In lieu of an abstract, here is a brief excerpt of the content:Walking the Bodhisattva Path/Walking the Christ PathU.S. Conference of Catholic BishopsCatholics and Buddhists brought together by Dharma Realm Buddhist Association, the San Francisco Zen Center, and the United States Conference of Catholic Bishops (USCCB) met 20-23 March 2003 in the first of an anticipated series of four annual dialogues. Abbot Heng Lyu, the monks and nuns, and members of the Dharma Realm Buddhist Association hosted the dialogue at the (...) City of Ten Thousand Buddhas in Talmage, California. The community had transformed their Confucius Hall into a chapel, with a large, newly constructed wooden cross on a wall above an altar facing out from a raised platform, so the Catholic delegation would have a space for worship. A special poster announcing the theme of the conference, "Walking the Bodhisattva Path/Walking the Christ Path," showing a Buddhist dharma wheel at one foot of a bridge and a Celtic cross at the other foot, had been created by the hosting community.The daily schedule for the dialogue did not interfere with the monastic routine at the City of Ten Thousand Buddhas, which included early morning chanting (4:00-5:00 am), daily rituals in the Buddha Hall, and the single large meal for monastics before noon to which all participants were invited. Catholics and Buddhists meeting in the dialogue began each day with an optional forty-five minutes of quiet sitting together in the monastery meditation hall, each following their own form of meditation and prayer. There was a Eucharist on Friday and Saturday evenings for the Catholics and to which the Buddhists were invited, and additional meals of breakfast and supper throughout the meeting at the City's Jyun Kang Restaurant.Fourteen Buddhists and fourteen Catholics participated in the formal discussions. Each participant spoke on what it means to them to follow Christ or the way of the Bodhisattvas or enlightened beings. In addition to speaking, each participant distributed a short passage, usually from a traditional text, on this topic. Several passages from Master Dogen's Shobogenzo and the Gospel of John were among those shared by the group. Other passages shared were taken from the Avatamsaka Sutra and the Shurangama Sutra and from the other gospels and the writings of Zen master HakuinZenji, the Dalai Lama, Korean Zen Master I-sahn Hye-yeon, the teachings of Master [End Page 247] Hsuan Hua, Shantideva, Origen, St. Ignatius, St. Benedict, the author of The Cloud of Unknowing, St. John of the Cross, and Chiara Lubich. A minute of silence separated the fifteen-minute segments for each presentation. With so large a group, most of the time in the formal sessions involved listening to series of presentations; however, during discussion periods the participants asked several challenging questions of one another.Some participants noted the tension they felt between belonging to a particular religious group and the need to let go of any consequential narrowness. The goal of the discussion was to promote understanding of the differences and similarities between Christianity and Buddhism. At one point, the participants noted several concerns about mutual enrichment: whether Buddhist methods can be separated from Buddhism and Christian methods from Christianity, and what specific aspects of Christian practice truly enrich Buddhists and which Buddhist practices likewise enrich Christians. Several times participants discussed their understanding of terms such as transformation, grace, the incarnation and the passion of Jesus, discernment of spirits, prayer, Buddha-nature, and related terms. There was consensus that Buddhists and Catholics can practically help one another regarding mental cultivation and religious precepts. They expressed this consensus in their choice of the topic forthe next dialogue, "Transformation of Hearts and Minds: Chan/Zen-Catholic Approaches to Precepts."The group agreed to reconvene in 2004 to respond to six presentations, three from each side, on different aspects of this topic.Chairing the sessions were Rev. Heng Sure for the Dharma Realm Buddhist Association, Rev. Taigen Dan Leighton for the San Francisco Zen Center, and Bishop John Webster for the United States Conference of Catholic Bishops. Their three organizations are the sponsors of the Northern California Chan/Zen-Catholic Dialogue. Chan and Zen are respectively the Chinese and Japanese terms for the meditation school... (shrink)

The goal of the paper is to develop and propose a general model of the state space of AI. Given the breathtaking progress in AI research and technologies in recent years, such conceptual work is of substantial theoretical interest. The present AI hype is mainly driven by the triumph of deep learning neural networks. As the distinguishing feature of such networks is the ability to self-learn, self-learning is identified as one important dimension of the AI state (...) class='Hi'>space. Another dimension is recognized as generalization, the possibility to go over from specific to more general types of problems. A third dimension is semantic grounding. Our overall analysis connects to a number of known foundational issues in the philosophy of mind and cognition: the blockhead objection, the Turing test, the symbol grounding problem, the Chinese room argument, and use theories of meaning. It shall finally be argued that the dimension of grounding decomposes into three sub-dimensions. And the dimension of self-learning turns out as only one of a whole range of “self-x-capacities” (based on ideas of organic computing) that span the self-x-subspace of the full AI state space. (shrink)

