Great Throughts Treasury

"Computational irreducibility tends to make infinite questions undecidable. The presence of universality implies that there must at some level be computational irreducibility… This means that today’s mathematics will be viewed as small and surprisingly uncharacteristic sample of what is possible. If a system is computationally irreducible this means that there is in effect a tangible separation between the underlying rules for the system and its overall behavior associated with the irreducible amount of computational work needed to go from one to the other. And it is this separation that the basic origin of the apparent freedom we see in all sorts of system lie – whether those systems are abstract cellular automata or actual living brains."

"The overall similarity between mathematics and nature must have a deeper origin, both involving processes that can be thought of as computations."

"Historically, we have assumed that it would only be a matter of time before everything would somehow be predicted by science."

"When it comes to more complex behavior mathematics has never in fact done well at explaining most of what we see every day in nature."

"A New Kind of Science gives us three very important facts: 1. Mathematics can only explain simple things. 2. You need a model to explain complicated things. 3. But - a simple model can explain complicated things. [paraphrased summary statement]"

"And at the lowest level what I expect is that even though the rules being applied are perfectly definite, the overall pattern of connections that will exist in the network corresponding to our universe will continually be rearranged in ways complicated enough to seem effectively random. Yet on a slightly larger scale such randomness will then lead to a certain average uniformity. And it is then essentially this that I believe is responsible for maintaining something like ordinary space - with gradual variations giving rise to the phenomenon of gravity. But superimposed on this effectively random background will then presumably also be some de some definite structures that persist through many updatings of the network. And it is these, I believe, that are what correspond to particles like electrons."

"And indeed in the end the PCE encapsulates both the ultimate power and the ultimate weakness of science. For it implies that all the wonders of the universe can in effect be captured by simple rules, yet it shows that there can be no way to know all the consequences of these rules, except in effect just to watch and see how they unfold."

"How can one tell in an actual experiment on some system in nature to what extent intrinsic randomness generation is really the mechanism responsible? …The clearest sign is a somewhat unexpected phenomenon: …if intrinsic randomness generation is…at work, then the precise details of the behavior can… be repeatable."

"In time I expect that the ideas of this book [A New Kind of Science] will come to pervade not only science and technology but also many areas of general thinking. And with this its methods will eventually become a standard part of education--much as mathematics is today. And in the end most of what now seems surprising and remarkable in the book will come to seem familiar and commonplace. But for me what has always been most important is the actual process of discovery. For I know of nothing as profoundly exciting as to glimpse for the first time some new and basic truth. And now that I have finished building the intellectual structure that I describe in this book it is my hope that those who read these words can share in the excitement I have had in making the discoveries that were involved."

"Over the years, I have watched with disappointment the continuing failure of most scientists and mathematicians to grasp the idea of doing computer experiments on the simplest possible systems… [Mathematicians] tend to add features to make their systems fit in with complicated and abstract ideas—often related to continuity—that exist in modern mathematics. …One might imagine that the best way to be certain about what could possibly happen in some particular system would be to prove a theorem… But in my experience… it is easy to end up making implicit assumptions that can be violated by circumstances one cannot foresee. And indeed, by now, I have come to trust the correctness of conclusions based on simple systematic computer experiments much more than I trust all but the simplest proofs."

"It took me more than a decade to come to terms with this result [that simple computer programs can produce complex behavior], and to realize just how fundamental and far-reaching its consequences are. In retrospect there is no reason the result could not have been found centuries ago, but increasingly I have come to view it as one of the more important single discoveries in the whole history of theoretical science. For in addition to opening up vast new domains of exploration, it implies a radical rethinking of how processes in nature and elsewhere work."

"I want to know the truth, however perverted that may sound."

"I guess the good news is that we didn't make any big mistakes in the design of earlier versions of Mathematica that we'd have to go back on now."

"The fact that the same symbolic programming primitives work for those as work for math kinds of things, I think, really validates the idea of symbolic programming being something pretty general."

"The thesis of A New Kind of Science is twofold: that the nature of computation must be explored experimentally, and that the results of these experiments have great relevance to understanding the natural world, which is assumed to be digital. Since its crystallization in the 1930s, computation has been primarily approached from two traditions: engineering, which seeks to build practical systems using computations; and mathematics, which seeks to prove theorems about computation (albeit already in the 1970s computing as a discipline was described as being at the intersection of mathematical, engineering, and empirical/scientific traditions. Wolfram describes himself as introducing a third major tradition, which is the systematic, empirical investigation of computational systems for their own sake. This is where the "New" and "Science" parts of the book's title originate. However, in proceeding with a scientific investigation of computational systems, Wolfram eventually came to the conclusion that an entirely new method is needed. In his view, traditional mathematics was failing to describe the complexity seen in these systems meaningfully. He suggests that each system consists of numerous more or less identical elements, even if there can be different types of elements in the same system, and that each element can only show a limited amount of states. Which state, depends on the state of the neighboring elements and the rules that determine how to respond to them. Through a combination of experiment and theoretical positioning, the book introduces a method that Wolfram argues is the most realistic way to make scientific progress with computational systems, casting A New Kind of Science as a "kind" of science, and allows its principles to be potentially applicable in a wide range of fields, like living organisms, ecology, society and traffic"

"The aphorism that the weather has a mind of its own may be less silly than you might imagine. The weather represents computations as sophisticated as anything in our brains."

