Much of biology has become mathematically inclined over the past 40 years. Early on, Biology departments started teaching their own mathematical analysis courses - in part because math departments refused to add pair-joining and pseudo-metrics to their course curricula. This has not been the case with other math-oriented majors - they require their undergraduates to take the junior level numerical analysis course, and their graduate students to take applied math analysis + numerical analysis II.

Want to help me make a new and better version in WL?

There are two additional items I should have considered in my earlier presentation of what I considered to be the key features of Mathematica/the Wolfram tech stack, but which I neglected to discuss properly.

The first is such an integral part of Mathematica that it is easy to overlook - and yet it is one of its core features and a cornerstone of the technology, i.e., the Notebook. This concept has been copied by others (e.g., Python), but no one has ever surpassed Wolfram in the functionality and flexibility of its notebook technology.

The second feature that is gradually becoming more and more important is computational data. This is something that I believe Stephen has, from the outset, regarded as being integral to the Mathematica product, as demonstrated by the integration of Wolfram Alpha and the development of the Entity Store concept.

My sense is that, at the beginning, when computational data is limited, this feature looks rather unimportant, and only over time does its real significance become apparent to the end user. An analogy might be made with Wikipedia. At the beginning, when articles were scarce and of questionable quality, the enterprise looked doomed to failure. But gradually, over time, more users discovered the site and perceived how useful a universal online encyclopedia might become. This drove the creation of more, higher-quality articles, which attracted more users, and so on. There is a concept in Systems Dynamics that describes this phenomenon, known as the "success to the successful" paradigm. This is what I believe is now beginning to happen with Mathematica/Wolfram Alpha, although the rate of development and user acceptance is slower than for Wikipedia as the technology is proprietary.

In my experience, a proprietary solution can be worth having. CDC had a great Algol compiler and the Cray Research Fortran compiler was the best bar none. Every now and then I see a bit of that style in the Mathematica error diagnostics.

I don't enjoy reading requests for a less proprietary WL platform in the same paragraphs with requests for internal details of algortihms. If you want to know what was method was used you can look it up in the documentation. Beyond that seems to be some sort of claim of entitlement that does not exist.

Fifty years ago you would have had to write, perhaps: "a technically educated person should be writing with a fountain pen, doing reports using a pocket calculator and hardly thinking at all about technical computing languages, unless they are computer scientists".

Fifty years ago I had a programmable HP calculator, was writing with a word processor on an HP portable, doing reports with Lotus 1-2-3 (spreadsheet) on the portable, and programming in octal, Algol, Macro (assembly language), Basic, and Fortran on time-sharing systems. I was not a computer scientist nor had I taken any C.S. courses in college. These activities were typical among my colleagues.

I wasn't referring to consumer products. We bought these systems from laboratory suppliers or the build-it yourself versions from Heathkit. Fountain pens were a rarity - we all used ballpoint.

We should leave Wolfram Alpha out of this discussion about programming languages. This is not a programming language at all, but a search or - if you like - answer engine. You can't do anything in it that could be called "programming".

You are right. Your question was "what is Mathematica good for".

I mistakenly narrowed that down to programming languages. Perhaps I was misled by the word "Mathematica". After all, I don't really know what the word means. What was obviously meant was "what is the Wolfram System good for"?

Of course, Wolfram Alpha is a very important and valuable part of the Wolfram System.

Every language is an algebra.

A couple of recent comments led me to review the original post and some of the replies from community members which, I have to say, I find rather interesting (although I suspect Stephen Wolfram would perhaps beg to differ). A thought on this subject that I have been entertaining for some time is the concept of Mathematica as a kind of computational Wikipedia. Perhaps, if it has any relevance at all, the comparison could be better made between Wikipedia and Wolfram Alpha, rather than Mathematica.

In any event, what I find interesting is that for a long time Wikipedia looked like a hopeless quest: in the early days many subjects were barely covered and the quality of the articles was often very poor. But then, amazingly, the whole idea took off and Wikipedia became the kind of indispensable reference that Jimmy Wales imagined it could become.

