The technological mediation of mathematics and its learning

This paper examines the extent to which mathematical knowledge, and its related pedagogy, is inextricably linked to the tools – physical, virtual, cultural – in which it is expressed. Our goal is to focus on a few exemplars of computational tools, and to describe with some illustrative examples, how mathematical meanings are shaped by their use. We begin with an appraisal of the role of digital technologies, and our rationale for focusing on them. We present four categories of digital tool-use that distinguish their differing potential to shape mathematical cognition. The four categories are: i. dynamic and graphical tools, ii. tools that outsource processing power, iii. new representational infrastructures, and iv. the implications of highbandwidth connectivity on the nature of mathematics activity. In conclusion, we draw out the implications of this analysis for mathematical epistemology and the mathematical meanings students develop. We also underline the central importance of design, both of the tools themselves and the activities in which they are embedded

A Connectionist Theory of Phenomenal Experience

When cognitive scientists apply computational theory to the problem of phenomenal consciousness, as many of them have been doing recently, there are two fundamentally distinct approaches available. Either consciousness is to be explained in terms of the nature of the representational vehicles the brain deploys; or it is to be explained in terms of the computational processes defined over these vehicles. We call versions of these two approaches vehicle and process theories of consciousness, respectively. However, while there may be space for vehicle theories of consciousness in cognitive science, they are relatively rare. This is because of the influence exerted, on the one hand, by a large body of research which purports to show that the explicit representation of information in the brain and conscious experience are dissociable, and on the other, by the classical computational theory of mind – the theory that takes human cognition to be a species of symbol manipulation. But two recent developments in cognitive science combine to suggest that a reappraisal of this situation is in order. First, a number of theorists have recently been highly critical of the experimental methodologies employed in the dissociation studies – so critical, in fact, it’s no longer reasonable to assume that the dissociability of conscious experience and explicit representation has been adequately demonstrated. Second, classicism, as a theory of human cognition, is no longer as dominant in cognitive science as it once was. It now has a lively competitor in the form of connectionism; and connectionism, unlike classicism, does have the computational resources to support a robust vehicle theory of consciousness. In this paper we develop and defend this connectionist vehicle theory of consciousness. It takes the form of the following simple empirical hypothesis: phenomenal experience consists in the explicit representation of information in neurally realized PDP networks. This hypothesis leads us to re-assess some common wisdom about consciousness, but, we will argue, in fruitful and ultimately plausible ways

Deep Neural Networks Rival the Representation of Primate IT Cortex for Core Visual Object Recognition

The primate visual system achieves remarkable visual object recognition performance even in brief presentations and under changes to object exemplar, geometric transformations, and background variation (a.k.a. core visual object recognition). This remarkable performance is mediated by the representation formed in inferior temporal (IT) cortex. In parallel, recent advances in machine learning have led to ever higher performing models of object recognition using artificial deep neural networks (DNNs). It remains unclear, however, whether the representational performance of DNNs rivals that of the brain. To accurately produce such a comparison, a major difficulty has been a unifying metric that accounts for experimental limitations such as the amount of noise, the number of neural recording sites, and the number trials, and computational limitations such as the complexity of the decoding classifier and the number of classifier training examples. In this work we perform a direct comparison that corrects for these experimental limitations and computational considerations. As part of our methodology, we propose an extension of "kernel analysis" that measures the generalization accuracy as a function of representational complexity. Our evaluations show that, unlike previous bio-inspired models, the latest DNNs rival the representational performance of IT cortex on this visual object recognition task. Furthermore, we show that models that perform well on measures of representational performance also perform well on measures of representational similarity to IT and on measures of predicting individual IT multi-unit responses. Whether these DNNs rely on computational mechanisms similar to the primate visual system is yet to be determined, but, unlike all previous bio-inspired models, that possibility cannot be ruled out merely on representational performance grounds.Comment: 35 pages, 12 figures, extends and expands upon arXiv:1301.353

The Semantic Web: Apotheosis of annotation, but what are its semantics?

This article discusses what kind of entity the proposed Semantic Web (SW) is, principally by reference to the relationship of natural language structure to knowledge representation (KR). There are three distinct views on this issue. The first is that the SW is basically a renaming of the traditional AI KR task, with all its problems and challenges. The second view is that the SW will be, at a minimum, the World Wide Web with its constituent documents annotated so as to yield their content, or meaning structure, more directly. This view makes natural language processing central as the procedural bridge from texts to KR, usually via some form of automated information extraction. The third view is that the SW is about trusted databases as the foundation of a system of Web processes and services. There's also a fourth view, which is much more difficult to define and discuss: If the SW just keeps moving as an engineering development and is lucky, then real problems won't arise. This article is part of a special issue called Semantic Web Update

Humanoid Theory Grounding

In this paper we consider the importance of using a humanoid physical form for a certain proposed kind of robotics, that of theory grounding. Theory grounding involves grounding the theory skills and knowledge of an embodied artificially intelligent (AI) system by developing theory skills and knowledge from the bottom up. Theory grounding can potentially occur in a variety of domains, and the particular domain considered here is that of language. Language is taken to be another “problem space” in which a system can explore and discover solutions. We argue that because theory grounding necessitates robots experiencing domain information, certain behavioral-form aspects, such as abilities to socially smile, point, follow gaze, and generate manual gestures, are necessary for robots grounding a humanoid theory of language