A continuous spatial automaton is analogous to a cellular automaton, except that the cells form a continuum, as do the possible states of the cells. After an informal mathematical description of spatial automata, we describe in detail a continuous analog of Conway’s “Life,” and show how the automaton can be implemented using the basic operations of ﬁeld computation.
(brain area) to small (dendritic) scales. Further, it is often useful to describe motor control and sensorimotor coordination in terms of external elds such as force elds and sensory images. We survey the basic concepts of eld computation, including both feed-forward eld operations and eld dynamics resulting from recurrent connections. Adaptive and learning mechanisms are discussed brie y. The application of eld computation to motor control is illustrated by several examples: external force elds associated with spinal neurons (Bizzi (...) & Mussa-Ivaldi 1995), population coding of direction in motor cortex (Georgopoulos 1995), continuous transformation of direction elds (Droulez & Berthoz 1991a), and linear gain elds and coordinate transformations in posterior parietal cortex (Andersen 1995). Next we survey some eld-based representations of motion, including direct, Fourier, Gabor and wavelet or multiresolution representations. Finally we consider brie y the application of these representations to constraint satisfaction, which has many applications in motor control. (shrink)
In this report we develop the basic properties of a set of functions analogous to the circular and hyperbolic functions, but based on L p circles. The resulting identities may simplify analysis in L p spaces in much the way that the circular functions do in Euclidean space. In any case, they are a pleasing example of mathematical generalization.
The title of my talk, “Living Neoplatonism,” is intentionally ambiguous, for it can refer, first, to Neoplatonism as a living philosophy rather than as a historical artifact embodied in the writings of Plotinus, Proclus, and the rest. And second, it can refer to the practice of living Neoplatonically as a modern way of life. But why Neoplatonism, as opposed to some other philosophy? From my perspective as a scientist I will explain why I think Neoplatonism is especially suited to provide (...) a spiritual complement to the contemporary scientific worldview, which is otherwise materialistic in orientation and ill-equipped to deal with many peoples’ spiritual concerns. (shrink)
I take the `hard problem' of consciousness to be to understand the relation between our subjective experience and the brain processes that cause it; that is, to reconcile our everyday feeling of consciousness with the scienti c worldview (MacLennan, 1995). This problem is hard because consciousness has unique epistemological characteristics, which must be accommodated by any attempted solution. I will summarize these characteristics; more detail can be found in Searle (1992, chs. 4, 5) and Chalmers (1995, 1996), whose positions, if (...) I have understood them correctly, are consistent with mine. 1 First, science is a public enterprise; it attains knowledge that is independent of the individual investigator by limiting itself to public phenomena. Ultimately it is grounded in shared experiences, for example, when we both look at a thermometer and read the same temperature. Traditionally science has accomplished its ends by focusing on the more public, objective aspects of phenomena (e.g. temperature as measured by a thermometer), and by ignoring the more private, subjective aspects (how warm it feels to me). In other words, science has restricted itself to facts about which it is easy to reach agreement among a consensus of trained observers. Although this restriction has aided scienti c progress, it prevents the scienti c study of consciousness, which is essentially private and subjective. 2 Second, scienError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapce's neglect of the subjective is also apparent in its reductive methods.. (shrink)
Daniel Dennett's theory of intentionality has much to recommend it. Nevertheless, it could be significantly strengthened by addressing the causes of intentionality, that is, the mechanisms undelying intentional behavior. I will discuss three problems that a causal theory of intentionality could alleviate: attribution of rationality (or optimality), alternatives to sentential representation, and directedness of consciousness.
