I have degrees in EE, math, and physics, and have worked in engineering disciplines for about 45 years. My work has included electronics engineering, embedded system programming, material science, optical engineering, and physical modeling. The first years were done with Fortran and a slide rule. The last 20 with computer tools which have become more capable each year.

For the past 10 years I have worked mostly as a consultant in optical design and physical systems modeling. I use Zemax OpticStudio for lens design and Comsol Multiphysics (an FEA tool) for mechanical, electrical, and thermal modeling. They are each better at what they do than Mathematica.

I have been a Mathematica user for a bit over 20 years -- since version 2.

First, if all Mathematica did was "encourage a kind of intellectual dilettantism," it would be well worth the price. I have been an intellectual dilettante my whole life. Always interested in some new thing, often around science and mathematics. For me, it's the only kind of life worth living. And Mathematica's contribution to this has been enormous. And this has been a very practical endeavor. These entertaining explorations have almost always found a use in some later for-pay endeavor.

However, to specifically address the question, here are some practical uses I have made of Mathematica:

1. Optical design tools rely heavily on optimization capabilities. But you cannot just give them some arbitrary starting point and expect good results. The first part of the job is to use prior work or first principals to define a design which roughly meets the various system requirements. Optical matrix methods are well suited to this. I use Mathematica to describe propagation of a few trays through a trial layout. I can even include unknowns in the symbolic description and solve for them to gain insight, or to solve roughly for element positions and focal lengths.
2. With the same tool set, once an optical design has been done, it needs to be toleranced to predict the performance which will actually be obtained given manufacturing uncertainties. I use Mathematica to analyze the output of Monte Carlo runs to develop survival statistics which relate yield to the choice of specifications and binning.
3. In the most recent release of OpticStudio, there is a .NET API. I will certainly continue to use the tools user interface for design work ans simple analysis. But I am now driving OpticStudio from Mathematica using NET/LINK for more complicated analyses of design behavior.
4. In another job a few years ago I was modeling a piece of semiconductor equipment as it goes through heating and cooling processes. For this purpose, the system was adequately describe as coupled heat reservoirs, each of uniform internal temperature, coupled by thermal conductivities. Its really an easy problem of 1st order diffeqs. A complicated tool could have been used, but I had a useful model, with T vs time graphs, in minutes.
5. For the same company, I spent a great deal of effort analyzing defectivity in a wafer-scale manufacturing process, seeking cause. Mathematica is a great tool for that.
6. I recently needed an amplifier capable of driving a 10 uF load capacitance from DC to 50kHz. I used Mathematica's control system tools to design a feedback network to stabilize a commercial audio power amp IC for those specifications. At the start of my career, this would have been done with Laplace transforms and a slide rule. With Mathematica, the step response of this design on my oscilloscope looked just like the graph output by Mathematica.

This is a few of the concrete uses I have made of Mathematica in the past few years. But it does not come close to describing it's true usefulness. It somehow gets opened up early in any piece of work, and not closed until the end of the day. One very important reason for that is that, at its heart, it is a symbolic math tool. For me, the first step in looking at most problems is to look at the mathematical dependencies. This means looking at equations and relationships in symbolic form. After that, even more intuition can be gained by trying real numbers and viewing these relationships graphically. Mathematica excels at both these tasks.

I also have had the experience of seeing Mathematica fail to take hold in an engineering group. It seemed to center around three issues. The first was a threshold between starting up and obtaining a result. I have seen a lot of engineers trying to get a simple graph for an afternoon meeting go to Excel in frustration. Or because they had trouble importing data to begin with. These issues have been getting better, but WRI needs to do a better job of providing start-up training online. The collection of training videos on the Wolfram site needs a complete overhaul. The second issue is making the transition from a procedural language (like C) to the functional programming paradigm. This is something that is difficult to demonstrate in a completed notebook -- it's about the process, not the end product. The third issue is the deplorable lack of debugging tools. This is especially a problem when you need to develop a large single function in a single cell. (a result of the functional paradigm) The usual suggestion is to insert print statements. REALLY? This is how we debugged our Fortran programs 40 years ago. Modern programming tools have had good debug environments -- single step, watch lists, breakpoints -- for ages. And engineers are used to this. This has been completely neglected by WRI. Workbench is poorly documented, and so lacking in commitment and support that you have to ask for a beta Workbench 3 just to get it working with version 10. There is a built-in debugger, which is also poorly documented. I inquired about it from tech support a year or two ago -- They suggested I use Workbench. This is a very important issue. New engineering users which encounter such a primitive debug environment will consider the tool disqualified. Mathematica is a great tool. Getting new users over the threshold, to the point where they see benefits, should be a priority.