"So the thing I realized rather gradually - I must say starting about 20 years ago now that we know about computers and things - there's a possibility of a more general basis for rules to describe nature."

"The thing that got me started on the science that I've been building now for about 20 years or so was the question of okay, if mathematical equations can't make progress in understanding complex phenomena in the natural world, how might we make progress?"

"The most important precedents deal with the whole idea of symbolic programming - the notion of setting up symbolic expressions that can represent anything one wants, and then having functions that operate on both their structure and content."

"There are a few very small incompatible changes - I really doubt most people will ever run into them."

"Well, the first thing to say is that we've worked hard to maintain compatibility, so that any program written with an earlier version of Mathematica can run without change in 3.0, and any notebook can be converted."

"There are various ways to state the Principle of Computational Equivalence, but probably the most general is just to say that almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication… The Principle of Computational Equivalence has vastly richer implications than the laws of thermodynamics or for that matter, than essentially any single collection of laws in science… But the Principle of Computational Equivalence asserts that in fact even if it comes from simple initial conditions almost all behavior that is not obviously simple will in the end correspond to computations of equivalent sophistication… even with a single very simple initial condition the actual evolution of a system will generate blocks that correspond to essentially all possible initial conditions. And this means that whatever behavior would be seen with a given overall initial condition, that same behavior will also be seen at appropriate places in the single pattern generated from a specific initial condition."

"You kind of alluded to it in your introduction. I mean, for the last 300 or so years, the exact sciences have been dominated by what is really a good idea, which is the idea that one can describe the natural world using mathematical equations."

"Wolfram's central thesis in his book, A New Kind of Science [paraphrase]: That three hundred years of mathematics based on the equals sign have failed to provide true insight into various complex systems in nature, and that algorithms based on the DO loop can succeed in this endeavor where mathematics has failed. The reason, claims Wolfram, is that deceptively simple algorithms can produce heretofore undreamed of levels of complexity. He claims that while frontier intellectual efforts such as chaos theory, fractals, AI, cybernetics and so forth have hinted at this concept for years, his decade of isolation studying cellular automata has taken the idea of simple algorithms or rules embodying universal complexity to the level of a new paradigm."

"As Mathematica was being developed, we showed it to Steve Jobs quite often. He always claimed he didn?t understand the math of it (though I later learned from a good friend of mine who had known Steve in high school that Steve had definitely taken at least one calculus course). But he made all sorts of ?make it simpler? suggestions about the interface and the documentation. With one slight exception, perhaps of at least curiosity interest to Mathematica aficionados: he suggested that cells in Mathematica notebook documents (now CDFs) should be indicated not by simple vertical lines?but instead by brackets with little serifs at their ends. And as it happens, that idea opened the way to thinking of hierarchies of cells, and ultimately to many features of symbolic documents."

"Bringing in an expert to do a function when you don't understand the function yourself is usually not a good idea. That's the place where the thing fails."

"At the time, all sorts of people were telling me that I needed to put quotes on the back cover of the book. So I asked Steve Jobs if he?d give me one. Various questions came back. But eventually Steve said, ?Isaac Newton didn?t have back-cover quotes; why do you want them?? And that?s how, at the last minute, the back cover of A New Kind of Science ended up with just a simple and elegant array of pictures."

"He'd [Steve Jobs] always push us to make use of his latest technology, be pleased when we did, and even after he was quite ill, occasionally intercede with remarkably detailed emails and phone calls. When the iPad came out, he was instrumental in the success of our Touch Press interactive book publishing company. And just the day before Steve died came the announcement of the iPhone 4S, and Siri, which uses our Wolfram|Alpha knowledge engine. The timing was so tragic. But it was a quintessential Steve Jobs move. To realize that people just want direct access to knowledge on their phones, without all the extra steps that people would usually assume have to be there. To those of us who spend our lives striving to build great technology, Steve Jobs will always be a remarkable inspiration, not only for his own technological achievements, but also for his great tenacity and dramatic ultimate success ? upon which so much more will yet be built."

"Could it be that some place out there in the computational universe, we might find our physical universe?"

"I had a very selfish reason for building Mathematica. I wanted to use it myself, a bit like Galileo got to use his telescope four hundred years ago. But I wanted to look, not at the astronomical universe, but at the computational universe."

"I think Computation is destined to be the defining idea of our future."

"I'm committed to seeing this project done. To see if within this decade we can finally hold in our hands the rule for our universe, and know where our universe lies in the space of all possible universes."

"It's always seemed like a big mystery how nature, seemingly so effortlessly, manages to produce so much that seems to us so complex. Well, I think we found its secret. It's just sampling what's out there in the computational universe."

"One of the things I always admired about Steve Jobs was his clarity of thought. Time and again, he would take a complex situation, understand its essence and use that understanding to make a bold and unexpected move."

"To me, Steve Jobs stands out most for his clarity of thought. Over and over again he took complex situations, understood their essence, and used that understanding to make a bold definitive move, often in a completely unexpected direction."