Information: A Definition

\documentclass[12pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts,amsthm} \usepackage{setspace} \usepackage{graphicx} \usepackage{hyperref} \usepackage{csquotes} \usepackage{natbib} \usepackage{times} % Custom commands \newcommand{\Hent}{\mathrm{H}} % entropy operator % Theorem environments \newtheorem{definition}{Definition} \newtheorem{proposition}{Proposition} \title{\textbf{Information as Distinction: The Foundational Difference that Drives Emergence}} \author{C.~L.~Vaillant \\ RG~Emergence} \date{October 2025} \begin{document} \maketitle \doublespacing \begin{abstract} \noindent This paper proposes a general definition of information as \emph{the minimal distinguishable difference that reduces uncertainty within a system}. Information is treated not as a substance but as a relational pattern---an identifiable distinction that can be registered by a system. From this foundation, intelligence, structure, and meaning are understood as emergent properties of recursive informational dynamics. The argument situates this view within the lineage of Shannon's mathematical theory of communication, Bateson's cybernetic epistemology, Wheeler's ``it from bit'' cosmology, and contemporary emergence theory. \end{abstract} \section{Introduction} Information underlies both the physical and cognitive domains. Whether in the spin of an electron, the firing of a neuron, or the choice within a decision space, reality presents itself as a field of possible states. The process by which a distinction is drawn within that field---the recognition of a difference that makes a difference---constitutes the most primitive act of information. In this sense, information is ontological as well as epistemic. It describes not only how observers come to know, but how being itself differentiates. This view aligns with the broader movement in systems and emergence theory that sees complexity, agency, and meaning as products of recursive informational organization. \section{Defining Information} \begin{definition}[General Form] \label{def:info} Information is the minimal distinguishable difference that reduces uncertainty within a system. \end{definition} Information is not an object or a medium but a relation among possible states. It exists only when three conditions obtain: \begin{enumerate} \item A \textbf{state space} \( S \): the set of possible configurations of a system. \item A \textbf{difference} \( \Delta s \): a distinguishable contrast between states in \( S \). \item An \textbf{observer or subsystem} \( O \): an entity capable of registering or responding to \( \Delta s \). \end{enumerate} Formally, we can express this relation as: \begin{equation} I(S, \Delta s, O) \Longleftrightarrow \exists \, \Delta s \in S : O \text{ registers } \Delta s \text{ and } \Hent(S) \downarrow, \end{equation} where \( \Hent(S) \) denotes the Shannon entropy---a measure of uncertainty over possible states of \( S \). Thus, information corresponds to the reduction of uncertainty through the recognition of a difference. \section{Formalization and Context} Shannon~(\citeyear{shannon1948}) defined the quantity of information as the expected reduction in uncertainty: \begin{equation} I = \Hent(X) - \Hent(X|Y), \end{equation} where \( \Hent(X) \) is the entropy of a random variable \( X \) and \( \Hent(X|Y) \) is the conditional entropy after observing \( Y \). Bateson~(\citeyear{bateson1972}) reformulated this notion qualitatively as ``a difference that makes a difference.'' Our present definition synthesizes these perspectives by grounding both in a minimal formal and phenomenological requirement: information as the smallest distinguishable change capable of reducing uncertainty for a system capable of registering it. \section{Recursive Dynamics and Emergence} \begin{proposition}[Recursive Information Process] Let \( I_t \) represent the informational state of a system at time \( t \). The recursive dynamic is given by \begin{equation} I_{t+1} = f(I_t, \Delta I_t), \end{equation} where \( f \) encodes how new distinctions \( \Delta I_t \) are integrated into prior informational structure. \end{proposition} Through iteration, such processes yield emergent structures: feedback loops stabilize into attractors, and uncertainty transforms into order. \subsection{Example: Adaptive Agent} Consider an adaptive agent learning from an environment \( E \). Each observation introduces a distinction \( \Delta s_t \) that updates the agent’s internal model: \begin{equation} M_{t+1} = M_t + g(\Delta s_t), \end{equation} where \( g \) represents an assimilation function. Over time, \( M_t \) converges toward a more predictive state, embodying the emergence of intelligence through recursive information refinement. \section{Discussion and Counterpoints} One might object that such a framework collapses into pancomputationalism---the claim that the universe is merely computing itself. However, the distinction-based model avoids this reduction by grounding computation in relational ontology: not all dynamics are informational, only those involving distinguishable and registered differences. Further, this view aligns with complexity theory’s emphasis on constraint and self-organization rather than universal computation. Information, as distinction, underwrites the generation of new levels of order without presupposing total computability. \section{Implications} This definition carries implications across multiple domains: \begin{itemize} \item \textbf{Physics:} Wheeler’s aphorism ``it from bit''~\citep{wheeler1990} finds grounding—physical reality arises from networks of distinctions. \item \textbf{Biology:} Life can be viewed as recursive information maintenance—autopoietic systems preserving distinctions against entropy~\citep{maturana1980,deacon2012}. \item \textbf{Cognition:} Intelligence becomes the continual refinement of informational structure through learning, compression, and transformation. \end{itemize} \section{Conclusion} Information, defined as the minimal distinguishable difference that reduces uncertainty, provides a unifying foundation for understanding emergence. It is the substrate of structure and the grammar of becoming. From elementary physics to collective intelligence, the world unfolds as a recursive play of differences organizing themselves into meaning. \bibliographystyle{plainnat} \begin{thebibliography}{} \bibitem[Shannon(1948)]{shannon1948} Shannon, C.~E. (1948). \newblock A mathematical theory of communication. \newblock \emph{Bell System Technical Journal}, 27(3), 379--423. \bibitem[Bateson(1972)]{bateson1972} Bateson, G. (1972). \newblock \emph{Steps to an Ecology of Mind}. \newblock Ballantine Books, New York. \bibitem[Wheeler(1990)]{wheeler1990} Wheeler, J.~A. (1990). \newblock Information, physics, quantum: The search for links. \newblock In W. Zurek (Ed.), \emph{Complexity, Entropy, and the Physics of Information}, Addison--Wesley. \bibitem[Maturana and Varela(1980)]{maturana1980} Maturana, H., \& Varela, F. (1980). \newblock \emph{Autopoiesis and Cognition: The Realization of the Living}. \newblock Reidel, Dordrecht. \bibitem[Deacon(2012)]{deacon2012} Deacon, T.~W. 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