Best regards,

David

"Intellectual dilettantism", I like that! I tend to call it "Geeky bagatelles".

I've sold around 300 copies of several Mathematica applications and done a little paid consulting - but it is far from a living. I know there are people who have appeared on this forum who earn money through Mathematica consulting. There are many more who use Mathematica in their work but many of them say it is very difficult to get others in their company to use it.

Then we should keep in mind that research and exploration, even if not profitable now, may have a significant impact on the future. "What is the use of a new baby?" and "Someday Sir you will tax it." and all that. Just from reason and knowledge of Mathematica's features it should have a greater impact.

But it's not having a significant impact. Just one example is the publication of physics documents. I haven't done a thorough search but I suspect that you won't find a single Mathematica notebook or CDF document on physics archives. Mathematica notebooks with their active and dynamic capabilities should, by all rights, blow static PDF documents out of the water! But Mathematica hasn't even gotten its toe into the water.

The real question then is: Why doesn't Mathematica have a greater penetration in science, education and the commericial world? This is a problem to be solved. If Wolfram Research doesn't see it as a problem then it won't be solved. Mathematica aquires more nice features with each version but this hasn't solved the problem. Here are some of my suggestions, for what they're worth.

1) Mathematica and the Wolfram Language are difficult to learn. This is mostly because of the sheer size of it. The core is pretty good and the edges are pretty ragged. WRI has done a lot with documentation, but they could still do more. One thing they might do is to give users freer access for adding to the documentation, maybe like Wikipedia. More practical examples might help a lot. Just formalistic usage and then jumping to a complicated "Neat Example" is not always sufficient. To give an example: while answering another recent posting on Wolfram Community I started using the ExtractArchive routine. This sounds very nice. It can extract a zip file and install an application on the user's computer. But then it turned out that it was very limited. There can be no files in the existing extraction folder else it completely fails. This means it cannot update an application, such as can be done if one downloads a zip file and unzips it with WinZip. I checked its options and found CreateIntermediateDirectories, but without any explanation of what that means. One can't delete the application before using ExtractArchive because there may be important files there that are not updated. One has to do a little research project to find if a routine works or not, and sometimes it doesn't. There are too many routines like this that just pretend to be useful but in fact are not.

2) Mathematica graphics is extremely difficult, cumbersome and convoluted! It is a major negative. I've communicated with many people who complained about it. Its main flaw is that it's designed from the top down with scads of plot routines with little correlation. Lots of choices but if you don't want to accept one of their choices you are pretty much stuck. The sad thing about this is that Mathematica opens the possibility of more innovation in graphical presentation but their paradigm chokes it off. My best selling application, Presentations, was based on the idea that in graphics "everything is a graphics primitive". It would extract the graphics primitives from higher level plots so that you could directly combine them with other primitives and graphical directives. The paradigm was more like drawing on a sheet of paper, one thing after another by specifying the primitives. Users seemed to like it a lot.

3) Mathematica is not as stable over versions as they nonchalantly claim. I think I read a claim that Mathematica 10 would run a Mathematica 2 notebook. Maybe, but not necessarily a Mathematica 7 notebook. The graphics paradigm comes to mind again. It is not at all uncommon to extract information from a graphic by extracting primitives. Roman Maeder was showing how to do it back in Mathematica 2. However, WRI keeps adding new graphical structures and changing the old ones and that breaks a lot of code that depended on it. Sometimes it might be worth the price but often it's just ad hoc. And sometimes WRI introduces new problems. An example is that Graphics3D has an option Lighting. It no longer works. Instead Lighting options must be put in the specification of each surface drawn. Serious Mathematica users cannot live with this kind of stuff.

4) "Mathematica somehow fails to live up to its apparently unlimited potential for encapsulating creative thought-product ..." Exactly, but there are only a very few who think this way. I've written an essay at my web site where the second sentence is: "This essay is directed to individuals who are using Mathematica to develop a base of knowledge and a set of capabilities centered on some subject matter." It's a way to accumulate knowledge and capability, preserve work and communicate with colleagues. Perhaps it is too difficult for most users. Although the basic layout has been designed by Wolfram Research, and its a good one, they have not entirely smoothed the path. For example, many users have difficulty with style sheets. Style sheets for an application should be in the application. Mathematica will find them. But they confuse users with an Install Style Sheet button that will not install the style sheet in the application. Before Mathematica 6 there was a documentation system that was available to everyone because it simply used a txt file - not as good as the present system but not bad either. The documentation system is no longer available to everyone but only those who have Premier Service. It uses Wolfram Workbench 3. This is not even a regular product but is obtained by special request. That indicates to me that there are probably not many users actually using it.