The impossibility theorem of Dekel, Lipman and Rustichini has been thought to demonstrate that standard state-space models cannot be used to represent unawareness. We first show that Dekel, Lipman and Rustichini do not establish this claim. We then distinguish three notions of awareness, and argue that although one of them may not be adequately modeled using standard state spaces, there is no reason to think that standard state spaces cannot provide models of the other two notions. (...) In fact, standard space models of these forms of awareness are attractively simple. They allow us to prove completeness and decidability results with ease, to carry over standard techniques from decision theory, and to add propositional quantifiers straightforwardly. (shrink)

This article provides a simple decision theoretic model in which elements of the world successively enter the decision maker’s scope and the state space expands over time, which is intended to be the closest correspondence to the standard subjective expected utility theory. We propose a dynamic consistency condition that after any expansion of the scope, the preference ranking should remain unchanged over acts to which the expansion is irrelevant. Together with other natural axioms, it characterizes a model in (...) which the decision maker’s belief extends over time in order that the marginal distribution of the new belief induced over the old state space coincides with the old belief. It is extended to encompass both expansion of scope and learning events, and we characterize the model with an additional property that the decision maker’s belief updating follows Bayes’ rule when she learns events. (shrink)

Jerry Fodor and Ernest Lepore [(1992) Holism: a shopper's guide, Oxford: Blackwell; (1996) in R. McCauley (Ed.) The Churchlands and their critics , Cambridge: Blackwell] have launched a powerful attack against Paul Churchland's connectionist theory of semantics--also known as state space semantics. In one part of their attack, Fodor and Lepore argue that the architectural and functional idiosyncrasies of connectionist networks preclude us from articulating a notion of conceptual similarity applicable to state space semantics. Aarre Laakso (...) and Gary Cottrell [(1998) in M. A. Gernsbacher & S. Derry (Eds) Proceedings of the 20th Annual Conference of the Cognitive Science Society, Mahway, NJ: Erlbaum; Philosophical Psychology ] 13, 47-76 have recently run a number of simulations on simple feedforward networks and applied a mathematical technique for measuring conceptual similarity in the representational spaces of those networks. Laakso and Cottrell contend that their results decisively refute Fodor and Lepore's criticisms. Paul Churchland [(1998) Journal of Philosophy, 95, 5-32 ] goes further. He uses Laakso and Cottrell's neurosimulations to argue that connectionism does furnish us with all we need to construct a robust theory of semantics and a robust theory of translation. In this paper I shall argue that whereas Laakso and Cottrell's neurocomputational results may provide us with a rebuttal of Fodor and Lepore's argument, Churchland's conclusion is far too optimistic. In particular, I shall try to show that connectionist modelling does not provide any objective criterion for achieving a one-to-one accurate translational mapping across networks. (shrink)

I present an annotated development of the basic ideas of the Geneva School approach to the foundations of physics and the structures which emerge as mathematical representations of the physically dual notions of state and property.

I present an annotated development of the basic ideas of the Geneva School approach to the foundations of physics and the structures which emerge as mathematical representations of the physically dual notions of state and property.

It has often been suggested that the meaning of terms is theory dependent. Bas van Fraassen has proposed a particular way of inferring which sentences are true in virtue of meaning, given a theory in so-called state-space format. I examine his claims by means of simple examples.

Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought of as a restricted subset of all potentially available probabilities. A recent publication (Fuchs and Schack, arXiv:0906.2187v1, 2009) advocates such a representation using symmetric informationally complete (SIC) measurements. Building upon this work we study how this subset—quantum-state space—might be characterized. Our leading (...) characteristic is that the inner products of the probabilities are bounded, a simple condition with nontrivial consequences. To get quantum-state space something more detailed about the extreme points is needed. No definitive characterization is reached, but we see several new interesting features over those in Fuchs and Schack (arXiv:0906.2187v1, 2009), and all in conformity with quantum theory. (shrink)

The fact that the space of states of a quantum mechanical system is a projective space (as opposed to a linear manifold) has many consequences. We develop some of these here. First, the space is nearly contractible, namely all the finite homotopy groups (except the second) vanish (i.e., it is the Eilenberg-MacLane space K(ℤ, 2)). Moreover, there is strictly speaking no “superposition principle” in quantum mechanics as one cannot “add” rays; instead, there is adecomposition principle by (...) which a given ray has well-defined projections in other rays. When the evolution of a system is cyclic, any representativevector traces out an open curve, defining an element of the holonomy group, which is essentially the (geometrical) Berry phase. Finally, for the massless case of the representations of the Poincaré group (the so-called “Wigner program”), there could be in principle arbitrarily multivalued representations coming from the Lie algebra of the Euclidean plane group. In fact they are at most bivalued (as commonly admitted). (shrink)

I present an annotated development of the basic ideas of the Geneva School approach to the foundations of physics and the structures which emerge as mathematical representations of the physically dual notions of state and property.

