Stephen says Wolfram Research will follow up an extremely successful 2009 with a remarkable “breakout year for our company” and the Mathematica community.

Over the next few weeks, we will be sharing video highlights and transcript excerpts with you from Stephen’s conference keynote.

In the first part of this series, Stephen shares the history and trajectory of Mathematica, including some insight on what he calls the “most dramatic” development for him.

Transcript Excerpt:

I know some of you have come to many conferences with us, but particularly for those who are new here today, I thought I’d say a little about our history, and how we’ve gotten to where we are today.

I started building SMP—which was a precursor to Mathematica—back in 1981, and inventing some of the ideas that would become the core of the Mathematica language.

We started creating Mathematica itself in 1986, just a little more than 23 years ago.

Mathematica 1.0 was released on June 23, 1988—with over 600 built-in functions.

I think it was very important that before I worked on designing Mathematica, I’d spent a decade doing basic science—eventually working on precursors of what became NKS.

I viewed the design of Mathematica as like a difficult basic science project.

A big puzzle to drill down and find the essence of everything. To find the underlying principles on which things are based.

So from the very beginning, we built Mathematica on principles—on powerful unifying ideas.

Sometimes that kind of intellectual purity is an impediment to progress.

But I’m happy to say that in the case of Mathematica, it’s been quite the opposite.

In fact, as the years have gone by, we’ve realized just how powerful the principles really are, and just how much we can build from them.

I suppose the first principle of Mathematica is unity. Everything has to fit together.

Partly that’s a matter of detailed design: of carefully thinking through every function that’s added to the system, to make sure it fits in well with everything else that’s there.

But what makes this possible is really a fundamental organizing idea: the idea of symbolic programming.

The idea that everything can be represented in a unified way, as a symbolic expression.

And then that this unity of representation makes it possible to have language primitives that apply in a unified way to everything.

Well, unity has been one guiding principle of Mathematica.

Another is automation.

The idea that once you tell Mathematica what you’re trying to achieve, it should as much as possible automate actually doing it.

So that, for example, it automatically picks what algorithms to use, automatically works out how best to present your results, and so on.

Automation is sort of a key engine of progress through the world of technology.

Because it’s what lets you not have to worry about one layer, and be able to work on a layer above.

And in a sense unity and automation together do something really remarkable: they make it possible to build a system in a sort of recursive exponential way—so that the whole becomes greater and greater than the sum of its parts.

Unity is what lets one part of the system readily make use of all the other parts.

Automation is what lets the parts that one can use to build more in the system be ever larger.

It’s really been a thrill over the years to see these principles play out in what can be achieved, and built, in Mathematica.

In some ways, it’s been a humbling experience, though.

Because it’s often taken a very long time—sometimes a decade—to see just what might now be possible, given what’s been built so far.

Even after all these years of experience, it’s hard to imagine what might be possible until you have the actual concrete technology—and often use it for several years.

And for example right now it’s certainly clear that that’s happening with Wolfram|Alpha technology.

Well, you know, when we first introduced Mathematica 1.0, we described it as “A System for Doing Mathematics by Computer”.

The structure that we had built—the whole symbolic programming system and so on—was certainly much more general than traditional mathematics.

But mathematics was an important core area for us, and is what we emphasized first.

So how did things evolve after Version 1.0 in June 1988?

Version 2.0 arrived in 1991, particularly rounding out some of the general language constructs.

Mathematica notebooks, with their cell structure and evaluatable content, had already been part of Mathematica 1.0.

But we’d thought of them as a separate layer—just an interface around the Mathematica kernel.

But at the beginning of the 1990s we realized that a great unification was actually possible.

That with our symbolic programming paradigm, we were able to represent notebook documents just like we represent everything else.

Treat them like data that we can generate, and manipulate, automatically.

Well, it took a few years to develop this idea—but the result was Mathematica 3.0, released in 1996.

Which introduced many key innovations. Like symbolic documents. And also generalizations of the very concept of a language.

We get a lot of satisfaction out of building definitive things.

Things that really nail an area; that capture it in a permanent way.

And for example the integrated math input and output system of Mathematica 3.0 is such a thing.

It’s a unique feature of Mathematica that’s possible because of the whole symbolic programming and symbolic document structure.

And even though 12 years have passed since it was first introduced, nothing like it exists elsewhere, or seems even vaguely on the horizon.

Well, we’d described Mathematica 1.0 as a “System for Doing Mathematics by Computer”.

Already by the time of Mathematica 2.0 that wasn’t a good description, and we invented—somewhat to my chagrin—the term “technical computing”.

