Pi on Earth, or Mathematics in the Real World

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like.

---

The intergenerational transmission of authoritarianism: The mediating role of parental goal promotion

This study examined the intergenerational transmission of adolescent authoritarian submission (Right-Wing Authoritarianism or RWA) and authoritarian dominance (Social Dominance Orientation or SDO). It was hypothesized that the type of goals that parents promote (i.e., conservation versus openness to change and extrinsic versus intrinsic goal promotion) would mediate any direct association between parents’ and adolescents’ authoritarian attitudes. This hypothesis was examined in a sample of middle adolescents and their parents. First, a significant parent–child concordance was found for RWA and SDO. Second, whereas parental RWA predicted parental promotion of conservation goals (rather than openness to change goals) as well as the promotion of extrinsic goals (rather than intrinsic goals), parental SDO predicted parental promotion of extrinsic goals only. Third, process analyses showed that, whereas parental conservation goal promotion mediates the relationship between parent and child RWA, parental extrinsic goal promotion mediates the relationship between parent and child SDO.     

Contaminants and the Path to Salvation: A Study of the Sarvāstivāda Hṛdaya Treatises

The Sa?gītiparyāya is the earliest Sarvāstivāda philosophical text that enumerates a series of contaminants (anu?aya), i.e. innate proclivities, inherited from former births, to do something of usually evil nature. This early list comprises seven such contaminants. As it is the contaminants that lead a worldling (p?thagjana) to doing volitional actions and thus to forming a karmic result (karmavipāka), these contaminants naturally also bear on the path to salvation. The gradual development of the peculiar Sarvāstivādin path to salvation necessitated a gradual refinement and reinterpretation of the original list of seven contaminants. Apart from a mere technical aspect, this reinterpretation also reflects the viewpoint of the Sautrāntika school of Buddhist philosophy on the nature of contaminants, i.e. their acceptance of a latent and an active state of the defilements, vis-à-vis the Vaibhā?ika viewpoint according to whom no such difference exists. Within Sarvāstivāda literature, the H?daya treatises illustrate this philosophical development.     

Geometric analogue of holographic reduced representation

Holographic reduced representations (HRRs) are distributed representations of cognitive structures based on superpositions of convolution-bound n-tuples. Restricting HRRs to n-tuples consisting of ±1, one reinterprets the variable binding as a representation of the additive group of binary n-tuples with addition modulo 2. Since convolutions are not defined for vectors, the HRRs cannot be directly associated with geometric structures. Geometric analogues of HRRs are obtained if one considers a projective representation of the same group in the space of blades (geometric products of basis vectors) associated with an arbitrary n-dimensional Euclidean (or pseudo-Euclidean) space. Switching to matrix representations of Clifford algebras, one can always turn a geometric analogue of an HRR into a form of matrix distributed representation. In typical applications the resulting matrices are sparse, so that the matrix representation is less efficient than the representation directly employing the rules of geometric algebra. A yet more efficient procedure is based on ‘projected products’, a hierarchy of geometrically meaningful n-tuple multiplication rules obtained by combining geometric products with projections on relevant multivector subspaces. In terms of dimensionality the geometric analogues of HRRs are in between holographic and tensor-product representations.     

Understanding sustainability

Proposed solutions to sustainability often bring different economic sectors into conflict; when a sustainable solution for one sector is non-sustainable for another it creates what we call the dilemma of sustainability. A recent example took place in the Columbia Basin of the Pacific Northwest, involving competing notions of sustainability by fisheries and the energy industry. Taking up some ideas of Eger and Lyotard, we criticize the constructivist approach which treats large ecosystems as constructions and the process of resolving conflicts of sustainability as one solely consisting of negotiations involving the trading off of interests. We propose instead to treat such conflict resolution via a Gadamerian-inspired hermeneutics that sees different economic sectors as having different interpretations of sustainability and that aims at common understanding.     

Understanding sustainability

Proposed solutions to sustainability often bring different economic sectors into conflict; when a sustainable solution for one sector is non-sustainable for another it creates what we call the dilemma of sustainability. A recent example took place in the Columbia Basin of the Pacific Northwest, involving competing notions of sustainability by fisheries and the energy industry. Taking up some ideas of Eger and Lyotard, we criticize the constructivist approach which treats large ecosystems as constructions and the process of resolving conflicts of sustainability as one solely consisting of negotiations involving the trading off of interests. We propose instead to treat such conflict resolution via a Gadamerian-inspired hermeneutics that sees different economic sectors as having different interpretations of sustainability and that aims at common understanding.     

