Big toy models

We pursue a model-oriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we aim for a ??big toy model??, in which both quantum and classical systems can be faithfully represented??as well as, possibly, more exotic kinds of systems. To this end, we show how Chu spaces can be used to represent physical systems of various kinds. In particular, we show how quantum systems can be represented as Chu spaces over the unit interval in such a way that the Chu morphisms correspond exactly to the physically meaningful symmetries of the systems??the unitaries and antiunitaries. In this way we obtain a full and faithful functor from the groupoid of Hilbert spaces and their symmetries to Chu spaces. We also consider whether it is possible to use a finite value set rather than the unit interval; we show that three values suffice, while the two standard possibilistic reductions to two values both fail to preserve fullness.     

A physicalist reinterpretion of ‘phenomenal’ spaces

This paper argues that phenomenal or internal metrical spaces are redundant posits. It is shown that we need not posit an internal space-time frame, as the physical space-time suffices to explain geometrical perception, memory and planning. More than the internal space-time frame, the idea of a phenomenal colour space has lent credibility to the idea of internal spaces. It is argued that there is no phenomenal colour space that underlies the various psychophysical colour spaces; it is parasitic upon physical and psychophysical colour spaces. The argumentation is further extended to other sensory spaces and generalised quality spaces.     

Can robots make good models of biological behaviour?

How should biological behaviour be modelled? A relatively new approach is to investigate problems in neuroethology by building physical robot models of biological sensorimotor systems. The explication and justification of this approach are here placed within a framework for describing and comparing models in the behavioural and biological sciences. First, simulation models--the representation of a hypothesis about a target system--are distinguished from several other relationships also termed "modelling" in discussions of scientific explanation. Seven dimensions on which simulation models can differ are defined and distinctions between them discussed: 1. Relevance: whether the model tests and generates hypotheses applicable to biology. 2. Level: the elemental units of the model in the hierarchy from atoms to societies. 3. Generality: the range of biological systems the model can represent. 4. Abstraction: the complexity, relative to the target, or amount of detail included in the model. 5. Structural accuracy: how well the model represents the actual mechanisms underlying the behaviour. 6. Performance match: to what extent the model behaviour matches the target behaviour. 7. Medium: the physical basis by which the model is implemented. No specific position in the space of models thus defined is the only correct one, but a good modelling methodology should be explicit about its position and the justification for that position. It is argued that in building robot models biological relevance is more effective than loose biological inspiration; multiple levels can be integrated; that generality cannot be assumed but might emerge from studying specific instances; abstraction is better done by simplification than idealisation; accuracy can be approached through iterations of complete systems; that the model should be able to match and predict target behaviour; and that a physical medium can have significant advantages. These arguments reflect the view that biological behaviour needs to be studied and modelled in context, that is, in terms of the real problems faced by real animals in real environments.     

Probability spaces,hilbert spaces,and the axioms of test theory

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.     

Psychophysics without physics: extension of Fechnerian scaling from continuous to discrete and discrete-continuous stimulus spaces

The computation of subjective (Fechnerian) distances from discrimination probabilities involves cumulation of appropriately transformed psychometric increments along smooth arcs (in continuous stimulus spaces) or chains of stimuli (in discrete spaces). In a space where any two stimuli that are each other's points of subjective equality are given identical physical labels, psychometric increments are positive differences ψ(x,y)-ψ(x,x) and ψ(y,x)-ψ(x,x), where x≠y and ψ is the probability of judging two stimuli different. In continuous stimulus spaces the appropriate monotone transformation of these increments (called overall psychometric transformation) is determined uniquely in the vicinity of zero, and its extension to larger values of its argument is immaterial. In discrete stimulus spaces, however, Fechnerian distances critically depend on this extension. We show that if overall psychometric transformation is assumed (A) to be the same for a sufficiently rich class of discrete stimulus spaces, (B) to ensure the validity of the Second Main Theorem of Fechnerian Scaling in this class of spaces, and (C) to agree in the vicinity of zero with one of the possible transformations in continuous spaces, then this transformation can only be identity. This result is generalized to the broad class of “discrete-continuous” stimulus spaces, of which continuous and discrete spaces are proper subclasses.     

