Is there a global theory for the shapes of fishes? It’s the kind of thing I might feel encouraged to ask by my explorations of simple programs and the forms they produce. But for most of the history of biology, it’s not the kind of thing anyone would ever have asked. With one notable exception: D’Arcy Wentworth Thompson.

And it’s now 100 years since D’Arcy Thompson published the first edition of his magnum opus On Growth and Form—and tried to use ideas from mathematics and physics to discuss global questions of biological growth and form. Probably the most famous pages of his book are the ones about fish shapes:

A week ago a new train station, named “Cambridge North”, opened in Cambridge, UK. Normally such an event would be far outside my sphere of awareness. (I think I last took a train to Cambridge in 1975.) But last week people started sending me pictures of the new train station, wondering if I could identify the pattern on it:

It’s now 15 years since I published my book A New Kind of Science—more than 25 since I started writing it, and more than 35 since I started working towards it. But with every passing year I feel I understand more about what the book is really about—and why it’s important. I wrote the book, as its title suggests, to contribute to the progress of science. But as the years have gone by, I’ve realized that the core of what’s in the book actually goes far beyond science—into many areas that will be increasingly important in defining our whole future. Continue reading

“In the next hour I’m going to try to make a new discovery in mathematics.” So I began a few days ago at two different hour-long Math Encounters events at the National Museum of Mathematics (“MoMath”) in New York City. I’ve been a trustee of the museum since before it opened in 2012, and I was looking forward to spending a couple of hours trying to “make some math” there with a couple of eclectic audiences from kids to retirees.

People usually assume that new discoveries aren’t things one can ever see being made in real time. But the wonderful thing about the computational tools I’ve spent decades building is that they make it so fast to implement ideas that it becomes realistic to make discoveries as a kind of real-time performance art. Continue reading

For as long as I can remember, my all-time favorite activity has been creating ideas and turning them into reality—a kind of “entrepreneurism of ideas”. And over the years—in science, technology and business—I think I’ve developed some pretty good tools and strategies for doing this, that I’ve increasingly realized would be good for a lot of other people (and organizations) too.

So how does one spread idea entrepreneurism—entrepreneurism centered on ideas rather than commercial enterprises? Somewhat unwittingly I think we’ve developed a rather good vehicle—that’s both a very successful educational program, and a fascinating annual adventure for me.

Twelve years ago my book A New Kind of Science had just come out, and we were inundated with people wanting to learn more, and get involved in research around it. We considered various alternatives, but eventually we decided to organize a summer school where we would systematically teach about our methodology, while mentoring each student to do a unique original project.

From the very beginning, the summer school was a big success. And over the years we’ve gradually improved and expanded it. It’s still the Wolfram Science Summer School—and its intellectual core is still A New Kind of Science. But today it has become a broader vehicle for passing on our tools and strategies for idea entrepreneurism.

This year’s summer school just ended last week. We had 63 students from 21 countries—with a fascinating array of backgrounds and interests. Most were in college or graduate school; a few were younger or older. And over the course of the three weeks of the summer school—with great energy and intellectual entrepreneurism—each student worked towards their own unique project. Continue reading

Today ten years have passed since A New Kind of Science (“the NKS book”) was published. But in many ways the development that started with the book is still only just beginning. And over the next several decades I think its effects will inexorably become ever more obvious and important.

Indeed, even at an everyday level I expect that in time there will be all sorts of visible reminders of NKS all around us. Today we are continually exposed to technology and engineering that is directly descended from the development of the mathematical approach to science that began in earnest three centuries ago. Sometime hence I believe a large portion of our technology will instead come from NKS ideas. It will not be created incrementally from components whose behavior we can analyze with traditional mathematics and related methods. Rather it will in effect be “mined” by searching the abstract computational universe of possible simple programs.