The issue of symbol grounding is not essentially different in analog and digital computation. The principal difference between the two is that in analog computers continuous variables change continuously, whereas in digital computers discrete variables change in discrete steps (at the relevant level of analysis). Interpretations are imposed on analog computations just as on digital computations: by attaching meanings to the variables and the processes defined over them. As Harnad (2001) claims, states acquire intrinsic meaning through their relation to the (...) real (physical) environment, for example, through transduction. However, this is independent of the question of the continuity or discreteness of the variables or the transduction processes. (shrink)
The central claim of computationalism is generally taken to be that the brain is a computer, and that any computer implementing the appropriate program would ipso facto have a mind. In this paper I argue for the following propositions: (1) The central claim of computationalism is not about computers, a concept too imprecise for a scienti c claim of this sort, but is about physical calculi (instantiated discrete formal systems). (2) In matters of formality, interpretability, and so forth, analog computation (...) and digital computation are not essentially di erent, and so arguments such as Searle's hold or not as well for one as for the other. (3) Whether or not a biological system (such as the brain) is computational is a scienti c matter of fact. (4) A substantive scienti c question for cognitive science is whether cognition is better modeled by discrete representations or by continuous representations. (5) Cognitive science and AI need a theoretical construct that is the continuous analog of a calculus. The discussion of these propositions will illuminate several terminology traps, in which it's all too easy to become ensnared. (shrink)
Images and Models. The distinction between models and images is treated briefly in JL&B (pp. 38, 93, 140), but four differences are described in Johnson-Laird (1983, esp. ch. 8). I'll argue that the distinction better treated a matter of degree than of kind.
Values are critical for intelligent behavior, since values determine interests, and interests determine relevance. Therefore we address relevance and its role in intelligent behavior in animals and machines. Animals avoid exhaustive enumeration of possibilities by focusing on relevant aspects of the environment, which emerge into the (cognitive) foreground, while suppressing irrelevant aspects, which submerge into the background. Nevertheless, the background is not invisible, and aspects of it can pop into the foreground if background processing deems them potentially relevant. Essential to (...) these ideas are questions of how contexts are switched, which defines cognitive/behavioral episodes, and how new contexts are created, which allows the efficiency of foreground/background processing to be extended to new behaviors and cognitive domains. Next we consider mathematical characterizations of the foreground/background distinction, which we treat as a dynamic separation of the concrete space into (approximately) orthogonal subspaces, which are processed differently. Background processing is characterized by large receptive fields which project into a space of relatively low dimension to accomplish rough categorization of a novel stimulus and its approximate location. Such background processing is partly innate and partly learned, and we discuss possible correlational (Hebbian) learning mechanisms. Foreground processing is characterized by small receptive fields which project into a space of comparatively high dimension to accomplish precise categorization and localization of the stimuli relevant to the context. We also consider mathematical models of valences and affordances, which are an aspect of the foreground. Cells processing foregound information have no fixed meaning (i.e., their meaning is contextual), so it is necessary to explain how the processing accomplished by foreground neurons can be made relative to the context. Thus we consider the properties of several simple mathematical models of how the contextual representation controls foreground processing. We show how simple correlational processes accomplish the contextual separation of foreground from background on the basis of differential reinforcement. That is, these processes account for the contextual separation of the concrete space into disjoint subspaces corresponding to the foreground and background. Since an episode may comprise the activation of several contexts (at varying levels of activity) we consider models, suggested by quantum mechanics, of foreground processing in superposition. That is, the contextual state may be a weighted superposition of several pure contexts, with a corresponding superposition of the foreground representations and the processes operating on them. This leads us to a consideration of the nature and origin of contexts. Although some contexts are innate, many are learned. We discuss a mathematical model of contexts which allows a context to split into several contexts, agglutinate from several contexts, or to constellate out of relatively acontextual processing. Finally, we consider the acontextual processing which occurs when the current context is no longer relevant, and may trigger the switch to another context or the formation of a new context. We relate this to the situation known as "breakdown" in phenomenology. (shrink)
It has been argued that neural networks and other forms of analog computation may transcend the limits of Turing-machine computation; proofs have been offered on both sides, subject to differing assumptions. In this article I argue that the important comparisons between the two models of computation are not so much mathematical as epistemological. The Turing-machine model makes assumptions about information representation and processing that are badly matched to the realities of natural computation (information representation and processing in or inspired by (...) natural systems). This points to the need for new models of computation addressing issues orthogonal to those that have occupied the traditional theory of computation. (shrink)
For all animals, color is an indicator of the substance and state of objects, for which purpose reflectance is just one among many relevant optical properties. This broader meaning of color is confirmed by linguistic evidence. Rather than reducing color to a simple physical property, it is more realistic to embrace its full phenomenology.