Projectible predicates in analogue and simulated systems

We investigate the relationship between two approaches to modeling physical systems. On the first approach, simplifying assumptions are made about the level of detail we choose to represent in a computational simulation with an eye toward tractability. On the second approach simpler, analogue physical systems are considered that have more or less well-defined connections to systems of interest that are themselves too difficult to probe experimentally. Our interest here is in the connections between the artifacts of modeling that appear in these two approaches. We begin by outlining an important respect in which the two are essentially dissimilar and then propose a method whereby overcoming that dissimilarity by hand results in usefully analogous behavior. We claim that progress can be made if we think of artifacts as clues to the projectible predicates proper to the models themselves. Our degree of control over the connection between interesting analogue physical systems and their targets arises from determining the projectible predicates in the analogue system through a combination of theory and experiment. To obtain a similar degree of control over the connection between large-scale, distributed simulations of complex systems and their targets we must similarly determine the projectible predicates of the simulations themselves. In general theory will be too intractable to be of use, and so we advocate an experimental program for determining these predicates.

the object of the natural history which I propose is...to give light to the discovery of causes and supply a suckling philosophy with its first food. Francis Bacon, The Great Instauration

     

Quantification for Peirce's preferred system of triadic logic

Without introducing quantifiers, minimal axiomatic systems have already been constructed for Peirce's triadic logics. The present paper constructs a dual pair of axiomatic systems which can be used to introduce quantifiers into Peirce's preferred system of triadic logic. It is assumed (on the basis of textual evidence) that Peirce would prefer a system which rejects the absurd but tolerates the absolutely undecidable. The systems which are introduced are shown to be absolutely consistent, deductively complete, and minimal. These dual axiomatic systems reveal an interesting elegance, independent of their historical motivation.     

Geometrical structures of photographic and stereoscopic spaces

Two experiments were conducted to investigate the geometrical structures of photographic and stereoscopic spaces. In Experiment 1, it was investigated how accurately photographic space reproduces real physical space, and the geometrical structure of photographic space was compared with that of visual space. As a result, the mapping function of distance between photographic and physical spaces (delta = ad(b)) shows that a and b range from 0.96-1.1 and 0.69-0.78. The mapping function of angle between photographic and physical spaces (phi = g phi(h)) shows that g and h range from 2.37-5.29 and 0.74-0.97. Further, photographic space has larger anisotropic property than visual space and photographic space may be hyperbolic. In Experiment 2, the geometrical structure of stereoscopic space was compared with that of visual space. It was found that stereoscopic space was almost the same as visual space.     

Preference and similarity: affective and cognitive judgement?

This study is a test of some implications of the distinction between cognitive and affective processes proposed by Zajonc (1980), applied to preference and similarity judgements. Preference and similarity data were obtained for male and female subjects judging male and female cinema actors. Preference judgements were assumed to be an example of primarily affective judgement, similarity was assumed to be mainly cognitive. It was furthermore assumed that men when judging women and women judging men should be more affectively involved than the opposite cases. It was predicted (a) that making preference judgements would be more confident, (b) that such judgements would be more stable, (c) that preference judgements would show a more simple structure than similarity judgements, and (d) that the difference between similarities and preferences with regard to the level of complexity indicated by the ratings should be especially large with large affective involvement. In support of Zajonc's model, preference judgements were given with more confidence by the subjects and they were more stable over time. There was a tendency for similarity spaces to be more complex for affectively involved subjects while the opposite occurred for preference ratings. Women seemed to react more negatively to the similarity task than men did.     

On the Meaning of Hilbert's Consistency Problem (Paris, 1900)

The theory that ``consistency implies existence' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems.     

Grounding,physicalism and necessity

Recent work on metaphysical grounding has suggested that physicalism can be characterised in terms of the mental facts being grounded in physical facts. It is often assumed that the full grounds of a fact metaphysically necessitate that fact. Therefore, it seems that if the physical grounds the mental, then the physical facts metaphysically necessitate the mental facts. Stefan Leuenberger argues that such a version of physicalism would be vulnerable to counterexamples. I shall outline a characterisation of grounding which appeals to a relation between grounding and the essences of properties instantiated in the grounded facts or in their grounds. If a grounded fact is such that its constituent property is essentially related to the properties instantiated in its grounds, or vice versa, then the grounded fact will be metaphysically necessitated by its full grounds. This characterisation of grounding not only avoids Leuenberger’s counterexamples, but has broader implications for characterising physicalism in terms of grounding.     

