Towards a new SymPy: part 2 – Polynomials
oscarbenjamin.github.ioThis is only tangentially related to the post, but one thing I feel sore about is the lack of consolidation of computer algebra efforts. A computer algebra system is one of the most difficult pieces of software to build and make/keep useful, yet it continues to be reinvented decade after decade. (All while Mathematica in the background continues to improve unboundedly.) Seriously, it's easy to pour person-centuries of time into a CAS and still not achieve state-of-the-art. We have Maxima, Axiom and its two (!) forks, SymPy, REDUCE, Symbolics.jl, Mathics, SageMath, FORM, Yacas, Xcas, etc., but there's been no clear leadership to really make for a clear and unequivocal open-source success story. Symbolics.jl is the latest one to join the pack, starting in 2021, while Maxima began its life in 1967—a whopping span of five decades. From this blog post and its predecessor, it's clear that SymPy needs an overhaul at its core. (I personally believe building a CAS on Python is a mistake and predestined for eventual failure.)
If I could wave my magic wand, I'd have Axiom (or one of its forks) be the "winner" that everybody rallies around, with a change in leadership around it. It has the most advanced math, and it was built by the titans of computer algebra research. To this day, despite being developed by a skeleton crew, it does math no other "more modern" CAS can do. And it's built on a sane language that will never die.
The world deserves a robust, supported, high-performing CAS. Computer algebra could be as ubiquitous to computation as ordinary arithmetic, but everything non-commercial just falls short for general use.
Honestly I think no single system of writing is powerful enough to handle all the different versions of algebra. Mathematics can be extremely concise but only by constantly omitting stuff or overriding the meaning of symbols.
There are some happy accidents like df/dx making sense as both a derivative and a division of 1-forms, but on the whole you can't pick a single notation and hope for it to work well in all situations, yet that seems to be what most algebra systems do out of necessity.
To fix this I reckon we'll need to standardize not the automated algebra part but rather get those systems to communicate using some standardized format. Which is nigh impossible because of the sheer variety of mathematical concepts, but one can hope.
I don't think it's as difficult as you think it is. It's still a lot of work to implement mind you, but not nearly impossible. I talked about Mathlib [0] in a reply to the grandparent comment, which is a library of mathematical theorems and proofs that has a big community effort behind it. The language they use to state theorems would help a lot in making mathematics machine-readable enough for a computer algebra system to be able to make sense of it.
I'm hoping to build a prototype for this sometime in Julia's `Symbolics.jl` CAS. It would be just a simple proof of concept to implement derivative and integral rules from Mathlib or some custom maths library, and get them into the CAS. Symbolics.jl seems to be a perfect fit for this since it's made to be extensible, and Julia's metaprogramming would be a welcome addition as well. Either way, I would love to see this become part of the CAS ecosystem.
> We have Maxima, Axiom and its two (!) forks, SymPy, REDUCE, Symbolics.jl, Mathics, SageMath, FORM, Yacas, Xcas, etc.
Can these efforts be combined into a high quality OS library? The license would permit use from commercial applications like MATLAB. There is precedent in libBLAS.
Isn't SageMath based on SymPy?
Sage uses sympy for many symbolic things (and maxima for various others). In particular, sage doesn't contain a complete reimplementation, but tries to glue these other tools together.
SageMath is a python frontend to a LARGE number of math backends, not just SymPy.
> And it's built on a sane language that will never die.
It looks like lisp... which I have to say is not used much these days.
It's been on my mind lately as well. I was trying out `symbolics.jl` (a CAS written in Julia), and it turned out that it didn't support symbolic integration beyond simple linear functions or polynomials (at least back then, things have changed now it seems). Implementing a generic algorithm for finding integrals is hard, but I was expecting more from that CAS since this seems to be implemented in most other CASs. The thing is that every single CAS that covers general maths knowledge will have to implement the same algorithm, while it's hard to do it even once!
I feel like at least a large part of the functionality of a general purpose CAS can be written down once, and every CAS out there could benefit from it, similar to what the Language Server Protocol did for programming tools. They also had to rewrite the same tool for some language multiple times because there are lots of editors out there, and the LSP cut the time investment down a lot. They did have to invest a large amount of time to get LSP up and running, and it'll have to be maintained, but I think it's orders of magnitudes more efficient than having every tool developed and maintained for every single (programming language, editor) pair out there.
Main problem is like you said how to write down mathematical knowledge in a way that all CASs can understand it. I've been learning about Mathlib lately [0], which seems like a great starting point for this. It is as far as I know one of the first machine readable libraries of mathematical knowledge; it has a large community which has been pushing it continuously forward for years into research-level mathematics and covering the entire undergraduate maths curriculum and it's still accelerating. If some kind of protocol can be designed to read from libraries like this and turn it into CAS code, that would be a major step towards making the CAS ecosystem more sustainable I think.
It's not exactly what you were talking about, as in, this would allow multiple CASs to co-exist and benefit from each other, but I think that's better than having one massive CAS that has a monopoly. No software is perfect, but having a diverse set of choices that are open source would be more than enough to satisfy everyone.
(I have posted about this before [1] on the Lean Zulip forum. It's open to everyone to read without an account)
[0] https://leanprover-community.github.io/ [1] https://leanprover.zulipchat.com/#narrow/stream/113488-gener...
Also Oscar.jl, Macaulay2, Singular.
Did you consider writing parts in Rust?