Those "specialist alternatives" often contain modules pieced together by graduate students with no background in numerical analysis, and consequently led to a proliferation of errors spanning decades. Then by demand of specialists, some of the erroneous approaches have found their way into commercial software. Ecology has its share of this, including fantasies such as Bayesian Clustering.

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

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.

I invested in WL a long time ago, and resist learning other languages because WL is so productive.

Yeah, I'm in the same boat. I suppose I feel isolated a little, but not in the 'peer review' way. One of the nice things about WL is that I can provide a single notebook that explains all the computational pieces in enough granular detail that anyone with basic skills should be able see that it does what we say it does. And I myself can do basic R, although that is hardly relevant because in my field R is used mainly for stats, via packages. What's needed is that I understand the stat methods and when they are appropriate. I'm not going to unpick the algorithms in, say, a linear mixed effects package.

I actually like coding my own versions of discipline-specific tools, such as those in the VEGAN package. It makes me feel more confident that I know what they are doing, and what the output means.

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

The functionalities you've listed only scratch the surface. For example, consider the interconnectivity to devices. Mathematica is used as a controller for laboratory apparatus, as a receiver for remote sensing applications, and so on. Take a stroll through the main Language Documentation webpage and you'll find more capabilities you missed.

With regard to LaTeX, it's remarkable that Wolfram provides any support at all to something that is not a product but rather an ever-growing collection of TeX macros. As Peter points out, there is serious support for TeX. You can also use the Mathematica "Copy As" (select, right click) to paste Mathematica output as TeX and paste it into your TeX document. If you need something fancier, then use your programming skills to generate TeX within Mathematica and write it to a TXT file.

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.

Mathematica started as a computational algebra system; a product with its own proprietary language and unique notebook system to integrate computational code with text and graphics.

It works very well as a sandbox computational environment for e.g. research, analyze, document and develop new models and computational methods (algorithms). It’s strengths are:

• Symbolic computation combined with numerical methods
• Functional and rule-based programming (and procedural)
• Large library of functions so you can write concise code
• Well-written documentation
• A notebook environment.

Mathematica is a fun system to work with and it delivers results quickly, even though its learning curve is steep (maybe because some of us came from procedural languages and there are many ways to achieve the same result).

I don’t agree with Jonathan on the use of Excel versus Mathematica. Mathematica is my go-to system to analyze data and models and yes it replaced Excel for me in most cases.

I used Excel for large datasets, with pivot tables, VBA and many worksheets. However, my code to analyze data became opaque or hidden in many worksheet formulas. As such, Excel became unworkable. Now I may use Excel to create a well-formed CSV file, which I import to create a Dataset and analyze it in a Mathematica notebook. I spent a lot of time to master Datasets and associations and now it pays off.

Mathematica's language i.e. the Wolfram Language, has transcended its original product: it has become the vehicle for all things computable. So Jonathan’s question “what is Mathematica for” is relevant.

I think Mathematica should serve a niche market, comprising of mostly researchers, analysts and maybe some system designers. It should become a vehicle of state-of-the-art computational functions and methods. I agree with Jonathan: WR should focus on finishing and polishing e.g. statistics and time series analysis, machine learning and a number of other application areas, so it becomes the go-to tool for research. If you have Mathematica, you shouldn’t need any other specialized application to do your research.

I think the WL should focus on a computational ecosystem with capabilities to develop, test and deploy code, create APIs and GUIs for different OS or distributed computing environments. Its symbolic computational capabilities and extensive library of functions make the WL an ideal language for education or data analysts.

Having said this, I see a number of improvement areas to facilitate the WL/Mathematica adoption:

Transparency of outstanding issues. If Mathematica becomes the indispensable tool for researchers, then more transparency about stability, reliability and accuracy becomes paramount. We need to be able to rely on the tool and hence we need insight in outstanding issues, bugs and workarounds. I hope access to redmine is a first step into more transparency.

Reduction of the number of functions to a select number of unique ones. Mathematica prides itself of its 5000-odd number of functions. However, I see a lot of redundancies i.e. new functions that do things slightly different from similar functions. I like to see a reduction of redundant functions. For example take the ListPlot function: we have ListLinePlot, DateListPlot, StepListPlot, DateStepPlot etc. they are all of the class “ListPlot” and with the new axis capabilities, they can all be integrated into one function. This saves us the effort to remember the name of each specialized function, it makes code maintenance and documentation easier for WR too.

Document implemented methods. I would like to see WR open up the black box of some functions in the documentation i.e. explanation of methods used or references to methods implemented.

Conclusion

I think the purpose of Mathematica should be the go-to tool for researchers and analysts. I agree with Jonathan that WR should focus on delivering a complete set state-of-the-art tools.

I think the Wolfram Language could become the computational ecosystem to develop, test and deploy code for all things computable. We have some miles to go there.

