actually confirm with precision what it is able to do.

fdej 8y ago

There are more specific cases where open source tooling outperforms Mathematica than where the opposite holds. This especially true when you move away from basic calculus and towards research math. For example, many number theorists don't care about Mathematica at all because it has virtually no functionality for modular forms and L-functions which Pari/GP and Sage on the other hand support very well. However, nothing open source currently comes close to Mathematica for symbolic and symbolic-numerical calculus (expression simplification, symbolic integration, solving PDEs, etc.). Also, one of Mathematica's main strengths is providing a lot of functionality in a neat package where diverse components are very easy for the user to glue together, with automagical interfaces where the user doesn't need to know what happens behind the scenes (or at least that's how Wolfram PR spins Mathematica being closed source). Meanwhile, Sage combines a lot of functionality but you can't go very far before you run into visible duct tape, API inconsistencies and things that don't quite work together automatically when you think they ought to. Its documentation is also not as good (Sage has very detailed documentation, but often not well structured). Many of the same issues that exist in open source software in other fields, really.