What environmental studies students think about mathematics and what could be done about it

Mathematics is playing an increasing role in environmental studies, which causes problems for many students because of their low mathematical ability. A survey among first year environmental studies students revealed a significant number both of negative experiences in learning mathematics and negative feelings towards the subject. A proposal to design an alternative curriculum which aims to link the maths more closely with environmental studies is outlined as an answer to the problem of negative attitudes to the required mathematics.     

Ethics in OR/MS: Past,present and future

The pervasiveness and impact on society and on every day human life of technology has led to a growing awareness that science and technology cannot be considered above or beyond the realm of value judgements and hence of ethics. This is especially true for Operations Research / Management Science (OR/MS), that particular science which is concerned with methodologies for scientifically deciding how to design and operate man-machine systems in an optimal way, usually under conditions requiring the allocation of scarce resources. Here we try to give a historical account of the growing interest for ethics within the OR/MS community from its birth to present days. Starting from attempts to define models and codes of ethical behaviour in our profession, the OR/MS community has arrived at more fundamental questions about the ethical responsibility it faces in a world of growing inequalities and in which the ever greater stress that human activities impose on the environment puts at risk the very survival of human kind.     

Ethics in OR/MS: past,present and future

The pervasiveness and impact on society and on every day human life of technology has led to a growing awareness that science and technology cannot be considered above or beyond the realm of value judgements and hence of ethics. This is especially true for Operations Research/Management Science (OR/MS), that particular science which is concerned with methodologies for scientifically deciding how to design and operate man-machine systems in an optimal way, usually under conditions requiring the allocation of scarce resources. Here we try to give a historical account of the growing interest for ethics within the OR/MS community from its birth to present days. Starting from attempts to define models and codes of ethical behaviour in our profession, the OR/MS community has arrived at more fundamental questions about the ethical responsibility it faces in a world of growing inequalities and in which the ever greater stress that human activities impose on the environment puts at risk the very survival of human kind. This paper appeared in 4OR 2, 95–110, 2004.     

Bit by bit, the discoverers decided what it was they had discovered

Conclusion  Is it possible to answer the questionWhat is the “birthday” of elliptic functions? Yes, but far from uniquely. But does the overabundance of possible answers occasioned by the inherent n?ivety of the question mean that such lines of inquiry are pointless for the historian? Can questions regarding the temporal origins of mathematical areas and the research to which they lead ever be useful or instructive?     

Understanding what a proof is: a classroom-based approach

Being unaware of the assumptions underlying a deductive argument is widespread among learners and is a major stumbling block to their understanding of proof. Thus, the basic idea of the present paper is that at some points in the course of secondary education there should be classroom-based interventions addressing this difficulty and making the axiomatic organization of mathematics a theme. Students should be made aware that there are axioms in mathematics, what their role is and how mathematicians come to agree about which axioms should be accepted. An axiom which is not yet accepted is simply a hypothesis. A hypothesis is evaluated by deductively drawing consequences and by investigating whether these consequences agree with experience or should be accepted for other reasons. The teaching intervention discussed in this paper exemplifies this idea by way of the example of ancient attempts at modelling the path of the sun, the so-called “anomaly of the sun”. It is investigated to what extent the teaching intervention fostered students’ understanding of the conditionality of mathematical/astronomical statements, that is, of the fact that the truth of these statements is dependent on the initial hypotheses.     

Mixing OR methods in practice: Past, present and future directions

Although mixing of OR methods is an area of increasing interest to the OR community, there has been little discussion regarding generic lessons that can be learnt from mixing methods in practice. The aim of this paper is to carry out such an analysis through considering generic lessons that may be associated with mixing methods, regardless of the methods chosen. To identify these lessons, published case studies on how OR methods have been mixed are analysed giving rise to a number of themes revealing the lessons. These themes include; the implications from the use of different facilitators/modellers, how methods have been mixed together, the nature of the modelling interventions, the client value and the rationale given for mixing methods. The paper discusses the lessons learnt in each of these themes and presents opportunities for future work.     

To be or not to be constructive,that is not the question

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out – mathematically speaking – for its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer–Heyting–Kolmogorov interpretation by which ‘there exists $x$’ intuitively means ‘an algorithm to compute $x$ is given’. A number of schools of constructive mathematics were developed, inspired by Brouwer’s intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an ‘excluded middle’. In particular, we challenge the ‘binary’ view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate ‘$x$ is standard’ typical of Nonstandard Analysis can be interpreted as ‘$x$ is computable’, giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald’s longstanding conjecture that classical Nonstandard Analysis is locally constructive. Finally, an alternative explanation of our results is provided by Brouwer’s thesis that logic depends upon mathematics.     

Rethinking tradeoffs and OR/MS methodology

The purpose of this paper is to explore the role of a process known as the ‘evaporating cloud’ (EC) as a way of enhancing OR/MS modelling. While other authors have described the use of the EC process to explore the tradeoffs in a traditional Economic Order Quantity model, this paper demonstrates the use of the EC with a facilities location problem, which was originally solved via mixed integer programming. We relate our findings to OR/MS modelling in general, and argue that the EC could contribute effectively to different phases of the problem-solving process, for example, aiding the problem-structuring phase and helping to find better solutions in many OR/MS modelling situations. We demonstrate that the EC is a valuable tool to use in modelling practice as a problem-structuring tool, as a way of dealing specifically with tradeoffs, and that it can therefore be a useful complement to OR/MS methodology, and contribute to OR/MS practice and teaching.