But after Mathematica 3.0 it was clear our paradigm and principles could still go much much further.

There were—as there are now—zillions of special-purpose systems that do little slices of what Mathematica can do.

And somehow there was the view that—even though it might not be as convenient to use—a special-purpose system must somehow always be more efficient than a general system.

Well, Mathematica 4 and 5 blew away that idea.

And proved that with our principles of unity and automation we could make a general system that was actually more efficient, not less, than special-purpose ones.

Numerics was a big example of this.

And the critical point was that numerics can be done most efficiently if it’s not just done with numerics.

If one can rely on robust, automated, symbolic analysis—or on-the-fly symbolic algorithm construction—in the middle of a piece of numerics, it’s possible to do it much, much more efficiently.

And the crucial point is that with the unity and automation of Mathematica, it becomes in a sense cheap to do all these fancy things as part of any old piece of numerics.

Unity and automation let us create ever larger building blocks—which we can use to build more and more, at a faster and faster rate.

And as we moved through Mathematica 4 and 5, and into the years beyond, our rate of algorithmic progress has been ever increasing, maybe even almost exponentially.

But there’ve been big new ideas as well.

Starting soon after Mathematica 3, we started wondering whether our symbolic paradigm could be used not just for what one might call static results, but also to create dynamic results.

It took a while. But in 2006 we finally finished it.

Really reinventing Mathematica. Creating the concept of dynamic interactivity. And in a sense taking Mathematica from a language for creating static results, to a language for creating interactivity.

We’d come up with Manipulate, and we’d figured out how to use the symbolic paradigm to represent controls and processes as well as functions and results.

I view Manipulate and its friends as a great feat of automation.

Taking the process of creating user interfaces from being one that takes large amounts of manual work, to something that one can specify instantly, and get carried out automatically.

It’s really changed the way I work.

Being able to let me create interactive interfaces all the time, in the middle of anything I’m doing.

Well, this arrived in Mathematica 6.

And actually Mathematica 6 was a huge release.

Our exponential curve of algorithmic development was beginning to be visible.

We’d gotten very serious in things like computational aesthetics—introducing yet another kind of automation.

And we’d introduced the idea of data paclets—of curated data that could automatically be accessed inside Mathematica.

And inside our own internal development process, we’d also done a huge amount of automation.

Building up a very sophisticated software system for building and testing Mathematica—a giant collection of Mathematica programs for creating Mathematica itself.

And we’d been sharpening some of our more secret weapons.

Like using NKS methods to discover algorithms just by searching the computational universe.

In a sense it was fitting that Mathematica 6 came 18 years after Mathematica 1.0 was born.

Because I felt it was really the time when Mathematica came of age.

When we could see just how limitless the possibilities are. And when we’d created systems that could really let us capture those possibilities.

Well, Mathematica 6 was a huge release. And we might have thought that after that, we’d rest for a while.

But no. Those of you who were here last year will remember that we had a big surprise at this conference.

18 months after Mathematica 6, we were able to release Mathematica 7.

Which was another huge release—with more than 500 new functions just in that one release. Almost as many functions as the total in Mathematica 1.0.

We introduced integrated image processing. Integrated parallelism. Symbolic charting. Boolean computation. Discrete calculus. Genomics. Weather. And lots and lots more.

A huge release.

Well, that pace of development has not just continued. It’s become even faster.

The pipeline that we have heading for Mathematica 8, Mathematica 9, and Mathematica 10 is really incredible.

I think already this morning there’s been a preview of a few things.

But the whole collection is really, really remarkable.

I suppose the thing that’s most dramatic to me is this.

In the early days of working on Mathematica I had all these to-do lists.

I knew it was a long-term project. That there was just a huge stack of things that we would have to build to make possible things in the later parts of the to-do list.

Where if it was going to be possible to automate that kind of computation in a robust way, it would have to rely on integrating a dozen other kinds of computation.

Well, here’s the exciting thing. In the next couple of Mathematica releases, we’re going to be able to get the end of that old to-do list of mine!

Before Mathematica reaches its quarter century, my whole original to-do list is going to be done.

It’s a great feeling—to see the whole arc of Mathematica development get us to this point. And to be able to see it go so much further from here.

Not surprisingly, we’ve seen increased interest in Mathematica’s financial applications stemming from the current economic struggles. Accurate models and analyses are in demand to determine the best way to get the world’s economy back on track and prevent future crises.

That’s a great match for Mathematica’s powerful capabilities.

To showcase how effective Mathematica is, we designed a portal to demonstrate its capabilities for financial applications. The new Mathematica Solution for Financial Risk Management web page, which I researched and created, is a comprehensive overview of what Mathematica can do in finance.