I think something similar may be happening with Mathematica (or, rather, Wolfram Alpha) as its scope has enlarged over time (and the quality of the reference material has always been excellent). My sense is that it is close to some kind of tipping point, on the verge of becoming the kind of indispensable computational resource that Stephen Wolfram had in mind from the outset.

Do others agree? Or take an altogether different view?

The universities I'm familiar with have site licenses, typically at the department or laboratory level. Also, the Mathematica pricing for students appears in line with textbook prices from my perspective.

The main hindrance I see for students is with department level inbred thinking that community-based free software is better in all regards. However, in biology these projects have become the largest source of research errors in modern history:

http://wireilla.com/papers/ijcsa/V12N4/12422ijcsa01.pdf

I'm aware of research units in university social sciences and liberal arts with Mathematica installations.

What about the cost of a Mathematica (desktop) student license in your country?

Given the prices of textbooks, I believe that is very reasonable.

Yes, I completely agree. I used MMA to help solve a very challenging problem in stochastic calculus many years ago and find it invaluable for scratch-pad type quant work, or more complex research in mathematical finance.

Indeed.

I had no idea! Is there a site you could point me to, in order to get a better understanding of the application?

Do you have any references for that? I'm curious.

Some comments from a long time programmer (coming up on my 50th anniversary) and Mathematica user (32 years):

First, Bernard Gress's comment that WL exists so that Stephen can get his projects done is truer than might be expected. Certainly, the best way to get feature 'x' implemented is to get Stephen interested.

Second, I think that there really is confusion about what Mathematica is for. I initially used it solely to solve maths problems, and for that it was, and is, superb. The first edition of the Mathematica book had the subtitle: "A system for doing mathematics by computer". Over the versions, the definition of what maths is has expanded, and a bunch of other stuff has been added.

The notebook interface is great for this type of investigation and interactive exploration.

However, it is rubbish at making stand-alone apps.

It is also far too hard to do computer-science or complicated programming. When I learned c, almost everything I needed to know was in the K&R book, some 125 pages or so, as I recall. Sure, you needed to understand the standard library, but the header files gave the information you needed. This level of documentation is missing in WL, although the general level of documentation otherwise is very good. With c, the step between the 'simple' stuff and a complex program was gradual and the simplicity of the core language made learning what to do relatively straight-forward. With Wolfram language, it is by no means clear what the core language is, and understanding it is not simple. This makes going beyond the basics much more difficult than it should be. At Apple in the early 2000s, they told developers that Xcode (and objective-c) made the regular stuff easy, and the hard stuff possible. With Mathematica, the mathematical functionality for interactive use is made very easy (compare to writing your own), but anything beyond that has been made much more difficult than it needs to be.

To give an example, it is not clear from the documentation how to pass a variable by reference -- if it is there, it is pretty well hidden.

Your comment about losing something when key people leave is very telling, especially for Mac users. When Theo Gray was involved, you could expect that Mathematica would make use of Apple technologies -- it was largely due to him that Mathematica remained a Mac program through the dark days. However, I am no longer feeling the love, specifically in the areas of UI and GPU utilization. As far as I can tell, Apple's neural engine is at least as good as anything based oil NVIDIA, yet it is not used at all for machine learning in Mathematica. Embracing other new OS features has been very slow -- it took far too long to move the front end to 64 bit, for example.

There is also the tendency to leave stuff at the 99% done level and move on to the next shiny object. There are simply too many bugs. There are too many similar functions that do pretty much the same thing, but which have subtle differences in the API or performance, and the documentation does not make that clear at all.

When I retired, I decided that I would let my skills in c (etc....) lapse in favor of doing all my coding in WL. I have come to regret this decision. When I use WL as part of my mathematical studies, I have (mostly) no problems. For anything else, it is a constant slog to achieve even moderate success.

