PM. This is a short essay, that has been included in the May 23 2018 update for this notebook and package:
http://community.wolfram.com/groups/-/m/t/1313302
Mathematics concerns patterns and can involve anything, so that we need flexibility in our tools when we do or use mathematics. In the dawn of mankind we used stories. When writing was invented we used pen and paper. It is a revolution for mankind, comparable to the invention of the wheel and the alphabet, that we now can do mathematics using a computer. Many people focus on the computer and would say that it is a computer revolution, but computers might also generate chaos, which shows that the true relevance comes from structured use.
I regard mathematics by computer as a two-sided coin, that involves both human thought (supported by tools) and what technically happens within a computer. The computer language (software) is the interface between the human mind and the hardware with the flow of electrons, photons or whatever (I am no physicist). We might hold that thought is more fundamental, but this is of little consequence, since we still need consistency that 1+1 = 2 in math also is 1+1 = 2 in the computer, and properly interfaced by the language that would have 1+1 = 2 too. The clearest expression of mathematics by computer is in "computer algebra" languages, that understand what this revolution for mankind is about, and which were developed for the explicit support of doing mathematics by computer.
The makers of Mathematica (WRI) might be conceptually moving to regarding computation itself as a more fundamental notion than mathematics or the recognition and handling of patterns. Perhaps in their view there would be no such two-sided coin. The brain might be just computation, the computer would obviously be computation, and the language is only a translator of such computations. The idea that we are mainly interested in the structured products of the brain could be less relevant.
Stephen Wolfram by origin is a physicist and the name "Mathematica" comes from Newton's book and not from "mathematics" itself, though Newton made that reference. Stephen Wolfram obviously has a long involvement with cellular automata, culminating in his New Kind of Science. Wolfram (2013) distinguishes Mathematica as a computer program from the language that the program uses and is partially written in. Eventually he settled for the term "Wolfram language" for the computer language that he and WRI use, like "English" is the language used by the people in England (codified by their committees on the use of the English language).
My inclination however was to regard "Mathematica" primarily as the name of the language that happened to be evaluated by the program of the same name. I compared Mathematica to Algol and Fortran. I found Wolfram's Addison-Wesley book title in 1991 & 1998 "Mathematica. A system for doing mathematics by computers" as quite apt. Obviously the system consists of the language and the software that runs it, but the latter might be provided by other providers too, like Fortran has different compilers. Every programmer knows that the devil is in the details, and that a language documentation on paper might not give the full details of actually running the software. Thus when there are not more software providers then it is only accurate to state the the present definition of the language is given precisely by the one program that runs it. This is only practical and not fundamental. In this situation there is no conflict in thinking of "Mathematica as the language of Mathematica". Thus in my view there is no need to find a new name for the language. I thought that I was using a language but apparently in Wolfram's recent view the emphasis was on the computer program. I didn't read Wolfram's blog in 2013 and otherwise might have given this feedback.
Wolfram (2017) and (2018) uses the terms "computational essay" and "computational thinking" while the latter is used such that he apparently intends this to mean something like (my interpretation): programming in the Wolfram Language, using internet resources, e.g. the cloud and not necessarily the stand-alone version of Mathematica or now also Wolfram Desktop. My impression is that Wolfram indeed emphasizes computation, and that he perhaps also wants to get rid of a popular confusion of the name "Mathematica" with mathematics only. Apparently he doesn't want to get rid of that name altogether, likely given his involvement in its history and also its fine reputation.
A related website is https://www.computerbasedmath.org (CBM) by Conrad Wolfram. Most likely Conrad adopts Stephen's view on computation. It might also be that CBM finds the name "Mathematica" disinformative, as educators (i) may be unaware of what this language and program is, (ii) may associate mathematics with pen and paper, and (iii) would pay attention however at the word "computer". Perhaps CBM also thinks: You better adopt the language of your audience than teach them to understand your terminology on the history of Mathematica.
I am not convinced by these recent developments. I still think: (1) that this is a two-sided coin (but I am no physicist and do no know about electrons and such), (2) that it is advantageous to clarify to the world: (2a) that mathematics can be used for everything, and (2b) that doing mathematics by computer is a revolution for mankind, and (3) that one should beware of people without didactic training who want to ship computer technology into the classroom. My suggestion to Stephen Wolfram remains, as I did before in (2009, 2015a), that he turns WRI into a public utility like those that exist in Holland - while it already has many characteristics of this. It is curious to see the open source initiatives that apparently will not use the language of Mathematica, now by WRI (also) called the Wolfram Language, most likely because of copyright fears even while it is good mathematics.
Apparently there are legal concerns (but I am no lawyer) that issues like 1+1 = 2 or [Pi] are not under copyright, but that choices for software can be. For example the use of h[x] with square brackets rather than parentheses h(x), might be presented to the copyright courts as a copyright issue. This is awkward, because it is good didactics of mathematics to use the square brackets. Not only computers but also kids may get confused by expressions a(2 + b) and f(x + h) - f(x). Let me refer to my suggestion that each nation sets up its own National Center for Mathematics Education. Presently we have a jungle that is no good for WRI, no good for the open source movement (e.g. R or https://www.python.org or http://jupyter.org), and especially no good for the students. Everyone will be served by clear distinctions between (i) what is in the common domain for mathematics and education of mathematics (the language) and (ii) what would be subject to private property laws (programs in that language, interpreters and compilers for the language) (though such could also be placed into the common domain).
Colignatus, Th. (2009, 2015a), Elegance with Substance,
(1) website: http://thomascool.eu/Papers/Math/Index.html
(2) PDF on Zenodo: https://zenodo.org/record/291974
Wolfram, S. (1991, 1998), Mathematica. A system for doing mathematics by computer, 2nd edition, Addison-Wesley
Wolfram, S. (2013), What Should We Call the Language of Mathematica?, http://blog.stephenwolfram.com/2013/02/what-should-we-call-the-language-of-mathematica/
Wolfram, S. (2017), What Is a Computational Essay?, http://blog.stephenwolfram.com/2017/11/what-is-a-computational-essay/
Wolfram, S. (2018), Launching the Wolfram Challenges Site, http://blog.stephenwolfram.com/2017/11/what-is-a-computational-essay/