And even at a visual level this will have obvious consequences. For today’s technological systems tend to be full of simple geometrical shapes (like beams and boxes) and simple patterns of behavior that we can readily understand and analyze. But when our technology comes from NKS and from mining the computational universe there will not be such obvious simplicity. Instead, even though the underlying rules will often be quite simple, the overall behavior that we see will often be in a sense irreducibly complex.

So as one small indication of what is to come—and as part of celebrating the first decade of A New Kind of Science—starting today, when Wolfram|Alpha is computing, it will no longer display a simple rotating geometric shape, but will instead run a simple program (currently, a 2D cellular automaton) from the computational universe found by searching for a system with the right kind of visually engaging behavior.

(This is the second of a series of posts related to next week’s tenth anniversary of A New Kind of Science. The previous post covered developments since the book was published; the next covers its future.)

“You’re destroying the heritage of mathematics back to ancient Greek times!” With great emotion, so said a distinguished mathematical physicist to me just after A New Kind of Science was published ten years ago. I explained that I didn’t write the book to destroy anything, and that actually I’d spent all those years working hard to add what I hoped was an important new chapter to human knowledge. And, by the way—as one might guess from the existence of Mathematica—I personally happen to be quite a fan of the tradition of mathematics.

He went on, though, explaining that surely the main points of the book must be wrong. And if they weren’t wrong, they must have been done before. The conversation went back and forth. I had known this person for years, and the depth of his emotion surprised me. After all, I was the one who had just spent a decade on the book. Why was he the one who was so worked up about it?

And then I realized: this is what a paradigm shift sounds like—up close and personal. Continue reading

(This is the first of a series of posts related to next week’s tenth anniversary of A New Kind of Science. The second covers what’s happened since it was published, and the third its future.)

On May 14, 2012, it’ll be 10 years since A New Kind of Science (“the NKS book”) was published.

After 20 years of research, and nearly 11 years writing the book, I’d taken most things about as far as I could at that time. And so when the book was finished, I mainly launched myself back into technology development. And inspired by my work on the NKS book, I’m happy to say that I’ve had a very fruitful decade (Mathematica reinvented, CDF, Wolfram|Alpha, etc.).

I’ve been doing little bits of NKS-oriented science here and there (notably at our annual Summer School). But mostly I’ve been busy with other things. And so it’s been other people who’ve been having the fun of moving the science of NKS forward. But almost every day I’ll hear about something that’s been being done with NKS. And as we approach the 10-year mark, I’ve been very curious to try to get at least a slightly more systematic view of what’s been going on.

A place to start is the academic literature, where there’s now an average of slightly over one new paper per day published citing the NKS book—with that number steadily increasing. The papers span all kinds of areas (here identified by journal fields):

Last weekend I decided to have a short break from all the exciting technological things we’re doing… and to give a talk at the Singularity Summit in New York City about the implications of A New Kind of Science for the future, technological and otherwise. Here’s the transcript:

Well, what I wanted to do here was to have some fun—and talk about the future.

That’s something that’s kind of recreational for me. Because what I normally do is work in the trenches just trying to actually build the future… kind of one brick at a time—or at least one big project at a time.

I’ve been doing this now for a bit more than 30 years, and I guess I’ve built a fairly tall tower. From which it’s possible to do and see some pretty interesting things. Continue reading

This week I’m giving a talk at a conference on Mathematics and Computation in Music (MCM 2011)… so I decided to collect some of my thoughts on such topics…

How difficult is it to generate human-like music? To pass the analog of the Turing test for music?

Though music typically has a certain formal structure—as the Pythagoreans noted 2500 years ago—it seems at its core somehow fundamentally human: a reflection of raw creativity that is almost a defining characteristic of human capabilities.

But what is that creativity? Is it something that requires the whole history of our biological and cultural evolution? Or can it exist just as well in systems that have nothing directly to do with humans?

In my work on A New Kind of Science, I studied the computational universe of possible programs—and found that even very simple programs can show amazingly rich and complex behavior, on a par, for example, with what one sees in nature. And through my Principle of Computational Equivalence I came to believe that there can be nothing that fundamentally distinguishes our human capabilities from all sorts of processes that occur in nature—or in very simple programs.