The literature on speculative bubbles in foreign exchange rates is voluminous, with much of it failing to reject the presence of bubbles in many exchange markets.1 Serious testing of this issue began with the work of Meese (1986), Evans (1986), and Woo (1987). Each used a different approach, and each found evidence failing to reject the presence of bubbles in at least some exchange markets.

The geometry of the state space of a finite-dimensional quantum mechanical system, with particular reference to four dimensions, is studied. Many novel features, not evident in the two-dimensional space of a single spin, are found. Although the state space is a convex set, it is not a ball, and its boundary contains mixed states in addition to the pure states, which form a low-dimensional submanifold. The appropriate language to describe the role of the observer is (...) that of flag manifolds. (shrink)

Most models of generational succession in sexually reproducing populations necessarily move back and forth between genic and genotypic spaces. We show that transitions between and within these spaces are usually hidden by unstated assumptions about processes in these spaces. We also examine a widely endorsed claim regarding the mathematical equivalence of kin-, group-, individual-, and allelic-selection models made by Lee Dugatkin and Kern Reeve. We show that the claimed mathematical equivalence of the models does not hold. *Received January 2007; revised (...) April 2008. †To contact the authors, please write to: Elisabeth Lloyd, Department of History and Philosophy of Science, 130 Goodbody Hall, Indiana University, Bloomington, IN 47405; e-mail: [email protected]; Richard Lewontin, Department of Organismic and Evolutionary Biology, Harvard University, 26 Oxford Street, Cambridge, MA 02138; Marcus Feldman, Department of Biological Sciences, Stanford University, Stanford, CA 94305; e-mail: [email protected]. (shrink)

Most models of generational succession in sexually reproducing populations necessarily move back and forth between genic and genotypic spaces. We show that transitions between and within these spaces are usually hidden by unstated assumptions about processes in these spaces. We also examine a widely endorsed claim regarding the mathematical equivalence of kin-, group-, individual-, and allelic-selection models made by Lee Dugatkin and Kern Reeve. We show that the claimed mathematical equivalence of the models does not hold.

Uniformities describing the distinguishability of states and of observables are discussed in the context of general statistical theories and are shown to be related to distinguished subspaces of continuous observables and states, respectively. The usual formalism of quantum mechanics contains no such physical uniformity for states. Using recently developed tools of quantum harmonic analysis, a natural one-to-one correspondence between continuous subspaces of nonrelativistic quantum and classical mechanics is established, thus exhibiting a close interrelation between physical uniformities for quantum states and (...) compactifications of phase space. General properties of the completions of the quantum state space with respect to these uniformities are discussed. (shrink)

In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent’s personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. The starting point (...) is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor. (shrink)

A proposal for an objective interpretation of probability is introduced and discussed: probabilities as deriving from ranges in suitably structured initial-state spaces. Roughly, the probability of an event on a chance trial is the proportion of initial states that lead to the event in question within the space of all possible initial states associated with this type of experiment, provided that the proportion is approximately the same in any not too small subregion of the space. This I (...) would like to call the “natural-range conception” of probability. Providing a substantial alternative to frequency or propensity accounts of probability in a deterministic setting, it is closely related to the so-called “method of arbitrary functions”. It is explicated, confronted with certain problems, and some ideas how these might be overcome are sketched and discussed. (shrink)

This article sketches an idealized strategy for the identification of neural correlates of consciousness. The proposed strategy is based on a state space approach originating from the analysis of dynamical systems. The article then focuses on one constituent of consciousness, phenomenal awareness. Several rudimentary requirements for the identification of neural correlates of phenomenal awareness are suggested. These requirements are related to empirical data on selective attention, on completely intrinsic selection and on globally unconscious states. As an example, neuroscientific (...) findings on synchronized γ activity are categorized according to these requirements. (shrink)

Over the past 20 years or so, a small but growing literature has emerged with the aim of modeling agents who are unaware of certain things. In this paper we compare two different approaches to modeling unawareness: the object-based approach of Board and Chung (Object-based unawareness: theory and applications. University of Minnesota, Mimeo, 2008) and the subjective-state-space approach of Heifetz et al. (J Econ Theory 130: 78-94,2006). In particular, we show that subjectivestate-space models (henceforth HMS structures) can (...) be embedded within object-based models (henceforth OBU structures), demonstrating that the latter are at least as expressive. As long as certain restrictions are imposed on the form of the OBU structure, the embedding can also go the other way. A generalization of HMS structures (relaxing the partitional properties of knowledge) gives us a full converse. (shrink)

