There is a benchmark of Sympy vs Mathematica at
https://www.12000.org/my_notes/CAS_integration_tests/reports...
The results were Mathematica failed to solve 1,523 problems, Sympy failed to solve 48,529.
So it has some catching up to do.
The results were Mathematica failed to solve 1,523 problems, Sympy failed to solve 48,529.
So it has some catching up to do.
Additionally, SageMath (which depends on SymPy) is the more comparable product (and is open source).
Why does that matter?
2) able to be around if a single CEO isn't around (Wolfram)
3) able to continue if the supporting company is not profitable anymore
4) possibility of greater oversight if popularity rises
5) extensible if one puts the effort into it
This is like opining that the best "car" out there is a gokart you can get complete schematics on, for all of these reasons. I think most of us would accept the argument that the better cars are the ones that pass metrics aimed at cars. In this analogy, the better algebra system is the one that does the most algebra.
Additionally, Mathematica is not really that expensive.
You can try to broaden it to saying it is a better vehicle for you. And, sure, for a lot of folks the cost will be important there. As a CAS, though, Mathematica is tough to beat.
I'm not arguing that SymPy is going to beat Mathematica on benchmarks. But if both of them meet your needs, and you like having money and/or control of the code, SymPy wins.
Similarly, a lamborghini is almost certainly an objectively faster car. Such that if you were discussing fast vehicles and someone pointed out that their ebike was good enough for them, it would be a statement out of nowhere that is not using the rubric for ordering that was being discussed. Are they wrong that the ebike is a better choice for them? Almost certainly not. Would it be valid to say that it is the best fast vehicle because of that? (I say this as someone that loves bikes and is fairly anti car...)
And there would be other rubrics that would shine light in either direction regarding python. Arguably, the stewardship of the language lost a lot of trust with people in the hilariously bad 2->3 migration. More so in how bad dependency management has become. Yes, you can roll your own, but people with large support contracts can almost certainly offload a lot of that to the team on Mathematica, if that is truly a concern.
(I could similarly cast shade on Mathematica, but I think my point is made. Yes, you can have a rubric that changes which is the better choice for a situation. No, there is no total ordering of correct choices.)
I also think the math systems will lag for more than just donations. The work to make a good CAS is pretty intense. A lot like a good SAT system. Or really anything that is deep in the weeds of computer science. A lot of us are so far removed from the math that they focus on, that it can be mind bending to try and get back into it. (Indeed, for a lot like me, we were probably never really great at it, in the first place.)
Mathematica and Matlab are interesting to consider, as they are likely very well integrated into older workflow systems from the mainframe era. In particular, I'd expect the high end simulations for car and vehicle designs are much more integrated with those than anything open source. And a lot of that is largely availability of what they are integrating with. Most of us do not have the science labs and all of the equipment that goes with it.
Which, I think, is a bad feedback loop on this. For folks without those labs, Mathematica/Matlab are prohibitively expensive. For those with the labs, they are probably a rounding error. And there is no real path from the current equilibrium to one that can get it to more people. (The old path was free access in college. But that is becoming less of a thing in modern programming jobs.)
I ask because I always see open source thrown around as if it's some paragon of quality and productiveness. In reality, the actual usefulness of a product is fairly independent of its open source status. And rarely does it matter all that much to a project that a software component is open source or not.
I have also modified and extended open source implementations in sage to work with cases I needed. And I've added some of this back to sage.
It is undeniable that Mathematica evaluates crazy integrals better than most other tools. But it will happily output complete nonsense. And you can't check!
Do you actually do this verification? How do you accomplish this? The software stacks are huge. Why do you trust other people over the people who develop Mathematica, who just happened to be paid?
It is also true that, just like with a generic math research paper, that I don't check every claim of every step of every implementation of every algorithm in the process. But checking is possible, and when we find errors (which we do frequently) we can look and try to explain what it happening.
But when we find errors in tools such as Mathematica, we cannot. We report the errors and then know nothing more. (And sometimes the errors are never fixed).
As an open source project depending on volunteers (or is it just the one major author?) I am impressed that sympy does as much as it does.
Not that I want to dispute this, but depending on what you meant, there is in fact such an algorithm: https://en.wikipedia.org/wiki/Risch_algorithm
Though often it is not implemented because it is quite complex (its details covering two thick books) and many of the special cases it covers rarely crop up in the real world, so the effort isn't worth it.
The caveat of Risch's algorithm is that it only "works" if the function you are trying to integrate has an elementary antiderivative. Many of the problems that Mathematica can solve (but SymPy fails at) involved special (i.e. non-elementary) functions.
I imagine the difference is even bigger in things like solving ODE's/PDE's.
Nice. The PhDs just need take care their contributions aren't misappropriated. See https://en.m.wikipedia.org/wiki/Rule_110
By comparison, open source projects are developed by people with a wide range of knowledge level and commitment, and you simply can't expect the quality to be the same.
I find that discussions on HN often fail to acknowledge that proprietary software is usually extremely good at their domain, and what companies put into UX, support and the development/feedback loop are actually very valuable.
Thank you for allowing me to disambiguate.
I have not met any developers for either of these products but I know that SymPy has a huge list of contributors for a project of its size. See: https://github.com/sympy/sympy/blob/master/AUTHORS
You may not be hearing about SymPy users because SymPy is not a monolithic product. It is a library. If you know mathematicians big into using Python, they are probably aware of SymPy as it is the main attraction when it comes to symbolic computation in Python. They wouldn't necessarily spit out a bunch of libraries in the same breath as "I use Python."
1261 is an impressive number of contributors. I am interested to see if I could round up some people to hack up some of these test failures.
In the same vein, I was expecting SymPy to be like 80% of Mathematica but the given benchmark says it's about 25%. So I was suprised.
And I'm not thinking about UX, support, etc. which are indeed not often very good because, I guess, people prefer to put their energy in things that have the bigger leverage.