I do have proposals for a solution:

1. Fix the bugs.

2. Market Mathematica (Wolfram|One, etc.) for what it does best -- interactive computation.

3. Fully implement the Wolfram engine so that it can be used as a resource when developing stand-alone code in pretty much any language for any platform. Specifically, this means making the engine work with iOS and iPadOS so that other developers can develop apps with a state-of-the-art UI and functionality, yet making use of the strengths of WL. (A similar case can be made for Android, as well.)

4. Make full use of the native capabilities of each hardware platform. This means making use of all the technologies in macOS (and iPad/iOS), Windows and Linux, although the most obvious mismatch is with macOS at the moment.

5. Institute a developer program (with nominal costs attached), so that developers will have access to essentially whatever or whoever tech support gets their undocumented answers from. I remember that the Wolfram Technology conference was originally called the Mathematica Developer's Conference.

6. As a corollary, provide some really state-of-the art developer tools. I tried using Eclipse, but compared to the tools that I used when developing Mac apps (Lightspeed c, Code Warrior, Xcode), the tool is clumsy and out-of-date. I have no experience in Java development or development on Linux or Windows; perhaps Eclipse is perfectly usable for these platforms. I know that discussing tools is likely to lead to a religious war -- after all there are those who think c++ is the best of all possible languages. However, if Wolfram Research wants to break out if its ghetto, it will need to integrate with modern IDEs, and Eclipse is not going to help with this (in my opinion, of course).

7. Consider making a new language. All older languages have had re-thinks about what the language means. C has had several successors, with different levels of success: c++, Objective-c, and Swift. While I am not fluent in Swift (yet) it seems to have captured the essence of what was good about c without the excesses of c++, and is easy to learn. A similar rethink of WL would be a really good idea. While many of the design objective of WL were good, not all were, and some of them are real howlers -- especially in the area of UI design and implementation. Done right, no old code would be obsolete (you can still use ANSI c in Swift), but things would be a lot easier both for typical users and experts. Of course, if Wolfram Engine were available for macOS and iPadOS, some enterprising person or group could re-implement WL from within Swift, so this might not be so far out after all. (Remember Swift is not proprietary -- and I am using that only as an example.)

When I first used Mathematica, I was developing (in-house) real time process control software for a large clinical lab. I would solve the mathematical problems and prototype graphs, etc., using Mathematica, then translate the results into c on a Macintosh. This combination worked extraordinarily well. If my suggestions were implemented, and I were doing the same thing today, I would do the prototyping in WL, then use the Wolfram Engine in a native program I would write in Swift for macOS and iPadOS that would make native use of Apple's Neural engine. This would result is a more useful and stable application than I could ever hope to do using WL alone.

So, to answer your question: what is Mathematica for?

In its current state, Mathematica is for interactive problem solving of primarily mathematical problems using the notebook interface.

I do not think it is effective when used for anything else (certainly by not anyone outside of Wolfram Research) : web deployment, etc. except as a way to publish notebooks interactively, seems to be a toy project.

With proper focus, as I outlined above, Wolfram Language could be a significant tool in the development and implementation of basically any type of scientific application.

Perhaps as you say the solution to the challenge facing WR is not to try to "fix" MMA to enable it to do production work, but to focus on the Wolfram engine instead...

Exactly. Of course, the Wolfram engine, as I understand it, is mostly the Kernel, plus some bits that ended up in the front end for historical reasons. So, it really isn't focusing on the Wolfram Engine as a separate project, but making sure that there is functional parity -- as well as full support for native hardware.

An interesting test case will be whether, as part of the port of the Wolfram Engine to Apple Silicon, it will also be available for iPadOS. If this turns out to be the case, then I can see a lot of apps that use the Wolfram Engine, but which use a custom UI. One of the issues (among many others) with porting Mathematica to iPad is the need to make the UI for the iPad version as close to the desktop as possible, a formidable task given two fundamentally different design principles at work.

I want to re-iterate that Mathematica and its notebook interface is perfect for a wide range of applications and probably is the best in class for the vast majority of Mathematica users. But it should not try to be all things to all people: the best hammer in the world is not much use if you need a wrench.