Multicomputation

A Fourth Paradigm for Theoretical Science. page

Need model of observer to determine state

The first, originating in antiquity, one might call the “structural paradigm”. Its key idea is to think of things in the world as being constructed from some kind of simple-to-describe elements—say geometrical objects—and then to use something like logical reasoning to work out what will happen with them. Typically this paradigm has no explicit notion of Time or dynamical change, though in its modern forms it often involves making descriptions of structures of relationships, usually built from logical or “flowchart-like” elements.

Many would say that modern exact science was launched in the 1600s with the introduction of what we can call the “mathematical paradigm”: the idea that things in the world can be described by mathematical equations—and that their behavior can be determined by finding solutions to these equations. It’s common in this paradigm to discuss time—but normally it’s just treated as a variable in the equations, and one hopes that to find out what will happen at some arbitrary time one can just substitute the appropriate value for that variable into some formula derived by solving the equations.

For three hundred years the mathematical paradigm was the state of the art in theoretical science—and immense progress was made using it. But there remained plenty of phenomena—particularly associated with complexity—that this paradigm seemed to have little to say about. But then—basically starting in the early 1980s—there was a burst of progress based on a new idea (of which, yes, I seem to have ultimately been the primary initiator): the idea of using simple programs, rather than mathematical equations, as the basis for models of things in nature and elsewhere.

Part of what this achieves is to generalize beyond traditional mathematics the kind of constructs that can appear in models. But there is something else too—and it’s from this that the full computational paradigm emerges. In the mathematical paradigm one imagines having a mathematical equation and then separately somehow solving it. But if one has a program one can imagine just directly taking it and running it to find out what it does. And this is the essence of the computational paradigm: to define a model using computational rules (say, for a cellular automaton) and then explicitly be able to run these to work out their consequences.

And one feature of this setup is that Time becomes something much more fundamental and intrinsic. In the mathematical paradigm it’s in effect just the arbitrary value of a variable. But in the computational paradigm it’s a direct Reflection of the actual Process of applying computational rules in a model—or in other words in this paradigm the passage of time corresponds to the actual progress of computation.

A major discovery is that in the computational universe of possible programs even ones with very simple rules can show immensely complex behavior. And this points the way—through the Principle of Computational Equivalence—to computational irreducibility: the phenomenon that there may be no faster way to find out what a system will do than just to trace each of its computational steps. Or, in other words, that the passage of time can be an irreducible process, and it can take an irreducible amount of computational work to predict what a system will do at some particular time in the future. (Yes, this is closely related not only to things like undecidability, but also to things like the Second Law of Thermodynamics.)

In the full arc of scientific history, the computational paradigm is very new. But in the past couple of decades, it’s seen rapid and dramatic success—and by now it’s significantly overtaken the mathematical paradigm as the most common source for new models of things. Despite this, however, fundamental physics always seemed to resist its advance. And now, from our Physics Project, we can see why.

Because at the core of our Physics Project is actually a new paradigm that goes beyond the computational one: a fourth paradigm for theoretical science that Stephen Wolfram is calling the multicomputational paradigm.

There’ve been hints of this paradigm before—some even going back a century. But it’s only as a result of our Physics Project that we’ve been able to start to see its full depth and structure. And to understand that it really is a fundamentally new paradigm—that transcends physics and applies quite generally as the foundation for a new and broadly applicable methodology for making models in theoretical science.

⇒ Multiway Systems and the Concept of the Multicomputation