5) The Wolfram Research business and pricing model gets ever more complicated. I know of one academic who, because of complications, just chucked Mathematica because they didn't have what they needed when they needed it. Another part of the business model is the question of who is going to develop the Applications: Wolfram Research or independent developers? I think it should be independent developers because Wolfram Research can't be experts in everything. By not strengthening and easing the core part of Mathematica for creating Applications they are losing this resource. When I looked into marketing some of my Applications at WRI they wanted 80% of the sales price, just for providing a blurb and download link. That's ridiculous, this isn't book publishing! I went with the Kagi store, which takes 10% and provided better service.

6) Diffuse effort. Instead of perfecting the "Application" model for building capability over time, Wolfram Research has been drawn into many other endeavors. Many of them have excellent merit but altogether it may be too much to chew at once. Some of them are a bit silly such as converting 140 character Wolfram Language programs for Twitter. And why should all Mathematica computing be drawn to the cloud, instead of just retrieving small pieces of information?

I challenge you to make the last graph of yours in Excel and in 10 'lines'! I think you just proved that it is really easy, all your examples are like 10 lines! I don't know any other software that can do that and still be reasonably readable and adjustable with such low amounts of lines.

Yes sometimes you have to wrap a Graphics around, and yes it is better to stay away from Epilog/Prolog if you combine some graphics, there is always room for improvement. But knowing that, i think it is very easy. I think your examples proof my point more than disproof it...

Sander, You show simple examples that work, but then if one deviates a little from them they don't work. here is a simple graphic done in Presentations using graphics primitives.

Draw2D[
 {{GrayLevel[0.8], Rectangle[{\[Pi]/4, -1/2}, {3 \[Pi]/4, 1/2}]},
  {Red, Draw[Sin[x], {x, 0, 2 \[Pi]}]},
  Text["Sin", {\[Pi]/2, 0.8}],
  {Darker@Green, Draw[Sin[x + \[Pi]/2], {x, 0, 2 \[Pi]}]},
  Text["Cos", {\[Pi]/2, 0.0}, {-1.5, 0}]},
 Frame -> True]

It's pretty simple. You just draw one thing after another. The curves are converted to graphics primitives. We can set up the applicative graphic directives before drawing the curves. To me this is intuitive. I don't know how others feel.

A newbie might not know he has to put directives in an option or that he has to wrap primitives in Graphics. But let's say he figures that out and learns about Epilog. He might try this:

Show[
 {Graphics[{GrayLevel[0.8], 
    Rectangle[{\[Pi]/4, -1/2}, {3 \[Pi]/4, 1/2}]}],
  Plot[Sin[x], {x, 0, 2 \[Pi]},
   PlotStyle -> Red,
   Epilog -> Text["Sin", {\[Pi]/2, 0.8}]],
  Plot[Sin[x + \[Pi]/2], {x, 0, 2 \[Pi]},
   PlotStyle -> Darker@Green,
   Epilog -> Text["Cos", {\[Pi]/2, 0.0}, {-1.5, 0}]]},
 Frame -> True]

It doesn't work. The Text items don't appear. Why not? I have no idea why they both are missing.

The newbie has to figure out how to do it as follows - with a lot of graphics level jumping.

Show[
 {Graphics[{GrayLevel[0.8], 
    Rectangle[{\[Pi]/4, -1/2}, {3 \[Pi]/4, 1/2}]}],
  Plot[Sin[x], {x, 0, 2 \[Pi]},
   PlotStyle -> Red],
  Graphics[Text["Sin", {\[Pi]/2, 0.8}]],
  Plot[Sin[x + \[Pi]/2], {x, 0, 2 \[Pi]},
   PlotStyle -> Darker@Green],
  Graphics[Text["Cos", {\[Pi]/2, 0.0}, {-1.5, 0}]]},
 Frame -> True]

Or this, which changes the AspectRatio used:

Show[
 {Plot[Sin[x], {x, 0, 2 \[Pi]},
   PlotStyle -> Red],
  Plot[Sin[x + \[Pi]/2], {x, 0, 2 \[Pi]},
   PlotStyle -> Darker@Green]},
 Prolog -> {GrayLevel[0.8], 
   Rectangle[{\[Pi]/4, -1/2}, {3 \[Pi]/4, 1/2}]},
 Epilog -> {Text["Sin", {\[Pi]/2, 0.8}], 
   Text["Cos", {\[Pi]/2, 0.0}, {-1.5, 0}]},
 AspectRatio -> 1/\[Pi],
 Axes -> None,
 Frame -> True]

Here is a case with a Lighting option in Show. Again, you showed a simple case that worked. But this does not work. The Lighting specification is not used.

Show[
 {Plot3D[Sin[x + y^2], {x, -3, 3}, {y, -2, 2},
   PlotStyle -> Lighter@Brown,
   Mesh -> None],
  ParametricPlot3D[{x, Sin[x + z], z}, {x, -3, 3}, {z, -2, 2},
   PlotStyle -> Green,
   Mesh -> None]},
 Lighting -> "Neutral",
 ImageSize -> 300
 ]

The Lighting option has to be inserted into both of the surface plots. Why? Why can't we have a one overall Lighting specification for the entire graphic?

Show[
 {Plot3D[Sin[x + y^2], {x, -3, 3}, {y, -2, 2},
   PlotStyle -> Lighter@Brown,
   Lighting -> "Neutral",
   Mesh -> None],
  ParametricPlot3D[{x, Sin[x + z], z}, {x, -3, 3}, {z, -2, 2},
   PlotStyle -> Green,
   Lighting -> "Neutral",
   Mesh -> None]},
 ImageSize -> 300
 ]

Maybe you consider these things as trivial bits of knowledge that the user should have known. But as David Keith wrote; "I have seen a lot of engineers trying to get a simple graph for an afternoon meeting go to Excel in frustration."

One thing they might do is to give users freer access for adding to the documentation, maybe like Wikipedia.

I would like to express my support to this idea. One example of another company doing this is the case of MySQL. Here you can see the official documentation for the SELECT statement and it is followed by user comments. Sometimes the comments end up being an essential to understanding the official documentation.

In regards to the original question of this thread, I work at an engineering company and I use Mathematica for:

• Fully automated reports generation and sending my email
• Development of tools for our internal use that run with the CDF Player (this requires the Enterprise license.)

And yes. I am sure that this could be done with different technologies, but I am also sure that it would take much longer to develop (if ever developed).

variability is a natural phenomenon. uncertainty is in the human mind. today it rains, tomorrow, it doesn't. that's variability. a forecast of 40% probability of rain illustrates uncertainty. Mathematica was used to "deconvolve" uncertainty from variability. http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.54746

2) Combining graphics is as easy as Show[{graphics1,graphics2,graphics3,..}] which is quite easy I would say ;) I've made many many Graphs in Mathematica, and never had a problem, and where necessary I made my own plot routine, which you can, really easily. I don't know any other products that supports such a customisable and wide variety of plotting routines! You can also write by default 'like a sheet of paper' right?

Graphics[{Blue,Rectangle[{0,0},{3,4}],Red,Point[xydata], Green, Line[xydata2]}]

I would say it is not 'extremely difficult' at all! very easy, just combine the primitives and plots you like using Show/Graphics/Graphics3D ! that's it!

3) I'm not sure what you are talking about, Lighting -> Neutral works as it should:

Graphics3D[Sphere[],Lighting->{{"Directional",Orange,{{0,0,4},{0,0,0}}}}]

and has always worked like this as far as I can recall. As far as extracting parts of graphic, I always remind myself this will be version specific, the primitives it creates might be different per implementation. So if you rely that it produces Rectangles, but now it gives rectangular Polygons (in another version), that should be easy to catch in subsequent version using a new rule. But I always try to stay away from these practices, it can be done, but that doesn't mean it should be done... You should customise directly using the plot routine and then combining with e.g. Show, or Epilog/Prolog.

4) Isn't the entire documentation available online?? http://reference.wolfram.com/language/ ?

Jonathan,

I hesitate to respond, because I'm not sure I understand your perspective. Your two criteria for "real world application" seem contrived on purpose to make it difficult to satisfy them. Your item (ii), for example, is self-contradictory. If there is no other application that does the job better than Mathematica, then why shouldn't Mathematica get credit? And item (i), is a very rigid, specific bias. Is Excel not useful in the real-world, because most users don't re-sell Excel "applications"?

Again, I'm not sure what your perspective is, but you seem to want to compare Mathematica (or Wolfram Language) to development languages/platforms that generate stand-alone applications. Perhaps that is a direction Wolfram is heading toward, but I've never viewed Mathematic in that light. Mathematica IS the product/application. There are myriad problems for which Mathematica offers a space for exploration, solution, invention, analysis, etc. At the end of all of that, if there is no stand-alone application to sell, then you dismiss it as not real-world and therefore (I infer) not particularly valuable. That's fine if that's your perspective, but why should I feel compelled to accept your perspective? Or, to put it bluntly, what's your point?

