The Concept of the Ruliad

Stephen Wolfram

The Concept of the Ruliad

The Entangled Limit of Everything

I call it the ruliad. Think of it as the entangled limit of everything that is computationally possible: the result of following all possible computational rules in all possible ways. It’s yet another surprising construct that’s arisen from our Physics Project. And it’s one that I think has extremely deep implications—both in science and beyond.

In many ways, the ruliad is a strange and profoundly abstract thing. But it’s something very universal—a kind of ultimate limit of all abstraction and generalization. And it encapsulates not only all formal possibilities but also everything about our physical universe—and everything we experience can be thought of as sampling that part of the ruliad that corresponds to our particular way of perceiving and interpreting the universe.

We’re going to be able to say many things about the ruliad without engaging in all its technical details. (And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) But to ground things here, let’s start with a slightly technical discussion of what the ruliad is.

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Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

Xerxes D. Arsiwalla (Pompeu Fabra University/Wolfram Research)

Jonathan Gorard (University of Cambridge/Wolfram Research)

How do spaces emerge from pregeometric discrete building blocks governed by computational rules? To address this, we investigate non-deterministic rewriting systems (multiway systems) of the Wolfram model. We formalize these rewriting systems as homotopy types. Using this new formulation, we outline how spatial structures can be functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopy types are constructed from rewriting rules. These correspond to morphisms of an n-fold category. Subsequently, the n→∞ limit of the Wolfram model rulial multiway system is identified as an ∞-groupoid, with the latter being relevant given Grothendieck’s homotopy hypothesis. We then go on to show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an (∞,1)-topos. This correspondence to higher categorical structures offers a new way to understand how spaces relevant to physics may result from pregeometric combinatorial models. The key issue we have addressed here is to formally relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it removes ad hoc assumptions about geometric attributes of a model such as an a priori background or pre-assigned geometric data. Instead, geometry is inherited functorially from globular structures. This is relevant for formally justifying different choices of underlying spacetime discretization adopted by various models of quantum gravity. Finally, we end with comments on how the framework of higher category-theoretic combinatorial constructions developed here, corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.

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Multicomputation with Numbers: The Case of Simple Multiway Systems

Stephen Wolfram

A Minimal Example of Multicomputation

Multicomputation is an important new paradigm, but one that can be quite difficult to understand. Here my goal is to discuss a minimal example: multiway systems based on numbers. Many general multicomputational phenomena will show up here in simple forms (though others will not). And the involvement of numbers will often allow us to make immediate use of traditional mathematical methods.

A multiway system can be described as taking each of its states and repeatedly replacing it according to some rule or rules with a collection of states, merging any states produced that are identical. In our Physics Project, the states are combinations of relations between elements, represented by hypergraphs. We’ve also often considered string substitution systems, in which the states are strings of characters. But here I’ll consider the case in which the states are numbers, and for now just single integers. Continue reading

Even beyond Physics: Introducing Multicomputation as a Fourth General Paradigm for Theoretical Science

Stephen Wolfram

Even Beyond Physics: Introducing Multicomputation as a Fourth General Paradigm for Theoretical Science

The Path to a New Paradigm

One might have thought it was already exciting enough for our Physics Project to be showing a path to a fundamental theory of physics and a fundamental description of how our physical universe works. But what I’ve increasingly been realizing is that actually it’s showing us something even bigger and deeper: a whole fundamentally new paradigm for making models and in general for doing theoretical science. And I fully expect that this new paradigm will give us ways to address a remarkable range of longstanding central problems in all sorts of areas of science—as well as suggesting whole new areas and new directions to pursue.

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How Inevitable Is the Concept of Numbers?

Stephen Wolfram

Based on a talk at Numerous Numerosity: An interdisciplinary meeting on the notions of cardinality, ordinality and arithmetic across the sciences.

Everyone Has to Have Numbers… Don’t They?

The aliens arrive in a starship. Surely, one might think, to have all that technology they must have the idea of numbers. Or maybe one finds an uncontacted tribe deep in the jungle. Surely they too must have the idea of numbers. To us numbers seem so natural—and “obvious”—that it’s hard to imagine everyone wouldn’t have them. But if one digs a little deeper, it’s not so clear.

It’s said that there are human languages that have words for “one”, “a pair” and “many”, but no words for specific larger numbers. In our modern technological world that seems unthinkable. But imagine you’re out in the jungle, with your dogs. Each dog has particular characteristics, and most likely a particular name. Why should you ever think about them collectively, as all “just dogs”, amenable to being counted?

