The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer

Towards an integrated discovery system

Previous research on machine discovery has focused on limited parts of the empirical discovery task. In this paper we describe IDS, an integrated system that addresses both qualitative and quantitative discovery. The program represents its knowledge in terms of qualitative schemas, which it discovers by interacting with a simulated physical environment. Once IDS has formulated a qualitative schema, it uses that schema to design experiments and to constrain the search for quantitative laws. We have carried out preliminary tests in the domain of heat phenomena. In this context the system has discovered both intrinsic properties, such as the melting point of substances, and numeric laws, such as the conservation of mass for objects going through a phase change

Automated Discovery in Econometrics

Our subject is the notion of automated discovery in econometrics. Advances in computer power, electronic communication, and data collection processes have all changed the way econometrics is conducted. These advances have helped to elevate the status of empirical research within the economics profession in recent years and they now open up new possibilities for empirical econometric practice. Of particular significance is the ability to build econometric models in an automated way according to an algorithm of decision rules that allow for (what we call here) heteroskedastic and autocorrelation robust (HAR) inference. Computerized search algorithms may be implemented to seek out suitable models, thousands of regressions and model evaluations may be performed in seconds, statistical inference may be automated according to the properties of the data, and policy decisions can be made and adjusted in real time with the arrival of new data. We discuss some aspects and implications of these exciting, emergent trends in econometrics.Automation, discovery, HAC estimation, HAR inference, model building, online econometrics, policy analysis, prediction, trends

On the automated extraction of regression knowledge from databases

The advent of inexpensive, powerful computing systems, together with the increasing amount of available data, conforms one of the greatest challenges for next-century information science. Since it is apparent that much future analysis will be done automatically, a good deal of attention has been paid recently to the implementation of ideas and/or the adaptation of systems originally developed in machine learning and other computer science areas. This interest seems to stem from both the suspicion that traditional techniques are not well-suited for large-scale automation and the success of new algorithmic concepts in difficult optimization problems. In this paper, I discuss a number of issues concerning the automated extraction of regression knowledge from databases. By regression knowledge is meant quantitative knowledge about the relationship between a vector of predictors or independent variables (x) and a scalar response or dependent variable (y). A number of difficulties found in some well-known tools are pointed out, and a flexible framework avoiding many such difficulties is described and advocated. Basic features of a new tool pursuing this direction are reviewed

Machine learning : techniques and foundations

The field of machine learning studies computational methods for acquiring new knowledge, new skills, and new ways to organize existing knowledge. In this paper we present some of the basic techniques and principles that underlie AI research on learning, including methods for learning from examples, learning in problem solving, learning by analogy, grammar acquisition, and machine discovery. In each case, we illustrate the techniques with paradigmatic examples

Complex systems science: expert consultation report

Executive SummaryA new programme of research in Complex Systems Science must be initiated by FETThe science of complex systems (CS) is essential to establish rigorous scientific principles on which to develop the future ICT systems that are critical to the well-being, safety and prosperity of Europe and its citizens. As the “ICT incubator and pathfinder for new ideas and themes for long-term research in the area of information and communication technologies” FET must initiate a significant new programme of research in complex systems science to underpin research and development in ICT. Complex Systems Science is a “blue sky” research laboratory for R&D in ICT and their applications. In July 2009, ASSYST was given a set of probing questions concerning FET funding for ICT-related complex systems research. This document is based on the CS community’s response.Complex systems research has made considerable progress and is delivering new scienceSince FET began supporting CS research, considerable progress has been made. Building on previous understanding of concepts such as emergence from interactions, far-from-equilibrium systems, border of chaos and self-organised criticality, recent CS research is now delivering rigorous theory through methods of statistical physics, network theory, and computer simulation. CS research increasingly demands high-throughput data streams and new ICT-based methods of observing and reconstructing, i.e. modelling, the dynamics from those data in areas as diverse as embryogenesis, neuroscience, transport, epidemics, linguistics, meteorology, and robotics. CS research is also beginning to address the problem of engineering robust systems of systems of systems that can adapt to changing environments, including the perplexing problem that ICT systems are too often fragile and non-adaptive.Recommendation: A Programme of Research in Complex Systems Science to Support ICTFundamental theory in Complex Systems Science is needed, but this can only be achieved through real-world applications involving large, heterogeneous, and messy data sets, including people and organisations. A long-term vision is needed. Realistic targets can be set. Fundamental research can be ensured by requiring that teams include mathematicians, computer scientists, physicists and computational social scientists.One research priority is to develop a formalism for multilevel systems of systems of systems, applicable to all areas including biology, economics, security, transportation, robotics, health, agriculture, ecology, and climate change. Another related research priority is a scientific perspective on the integration of the new science with policy and its implementation, including ethical problems related to privacy and equality.A further priority is the need for education in complex systems science. Conventional education continues to be domain-dominated, producing scientists who are for the most part still lacking fundamental knowledge in core areas of mathematics, computation, statistical physics, and social systems. Therefore:1. We recommend that FET fund a new programme of work in complex systems science as essential research for progress in the development of new kinds of ICT systems.2. We have identified the dynamics of multilevel systems as the area in complex systems science requiring a major paradigm shift, beyond which significant scientific progress cannot be made.3. We propose a call requiring: fundamental research in complex systems science; new mathematical and computational formalisms to be developed; involving a large ‘guinea pig’ organisation; research into policy and its meta-level information dynamics; and that all research staff have interdisciplinary knowledge through an education programme.Tangible outcomes, potential users of the new science, its impact and measures of successUsers include (i) the private and public sectors using ICT to manage complex systems and (ii) researchers in ICT, CSS, and all complex domains. The tangible output of a call will be new knowledge on the nature of complex systems in general, new knowledge of the particular complex system(s) studied, and new knowledge of the fundamental role played by ICT in the research and implementation to create real systems addressing real-world problems. The impact of the call will be seen through new high added-value opportunities in the public and private sectors, new high added-value ICT technologies, and new high added-value science to support innovation in ICT research and development. The measure of success will be through the delivery of these high added-value outcomes, and new science to better understand failures

Theory formation by abduction : initial results of a case study based on the chemical revolution

Abduction is the process of constructing explanations. This chapter suggests that automated abduction is a key to advancing beyond the "routine theory revision" methods developed in early AI research towards automated reasoning systems capable of "world model revision" — dramatic changes in systems of beliefs such as occur in children's cognitive development and in scientific revolutions. The chapter describes a general approach to automating theory revision based upon computational methods for theory formation by abduction. The approach is based on the idea that, when an anomaly is encountered, the best course is often simply to suppress parts of the original theory thrown into question by the contradiction and to derive an explanation of the anomalous observation based on relatively solid, basic principles. This process of looking for explanations of unexpected new phenomena can lead by abductive inference to new hypotheses that can form crucial parts of a revised theory. As an illustration, the chapter shows how some of Lavoisier's key insights during the Chemical Revolution can be viewed as examples of theory formation by abduction