I've seen Mathematic in contexts that I consider real-world: Public school education, Private professional training, Generating assets for book publishing, Engineering problem solving, Alternative to Excel, Alternative to PowerPoint, Data analysis, Prototyping.

These observed real-world usages are mostly "accidental", in that I'm not really in a position to hunt for Mathematica in the real-world, and so I suspect that others could expand this list greatly. I don't think anyone is making a claim that Mathematica is a silver bullet that solves all real-world problems, nor that Mathematica's breadth of functionality is such that no other applications will ever be needed. So, again, I'm curious about the point you're trying to make, what actual relevant question you're trying to ask, or what methodology suggests your specific criteria.

You can produce Wolfram Language stand-alones applications that run without license via wolframscript. Apparentely very few people know it, and even less people know how to do it. I run my wolframscripts everywhere I can with no license.

David Keith's post is illuminating in terms of describing his successful application of Mathematica in his work. Several of his points chime with my own experience, where I have found Mathematica to be a great prototyping tool. I think the key to Mathematica's usefulness lies in its core capabilities, so aptly described by David in his post, as follows:

One very important reason for that is that, at its heart, it is a symbolic math tool. For me, the first step in looking at most problems is to look at the mathematical dependencies. This means looking at equations and relationships in symbolic form. After that, even more intuition can be gained by trying real numbers and viewing these relationships graphically. Mathematica excels at both these tasks.

Likewise, Bianca Eifert also pointed out the usefulness of Mathematica's graphical capabilities for the preparation of figures for publication.

To define Mathematica's core capabilities only in terms of symbolic logic and symbolic manipulation, as I have done above, appears to be too narrow. David and Bianca are quite right to highlight Mathematica's graphical capabilities also, which are indeed exceptional.

I am tempted also to add Mathematica's facility for creating dynamic user interfaces. These capabilities, too, are outstanding. Where this falls down, however, is with the difficulties presented in getting such content in front of a non-Mathematica user. CDF is a great attempt at a solution to this problem, but too often fails in practice (and anyway requires the user to download and install additional software). Web and cloud-based solutions may be a better attempt, but entail further investment and development effort.

In summary, while no doubt WR would argue strongly in favor of Mathematica's capabilities across a much broader spectrum of applications, the product appears most effective in providing solutions for problems that entail:

• Rapid prototyping
• Manipulating mathematical symbols and equations
• Graphing the results

I am somewhat uncomfortable with the limitation imposed by the first of these points, because it plays down Mathematica's capability to produce extremely polished, industrial strength solutions (see several of the Applications described on the WR web site for examples). However, I suspect that for the great majority of Mathematica users, the main benefit lies in prototyping rather than producing the kind of large-scale application more usually developed in languages like C++, or Java. Very happy to hear opposing views on this point, however.

As far as Matlab vs Mathematica, it should be noted that there are free online courses that teach Matlab and others which use Matlab. In many of these courses, a temporary license is provided. I've suggested to the sales team that Wolfram Research do something similar but have been told that the videos provided on the web site are just as useful. I don't agree with that statement.

Jack, ditto:

Are you able to give some specific example of how Wolfram script enables you to deploy stand-alone applications? I'd like to dig into it a little more, as it may be that I (and others) are missing a trick...

What about the annoyance of having cells, structures, and arrays (matrices?) which are just 'list' in Mathematica.

I suppose the free cloud plans may be useful for someone who just wants to try out the language...

It is quite limited indeed; I, myself, have used it throughout my bachelor, master, PhD and now postdoc. But I was already using before in high school too. But I'm one of the few who uses Mathematica, 'everyone' is obsessed with Matlab! And also books tend towards Matlab... I think it is because a lot of people think Mathematica is only for equation solving, and have no idea it can do (almost) everything Matlab can do.

I find your definition of "real world" kind of weird:

"money changed hands" surely does not equal real-world. Because now you exclude basically all academics! A HUGE number of academics do there analysis using Mathematica/Matlab/...... without getting money for it! It is to publish something. But not directly getting money.

I'm one of the very few in the group that I'm working in that uses Mathematica, but if I compare speed of building software to do analysis, Mathematica trumps any Matlab implementation. Nearly all the code is shorter, more legible, and much faster to implement. I generally have little programs done while the others are still struggling to import some data or so.... Of course that varies a lot on what you do exactly!