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.

Transreal proof of the existence of universal possible worlds

Transreal arithmetic is total, in the sense that the fundamental operations of addition, subtraction, multiplication and division can be applied to any transreal numbers with the result being a transreal number [1]. In particular division by zero is allowed. It is proved, in [3], that transreal arithmetic is consistent and contains real arithmetic. The entire set of transreal numbers is a total semantics that models all of the semantic values, that is truth values, commonly used in logics, such as the classical, dialetheaic, fuzzy and gap values [2]. By virtue of the totality of transreal arithmetic, these logics can be implemented using total, arithmetical functions, specifically operators, whose domain and counterdomain is the entire set of transreal numbers

A proof of van Aubel's theorem using orthogonal vectors

We show how two linearly independent vectors can be used to construct two orthogonal vectors of equal magnitude in a simple way. The proof that the constructed vectors are orthogonal and of equal magnitude is a good exercise for students studying properties of scalar and vector triple products. We then show how this result can be used to prove van Aubel's theorem that relates the two line segments joining the centres of squares on opposite sides of a plane quadrilateral.

Improving the health forecasting alert system for cold weather and heat-waves in England: a proof-of-concept using temperature-mortality relationships

Objectives In this study a prototype of a new health forecasting alert system is developed, which is aligned to the approach used in the Met Office’s (MO) National Severe Weather Warning Service (NSWWS). This is in order to improve information available to responders in the health and social care system by linking temperatures more directly to risks of mortality, and developing a system more coherent with other weather alerts. The prototype is compared to the current system in the Cold Weather and Heatwave plans via a case-study approach to verify its potential advantages and shortcomings. Method The prototype health forecasting alert system introduces an “impact vs likelihood matrix” for the health impacts of hot and cold temperatures which is similar to those used operationally for other weather hazards as part of the NSWWS. The impact axis of this matrix is based on existing epidemiological evidence, which shows an increasing relative risk of death at extremes of outdoor temperature beyond a threshold which can be identified epidemiologically. The likelihood axis is based on a probability measure associated with the temperature forecast. The new method is tested for two case studies (one during summer 2013, one during winter 2013), and compared to the performance of the current alert system. Conclusions The prototype shows some clear improvements over the current alert system. It allows for a much greater degree of flexibility, provides more detailed regional information about the health risks associated with periods of extreme temperatures, and is more coherent with other weather alerts which may make it easier for front line responders to use. It will require validation and engagement with stakeholders before it can be considered for use.

A Taylor Model Based Description of the proof stress of magnesium AZ31 during hot working

A series of hot-compression tests and Taylor-model simulations were carried out with the intention of developing a simple expression for the proof stress of magnesium alloy AZ31 during hot working. A crude approximation of wrought textures as a mixture of a single ideal texture component and a random background was employed. The shears carried by each deformation system were calculated using a full-constraint Taylor model for a selection of ideal orientations as well as for random textures. These shears, in combination with the measured proof stresses, were employed to estimate the critical resolved shear stresses for basal slip, prismatic slip, ⟨c+a⟩ second-order pyramidal slip, and { } twinning. The model thus established provides a semianalytical estimation of the proof stress (a one-off Taylor simulation is required) and also indicates whether or not twinning is expected. The approach is valid for temperatures between ∼150 °C and ∼450 °C, depending on the texture, strain rate, and strain path.

Development of a cooling fabric from conducting polymer coated fibres: proof of concept

It is supposed that there should be a thermal electric effect if a dc current is applied across two dissimilar conducting polymers, similar to so called “Peltier effect” in metals or semiconductors. However, this hypothesis has not been tested on conducting polymers and using these materials to make cooling fabrics has never been attempted before. Polypyrrole coated fabrics were used to test the hypothesis in this preliminary study. Seebeck and the Peltier effects were proven to exist. However, thermoelectricity effect between two conducting polymer coated fabric samples was only about 10 μV/°C. Cooling effect by conductive polymer powder was achieved but performance was unsteady due to electrical degradation of the conducting polymer. Nevertheless, the concept was demonstrated and the development of a cooling fabric is possible.

Scientific proof that humans enjoy punishing wrongdoers: The implications for punishment and sentencing

Recent developments in brain science confirm that as a race we are in fact a punitive lot. Human beings actually derive pleasure from inflicting punishment on wrongdoers. We are wired in such a way that the part of our brain that reports pleasure is activated when we punish norm violators. This is even when punishment has no tangible or demonstrable benefits. However, we are not slaves lo our emotions. Another region of our brain 'kicks-in' if punishment becomes self-defeating, in that it conflicts with our other interests. The implications of this research for punishment theory and the practice of sentencing are discussed in this paper. The findings give qualified support to the theory known as intrinsic retributivism, but do not suggest it is the soundest theory of punishment. This is because we stop punishing when it comes at a cost to us. The good feeling that punishment invokes in punishers is another consequential consideration in favour of the utilitarian theory of punishment. However, it is not clear that the utilitarian calculus is necessarily affected by the findings. The main implication of the research findings relates to the relevance of public opinion to sentencing practice. The findings support the view that public sentiment, which seems to support increasingly tougher sanctions, can be curtailed of the public are informed that punishment comes of a cost to community.