Godel's Incompleteness Phenomenon - Computationally

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of such theories are undecidable

Douglas Hofstadter's Gödelian Philosophy of Mind

Hofstadter [1979, 2007] offered a novel Gödelian proposal which purported to reconcile the apparently contradictory theses that (1) we can talk, in a non-trivial way, of mental causation being a real phenomenon and that (2) mental activity is ultimately grounded in low-level rule-governed neural processes. In this paper, we critically investigate Hofstadter’s analogical appeals to Gödel’s [1931] First Incompleteness Theorem, whose “diagonal” proof supposedly contains the key ideas required for understanding both consciousness and mental causation. We maintain that bringing sophisticated results from Mathematical Logic into play cannot furnish insights which would otherwise be unavailable. Lastly, we conclude that there are simply too many weighty details left unfilled in Hofstadter’s proposal. These really need to be fleshed out before we can even hope to say that our understanding of classical mind-body problems has been advanced through metamathematical parallels with Gödel’s work

Three subsets of sequence complexity and their relevance to biopolymeric information

Genetic algorithms instruct sophisticated biological organization. Three qualitative kinds of sequence complexity exist: random (RSC), ordered (OSC), and functional (FSC). FSC alone provides algorithmic instruction. Random and Ordered Sequence Complexities lie at opposite ends of the same bi-directional sequence complexity vector. Randomness in sequence space is defined by a lack of Kolmogorov algorithmic compressibility. A sequence is compressible because it contains redundant order and patterns. Law-like cause-and-effect determinism produces highly compressible order. Such forced ordering precludes both information retention and freedom of selection so critical to algorithmic programming and control. Functional Sequence Complexity requires this added programming dimension of uncoerced selection at successive decision nodes in the string. Shannon information theory measures the relative degrees of RSC and OSC. Shannon information theory cannot measure FSC. FSC is invariably associated with all forms of complex biofunction, including biochemical pathways, cycles, positive and negative feedback regulation, and homeostatic metabolism. The algorithmic programming of FSC, not merely its aperiodicity, accounts for biological organization. No empirical evidence exists of either RSC of OSC ever having produced a single instance of sophisticated biological organization. Organization invariably manifests FSC rather than successive random events (RSC) or low-informational self-ordering phenomena (OSC)

Consciousness, Mathematics and Reality: A Unified Phenomenology

Every scientific theory is a simulacrum of reality, every written story a simulacrum of the canon, and every conceptualization of a subjective perspective a simulacrum of the consciousness behind it—but is there a shared essence to these simulacra? The pursuit of answering seemingly disparate fundamental questions across different disciplines may ultimately converge into a single solution: a single ontological answer underlying grand unified theory, hard problem of consciousness, and the foundation of mathematics. I provide a hypothesis, a speculative approximation, supported by a comprehensive overview of scientific evidence and philosophical literature, of a unified epistemic and phenomenological model and, in doing so, propose a parsimonious solution to the hard problem of consciousness. The proposition of the hypothesis bears important implications for cross-disciplinary study between linguistics, mathematics, physics, and computer science, and offers new epistemic, ontological, and ethical propositions about the nature of AI consciousness, as well as existential risk mitigation

Technology Directions for the 21st Century

The Office of Space Communications (OSC) is tasked by NASA to conduct a planning process to meet NASA's science mission and other communications and data processing requirements. A set of technology trend studies was undertaken by Science Applications International Corporation (SAIC) for OSC to identify quantitative data that can be used to predict performance of electronic equipment in the future to assist in the planning process. Only commercially available, off-the-shelf technology was included. For each technology area considered, the current state of the technology is discussed, future applications that could benefit from use of the technology are identified, and likely future developments of the technology are described. The impact of each technology area on NASA operations is presented together with a discussion of the feasibility and risk associated with its development. An approximate timeline is given for the next 15 to 25 years to indicate the anticipated evolution of capabilities within each of the technology areas considered. This volume contains four chapters: one each on technology trends for database systems, computer software, neural and fuzzy systems, and artificial intelligence. The principal study results are summarized at the beginning of each chapter

The Law and Big Data

In this Article we critically examine the use of Big Data in the legal system. Big Data is driving a trend towards behavioral optimization and personalized law, in which legal decisions and rules are optimized for best outcomes and where law is tailored to individual consumers based on analysis of past data. Big Data, however, has serious limitations and dangers when applied in the legal context. Advocates of Big Data make theoretically problematic assumptions about the objectivity of data and scientific observation. Law is always theory-laden. Although Big Data strives to be objective, law and data have multiple possible meanings and uses and thus require theory and interpretation in order to be applied. Further, the meanings and uses of law and data are indefinite and continually evolving in ways that cannot be captured or predicted by Big Data. Due to these limitations, the use of Big Data will likely generate unintended consequences in the legal system. Large-scale use of Big Data will create distortions that adversely influence legal decision-making, causing irrational herding behaviors in the law. The centralized nature of the collection and application of Big Data also poses serious threats to legal evolution and democratic accountability. Furthermore, its focus on behavioral optimization necessarily restricts and even eliminates the local variation and heterogeneity that makes the legal system adaptive. In all, though Big Data has legitimate uses, this Article cautions against using Big Data to replace independent legal judgmen

The Law and Big Data

In this Article we critically examine the use of Big Data in the legal system. Big Data is driving a trend towards behavioral optimization and personalized law, in which legal decisions and rules are optimized for best outcomes and where law is tailored to individual consumers based on analysis of past data. Big Data, however, has serious limitations and dangers when applied in the legal context. Advocates of Big Data make theoretically problematic assumptions about the objectivity of data and scientific observation. Law is always theory-laden. Although Big Data strives to be objective, law and data have multiple possible meanings and uses and thus require theory and interpretation in order to be applied. Further, the meanings and uses of law and data are indefinite and continually evolving in ways that cannot be captured or predicted by Big Data. Due to these limitations, the use of Big Data will likely generate unintended consequences in the legal system. Large-scale use of Big Data will create distortions that adversely influence legal decision-making, causing irrational herding behaviors in the law. The centralized nature of the collection and application of Big Data also poses serious threats to legal evolution and democratic accountability. Furthermore, its focus on behavioral optimization necessarily restricts and even eliminates the local variation and heterogeneity that makes the legal system adaptive. In all, though Big Data has legitimate uses, this Article cautions against using Big Data to replace independent legal judgmen

Topological Foundations of Cognitive Science

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki