The Philosophy of Mathematics

The philosophy of mathematics considers what is behind the math that we do. What is mathematics? Is it some cosmic truth we discover, or is it created by humans? Do mathematical objects such as numbers and functions really exist, or are they just symbols we have invented? Two of the great debates in the history of mathematical philosophy center around ontology and epistemology. Where did mathematics come from? How do we know that it is true? Where did mathematics come from? Is it discovered or created? Ontological questions are concerned with the nature and status of mathematical objects. Some people believe that the numbers, functions, and other mathematical objects that we talk about actually exist, either in the world or “out there” somewhere. There are things about them that we need to learn. Statements like 2 + 2 = 4 are true whether we know about them or not, and they are waiting to be discovered. This is called independent truth. It can be likened to the perennial question, “If a tree falls in the forest with no one around, does it make a sound?” Here, “If there are no humans to study mathematics, does mathematical truth still exist?” Those who believe in independent truth say yes. Others believe that mathematical symbols have been made up to make our calculations possible, and that mathematical truth is a human invention. These debates can get very heated, sometimes taking on an almost religious tone. To some, denying the existence of independent truth is dangerously close to denying the role of God as creator of all. How do we know that mathematics is true? How do we know that what we think is true really is? Epistemological questions address issues of mathematical justification and knowledge. In the mathematical community, the primary means of justification is proof. A proof begins with a set of axioms, and generates a sequence of statements, through inference, that ends in the given proposition. The debate that arises surrounds the nature of the knowledge that results. The traditional philosophy of mathematics has been one which asserts that mathematics offers absolute certainty through proof. This certainty is called absolute truth. But how can we be sure that what we have come to know is true? Some philosophers say we are sure because we proved it using deduction and reasoning. Others say we are sure because of the way mathematics models scientific phenomena. There is also a group of philosophers who believe that we can’t be sure that what we know is actually true, but that it doesn’t matter. In this report, I will summarize several positions in the philosophy of mathematics, both historical and contemporary. The diversity of opinions provides an intriguing look at the world of mathematics. These philosophers offer a sense that there is more going on behind “ x +5 = 8” than meets the eye

The impact of limiting long term illness on internal migration in England and Wales: New evidence from census microdata

Previous research has suggested that poor health is associated with reduced migration; this knowledgestems from models based on past censuses, or longitudinal studies which imply that the factors influencing migration are the same between those in good and poor health. This paper addresses these issuesby utilising health-stratified analyses on the 2011 Census Individual Secure Sample for England and Wales.Multilevel models predict the odds of moving for working age adults, controlling for key predictors of migration, estimating the effect of health status on the odds of moving and the destination specific variance in migration. We find that those in poor health are less likely to move, after controlling for individual level characteristics. In contrast with expectations, economic inactivity, marriage and being in African, Caribbean, Black, Other or Mixed ethnic groups were not significant predictors ofmigration among the unhealthy sample, but were for the healthy sample. We conclude that migration is health-selective and propose implications for understanding area level concentrations of poor health in England and Wales

A conceptual framework to assess the impact of training on equipment cost and availability in the military context

Designing military support is challenging and current practices need to be reviewed and improved. This paper gives an overview of the Industry current practices in designing military support under Ministry of Defence/Industry agreements (in particular for Contracting for Availability (CfA)), and identifies challenges and opportunities for improvement. E.g. training delivery was identified as an important opportunity for improving the CfA in-service phase. Thus, an innovative conceptual framework is presented to assess the impact of training on the equipment availability and cost. Additionally, guidelines for improving the current training delivery strategies are presented, which can also be applied to other Industry contexts

Beyond employability: a more radical role for Higher Education in developing workplace capabilities

The relationship between higher education and the workplace might be imagined from two contrasting views – an employability viewpoint, and a workplace capability viewpoint. The employability viewpoint regards higher education institutions (HEIs) as one of many providers of training for enabling working people (principally employees) to enhance career progression whilst simultaneously enhancing economic performance as set and measured by external bodies (principally host employers and/or professional bodies). Alternatively, the workplace capability viewpoint, regards HEIs as providers of ongoing learning that transform workplace practices both at individual and institutional levels. Contrasting these two systems of HE support can identify opportunities and constraints for a more productive interface with the workplace. The two viewpoints have emerged from a series of three successive projects carried out since 2014 by academics, alumni and employers associated with the Applied Systems Thinking in Practice (ASTiP) group at the Open University. The third and current project entitled Changing the way the game is played: Transforming PG Curriculum praxis and workplace capabilities focuses on two orders of support – (i) curriculum support for developing workplace capabilities, and (ii) capabilities to support curriculum development. The paper explores some significant opportunities and constraints towards a capabilities approach for promoting wider social wellbeing

Housing works : assessing the impact of housing association employment support

Social housing providers in the UK have a long history of delivering support to their residents beyond housing, including providing employment related assistance. A small but growing area of research investigates the role of employment support outside the mainstream welfare system. However, little is known about the support provided, the impact it has on individuals’ prospects in the labour market, and related outcomes for housing providers, the taxpayer and wider society. As increasing numbers of people are expected to engage in work search and other work preparation activities in exchange for continued receipt of welfare benefits, the impact of housing provider employment support needs to be better understood. The membership of Give us a Chance (GUAC) have, as housing associations, been involved in a range of initiatives to improve the employment and life chance outcomes of their residents, including working with employment support organisations and the voluntary and community sector. However, there is considerable variation in the way that housing association employment support is currently provided, and a wide range of tools and measures are used to show the value of these initiatives, from specific organisational measures through to more generic approaches, such as the Housing Associations’ Charitable Trust (HACT) measures and the New Economy Cost Benefit Analysis (CBA) model. Hence, it is often difficult to compare the value of projects based on different approaches and measured using different tools. This research had two key aims: to contribute to an understanding of what helps people to move closer to, enter or sustain employment and to demonstrate the value of housing association investment in employment related support for residents. The project had the following linked objectives: 1. Provide a review of housing association employment related support activity; 2. Undertake a review of the impact measures currently used for employment related support across the social housing sector; and 3. Provide an understanding of ‘what works’ in terms of employment related support and the measures used among GUAC members