Comparison requirements and attention in identical-nonidentical stimulus discriminations

Perceptual matching data show several puzzling effects. Particularly problematic are the disparities between the processing rates for same and different stimuli--the fast-same effect--and between the processing rates for two same-different judgment tasks that are related as mirror images--the task effect. Current models have difficulty accounting simultaneously for both effects. Central to these models is a stimulus comparison process that derives relative judgments of sameness and difference from tests of the congruence of stimulus representations. A contrasting view holds that same-different judgments can be modeled as absolute, rather than relative, judgments. This latter view is shown to be supported by experimental data. Reaction times for judgments of identical letter strings increase with string length at the same rate whether judgments are based on all the information in the strings or just the information in a single pair of component letters. The data show that stimulus comparisons of the sort described by previous models are not involved in these judgments. An attentional model accounts for the data and for the fast-same and task effects as well.     

Parfit's unified theory of morality

Conclusions Insofar as much of commonsense morality is solely rights-focused, insofar as our commonsense duties to our M-relations often involve comparative and not noncomparative goods, and insofar as commonsense morality is a morality of rule following, to those extents Parfit's claim that commonsense morality is self-defeating does not apply. Furthermore, even if Parfit's claim that morality is self-defeating does have a substantive basis and so we amend M as Parfit recommends, because Parfit's revisions of M fail to move M toward C we will be as far from a unification of M and C as we were at the beginning of the project. In short, although there may be a unity to morality, Parfit has not uncovered it.     

An analysis of generative representation systems

Three ideas are basic to generative theory: (a) Subjects are assumed to attend to the relations among stimuli, extracting the transformations relating pairs of stimuli; (b) the set of abstracted transformations is decomposed or reduced to an elementary set of generators; (c) subjects use the elementary generators as the basis for judging similarity. The purpose of this paper is to illustrate these ideas with an experiment in which subjects were asked to rate the similarity between stimulus pairs. The stimulus materials consisted of the permutations of a 4-item pattern with the properties of a dihedral group which insured the existence of sets of elementary transformations. Three analytic techniques were used to determine the generator set of transformations abstracted by subjects. The first analysis consisted of a monotonic regression between dissimilarity ratings and the number of elementary generators of a given permutation. The residual variance of this monotone regression, suitably normalized, was used as a quantitative goodness-of-fit measure. For the stochastic analysis, cumulative distributions of dissimilarity ratings were obtained for permutations requiring one, two, or three generators. The idea was that permutations requiring fewer generators should be associated with distributions of lower dissimilarity values (higher similarity scores) as compared to permutations predicted to be transformationally more complex. The final analysis, a multidimensional scaling of dissimilarity ratings, converted subjects' ratings into spatial structures to determine whether individual subjects' ratings exhibited the predicted spatial arrangement. The monotone regression and stochastic analyses abstracted similar generator sets for individual subjects, some of which provided perfect fits to the data. Although the scaling analysis yielded similar estimates of generators, for some subjects, transformations with the same number of generators yielded unequal “cognitive” distances resulting in some-what deformed spatial structures for these subjects. It was concluded that the results generally supported a generative model as an approximation to subjects' representations of interstimulus relationships.     

Cognitive, Affective and Behavioral Correlates of Internalization of Regulations for Religious Activities

The main goal of this study was to examine the relationship between different types of extrinsic motivation for religious behaviors as conceptualised within self-determination theory (Ryan & Deci, 2000) and Wulff's (1991) framework of literal versus symbolic approaches of religious contents. Results from a Belgian sample of active believers (N=186) show that the internalization of one's reasons for performing religious behaviors was positively associated with an open, symbolic interpretation of religious belief contents and a stronger adherence to Christian beliefs. Moreover, internalization was also positively related to general well-being and frequency of prayer but unrelated to church attendance. It is concluded that individuals who engage in religious behaviors because of its perceived personal significance will show more cognitive flexibility and open-mindedness towards Christian belief contents, a stronger adherence to this message, higher well-being and more frequent engagement in specific religious behaviors.