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Mathematica has an army of mathematicians working on edge cases that only other mathematicians would care about. It's essentially impossible for any open source

Mathematica has an army of mathematicians working on edge cases that only other mathematicians would care about. It's essentially impossible for any open source software to match what mathematica does.

bowsamic 4y ago
Yep, this is why no other software can compete. People don’t realise that mathematica consists of huge lists of hard coded cases for different mathematical problems. It’s not like they managed to find the general formula for everything. And there are functions that I rely on that literally no other software has, that I haven’t even managed to find any papers that would even hint at their implementation, that magically work for almost any input. For example, MinimalStateSpace seems to be able to find the minimum required form for basically any input state space, which is highly non trivial. It’s a huge amount of work going into niche mathematical problems that even most people in STEM won’t have heard of. I don’t see how others can compete
- smithmayowa 4y ago
  This is so true for instance mathematica's treatment on numerical integration is the most complete I have seen out there so many methods that work on very minute and specific type of functions, this is even more so when you're working with functions that have singularities in them, and therefore need to be treated piecewisely, or functions that are highly oscillating. (https://reference.wolfram.com/language/tutorial/NIntegrateIntegrationStrategies.html https://reference.wolfram.com/language/tutorial/NIntegrateIn...)
  - bowsamic 4y ago
    Yeah and tbh that's even the simple stuff. Those are widely needed, very common situations in practical mathematical/physical calculations. The thing that is really crazy is that the stuff that literally doesn't even exist in any other software, but is really necessary, such as in my case a full symbolic control systems library, is still absurdly complete. That's why nothing can compete. Not only does Mathematica have an extremely complete implementation of common functionality, it even has an extremely complete implementation of very niche functionality. I don't think that the people who propose OSS solutions really appreciate this, probably because they aren't professional mathematicians/physicists
- bigbacaloa 4y ago
  Much other software can and does compete. Professional mathematicians use any number of other programs as well as simply writing their own.
  - bowsamic 4y ago
    No, it can't compete, because there will always be things that other software cannot do that _only_ Mathematica can. That's why sympy is a non-starter for many mathematicians and physicists
    - physPop 4y ago
      Maple is pretty close, better in some areas worse in others.
zozbot234 4y ago
Check out https://github.com/leanprover-community/mathlib https://github.com/leanprover-community/mathlib . Sure, it's not really a CAS but CAS algorithms could be added to it where applicable, since Lean is a constructive system and can thus express formally verified computations.
- Smaug123 4y ago
  The existence of https://github.com/robertylewis/mathematica https://github.com/robertylewis/mathematica makes me think it's more likely to go in the other direction, where you treat powerful tools as untrusted oracles which produce certificates that Lean then verifies.
  - zozbot234 4y ago
    Sure that's possible, but "untrusted oracles" are only useful if they provide easily-verified certificates. Some CAS problems are amenable to that approach, but not all. There's no difference in this case between Mma and, e.g. an external SAT/SMT solver.

bowsamic 4y ago

Yep, this is why no other software can compete. People don’t realise that mathematica consists of huge lists of hard coded cases for different mathematical problems. It’s not like they managed to find the general formula for everything. And there are functions that I rely on that literally no other software has, that I haven’t even managed to find any papers that would even hint at their implementation, that magically work for almost any input. For example, MinimalStateSpace seems to be able to find the minimum required form for basically any input state space, which is highly non trivial. It’s a huge amount of work going into niche mathematical problems that even most people in STEM won’t have heard of. I don’t see how others can compete

smithmayowa 4y ago
This is so true for instance mathematica's treatment on numerical integration is the most complete I have seen out there so many methods that work on very minute and specific type of functions, this is even more so when you're working with functions that have singularities in them, and therefore need to be treated piecewisely, or functions that are highly oscillating. (https://reference.wolfram.com/language/tutorial/NIntegrateIntegrationStrategies.html https://reference.wolfram.com/language/tutorial/NIntegrateIn...)
- bowsamic 4y ago
  Yeah and tbh that's even the simple stuff. Those are widely needed, very common situations in practical mathematical/physical calculations. The thing that is really crazy is that the stuff that literally doesn't even exist in any other software, but is really necessary, such as in my case a full symbolic control systems library, is still absurdly complete. That's why nothing can compete. Not only does Mathematica have an extremely complete implementation of common functionality, it even has an extremely complete implementation of very niche functionality. I don't think that the people who propose OSS solutions really appreciate this, probably because they aren't professional mathematicians/physicists
bigbacaloa 4y ago
Much other software can and does compete. Professional mathematicians use any number of other programs as well as simply writing their own.
- bowsamic 4y ago
  No, it can't compete, because there will always be things that other software cannot do that _only_ Mathematica can. That's why sympy is a non-starter for many mathematicians and physicists
  - physPop 4y ago
    Maple is pretty close, better in some areas worse in others.

zozbot234 4y ago

Check out https://github.com/leanprover-community/mathlib https://github.com/leanprover-community/mathlib . Sure, it's not really a CAS but CAS algorithms could be added to it where applicable, since Lean is a constructive system and can thus express formally verified computations.

Smaug123 4y ago
The existence of https://github.com/robertylewis/mathematica https://github.com/robertylewis/mathematica makes me think it's more likely to go in the other direction, where you treat powerful tools as untrusted oracles which produce certificates that Lean then verifies.
- zozbot234 4y ago
  Sure that's possible, but "untrusted oracles" are only useful if they provide easily-verified certificates. Some CAS problems are amenable to that approach, but not all. There's no difference in this case between Mma and, e.g. an external SAT/SMT solver.