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Homotopies in Multiway (Non-Deterministic) Rewriting Systems as n-Fold Categories

Xerxes D. Arsiwalla (Pompeu Fabra University/Wolfram Research)

Jonathan Gorard (University of Cambridge/Wolfram Research)

Hatem Elshatlawy (RWTH Aachen University/Wolfram Research)

We investigate the algebraic and compositional properties of multiway (non-deterministic) abstract rewriting systems, which are the archetypical structures underlying the formalism of the so-called Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where these homotopic maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order n may naturally be formalized as an n-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ∞-groupoid. Via Grothendieck’s homotopy hypothesis, this ∞-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of multiway rewriting systems may potentially be used for making formal connections to homotopy spaces upon which models of physics can be instantiated.

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The Problem of Distributed Consensus

Stephen Wolfram

In preparation for a conference entitled “Distributed Consensus with Cellular Automata and Related Systems” that we’re organizing with NKN (= “New Kind of Network”) I decided to explore the problem of distributed consensus using methods from A New Kind of Science (yes, NKN “rhymes” with NKS) as well as from the Wolfram Physics Project.

A Simple Example

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BlockRandom[SeedRandom[77]; 
 Module[{pts = 
    RandomPointConfiguration[HardcorePointProcess[0.09, 2, 2], 
      Rectangle[{0, 0}, {40, 40}]]["Points"], clrs}, 
  clrs = Table[
    RandomChoice[{.65, .35} -> {Hue[0.15, 0.72, 1], Hue[
       0.98, 1, 0.8200000000000001]}], Length[pts]]; 
  Graphics[{EdgeForm[Gray], 
    Table[Style[Disk[pts[[n]]], clrs[[n]]], {n, 
      Range[Length[pts]]}]}]]]

Consider a collection of “nodes”, each one of two possible colors. We want to determine the majority or “consensus” color of the nodes, i.e. which color is the more common among the nodes.

One obvious method to find this “majority” color is just sequentially to visit each node, and tally up all the colors. But it’s potentially much more efficient if we can use a distributed algorithm, where we’re running computations in parallel across the various nodes.

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Fast Automated Reasoning over String Diagrams using Multiway Causal Structure

Jonathan Gorard (University of Cambridge/Wolfram Research)

Manojna Namuduri (Wolfram Research)

Xerxes D. Arsiwalla (Pompeu Fabra University/Wolfram Research)

We introduce an intuitive algorithmic methodology for enacting automated rewriting of string diagrams within a general double-pushout (DPO) framework, in which the sequence of rewrites is
chosen in accordance with the causal structure of the underlying diagrammatic calculus. The combination of the rewriting structure and the causal structure may be elegantly formulated as a weak 2-category equipped with both total and partial monoidal bifunctors, thus providing a categorical semantics for the full multiway evolution causal graph of a generic Wolfram model hypergraph rewriting system. As an illustrative example, we show how a special case of this algorithm enables highly efficient automated simplification of quantum circuits, as represented in the ZX-calculus.

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Why Does the Universe Exist? Some Perspectives from Our Physics Project

Stephen Wolfram

Why Does the Universe Exist? Some Perspectives from Our Physics Project

What Is Formal, and What Is Actualized?

Why does the universe exist? Why is there something rather than nothing? These are old and fundamental questions that one might think would be firmly outside the realm of science. But to my surprise I’ve recently realized that our Physics Project may shed light on them, and perhaps even show us the way to answers.

We can view the ultimate goal of our Physics Project as being to find an abstract representation of what our universe does. But even if we find such a representation, the question still remains of why that representation is actualized: why what it represents is “actually happening”, with the actual stuff our universe is “made of”.

It’s one thing to say that we have a rule or program that can reproduce a representation of what our universe is doing. But it seems very different to say that the rule or program is “actually being run” and is “actually generating” the “physical reality” of our universe.

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The Wolfram Physics Project:
A One-Year Update

Stephen Wolfram

Upcoming livestream

The Wolfram Physics Project: A One-Year UpdateThe Wolfram Physics Project: A One-Year Update

How’s It Going?

When we launched the Wolfram Physics Project a year ago today, I was fairly certain that—to my great surprise—we’d finally found a path to a truly fundamental theory of physics, and it was beautiful. A year later it’s looking even better. We’ve been steadily understanding more and more about the structure and implications of our models—and they continue to fit beautifully with what we already know about physics, particularly connecting with some of the most elegant existing approaches, strengthening and extending them, and involving the communities that have developed them.

And if fundamental physics wasn’t enough, it’s also become clear that our models and formalism can be applied even beyond physics—suggesting major new approaches to several other fields, as well as allowing ideas and intuition from those fields to be brought to bear on understanding physics.

Needless to say, there is much hard work still to be done. But a year into the process I’m completely certain that we’re “climbing the right mountain”. And the view from where we are so far is already quite spectacular.

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