Local axioms in disguise: Hilbert on Minkowski diagrams

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski??s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating ??rigorously?? with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as ??drawn formulas??, and formulas as ??written diagrams??, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski??s diagrammatic methods in number theory prompted Hilbert??s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert??s assessment of Minkowski??s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.     

Absolute error revisited: an accuracy indicator in disguise

When target accuracy is defined as the probability that an individual will respond to an accuracy task within a fixed distance around the target, then the composite error measures, E and AE, are shown to be fairly strong indicators of target accuracy in a relative sense. When AE and E are compared, AE is shown to be an even stronger accuracy indicator than E for most reasonable accuracy requirements. This, plus the fact that AE has certain desirable properties in ANOVA procedures, suggests that AE is a good, composite measure of target accuracy and should be analyzed first to determine if target accuracy differences exist. Subsequent analyses of bias and/or variability are then recommended.     

The learned helplessness hypothesis: A framework in disguise

This article focuses on the theoretical statements of the reformulated helplessness hypothesis (Abramson, Seligman, & Teasdale, 1978). The argument is that the reformulated hypothesis should be regarded primarily as a psychological framework. However, most interpretations of the hypothesis have treated it as a specific, experimental model. Consequently, some of the more essential ideas presented in the original helplessness statements have been criticized inappropriately. If the helplessness position is acknowledged as a framework, the possibility of resolving some of these questions will be more realistic. Examples of research strategies which rely on interdisciplinary approaches are presented as potential means to this end.     

Worldviews and individual vulnerability to suicide: The role of social axioms

Research investigating the role of generalized beliefs about the world or worldviews is relatively scarce in the suicide literature. Two studies, using Hong Kong Chinese samples, examined how worldviews, as assessed by the Social Axioms Survey (SAS), were linked with individual vulnerability to suicide. In Study 1, we investigated the relationships of social axioms with various suicide indicators in cognitive, emotional and interpersonal domains, viz., suicidal ideation, negative self‐esteem, psychache, burdensomeness and thwarted belongingness. Results from canonical correlation analysis showed that beliefs along the axiom dimensions of social cynicism, reward for application, and social complexity were linked to these suicide indicators. In Study 2, we tested the interplay of worldviews and personality traits in the prediction of suicidal thoughts. Hierarchical regression results demonstrated the predictive power of social axioms over and above that provided by the Big Five personality dimensions. Moreover, a significant interaction was observed between belief in reward for application and negative life events in predicting suicidal ideation, showing that reward for application buffered the effect of negative life events on suicidal ideation. Based on these results, we discussed the significance of worldviews as a consideration in suicide research and their implications for clinical assessment and intervention. Copyright © 2010 John Wiley & Sons, Ltd.     

On necessity and commonsense: A discussion of central axioms in new approaches to lay explanation

Recent critical attempts to modify or replace attribution theory have focused particularly on two issues: the relationship of the reason—cause distinction to attribution theories and actor/observer differences in attribution, and the emphasis in attribution theories on inferential, as opposed to self-presentational, processes. In dealing with these two issues, some critics also point out ethnocentric and ideological influences on attribution theories. The issues raised are important, as both intentional (reason) explanations and self-presentational factors have been under-represented in attribution theories. But the new attempts to build these factors into theories of lay explanation are not wholly satisfactory. Authors make several stipulations about reasons and causes that are unsupportable, and they employ arguments about differences between actors and observers that are unsatisfactory in other respects. Nonetheless a number of their claims can be reconceptualized and supported by using definitions and parameters that do not confound the variables being examined. Claims that attribution theories have ignored self-presentational factors underestimate the extent to which self-presentational factors are being examined in attribution studies, but the proposition that self-presentational factors account for most aspects of explanations is unconvincing. Authors are also mistaken to assume that ethnocentric or ideological factors are particular to attribution theories. The errors, exaggerations and other difficulties in the authors' arguments do not nullify the attempt to expand or transcend attribution theory. Where flaws in the arguments are pointed out, alternative ways of tackling the same issues are referred to or proposed.     

Empirical evaluation of axioms fundamental to Stevens's ratio-scaling approach: I. Loudness production

Stevens's direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens's central axioms of commutativity and multiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens's (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.