But what about music? Some people used their belief that “no simple program will ever create great music” to argue that there must be something wrong with my Principle of Computational Equivalence.

So I became curious: is there really something special and human about music? Or can it in fact be created perfectly well in an automatic, computational way?

Wolfram|Alpha, A New Kind of Science, and even Mathematica all have aspects that are philosophy projects. Each of them, in different ways, informs questions in philosophy—and are themselves informed by philosophical ideas and discoveries.

Indeed, the very fact that I decided Wolfram|Alpha might be a possible project was the result of what amounts to a philosophical realization that I learned from A New Kind of Science: there is no bright line that identifies “intelligence”; it is all just computation.

I don’t get to talk much about philosophy. But here is a recording of a keynote speech I was recently asked to give about “computing and philosophy”.

I spent a decade of my life writing A New Kind of Science. Most of that time was devoted to discovering the science in the book. But another part was spent figuring out how to present the science in the best possible way—using words and pictures.

It took a lot of technology to do that presentation. On the software side, the biggest part was using Mathematica to create elaborate algorithmic diagrams—thousands of them. But then came the question of how to actually deliver everything. And back in 2002 when A New Kind of Science was published, the only real possibility was to print a book on paper, using the very best printing technology of the time.

The actual print production process was quite an adventure—going right to the edge of what was possible. But in the end we got many compliments on the object we produced. And from that time to this, that 5.5 lb (2.5 kg) lump of paper has been the definitive representation of my decade-plus of intellectual work.

But today I’m excited to be able to say that there’s something new and in some ways even better: a full version on the iPad.

(This post was originally published on the Wolfram Blog.)

May 14, 2009 marks the 7^{th} anniversary of the publication of A New Kind of Science, and it has been my tradition on these anniversaries to write a short report on the progress of NKS.

It has been fascinating over the past few years to watch the progressive absorption of NKS methods and the NKS paradigm into countless different fields. Sometimes there’s visible mention of NKS, though often there is not.

There has been an inexorable growth in the use of the types of models pioneered in NKS. There has been steadily increasing use of the kinds of computational experiments and investigations introduced in NKS. And the NKS way of thinking about computation and in terms of computation has become steadily more widespread. Continue reading

(This post was originally posted on the NKS Forum.)

The following are some remarks that I made on the Foundations of Math (FOM) mailing list in connection with discussion of the Wolfram 2,3 Turing Machine Prize. Though much of what I say is well understood in the NKS community, I thought it might nevertheless be of interest here.

Several people forwarded me the thread on this mailing list about our 2,3 Turing machine prize.

I’m glad to see that it has stimulated discussion. Perhaps I can make a few general remarks.

What do we learn from simple universal Turing machines?

John McCarthy wrote:

In the 1950s I thought that the smallest possible (symbol-state product) universal Turing machine would tell something about the nature of computation. Unfortunately, it didn’t.

I suspect that what was imagined at that time was that by finding the smallest universal machines one would discover some “spark of computation”—some critical ingredient in the rules necessary to make universal computation possible. (At the time, it probably also still seemed that there might be a “spark of life” or a “spark of intelligence” that could be found in systems.)

I remember that when I first heard about universality in the Game of Life in the early 1970s, I didn’t think it particularly significant; it seemed like just a clever hack.

I had searched the computational universe for the simplest possible universal Turing machine. And I had found a candidate—that my intuition told me was likely to be universal. But I was not sure.

And so as part of commemorating the fifth anniversary of A New Kind of Science on May 14 this year, we announced a $25,000 prize for determining whether or not that Turing machine is in fact universal.

I had no idea how long it would take before the prize was won. A month? A year? A decade? A century? Perhaps the question was even formally undecidable (say from the usual axioms of mathematics).