State spaces are, in the most general sense, sets of entities that contain information. Examples include states of dynamical systems, processes of observations, or possible worlds. We use domain theory to describe the structure of positive and negative information in state spaces. We present examples ranging from the space of trajectories of a dynamical system, over Dunn’s aboutness interpretation of fde, to the space of open sets of a spectral space. We show that these information (...) structures induce so-called hype models which were recently developed by Leitgeb. Conversely, we prove a representation theorem: roughly, hype models can be represented as induced by an information structure. Thus, the well-behaved logic hype is a sound and complete logic for reasoning about information in state spaces. As application of this framework, we investigate information fusion. We motivate two kinds of fusion. We define a groundedness and a separation property that allow a hype model to be closed under the two kinds of fusion. This involves a Dedekind–MacNeille completion and a fiber-space like construction. The proof-techniques come from pointless topology and universal algebra. (shrink)

The relationship whereby one physical theory encompasses the domain of empirical validity of another is widely known as “reduction.” Elsewhere I have argued that one influential methodology for showing that one physical theory reduces to another, associated with the so-called “Bronstein cube” of theories, rests on an oversimplified and excessively vague characterization of the mathematical relationship between theories that typically underpins reduction. I offer what I claim is a more precise characterization of this relationship, which here is based on a (...) more basic notion of reduction between distinct models of a single physical system. Reduction between two such models, I claim, rests on a particular type of approximation relationship between group actions over the models’ state spaces, characterized by a particular function between the model state spaces and a particular subset of the more encompassing model’s state space. Within this approach, I show formally in what sense and under what conditions reduction is transitive, so that reduction of a model 1 to another model 2 and reduction of model 2 to a third model 3 entails direct reduction of model 1 to model 3. Building on this analysis, I consider cases in which reduction of a model 1 to a model 3 may be effected via distinct intermediate models 2a and 2b, and motivate a set of formal consistency requirements between distinct “reduction paths” having the same models as their “end points”. These constraints are explicitly shown to hold in the reduction of a model of non-relativistic classical mechanics to a model of relativistic quantum mechanics, which may be effected by a composite reduction that proceeds either via a model of non-relativistic quantum mechanics or a model of relativistic classical mechanics. I offer some brief speculations as to whether and how this sort of consistency requirement might serve to constrain the reductions relating other theories and models, including the relationship that the Standard Model and general relativity must bear to any viable unification of these frameworks. (shrink)

We review the relation between spacetime geometries with trace-torsion fields, the so-called Riemann–Cartan–Weyl (RCW) geometries, and their associated Brownian motions. In this setting, the drift vector field is the metric conjugate of the trace-torsion one-form, and the laplacian defined by the RCW connection is the differential generator of the Brownian motions. We extend this to the state-space of non-relativistic quantum mechanics and discuss the relation between a non-canonical quantum RCW geometry in state-space associated with the gradient (...) of the quantum-mechanical expectation value of a self-adjoint operator given by the generalized laplacian operator defined by a RCW geometry. We discuss the reduction of the wave function in terms of a RCW quantum geometry in state-space. We characterize the Schroedinger equation in terms of the RCW geometries and Brownian motions. Thus, in this work, the Schroedinger field is a torsion generating field, both for the linear and non-linear cases. We discuss the problem of the many times variables and the relation with dissipative processes, and the role of time as an active field, following Kozyrev and a recent experiment in non-relativistic quantum systems. We associate the Hodge dual of the drift vector field with a possible angular-momentum source for the phenomenae observed by Kozyrev. (shrink)

Probabilistic theories are a natural framework to investigate the foundations of quantum theory and possible alternative or deeper theories. In a generic probabilistic theory, states of a physical system are represented as vectors of outcomes probabilities and state spaces are convex cones. In this picture the physics of a given theory is related to the geometric shape of the cone of states. In quantum theory, for instance, the shape of the cone of states corresponds to a projective space (...) over complex numbers. In this paper we investigate geometric constraints on the state space of a generic theory imposed by the following information theoretic requirements: every non completely mixed state of a system is perfectly distinguishable from some other state in a single shot measurement; information capacity of physical systems is conserved under making mixtures of states. These assumptions guarantee that a generic physical system satisfies a natural principle asserting that the more a state of the system is mixed the less information can be stored in the system using that state as logical value. We show that all theories satisfying the above assumptions are such that the shape of their cones of states is that of a projective space over a generic field of numbers. Remarkably, these theories constitute generalizations of quantum theory where superposition principle holds with coefficients pertaining to a generic field of numbers in place of complex numbers. If the field of numbers is trivial and contains only one element we obtain classical theory. This result tells that superposition principle is quite common among probabilistic theories while its absence gives evidence of either classical theory or an implausible theory. (shrink)