On the page, you’ll find tutorials, articles, documentation, and other resources to get you started. You can also view telling case-study videos and written descriptions of how finance professionals use Mathematica to get their jobs done.

For example, in one user story Fannie Mae economist Bernard Gress details how he created new mortgage-forecasting models utilizing Mathematica. In another, Alan Savoy of nGenera discusses combining Mathematica and webMathematica to build an online financial analysis tool.

One of the things I enjoyed personally about the project was going through the financial category of the Wolfram Demonstrations Project. There are currently nearly 200 financial Demonstrations—some of them tools created by finance professionals to use in their work, some of them created by educators to teach certain concepts (I admit I found those really useful), and some written by developers to show off various things Mathematica can do in the world of finance. And all the code used to create them is freely available, making them a great starting point for your own explorations.

I’d like to share some Demonstrations I think you’ll enjoy.

The Constant Risk Aversion Utility Functions Demonstration by Seth Chandler is interesting because it requires Mathematica to solve a second-order differential equation symbolically in real time whenever the parameters are changed, to plot utility functions that exhibit constant risk aversion under the Arrow-Pratt measure.

You can choose whether to show the constant absolute risk aversion curve or the constant relative risk aversion curve, vary the risk aversion coefficient, and set the coefficient’s value at two points along the curve.

Jason Cawley’s Credit Risk Demonstration looks at 30 years of actual data on corporate bond performance, then creates a Markov chain model of credit risk. Using Mathematica’s ability to manipulate lists of data and create visualizations, the Demonstration shows how the ratings of 100 simulated bonds change over time.

When using the tool, you can adjust the maturity, the length of call protection, the starting rating, and the subordination, or choose a new random seed for the simulation.

Robustness of the Longstaff-Schwartz LSM Method of Pricing American Derivatives by Andrzej Kozlowski uses the Longstaff-Schwartz least squares Monte Carlo method of computing the value of an American put option, approximated by a Bermudan option with 50 exercise times. It’s an intriguing Demonstration because it shows how Mathematica Demonstrations can be used to study problems in current financial research; the author references two journal articles.

In it you can select three kinds of bases and can vary several parameters including degree and number of paths of the Monte Carlo sample. The Demonstration produces a plot of the expected cash flow vs. stock price.

These are just a few of the Demonstrations that show how financial professionals use Mathematica to model, compute, and analyze.

When you get a moment, check out the Mathematica Solution for Financial Risk Management website. Let us know what you think, and also let us know how you’re using Mathematica in your financial work—leave your comments below.

I always analyze and explore these problems in Mathematica. Being a kernel developer, I see whether Mathematica can indeed find a solution. This last issue has challenging problems, and it was particularly gratifying to observe that Mathematica could solve them right out of the box. So here are my solutions to three of the paraphrased problems:

Problem 11457, by M. L. Glasser:

Solution in Mathematica: Relaxing assumptions on a and b to avoid degenerate cases:

Problem 11456, by R. Mortini:

Solution in Mathematica:

Problem 11449, by M. Bataille:

Solution in Mathematica: As the expression is left unchanged by the variable’s rescaling, and as c is positive, let us set c==1.

Hence the minimal value of 9/8 is attained for a==b==c.

Because the expression is left invariant by interchanging any of the variables, the maximum value of 2 is attained for a==b==c/2 or a==c==b/2 or b==c==a/2.

Also today, Wolfram Research developers spoke on the growing capabilities of Mathematica, Wolfram|Alpha, and the new Wolfram|Alpha API. In addition, our users covered a great range of topics with presentations from teachers and their students, researchers, and business professionals.

One of the nicest things about the conference is having developers and users in such close proximity. Both groups take advantage of the time between talks and during meals to discuss project needs and problems, and try to work out best solutions. There is also a lot of creative thinking about the future.

We look forward to sharing more about this year’s conference with you.

Along with many Wolfram Research developers, Mathematica users from all over the world have been speaking on the myriad ways they explore those same features and technologies in their projects. With so many interesting topics, choosing which talks to attend can be difficult.

The afternoon started with Stephen Wolfram’s keynote speech. The talk covered a timeline of growth for Mathematica, Wolfram Research, and Wolfram|Alpha. The recent Wolfram|Alpha API and iPhone application were discussed, as well as items like the Wolfram|Alpha corporate and professional versions that we can look forward to using in the future.

Attendees represent everyone from new users to longtime enthusiasts. We’re excited to see all of you and are looking forward to the rest of the conference.

