I have a theory, which I am in the process of writing about in a blog post, that other than in applications in mathematics (symbolic logic), Mathematica's primary usefulness is in encouraging a kind of intellectual dilettantism. (Of course, one man's dilettantism is another man's New Kind of Science: i.e. the speculative endeavors of a fertile, creative intellect might readily be mistaken for actual (scientific) achievement by less agile minds.)
I don't want this conjecture to be true: I'm as much a buyer of the hype around Mathematica as the next man. But I have noticed that, in practice, Mathematica somehow fails to live up to its apparently unlimited potential for encapsulating creative thought-product across an almost unlimited span of human intellectual endeavor. In my own work in finance, for instance, it has generally proved much less useful than other products such as Matlab. And, in general, when I look at the examples cited by Wolfram in its "customer stories", my reaction to many of them is: "Sure, you can use Mathematica to do that. But why would you, when there are much better alternatives available?". To take one such customer story from the field of 3D CAD, I don't understand why anyone would prefer to use Mathematica for such a task, rather than a specialist product like Solidworks.
I am aware of applications where the use of Mathematica is fully justified. In my own work, I have used Mathematica to price complex derivatives products, a field in which it excels. Likewise, I am somewhat familiar with Phil Zecker's work at EQA Partners, where he produced an outstanding risk management solution using Mathematica. In both these cases, however, there is no standard, specialized alternative offering in the field of risk management, as there are, for example, in engineering, CAD, or music. In these areas, it seems to me, Mathematica is like a Swiss Army Knife: sure you can use it to dabble in almost anything; but I have screwdrivers, knives and corkscrews that do a better job for their specific purpose.
In any event I am looking for counter-examples of real-world applications to refute my hypothesis about Mathematica. By "real-world" I mean specifically applications in which:
(i) money changed hands (e.g. a commercial product was sold, or consulting fee earned) ; and
(ii) an alternative solution was considered and Mathematica preferred for specific reasons ("it was the only software we could afford or that was available to tackle the job", is not a valid reason to qualify the application as real-world, according to this definition)
The secret weapon that has allowed us, and no one else, to assemble such a vast library of algorithms, in such a diverse range of fields, is Mathematica.from the-secret-behind-the-computational-engine-in-wolframalpha also half of Mathematica itself is written using Mathematica. I think about 50% is C/C++ and the other half is in Mathematica. This might have changed in recent versions. I remember reading this sometime ago. $\endgroup$