A semantical investigation on Brouwer-Zadeh logic

Conclusion In the standard approach to quantum mechanics, closed subspaces of a Hilbert space represent propositions. In the operational approach, closed subspaces are replaced by effects that represent a mathematical counterpart for properties which can be measured in a physical system. Effects are a proper generalization of closed subspaces. Effects determine a Brouwer-Zadeh poset which is not a lattice. However, such a poset can be embedded in a complete Brouwer-Zadeh lattice. From an intuitive point of view, one can say that these structures represent a natural logical abstraction from the structure of propositions of a quantum system. The logic that arises in this way is Brouwer-Zadeh logic. This paper shows that such a logic can be characterized by means of an algebraic and a Kripkean semantics. Finally, a strong completeness theorem for BZL is proved.     

Self-report and stability of physical symptoms by adolescents

Verbal reports of physical complaints and their relationship to school-related stress was investigated in two adolescent samples. It was hypothesized that symptom reporting is a function of age and sex, and is connected to psychological factors such as test (state) and trait anxiety. In addition, the consistency of symptoms reported during the course of one year was evaluated. It was found that, overall, the two adolescent groups complained of more physical symptoms than did Br?hler's (1978) adult sample. Differences between males and females in reporting symptoms were apparent only in the older adolescent group (ages 14 to 15); females reported more physical complaints. The positive correlation between body symptoms and test-trait anxiety was highly significant. During the year, few adolescents exhibited well-defined patterns of body complaints. On the contrary, many complaints seemed to be transient phenomena, although the number was high. In this respect, adolescence plays an important role in the self-reporting evaluation and development of physical complaints.     

Time,Quantum Mechanics,and Probability

A variety of ideas arising in decoherence theory, and in the ongoing debate over Everett's relative-state theory, can be linked to issues in relativity theory and the philosophy of time, specifically the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995; 1996a); in what follows we shall consider the same circle of ideas, but specifically in relation to the interpretation of probability, and its identification with relations in the Hilbert Space norm. The familiar objection that Everett's approach yields probabilities different from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as defined by Everett's theory, and that this can be transparently interpreted in terms of the ordinary notion of the chance occurrence of an event, as witnessed in the present. It is in this sense that the theory has empirical content.     

The importance of physicalism in the philosophy of religion

First, some say that core physicalism is not anti-religion. I argue that this seems to be incorrect. Physical completeness is a core element of contemporary physicalism; (the evidence for physical completeness is strong); and physical completeness both logically and not strictly logically rejects many central religious views. Consequently, there is a sense in which core physicalism is, in an important way, anti-religion. Second, physical completeness positively supports one significant religious view; and physical completeness permits one to hold two others. The view that physical completeness supports states that there is no natural grounding of the ordinarily taken boundary of the human body. The two views that physical completeness permits one to hold state that a person can be contrastlessly blissful in an ongoing way, and that a person can experience something like light circulating through the ordinary body in an ongoing way. It is further maintained that physicalism allows religious systems to develop in new forms.     

Secularism and securitisation: the imaginary threat of religious minorities in Canadian public spaces

ABSTRACT

From the assumed physical threat of a ceremonial Kirpan in an elementary school carried by a Sikh child, to the fictional possibility of rich, Arab, Muslim University students utilising their implicitly understood patriarchal power to subjugate all women from access to common swimming pools, Canada has become increasingly replete with examples of using religious minorities as a danger to secure public spaces for societies most privileged. Since 9/11, this has become a far too common public discourse on maintaining close surveillance, scrutiny and regulations for those religious and racialised Canadian minorities associated with the ‘war on terror’. Promoting public spaces, especially public-school spaces, as ‘secular’ has become the argument of supposed non-bias in ensuring safety and equality for the wider population, all the while leaving many of those used as an example of threat to wonder if the ultimate intent is to preserve white, Christian (and Christian cultural) privilege. This article proposes to examine cases since 9/11 that have problematised racialised groups associated with the terrorism in public schooling to the benefit of maintaining ‘Old Stock’ status quo.