But today I am thrilled to be able to announce that after only five months the prize is won—and we have answer: the Turing machine is in fact universal!

Alex Smith—a 20-year-old undergraduate from Birmingham, UK—has produced a 40-page proof.

I’m pleased that my intuition was correct. But more importantly, we now have another piece of evidence for the very general Principle of Computational Equivalence (PCE) that I introduced in A New Kind of Science.

We are also at the end of a quest that has spanned more than half a century to find the very simplest universal Turing machine.

Here it is. Just two states and three colors. And able to do any computation that can be done.

(This post was originally published on the Wolfram Blog.)

I don’t have much time for hobbies these days, but occasionally I get to indulge a bit. A few days ago I did a videoconference talking about one of my favorite hobbies: hunting for the fundamental laws of physics.

Physicists often like to think that they’re dealing with the most fundamental kinds of questions in science. But actually, what I realized back in 1981 or so is that there’s a whole layer underneath.

There’s not just our own physical universe to think about, but the whole universe of possible universes.

If one’s going to do theoretical science, one had better be dealing with some kind of definite rules. But the question is: what rules?

Nowadays we have a great way to parametrize possible rules: as possible computer programs. And I’ve built a whole science out of studying the universe of possible programs–and have discovered that even very simple ones can generate all sorts of rich and complex behavior.

Well, that’s turned out to be relevant in modeling all sorts of systems in the physical and biological and social sciences, and in discovering interesting technology, and so on.

But here’s my big hobby question: what about our physical universe? Could it be operating according to one of these simple rules?

(This post was originally published on the Wolfram Blog.)

When I hear about something like Wednesday’s bridge collapse, I immediately wonder whether any of the science I’ve worked on can be of any help.

Bridge design is one of the classic—almost iconic—successes of traditional mathematical science.

And when I first talked about A New Kind of Science, a not uncommon reaction was precisely, “But can it help build better bridges?”

Well, as a matter of fact, I rather suspect it can.

Bridges have a long history. Early on, only a few types seem to have been used. But with the arrival of iron structures in the 1800s there was a kind of “Cambrian explosion” of different types of truss bridges:

But what is the very best bridge structure, say from the point of view of robustness? There’s a huge universe of possibilities. But so far, only a tiny corner has been explored–and that mostly in the 1800s.

(This post was originally published on the Wolfram Blog.)

Our 2007 NKS Summer School started about two weeks ago, and one of my roles there was to show a little of how NKS is done.

In the past, it would have been pretty unrealistic to show this in any kind of live way. But with computer experiments, and especially with Mathematica, that’s changed. And now it’s actually possible to make real discoveries in real time in front of live audiences.

I’ve done a few dozen “live experiments” now (here is an account of one from 2005). My scheme is as follows. Sometime between a few hours and a few minutes before the live experiment, I come up with a topic that I’m pretty sure hasn’t been studied before. Then I make sure to avoid thinking about it until I’m actually in front of the live audience.

Then, once the experiment starts, I have a limited time to discover something. Just by running Mathematica. Preferably with a little help from the audience. And occasionally with a little help from references on the web.

Every live experiment is an adventure. And it seems like almost every time, at around the halfway point, things look bad. We’ve tried lots of things. We’ve opened lots of threads. But nothing’s coming together.

But then, somehow, things almost always manage to come together. And we manage to discover something. That’s often pretty interesting. (There are still papers coming out now based on the live experiment I did at our very first Summer School, back in 2003).

I usually make my first live experiment at each Summer School be a piece of “pure NKS”: an abstract investigation of some simple program out in the computational universe.

This year I decided to take a look at an “old chestnut” that I’d recently been reminded about: a simple program (though it wasn’t thought of that way then) that was actually first investigated all the way back in 1920. Continue reading

(This post was originally published on the Wolfram Blog.)

It is perhaps ironic that two weeks after releasing what is probably the single most complex computational system ever constructed, we are today announcing a prize for the very simplest of computational systems.