Late last week we introduced the Wolfram|Alpha Webservice API, allowing outside developers to call Wolfram|Alpha from their websites or application programs.

Then yesterday we released the first mobile implementation of Wolfram|Alpha, in the form of an iPhone app.

Tomorrow, we’re doing something completely different: Wolfram|Alpha Homework Day—a 14-hour live webcast event for students and educators.

Oh, and starting on Thursday is the International Mathematica User Conference, which will show many advances in the core Mathematica technology on which Wolfram|Alpha is based.

In May we launched Wolfram|Alpha as a major website. But the website is really just the tip of an iceberg. Wolfram|Alpha is a whole technology that can be deployed and used in all sorts of ways.

This week we’re seeing some of the diversity of what can be done. The API as a tool for developers. The iPhone app as a consumer product. And Homework Day as a community event that Wolfram|Alpha enables.

My approach to managing R&D is to maintain a portfolio of ongoing projects, short-term and long-term. There’s an amazing amount in the pipeline for Wolfram|Alpha, which will emerge on timescales from weeks to years.

I’m very proud of the achievements of our various teams that we’re seeing this week. The diversity of what’s becoming possible with Wolfram|Alpha is surprising even us. There are a lot of very exciting things to come…

Mathematica and Wolfram|Alpha are great resources for both teachers and students. Using the two together is a good way to explore topics in more depth. This video shows a few examples of how you can utilize Mathematica and Wolfram|Alpha in your own classroom.

In August, we traveled to ACS in beautiful Washington, DC, USA. The ACS meeting brought together the largest scientific society and its members’ families, colleagues, and students. It provided an ideal venue to demonstrate Mathematica’s capabilities in chemistry and chemical engineering. We demonstrated a broad range of features, including Mathematica 7’s fully curated chemical, genomic, and proteomic data, built-in parallelization capabilities, and unsurpassed modeling and visualization capabilities. The ability to visualize any data as well as update it on the fly has bridged a gap many researchers and scientists have had to work around when using other tools. You can even rapidly develop and test algorithms as well as generate accurate structural renderings in 2D and 3D using the integrated data, which is easy to retrieve programmatically.

One of the examples we showed at the Wolfram booth was how to return all property information for a single chemical. To do so, we wrote a short function that collected all the properties for a chemical, added a column with the property names, removed any missing data—some properties are just for liquids, some for solids, and so on—and created output as a styled table that could be used as a chart in a laboratory. For example:

And here’s its output (click to see the whole thing):

The meeting had over 14,000 people in attendance, and at the Wolfram booth we were busy for all the meeting days—at least 20% of the attendees spent time with us. Looks like San Francisco, California will be the place for the Spring 2010 ACS Meeting in March. You know we’ll be there, and it is certain we’ll have new stuff to talk about. Mark your calendars and we’ll see you then!

In my spare time, I queried the SDSS website, which is database driven, and in eight separate queries I was able to get all galaxies in the survey out to a redshift of 0.5. According to Wolfram|Alpha, this corresponds to looking back in time 5.02 billion years ago, or a distance of 6.14 billion light years, when the light we’re now seeing from the most distant galaxies started its journey here. That’s a billion years before our solar system formed. It’s taken this long for the light to reach us.

This animation is of only a tiny fraction of the visible universe, but still more than enough to give you a glimpse of the size and scale of the universe.

Each point represents a galaxy. Our galaxy would be at the very center (no, we aren’t the center of the universe—just the center of this animation). As you can see, galaxies tend to cluster along a web-like structure with “voids” interspersed. The survey images the sky in slices, which form the wedge-like pieces in the animation. These are the areas of the sky imaged so far.

The well-known Hubble Ultra Deep Field, while imaging galaxies to a much greater distance (about 13 billion years ago and redshifts between 7 and 12), covers only a tiny pinprick of the sky (11 arcminutes) and so doesn’t show the structure that this animation does.

This was achieved in Mathematica with a very tiny amount of code:

Here proj is just the collection of triples {x, y, z} obtained from the SDSS data. I then just varied the ViewPoint and exported each frame. Find out more about using Mathematica in the field of astronomy on our Astronomy Solution page.

In this video, Ulrey describes how Mathematica’s graphical and visualization capabilities play a crucial role in developing models to analyze and test the safety of new flight operations. “It puts the whole conversation of whether it’s safe on a firm quantitative-model basis that enables people to make decisions about whether to go forward,” says Ulrey. “They have much better insight and they have confidence in the results.”

Join us at this year’s International Mathematica User Conference, where Ulrey will be a speaker. Watch more of our Portraits of Success videos.