I'm not sure quite what you want here. You seem to be distinguishing between Mathematica and Wolfram Language, but I think that's primarily a BUSINESS distinction, not a technology distinction. That is, Wolfram offers a variety of products (eg Wolfram Alpha, Wolfram Player, or various high-school-targeted mini-Mathematica's) that cost less than MMA, and some (like various engineering or finance simulators) that cost a lot more. But they all basically run on Wolfram Language, so which you use depends on how much a level of functionality is worth to you.
As for MMA, I use it in multiple ways (and like any single human, I use probably 1% of its capacities!)
One obvious path is my Volumes 1..7 describing how Apple M-series chips work.
I've written a few hundred pages in LaTeX (via LyX) and MMA is *mostly* much easier to use for technical writing. LyX is not bad, but MMA makes it much easier to embed graphics or URLs without going out of your way. OTOH LyX makes it easier to embed footnotes (if you care about) and *much* easier to create a "structured" document (ie section 1, section 2, subsection 2.1, etc). It's easy to get the structure w/ MMA, but a real hassle to jump between structural elements unless you add some outside functionality [which has to be re-initialized every time you re-open a document].
But putting aside the big structure stuff, inputting math in MMA is much easier (so much so that I rejiggered my LyX to allow for MMA-style input of symbols, greek letters, sub-and super-scripts, etc). You can even create a full DSL (input method, display method, etc) for a particular domain. I did this for quantum mechanics so that I could enter proper QM (operator manipulation, not just numerics) that looked like math, not like a programming language, and it worked remarkably well, allowing me ultimately to prove (or rather have MMA do all the hard work of proof!) that the Kepler force law has a "four dimensional spherical symmetry" which is ultimately why hydrogen orbitals with different angular momenta [of course in the Schrodinger approximation] have exactly the same energies, the QM equivalent of the fact that Kepler orbits are perfect ellipses. (Why this remarkable? Well, simulate the classical orbits for a different force law, eg F=1/r^3. You'll get something like a spirograph pattern. The radial motion is periodic, and the angular motion is periodic, but they have DIFFERENT periods.
None of this is novel, I just wanted to explore it my way. If it interests you, you can find pointers here:
Hidden Symmetries of the Hydrogen Atom
Here’s the math colloquium talk I gave at Georgia Tech this week: • Hidden symmetries of the hydrogen atom. Abstract. A classical particle moving in an inverse square central force, like…
johncarlosbaez.wordpress.com
This allows you to write craziness like this below. (Everything in orange-brown is a quantum operator, with all that implies. The money shot, after lots of definitions, is that the R operators [rotations in the 4th dimension] behave and commute like the traditional L operators, so we have an additional "rotation-like" symmetry.
The fact that the + and - operations are too large and have a slightly different shade of orange is a bug. At the time I did this a few years ago I thought it was a bug in MMA, so gave up on trying to fix it. But it's persisted for a few years, so at some point (hopefully helped out by Wolfram Assistant!) I'll probably get into the guts of the DSL and try to fix it.
But the real thing with MMA is that it makes exploring any problem so damned easy! You can see how my M1 PDFs I could easily import data from a file as a multi-rank array, easily extract sub-arrays, easily plot those, change plot options, and see patterns. I could easily create a simulation for my hypothesis for how L1 TLB worked and compare the simulator results to real-world measured values. When playing with physics I can write down a differential equation then maybe solve it exactly. Or have MMA solve it numerically. Either way I can then again easily plot the results, possibly in 3D. Or I can compose them in some other fashion (eg threat the solution of differential equation as a generic function to be used for some other task, anywhere a function can be used). I can just as easily create Manipulate's, ie custom panels with controls, so that I can modify various parameters and see how something changes as the parameters are moved around etc etc.
For example I wanted to write an article on differential forms (the geometry more than the algebra). These are a generalization of what you think of as grad, div, and curl, but
- they work in any number of dimensions not just three
- once you see them explained geometrically, it's obvious why there are "natural" differential operators on fields, while other expression you might write down are not natural (ie don't make geometric sense)
- why there's a relationship (in any number of dimensions) between a particular differential operator and a particular line/area/volume type integral (eg Stokes' or Gauss' theorem).
I can't imagine any other software where I could fairly easily just create one graph/image after another of the necessary images (things like a network of curved co-ordinate axes, with the quadrants between some of them colored, or things that look like stacked egg crates of different sizes) with varying transparency for different elements, or parts of the 3D image cut out so you can see inside, or suchlike, something like this:
You aren't expected to understand the above diagram without help! but even so you can fairly easily see that it's constructed by "sheets" that are interleaved, a yellow set with some transparency, and a red set, with the beginning on each red sheet marked by a blue line.
Believe it or not, this picture (properly understood and explained!) is a visual proof of Stokes' theorem. Imagine say a circle drawn on the diagram cutting through the sheets. The count of blue lines piercing the interior of the circle is effectively the integral over the disk of the curl, while the number of sheets pierced by the circle (taking account of "negative" vs positive" piercing depending on the orientation of the circle relative to the sheet) is the line integral around the boundary of the disk. And the two are equal -- numbers of sheets pierced equals number of blue lines representing the start of a new sheet not connected to a previous sheet...
Done right this generalizes to any number of dimensions in any dimensional space!
This was all worked out by Cartan in the early 20th C and written up in his usual (UTTERLY incomprehensible, totally algebraic!) style. It was made more comprehensible with some geometric explanation in the 50s and 60s, but still hasn't, IMHO, been explained well at the undergrad level.
There are plenty of other tools superficially in this vein, from free (eg R or Octave) to cheaper (eg Matlab). But none of them have anything close to both the depth of MMA and the deep thought that has gone into it, not just the functionality, but also the composability, and even "trivial" things like the names of functions or the default colors used by graphs and in the UI. You may (or may not) know that for years now Wolfram's internal review sessions have been recorded and uploaded to YouTube: this URL will give you a (somewhat randomly ordered list) you can explore
There are about 900 of these now, each one a review session of Stephen Wolfram with some group about the ongoingd design of some feature. It's worth watching at least a few of them. Sometimes the discussion goes off into the mathematical (or MMA internals) weeds, but mostly it's about "I tried this new functionality. It works well in this way. But it's dumb in that way. The UI is inconsistent in these ways. I don't like the word you used for this function because it can be confused with <piece of functionality that was added 12 years ago>." etc etc. It's very impressive how organized it is, and how competent everyone involved is.
Is this worth about $200 a year to you? I don't know, maybe not. It's worth it to me!
Really it boils down to how often a year do you want mathematical functionality? To draw a complicated 3D graph or 3D image? To solve a differential equation? To create a stochastic simulation? Do you find something like Excel or Numbers is adequate for those tasks or not even close?
Even if my sort of "physics" math doesn't appeal to you, there's a massive amount of image processing in there that some find useful! Or the ability to create and experiment with, say, neural networks. Obviously you can, eg, create neural networks in Python for free. But can you easily extract elements along the way to draw graphs to test, eg, hypotheses you may have about how activations are changing during training? Or how they correlate from one layer to the next? Or easily to run two nets side by side, one slightly modified relative to the other, and compare how activations differ in the two cases?
I don't follow the ins and outs of exactly what plans are available and how they differ; but I know that you can get the basic Home & Hobby plan for like $200 a year, and if you can legitimately show you're a student (don't know how strict they are about you having to prove this) for $75 a year.
