One hopes...
...that Apple is actually paying for this traffic.
A quarter of traffic to the intelligent computational engine Wolfram Alpha comes from Siri, Stephen Wolfram said in a New York Times article. The robot-loving physicist (who once said that all science needed to know could be found by studying the behaviour of cellular automata) launched the knowledgable super-brain in 2009. …
Yeah, we got an idiot teaching course on ODE "with Mathematica". What I do know is that Mathematica of Wolfram Research is extremely expensive. Being proprietary and non-free software it contradicts the principles of research. pari-gp, octave, maxima, R, gnuplot, gnu gsl -- even dc/bc and emacs calc-el can do more.
Does it do the job you need it to do, with less effort than required by alternatives? Then it is perfectly appropriate. Otherwise, the free tools you suggested are fine. Turning tool choice into a holy war seems rather pointless; making irrational decisions based on ideology should not be what research is about either. Stick to the facts, and let the GNU folk engage in the license zealotry.
The situation is that Mathematica if installed on collegian computers gets there via lobbying not by any other way. It is like with other types of software: people get used to it and do not know the free alternatives. And yes, in many cases, the non-lobbied counterparts turn out to be superior. The "zealotry" is simply against this type (and more ) of corruption.
And, yes you don't heavily mix an ODE course with any CAS (be it Mathematica, Octave, or anything else).
Please post specific evidence of said corruption. And quantitative,objective evidence of the "many" cases where 'non-lobbied' counterparts are better. (Remember, that doesn't exclude paid-for stuff. There's no assumed link to lobbying, despite your standard as yet unsubstantiated zealot claims.) If you can't answer both these, stick to editing Wikipedia articles. I'll keep an eye out for anything on these topics plastered with "needs citation" on every sentence.
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It's called "grants" nowadays. Similarly, do I have to mention all 10^10 cases when a professor requires edition (n+1) textbook only for his/her course, while the earlier editions are not allowed (esp., when exercises are being reshuffled) ?
And yes, whatever they do here with Mathematica, can easily be done with Maxima (symbolic) and gp (numerically). Base conversion is better to like this in the CL
dc -e '123245744o10 100 ^p'
000000814 025973445 064772530 065857018 091827062 122790577 118492576 089245045 100763723 067655994 119741617 070543017 109620944
gp is better in polynomials and algebra . You tell me what Mathematica can do and other CAS can't?
Please list those alternatives to Mathematica's symbolic computing engine again? R? Gnuplot?! Octave!? "even emacs calc-el can do more"?? Ah, indeed, Macsyma is in your list, and does part of what Mathematica can on that topic.
The simple statement is that you need to have the entire above list to cover most topics handled by Mathematica. So it's all different interfaces and languages, and on the whole each to less depth.
R would be a possible exception, but then R is an emulation of S.
If there are any R proponents here, I'm sure they will dispute this. R is indeed a re-implementation of S, but it has evolved hugely. AFAIK, development in S stopped sometime in the mid 1990's (I was using it in about 1987). I don't even know if S is in the AT&T Toolchest any more.
Of all of the tools listed in the OP, R/S is about the only tool that covers a significant proportion of what Mathematica can, and is the only one that can (creditably) claim to have pre-dated it. All of the others have been produced for part of what Mathematica can by those who could not afford it.
The only other thing that I came across that was used to do similar things was SPSS, but that was originally not a package in itself, but allowed you to do data manipulation more easily from inside other programs. I think that it has changed since I used it in 1980!