At last, the adoption of Mathematica and WL could be facilitated if:

• We have more transparency of outstanding issues and bugs.
• The number of "redundant" functions has been reduced to a number of unique ones.
• We see more documentation to help us look inside the black box of some functions.

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.

Much has been said here that I understand and agree with. Especially the uselessness for real production systems. I myself have discovered WL only as about the 20th programming language since I am a pensioner. Before that it was always about large systems in FORTRAN, PL/1, C, C++, Java as a system architect and developer.

Today I program just for fun, and for that WL is great. For me especially because of the symbolic and mathematical capabilities and graphics. And I love notebooks that contain code and its documentation at the same time. Now I do everything simple with EXCEL VBA and everything else with WL.

I would like to add two important points that I think hinder the success of WL:

• The impenetrable jumble of products and licenses, i.e. the marketing.
• The price. For people who just want to get to know the product, especially students, a license is way too expensive (the cloud is not a way out, in my opinion you really need a desktop version with local files).

The second point is existential. Students can do all their tasks for free with things like Microsoft Visual Studio, Eclipse, C#, C++, Java, R, Python and the like, but nothing with WL because they can't afford a license.

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.

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.

In fact Wolfram has an embeddable solution for iOS: It's called CDFKit and it was presented many times in Wolfram Technology Conferences by Rob Raguet-Schofield and Jason Harris. CDFKit can be integrated as an iOS-Framework in Objective-C or Swift projects.

The availability of CDFKit seems to be bound to a purchase of an Enterprise Version of Mathematica (which is costly) and maybe a special contractual arrangement (to publish/distribute a certain number of apps).

This is of cause way too complicated: availability and usage should be as easy as free Wolfram Engine.

I agree what was said earlier: Apple Silicon (CPU, GPU, Neural-Engine) should be exploited and optimized much more for special tasks as AI, ML and -- if possible -- general compute. Maybe CDFKit could be adapted also as a "lightweight" Wolfram Engine for macOS.

It would be quite easy for Wolfram Research to develop a similiar "educational app" like Apple's Swift Playgrounds for iPadOS. A mini-REPL on the left with WL syntax coloring and CDF-Output incl. DynamicModule/Manipulate on the right (using CDFKit). This would be a much better experience than the Wolfram Cloud App.

The ultimate question remains: Where is Mathematica for the iPad?

Having said that, the Wolfram Engine would not be regarded as a complete solution by many:

(i) production systems developed using the WE require a production license and distribution of the WE runtime module

(ii) it entails development of the system in another programming language (which calls the necessary functionality in the WE).

In other words, in this mode of usage the WL alone is insufficient to create a production system. It is necessary to be proficient in another programming language such as C++ or Python to create the complete application. From that perspective, the WE is just another library (albeit a powerful one).

Adam,

I think my view is quite well aligned with yours: Mathematica is unparalleled in its facility for doing mathematics and its interactive nature makes it an outstanding choice for doing research, experimentation and prototyping. Which thereby also makes it an excellent pedagogical tool.

If we were to draw a line here and say: "this is what Mathematica is for", would that be so bad? Does its current inadequacy as a platform for developing production systems render it useless? No of course not, not at all. Why should it? After all, I have never heard anyone complain about R's shortcomings in the same area - why should Mathematica be held to a higher standard?

Well, I think there may be a couple of reasons (in addition to cost, of course):

Firstly, if we dispense with the notion that Mathematica can ever become a platform for doing series development work in a production environment, it means that you have to become highly proficient in at least one other language. And given the scope and complexity of WL, and of the other language one may be obliged to learn (e.g. Python, Java, C# or C++ ), a developer can be forgiven for thinking that they may as well save some time and effort and just focus on the second language, which can do everything (although perhaps not as elegantly as WL, in some areas).

Second, Stephen Wolfram's boundless enthusiasm and ambition for the WL leads one to hope that perhaps, eventually, a way will be found to remedy the WL's deficiencies and make it suitable for development of production systems. This is, after all, what Mathworks appears to have achieved with Matlab/Simulink, enabling Matlab code to be translated into production-suitable C++.

However, I suspect this is just a pipe dream: there are all kinds of structural reasons why a similar translations capability would be next to impossible to achieve in WL (beyond the very limited extant ability to compile simple functions).

Hi Kay, yes indeed it might that the premise is wrong (and in my original post I gave one possible explanation for the downtrend that is innocuous). In this case, however, I think that perhaps the chart for Facebook accurately reflects the slowdown in usage growth of the platform:

Source: Statista

So, while it certainly doesn't mean Facebook is dying yet, it tells us that growth is slowing, which may ultimately mean the same thing.