In the first part of this commentary I argue that a neurophenomenological analysis of color reveals additional asymmetries that preclude undetectable color transformations, without appealing to weak arguments based on Basic Color Categories (BCCs); that is, I suggest additional factors that must be included in “an empirically accurate model of color experience,” and which break the remaining asymmetries. In the second part I discuss the “isomorphism constraint” and the extent to which we may predict the subjective quality of experience from (...) its neurological correlates. Protophenomena are discussed as a way of capturing in a relational structure all of qualitative experience except for the bare fact of subjectivity. (shrink)
I discuss neuroscientific and phenomenological arguments in support of Millikan's thesis. I then consider invariance as a unifying theme in perceptual and conceptual tracking, and how invariants may be extracted from the environment. Finally, some wider implications of Millikan's nondescriptionist approach to language are presented, with specific application to color terms.
The idea of a calculus or discrete formal system is central to traditional models of language, knowledge, logic, cognition and computation, and it has provided a unifying framework for these and other disciplines. Nevertheless, research in psychology, neuroscience, philosophy and computer science has shown the limited ability of this model to account for the flexible, adaptive and creative behavior exhibited by much of the animal kingdom. Promising alternate models replace discrete structures by structured continua and discrete rule-following by continuous dynamical (...) processes. However, we believe that progress in these alternate models is retarded by the lack of a unifying theoretical construct analogous to the discrete formal system. In this paper we outline the general characteristics of continuous formal systems (simulacra), which we believe will be a unifying element in future models of language, knowledge, logic, cognition and computation. Therefore, we discuss syntax, semantics, inference and computation in the context of continuous formal systems. In addition, we address an issue that the discrete models were inadequate to address: the gradual emergence of (approximately) discrete structures from a continuum. This is relevant to the emergence of linguistic structures, including semantics and syntax, and to the emergence of rule-like regularities in behavior. (shrink)
The principal problem of consciousness is how brain processes cause subjective awareness. Since this problem involves subjectivity, ordinary scientific methods, applicable only to objective phenomena, cannot be used. Instead, by parallel application of phenomenological and scientific methods, we may establish a correspondence between the subjective and the objective. This correspondence is effected by the construction of a theoretical entity, essentially an elementary unit of consciousness, the intensity of which corresponds to electrochemical activity in a synapse. Dendritic networks correspond to causal (...) dependencies between these subjective units. Therefore, the structure of conscious experience is derived from synaptic connectivity. This parallel phenomenal/neural analysis provides a framework for the investigation of a number of problems, including sensory inversions, the unity of consciousness, and the nature of nonhuman consciousness. (shrink)
The central claim of computationalism is generally taken to be that the brain is a computer, and that any computer implementing the appropriate program would ipso facto have a mind. In this paper I argue for the following propositions: (1) The central claim of computationalism is not about computers, a concept too imprecise for a scientific claim of this sort, but is about physical calculi (instantiated discrete formal systems). (2) In matters of formality, interpretability, and so forth, analog computation and (...) digital computation are not essentially different, and so arguments such as Searle''s hold or not as well for one as for the other. (3) Whether or not a biological system (such as the brain) is computational is a scientific matter of fact. (4) A substantive scientific question for cognitive science is whether cognition is better modeled by discrete representations or by continuous representations. (5) Cognitive science and AI need a theoretical construct that is the continuous analog of a calculus. The discussion of these propositions will illuminate several terminology traps, in which it''s all too easy to become ensnared. (shrink)
In this commentary on Harnad's "Grounding Symbols in the Analog World with Neural Nets: A Hybrid Model," the issues of symbol grounding and analog (continuous) computation are separated, it is argued that symbol graounding is as important an issue for analog cognitive models as for digital (discrete) models. The similarities and differences between continuous and discrete computation are discussed, as well as the grounding of continuous representations. A continuous analog of the Chinese Room is presented.