But today is the fifth anniversary of the publication of A New Kind of Science, and to commemorate this, we have decided to establish the first NKS prize.

The prize is related to a core objective of the basic science of NKS: to map out the abstract universe of possible computational systems.

We know from NKS that very simple programs can show immensely complex behavior. And in the NKS book I formulated the Principle of Computational Equivalence that gives an explanation for this discovery.

That principle has many predictions. And one of them is that the ability to do general-purpose computing—to be capable of universal computation—should be common even among systems with very simple rules.

Today’s CPUs have millions of components. But the Principle of Computational Equivalence implies that all kinds of vastly simpler systems should also support universal computation.

The NKS book already gives several dramatic examples. But the purpose of the prize is to determine the boundary of universal computation for a particularly classic type of computational system: Turing machines. Continue reading

(This post was originally published on the Wolfram Blog.)

New technology is often what has driven the creation of new science. And so it has been with Mathematica.

One of the main reasons I originally started building Mathematica was that I wanted to use it myself.

And having Mathematica was a bit like having one of the first telescopes: I could point it somewhere, and immediately see all sorts of new things that had never been seen before.

Much has been discovered with Mathematica in almost every area of science.

But my particular interest has been to create a new kind of science that is uniquely made possible by Mathematica: a science based on exploring the computational universe.

We are used to creating computer programs for particular purposes. But as a matter of basic science we can ask about the universe of all possible programs.

And with Mathematica it becomes easy to explore this.

A quarter of a century ago I had begun my exploration of the computational universe—and had glimpsed some remarkable phenomena.

Then, when Mathematica was built, I went back and started a systematic study of the computational universe.

The results were remarkable. Wherever I looked—even among the simplest of programs—I saw all sorts of complex and interesting behavior. And from what I found I could make progress on a remarkable range of longstanding questions across all sorts of areas.

For eleven years I worked to develop this. And finally, on May 14, 2002, I published what I had done in my book A New Kind of Science.

(This post was originally published on the NKS Forum.)

I sent the following today to our NKS mailing list:

—

Today [May 14, 2004] marks the second anniversary of the release of A New Kind of Science. And I’m very happy to be able to report that NKS is continuing to develop extremely well.

A wonderful community is forming around the ideas of NKS. The pace of research and applications is steadily building—with an average of about one new paper now appearing every day. NKS classes and courses are being taught. And several times each week we hear about an ambitious new initiative based on NKS—in technology, or art, or business or somewhere else.

We’re trying to do our part to help. Earlier this year we released the online version of the complete book. We launched the NKS Forum. We just sponsored the second annual conference: NKS 2004. And we’re working hard to make wolframscience.com the best possible reference source and meeting place for the NKS community. Continue reading

(This post was originally published on the NKS Forum.)

At the NKS 2004 conference I did my now-traditional “live computer experiment”. This time the topic I picked came from a question someone asked at the minicourse before the conference: does increasing the “range” of a cellular automaton have a big effect on its behavior?

I decided to investigate a simple version of this question.

In an ordinary r=1 cellular automaton, the new color of a particular cell depends on the previous colors of cells with offsets -1, 0, 1. The question I asked was then: what happens if the offsets are larger?

In the simplest non-trivial cellular automata, the color of a cell depends on the previous colors of two cells. In the ordinary short-range case, the cells have offsets -1, 1. But now we can ask what happens if instead they have offsets -1, m. Continue reading

(This post was originally published on the NKS Forum.)

There’s a certain complexity to many of the characteristic forms used in Christmas images: snowflakes, Christmas trees, frost patterns, etc.

And as in so many other cases, it’s rather easy to capture the essence of these forms using very simple cellular automaton rules.

So that means it’s easy to use cellular automaton rules to make Christmas-like images.

Well, going through some of my archives recently, I was reminded that I did that almost exactly twenty years ago—for Christmas 1983. (The actual file date is November 22, 1983.) Continue reading