In term of your other points, I think we are saying the same thing: Mathematica has come so far as to be considered excellent in many fields, but seems to lack the last x% of functionality to render it truly outstanding (like design of experiments in statistics, in your example). Perhaps that is by design. A rational argument might be presented along the following lines:

Usage growth Mathematica is slowing. We (i.e WR) can't reverse that trend by providing more functionality to existing users who are already committed to our product. We have to find new users and bring them on board. To do that we have to provide new functionality that will appeal to potential users who haven't tried Mathematica before. That argues in favor of adding more breadth to the product scope, rather than more depth, beyond a certain level.

If this argument were to be accepted, it would mean that WR would be constantly developing functionality in a field up to a certain level - typically below what is required by a professional at the forefront of the field - before abandoning it to focus on developing new functionality in other areas.

It seems to me that WR needs a "finish the job" Tzar, whose function would be to identify the critical functionalities missing from key subject matter areas and with the authority to prioritize them for development.

In many cases the gaps in necessary functionality could be filled with a handful of new functions. But for some fields (e.g. machine learning) the list would be very long indeed.

Philipp,

Thanks for your thoughtful input. One can only hope that someone at WR is paying attention...

Perhaps the single biggest shortcoming in Mathematica, compared to, say, Matlab or Python is in regard to deployment in production. I have developed a couple of production systems IN WL, but my sense in each case was that I was pushing the envelope in terms of Mathematica's core capabilities. I always had to be ready to handle bug fixes and crash-recovery scenarios. Unlike other languages, which also had their fair share of bugs and crashes along the route to deployment, there never really seemed to come a time when Mathematica could be expected to work seamlessly - there always seemed to be another problem waiting in the wings. Similar systems developed in C#, C++, Python or Matlab would always (eventually) get to a point where they were reliable enough to deploy in a production setting, with a reasonable expectation of reliable performance. But I can't say the same for systems I have developed in Mathematica.
Now, it could be that this is just my lack of experience/expertise in developing production ready systems in WL. But I also think that it is perhaps one of the key limitations of the language. It's not to do with the language being interpreted - so are Matlab and Python - but some other aspect of the language architecture (e.g. memory management) that renders it unsuitable for deployment in production. I wish it were otherwise; but I can live with that limitation, providing I know it exists.

In any case, while the title of the post is about what Mathematica is For, it is perhaps also useful to produce a list of things that Mathematica is NOT suitable for, which from my perspective would include:

1) Anything that can be done more easily in Excel

2) Anything requiring deployment in production

What about other users' experience? Can anyone else offer an alternative view of Mathematica's capabilities in relation to development and deployment in a production setting?

Are there any other limitations you see Mathematica as having?

Yes, it would include standalone applications designed for general use and not requiring user familiarity with Mathematica (or possibly even a Mathematica runtime module). In Matlab, for example, applications developed in Matlab can be compiled as standalone C++ applications - a capability that Mathworks has worked long and hard on to produce.

But I also mean WL applications that are designed for use in the WL by Mathematica users (e.g. in package form, in a Mathematica notebook). Even here, as I said, it is challenging to produce an application for use in production, where application errors and bugs are very infrequent and remedied relatively easily. In most other languages bugs tends to get ironed out eventually, whereas my experience is that a Mathematica application is much harder to stabilize and often continues to surface problems. This is not always the case: for example my MATH-TWS application (which connects Mathematica to IB TWS) has proved very reliable. However, the great majority of that application is written in C++ and interfaces to the IB TWS api. So it doesn't really count as a true Mathematica application in the sense I mean here.

Hi Philipp,

Wolfram Application Server was announced today. Perhaps it will address some of the issues we face with deploying and supporting WL-based applications in production environments. There is no mention of pricing / licensing but I suspect it will likely be out of reach for most startups / small companies.

@Philipp Winkler, about bugs, the recently released Wolfram Research Issue Tracker:

https://redmine.wolfram.com

is a good sign for the path forward on this.

I think Mathematica is far behind Python. Since I have used Python, I seldom use Mathematica. Python is much more powerful than Mathematica in data processing.

@Wenguang Wang, maybe it is because you did not have a chance to get to the Mathematica's full potential. I heard a different story. My friend uses Mathematica extensively, data science, analytics, general coding, etc. Then she had to use python for some of her new work tasks. She got very frustrated with how not-smooth the experience was with some workflows and the front end in python. To me hybrid numeric-symbolics and fundamental symbolic nature of Mathematica and its functional style coding architecture seems far superior than python. Some similar user opinions here: https://mathematica.stackexchange.com/questions/86058

100% agree with your comments about Mathematica being enjoyable .... one of the great things about it is that it so easily allows you to "play" with concepts and ideas, and even pick up new ones from completely different fields. In that regard WL is unlike any other language I have programmed in (with the possible exception of APL). There is a great deal of tedium involved in programming in most other languages (although I quite enjoy working in R).

