[omaranto]

Results obtained with are better because Sage is open: in a mathematical research paper you can say:

"This reduces the problem to computing blah, which we did using the following Sage code. The function foo used here uses the algorithm of X and Y as described in their paper [XY2006]"

You can't say:

"This reduces the problem to calculating blah which the following Mathematica code computes using an unspecified algorithm for which there is no accompanying paper proving correctness."

Of course a paper proving that an algorithm is correct can contain errors, and also even if the proof of correctness is fine the actual implementation can contain bugs. But if you have no way of knowing how something is computed and whether anybody at any point in time even tried to prove mathematically that the method used is correct, you have no moral authority to rely on the result. That's just the standard adopted in mathematics: you can depend on results you have good reason to believe are true and are documented in the literature; you can't rely on stuff that's not written about. I don't know how citing results works in other areas, but that's how it is in mathematics (at least in the fields of mathematics I'm familiar with).

[szhorvat]

I guess it depends on one's field.

I've never cited a software I didn't write myself, simply saying trust this software, here's the reference. I wouldn't trust a result from Sage any better than one from Mathematica, simply based on which system produced it. I'd trust that x is a solution of an equation if substituting it back verifies it. I'll trust that two graphs are isomorphic if the software gives me a vertex permutation that makes the adjacency matrices identical. It doesn't matter how that isomorphism was computed.

When publishing work, I'll aim to make it verifiable this way.

I believe the vast majority of the use of these systems is not of the type when one needs to blindly trust the software and refer back to it in the paper. At least in my field (physics) it isn't. Yet I use programs like this daily, and I clearly depend on them for my work.

Most of the functionality available in Mathematica (or, I'd argue, most similar systems) are not of the type that one needs to cite. They either use standard and well known algorithms that are available in a multitude of systems (do you cite the methods for matrix multiplication or eigenvalue computation, and would it make a difference?), or the results are much easier to verify than to compute.

In those cases when I need to rely on a published method, like you mention, the method is very unlikely to be a built-in part of any system. So I either need to re-implement it, or use the original code of the authors. If the authors implemented their method in Mathematica instead of Python, does that make their program less reliable? No. It's still a published method, anyone can verify it.

My point is that I hear this argument about Sage very often, and the typical generalization is: "if you used Mathematica for your research, that's wrong, because it's not verifiable". This is a fallacy. It completely ignores how these software are used in practice, and implies that results from open source software are somehow magically reliable (they're not) and don't need verification (they do).

I've yet to come across a situation where the argument does apply at all: point me to a paper which goes truly wrong by citing Mathematica/MATLAB/Maple/etc this way.

[szhorvat]

Note that I'm not saying that one can claim that a result is correct, a theorem is true, etc. based on the fact that some undocumented algorithm produced it. That's clearly unacceptable.

Nor am I saying that it's never necessary to rely on an algorithm to get such a result.

What I'm saying that when people use Mathematica or other closed source systems, they do not usually commit these mistakes.

Also note that Mathematica programs can be open source and documented (many are). Several built-in packages have accessible and documented source code (e.g. Combinatorica). There's nothing wrong with using these to obtain such a result, and cite the (public and documented) program used to create it.

[omaranto]

I think we're in complete agreement: most of the time you don't need a citation for a program you use because the result can be easily verfied. And I agree that what you call a fallacy is a fallacy, I was just pointing out that open source can be citable in a way that closed source isn't and that that is an advantage.

[jevinskie]

People are apt to discount Mathematica completely. I don't see any problem using Mathematica to generate some results because the author prefers Mathematica over other offerings. However, it would give the results much more credence if the Mathematica-obtained result was then replicated using open software.

I would expect that the difficulty of the port could vary widely between different use cases.