arXiv subjects
Publications and source records attributed to Stephen Wolfram.
An extensive survey is given of the historical origins of the Second Law of thermodynamics, illustrated by excerpts from many original sources, and with biographical information about key contributors. The major strands of conceptual development of the Second Law are traced, in particular showing how persistent beliefs about it emerged.
Both metamathematics and physics are posited to emerge from samplings by observers of the unique ruliad structure that corresponds to the entangled limit of all possible computations. The possibility of higher-level mathematics accessible to humans is posited to be the analog for mathematical observers of the perception of physical space for physical observers. A physicalized analysis is given of the bulk limit of traditional axiomatic approaches to the foundations of mathematics, together with explicit empirical metamathematics of some examples of formalized mathematics. General physicalized laws of mathematics are discussed, associated with concepts such as metamathematical motion, inevitable dualities, proof topology and metamathematical singularities. It is argued that mathematics as currently practiced can be viewed as derived from the ruliad in a direct Platonic fashion analogous to our experience of the physical world, and that axiomatic formulation, while often convenient, does not capture the ultimate character of mathematics. Among the implications of this view is that only certain collections of axioms may be consistent with inevitable features of human mathematical observers. A discussion is included of historical and philosophical connections, as well as of foundational implications for the future of mathematics.
A categorized bibliography of combinators is given, providing what is believed to be a largely complete coverage of publications from the origination of combinators in 1920 to the present day.
Integer iteration rules such as n |-> {a n + b, c n +d} are studied as minimal examples of the general process of multicomputation. Despite the simplicity of such rules, their multiway graphs can be complex, exhibiting, for example, emergent geometry and difficult questions of confluence. Generalizations to rules involving non-integers and other functions are also considered. Connections with physics and with various number-theoretic and other questions are made.
Combinators were a key idea in the development of mathematical logic and the emergence of the concept of universal computation. They were introduced on December 7, 1920, by Moses Schönfinkel. This is an exploration of the personal story and intellectual context of Moses Schönfinkel, including extensive new research based on primary sources.
A survey is given of approaches to the problem of distributed consensus, focusing particularly on methods based on cellular automata and related systems. A variety of new results are given, as well as a history of the field and an extensive bibliography. Distributed consensus is of current relevance in a new generation of blockchain-related systems.
As an example of empirical metamathematics, we present a detailed study of the dependency structure of the 465 theorems in Euclid's Elements, finding empirical signatures of concepts such as the power of a theorem. We apply similar methods to a more exhaustive study of possible theorems in logic, as well as to the analysis of dependency structures in projects to formalize modern pure mathematics. We discuss the process of identifying both intrinsic features of metamathematical space, and features of its exploration through the historical progress of mathematics.
We give a modern computational introduction to the S,K combinators invented by Moses Schönfinkel in 1920, and present a variety of new results and ideas about combinators. We explore the spectrum of behavior obtained with small combinator expressions, showing a variety of approaches to analysis and visualization. We discuss the implications of evaluation strategies, and of multiway systems representing all possible strategies. We show how causal graphs introduced in recent models of fundamental physics can be applied to combinators, as well as describing how combinators introduce a new form of treelike separation. We give a variety of new results on minimal combinator expressions, as well as showing how empirical computation theory and computational complexity theory can be done with combinators. We also suggest that when viewed in terms of ongoing computation, the S combinator alone may be capable of universal computation.
Empirical, theoretical and historical aspects of Post's "problem of tag" from 1921 are explored. Evidence of strong computational irreducibility is found. Despite their deterministic origin, the lengths of successive sequences generated seem to closely approximate random walks. All 10^25 smallest initial conditions are found to eventually halt, although sometimes in > 6*10^11 steps. Implications of the Principle of Computational Equivalence are discussed, along with examples of identifiable computational capabilities of tag systems. Various minimal examples of complex behavior are found, including a less-biased analog of the 3n+1 Collatz problem. There is also discussion of the history of Emil Post and of tag systems in the context of ideas about the foundations of mathematics and computation.
Multiway Turing machines (also known as nondeterministic Turing machines or NDTMs) with explicit, simple rules are studied. Even very simple rules are found to generate complex behavior, characterized by complex multiway graphs, that can be visualized in multispace that combines "tape" and branchial space. The threshold for complex behavior appears to be machines with just s = 1 head states, k = 2 tape colors and p = 3 possible cases, and such machines may potentially be universal. Other characteristics of multiway Turing machines are also studied, including causal invariance, cyclic tapes and generalized busy beaver problems. Multiway Turing machines provide minimal examples of a variety of issues encountered in both concurrent computing and the theory of observers in quantum mechanics, especially in our recent models of physics.
We discuss the role of combinators in the development of the modern conception of computation over the course of the past century. We describe how ideas about formalism and mathematical logic led to the introduction of combinators in 1920 as an extension of the discovery of Nand as a basis for basic logic. We then discuss how combinators informed lambda calculus and symbolic computation, and their relationship to the development of practical computation. We finally describe recent views of combinators in terms of the computational universe of possible programs, and a recent approach to the fundamental theory of physics.
As an example of the concept of rulial space, we explore the case of simple Turing machines. We construct the rulial multiway graph which represents the behavior of all possible Turing machines with a certain class of rules. This graph (which is a Cayley graph of a "Turing machine group") gives a map of the space of non-deterministic Turing machines. We investigate the subgraph formed by deterministic machines, and explore the relationship to the P vs. NP problem. We also consider the implications of features of rulial space for physics, including estimating the maximum speed \r{ho} in rulial space, relations between rulial black holes and computational reducibility, and interpretations of hypercomputation.
A class of models intended to be as minimal and structureless as possible is introduced. Even in cases with simple rules, rich and complex behavior is found to emerge, and striking correspondences to some important core known features of fundamental physics are seen, suggesting the possibility that the models may provide a new approach to finding a fundamental theory of physics.