Conjecture, proof, and sense in Wittgenstein's philosophy of mathematics

One of the key tenets in Wittgenstein’s philosophy of mathematics is that a mathematical proposition gets its meaning from its proof. This seems to have the paradoxical consequence that a mathematical conjecture has no meaning, or at least not the same meaning that it will have once a proof has been found. Hence, it would appear that a conjecture can never be proven true: for what is proven true must ipso facto be a different proposition from what was only conjectured. Moreover, it would appear impossible that the same mathematical proposition be proven in different ways. — I will consider some of Wittgenstein’s remarks on these issues, and attempt to reconstruct his position in a way that makes it appear less paradoxical.

Military doctrine, command philosophy and the generation of fighting power: genesis and theory

Military doctrine is one of the conceptual components of war. Its raison d’être is that of a force multiplier. It enables a smaller force to take on and defeat a larger force in battle. This article’s departure point is the aphorism of Sir Julian Corbett, who described doctrine as ‘the soul of warfare’. The second dimension to creating a force multiplier effect is forging doctrine with an appropriate command philosophy. The challenge for commanders is how, in unique circumstances, to formulate, disseminate and apply an appropriate doctrine and combine it with a relevant command philosophy. This can only be achieved by policy-makers and senior commanders successfully answering the Clausewitzian question: what kind of conflict are they involved in? Once an answer has been provided, a synthesis of these two factors can be developed and applied. Doctrine has implications for all three levels of war. Tactically, doctrine does two things: first, it helps to create a tempo of operations; second, it develops a transitory quality that will produce operational effect, and ultimately facilitate the pursuit of strategic objectives. Its function is to provide both training and instruction. At the operational level instruction and understanding are critical functions. Third, at the strategic level it provides understanding and direction. Using John Gooch’s six components of doctrine, it will be argued that there is a lacunae in the theory of doctrine as these components can manifest themselves in very different ways at the three levels of war. They can in turn affect the transitory quality of tactical operations. Doctrine is pivotal to success in war. Without doctrine and the appropriate command philosophy military operations cannot be successfully concluded against an active and determined foe.

Analytic Benchmark Solutions for Open-Channel Flows

An important test of the quality of a computational model is its ability to reproduce standard test cases or benchmarks. For steady open–channel flow based on the Saint Venant equations some benchmarks exist for simple geometries from the work of Bresse, Bakhmeteff and Chow but these are tabulated in the form of standard integrals. This paper provides benchmark solutions for a wider range of cases, which may have a nonprismatic cross section, nonuniform bed slope, and transitions between subcritical and supercritical flow. This makes it possible to assess the underlying quality of computational algorithms in more difficult cases, including those with hydraulic jumps. Several new test cases are given in detail and the performance of a commercial steady flow package is evaluated against two of them. The test cases may also be used as benchmarks for both steady flow models and unsteady flow models in the steady limit.

Numerical computation of an analytic singular value decomposition of a matrix valued function

This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

Effects of psychotherapy for anxiety in children and adolescents: a meta-analytic review

This paper provides a comprehensive quantitative review of high quality randomized controlled trials of psychological therapies for anxiety disorders in children and young people. Using a systematic search for randomized controlled trials which included a control condition and reported data suitable for meta-analysis, 55 studies were included. Eligible studies were rated for methodological quality and outcome data were extracted and analyzed using standard methods. Trial quality was variable, many studies were underpowered and adverse effects were rarely assessed; however, quality ratings were higher for more recently published studies. Most trials evaluated cognitive behavior therapy or behavior therapy and most recruited both children and adolescents. Psychological therapy for anxiety in children and young people was moderately effective overall, but effect sizes were small to medium when psychological therapy was compared to an active control condition. The effect size for non-CBT interventions was not significant. Parental involvement in therapy was not associated with differential effectiveness. Treatment targeted at specific anxiety disorders, individual psychotherapy, and psychotherapy with older children and adolescents had effect sizes which were larger than effect sizes for treatments targeting a range of anxiety disorders, group psychotherapy, and psychotherapy with younger children. Few studies included an effective follow-up. Future studies should follow CONSORT reporting standards, be adequately powered, and assess follow-up. Research trials are unlikely to address all important clinical questions around treatment delivery. Thus, careful assessment and formulation will remain an essential part of successful psychological treatment for anxiety in children and young people.

Forecasting VaR using analytic higher moments for GARCH processes

It is widely accepted that some of the most accurate Value-at-Risk (VaR) estimates are based on an appropriately specified GARCH process. But when the forecast horizon is greater than the frequency of the GARCH model, such predictions have typically required time-consuming simulations of the aggregated returns distributions. This paper shows that fast, quasi-analytic GARCH VaR calculations can be based on new formulae for the first four moments of aggregated GARCH returns. Our extensive empirical study compares the Cornish–Fisher expansion with the Johnson SU distribution for fitting distributions to analytic moments of normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets, for the purpose of deriving accurate GARCH VaR forecasts over multiple horizons and significance levels.

On collective intentions: collective action in economics and philosophy

Philosophers and economists write about collective action from distinct but related points of view. This paper aims to bridge these perspectives. Economists have been concerned with rationality in a strategic context. There, problems posed by “coordination games” seem to point to a form of rational action, “team thinking,” which is not individualistic. Philosophers’ analyses of collective intention, however, sometimes reduce collective action to a set of individually instrumental actions. They do not, therefore, capture the first person plural perspective characteristic of team thinking. Other analyses, problematically, depict intentions ranging over others’ actions. I offer an analysis of collective intention which avoids these problems. A collective intention aims only at causing an individual action, but its propositional content stipulates its mirroring in other minds.