Absence of simulation evidence for critical depletion in slit-pores

Recent Monte Carlo simulation studies of a Lennard-Jones fluid confined to a mesoscopic slit-pore have reported evidence for ``critical depletion'' in the pore local number density near the liquid-vapour critical point. In this note we demonstrate that the observed depletion effect is in fact a simulation artifact arising from small systematic errors associated with the use of long range corrections for the potential truncation. Owing to the large near-critical compressibility, these errors lead to significant changes in the pore local number density. We suggest ways of avoiding similar problems in future studies of confined fluids.Comment: 4 pages Revtex. Submitted to J. Chem. Phy

In-situ X-ray structure measurements on aerodynamically levitated high temperature liquids

High energy, high flux X‐ray sources enable new measurements of liquid and amorphous materials in extreme conditions. Aerodynamic levitation in combination with laser beam heating can be used to access high purity and non‐equilibrium liquids at temperatures up to 3000 K. In this work, a small aerodynamic levitator was integrated with high energy beamline 11 ID‐C at the Advanced Photon Source. Scattered X‐rays were detected with a Mar345 image plate. The experiments investigated a series of binary in the CaO‐Al2O3, MgO‐SiO2, SiO2‐Al2O3 metal oxide compositions and pure SiO2. The results show that the liquids exhibit large changes in structure when the predominant network former is diluted. Measurements on glasses with the same compositions as the liquids suggest that significant structural rearrangement consistent with a fragile‐strong transition occurs in these reluctant glass forming liquids as they vitrify

Structural studies and polymorphism in amorphous solids and liquids at high pressure

When amorphous materials are compressed their structures are expected to change in response to densification. In some cases, the changes in amorphous structure can be discontinuous and they can even have the character of first-order phase transitions. This is a phenomenon referred to as polyamorphism. Most evidence for polyamorphic transitions between low and high density liquids or analogous transformations between amorphous forms of the same substance to date has been indirect and based on the changes in thermodynamic and other structure-related properties with pressure. Recent studies using advanced X-ray and neutron scattering methods combined with molecular dynamics simulations are now revealing the details of structural changes in polyamorphic systems as a function of pressure. Various "two state'' or "two species'' models are used to understand the anomalous densification behaviour of liquids with melting curve maxima or regions of negative melting slope. Thermodynamic analysis of the two state model leads to the possibility of low- to high-density liquid transitions caused by differences in bulk thermodynamic properties between different amorphous forms and on the degree of cooperativity between low- and high-density structural configurations. The potential occurrence of first-order transitions between supercooled liquids is identified as a critical-like phenomenon. In this tutorial review we discuss the background to polyamorphism, incorporating the experimental observations, simulation studies and the two-state models. We also describe work carried on several systems that are considered to be polyamorphic

High, clustered, nucleotide diversity in the genome of Anopheles gambiae revealed through pooled-template sequencing: implications for high-throughput genotyping protocols

<p>Abstract</p> <p>Background</p> <p>Association mapping approaches are dependent upon discovery and validation of single nucleotide polymorphisms (SNPs). To further association studies in <it>Anopheles gambiae </it>we conducted a major resequencing programme, primarily targeting regions within or close to candidate genes for insecticide resistance.</p> <p>Results</p> <p>Using two pools of mosquito template DNA we sequenced over 300 kbp across 660 distinct amplicons of the <it>An. gambiae </it>genome. Comparison of SNPs identified from pooled templates with those from individual sequences revealed a very low false positive rate. False negative rates were much higher and mostly resulted from SNPs with a low minor allele frequency. Pooled-template sequencing also provided good estimates of SNP allele frequencies. Allele frequency estimation success, along with false positive and negative call rates, improved significantly when using a qualitative measure of SNP call quality. We identified a total of 7062 polymorphic features comprising 6995 SNPs and 67 indels, with, on average, a SNP every 34 bp; a high rate of polymorphism that is comparable to other studies of mosquitoes. SNPs were significantly more frequent in members of the cytochrome p450 mono-oxygenases and carboxy/cholinesterase gene-families than in glutathione-S-transferases, other detoxification genes, and control genomic regions. Polymorphic sites showed a significantly clustered distribution, but the degree of SNP clustering (independent of SNP frequency) did not vary among gene families, suggesting that clustering of polymorphisms is a general property of the <it>An. gambiae </it>genome.</p> <p>Conclusion</p> <p>The high frequency and clustering of SNPs has important ramifications for the design of high-throughput genotyping assays based on allele specific primer extension or probe hybridisation. We illustrate these issues in the context of the design of Illumina GoldenGate assays.</p

Exploring the length scales, timescales and chemistry of challenging materials (Part 2)

This themed issue explores the different length and timescales that determine the physics and chemistry of a variety of key of materials, explored from the perspective of a wide range of disciplines, including physics, chemistry materials science, Earth science and biochemistry. The topics discussed include catalysis, chemistry under extreme conditions, energy materials, amorphous and liquid structure, hybrid organic materials and biological materials. The issue is in two parts, with this second set of contributions exploring hybrid organic materials, catalysis low-dimensional and graphitic materials, biological materials and naturally occurring, super-hard material as well as dynamic high pressure and new developments in imaging techniques pressure. This article is part of the theme issue 'Exploring the length scales, timescales and chemistry of challenging materials (Part 2)'