That said, I suspect that some of this is because I am not usually trying to build production systems in WL, a process that inevitably involves a lot of humdrum, laborious, house-keeping code. I use Mathematica for R&D, and in that context it's an ideal environment - fast, powerful, broad in scope and highly interactive. And, like many users, I love to dabble in all kinds of interesting topic areas that Mathematica opens up.

Your op is conversational/general (i like the content and agree, so i upvoted it) and i wouldn't wonder if it gets deleted. Nothing will come out of it (the discussion, if there is any) other than us writers/readers who agree with you feeling better because we said our piece and got the sore topic off our chest, hopefully without ranting. Anyway…

I am in the same boat, having expressed my existential angst for the product from the very start, at the time of registration see the About Me page.

"Success is the sum of right decisions." — If the popularity continues to trend downward, more and more senior key figures "turn their backs" ( @Michael, @Paul, @Roman, et al. because of age, capacity, retirement, or ultimately, expiration), and in 15yrs the product is no more than a relic, then that's fate. To me, $\text{Wolfram L}$ will still be the one and only programming language I am willing to accept; i never gave FREE $\text{Python}$ a serious or fair chance, because i am stubborn and, erh, i am no programmer (and have no interest in becoming one)!

I am no programmer (by profession, by heart, by need, by desire, by whatever) and have no interest in becoming one, also because of my very restricted talents hehe, yet i can write some simple functional code, create updated tables/graphs with a press of a key, and i do enjoy this ability of mine! All thanks to Mathematica. Honestly, I do miss the GUI features of M$ Word, Excel, Corel, Maple but i put up with these shortcomings because my enjoyment of the language is so much stronger. No software package or programming language is perfect or complete but imho Mathematica is the #1 contender in the all-rounder category. Nothing wrong with knowing how to draw maximum use out of a SAK Swiss Army Knife! And best of all, i have access to it all the time:

1. on the road: through the Android Wolfram Cloud App, on my phone
2. on the road: through the Wolfram Cloud web browser access (phone, computer, inet cafe, etc)
3. on the road: through the VNC Cloud access (e.g. on my phone) to my Raspberry Pi at home -> my preferred mobile access
4. at home: through the VNC Wi-Fi access (e.g. on my phone) to my Raspberry Pi at home
5. at home: through the VNC Wi-Fi access (on my PC) to my Raspberry Pi at home
6. at home: through eventual local installations on PC at home

The VNC access is possible only because my Raspberry Pi is non-stop powered by mains 24/7*365, a 2.0 Watts device.

Q: $\text{"So …, What is Mathematica For?"}$

A: $\text{"Well …, it is For Me! Basta ya."}$ — ©2021 Raspi. hh

My use of Raspberry Pi? Mainly for having the alternative mobile access to Mathematica (for mobile coding, or as calculator) and to my personal locally saved Mathematica notebook files. The web browser access and the Android app access are convenient and very good to have but they are not full versions of a Mathematica installation. So through the VNC access (does everyone here know what a VNC access is, oh please?) you're on the road and have access to a full local Mathematica installation, namely the one on your Raspberry Pi sitting at home near the Wi-Fi router. Seriously, this is reason enough and reason alone to buy and setup a Raspberry Pi, and shame on whomever that there is no youtube video showcasing how Mathematica can be used this way on a phone while you're on the road, apart from browser access and Android app access! Shame. The youtube title should be "How To Use Mathematica On Your Phone Through VNC Access" and it should demo both the VNC Wi-Fi access and the VNC Cloud access. No, i am not gonna do such a video, not my job. It's Wolfram marketing department's job :P

I also used the Raspberry Pi for logging electronic devices, like a logging multimeter, a logging charger/discharger, or other similar eletronic devices which offer a USB connection (or serial connection) for logging purposes. You can let the Pi do the logging for days and weeks, non-stop! There's logging software for Linux/Raspbian, wonderful stuff.

And i don't mind letting the Pi stream music non-stop from a streaming music subscription website, looping an online playlist infinitely. I only need to turn on the speakers and there's music!

Of course, the Pi is not a replacement of a full PC. One can watch a youtube or a HD video with VLC media player but you can't do both at the same time. In summary, my personal main uses are: 1) having access to a full local Mathematica installation (granted the Wi-Fi or mobile internet is working), 2) logging electronic devices, 3) 24/7 streaming of music. There is so much more, purposeful things, one could do with a Pi, but as you can see, I keep it simple, basic, boring.

The only difference is that a 2.0W RPi is constantly powered on, in its booted state. A 400W PC usually is turned off for the night or when you're travelling. Other than that, there is no difference between VNC access to RPi or VNC access to home PC, you're right. A full-fledged PC is more powerful than a small RPi, of course. Do you have your PC (or Mac) turned on for 24 7 365, see? With a RPi, one does so, for the sake of nonstop remote access to Wolfram L full installation, at slow performance.