In Praise of Unicorns

The general thrust of this paper will be to put the recent ‘King is Dead’ controversy into its proper perspective, to review the consequences of Russell Ackoff's criticisms of the 1978 Conference, to look with a critical yet benevolent eye on university courses in the light of these criticisms and to suggest a menu of unsolved problems which might indicate exciting developments in our subject.The paper will start with a discussion of the three disasters which have hit Operational Research in the last twenty years. It is the contention of the speaker that the ‘King is Dead’ controversy was largely a controversy about nothing of significance and, by sapping the energies of those who could otherwise have been constructively employed, was itself one of these disasters (the other two can be left as exercises for the reader).The criticisms of Russell Ackoff were of a much more substantial nature. The author does not agree completely either with Ackoff's diagnosis or his suggestion of the way the patient should be treated. The paper will suggest ways in which the universities have reacted to these criticisms and the extent to which they are moving away from the mind-bending dullness and boredom which seems to run through so much Operational Research teaching. The place of universities in contemporary society will be discussed with particular emphasis on research and on the need to serve the community.We shall discuss an outline of some of the major unsolved problems which at present are largely ignored by Operational Research scientists. None of these problems involves or needs the development of formal mathematics and for that reason they are much more difficult than the normal problems on which mathematicians give their advice. The paper will try to convey the sense of intellectual excitement which is needed to be reintroduced into Operational Research, particularly in order to move away from the nervous, defensive and apologetic posture assumed by so many Operational Research scientists as a reaction to recent criticisms.     

‘Re’-presenting qualitative data from multilingual mathematics classrooms

In this paper I consider what it means to ‘re’-present qualitative data from multilingual mathematics classrooms. I draw from a recent study that focused on language practices in multilingual mathematics classrooms to explore the different levels involved in the ‘re’-presentation of multilingual data. The purpose of the paper is not to discuss the details of the study but to use data from the study to raise the awareness of the conceptual underpinnings of data re-presentation in mathematics education research. I use the data to show one perspective to ‘re’-presentation of multilingual data. The main argument of the paper is that ‘re’-presentation of multilingual data is not just talk written down, it is inevitably a process of selection and is informed by theory, research questions, tools of analysis and the purposes of re-presenting the data.     

Are Bare Particulars Constituents?

In this article I examine an as yet unexplored aspect of J.P. Moreland’s defense of so-called bare particularism — the ontological theory according to which ordinary concrete particulars (e.g., Socrates) contain bare particulars as individuating constituents and property ‘hubs.’ I begin with the observation that if there is a constituency relation obtaining between Socrates and his bare particular, it must be an internal relation, in which case the natures of the relata will necessitate the relation. I then distinguish various ways in which a bare particular might be thought to have a nature and show that on none of these is it possible for a bare particular to be a constituent of a complex particular. Thus, Moreland’s attempt to resurrect bare particulars as ontologically indispensable entities is not wholly without difficulties.     

Perdurance, Endurance, and ‘Having a Property Atemporally’

In this paper, I argue that both perdurance theory and the ‘relations-to-times’ endurantist view rely on an atemporal notion of property instantiation and relation bearing. I distinguish two possible meanings of ‘atemporal’ which result in two different understandings of what it is for an object to have a property or to bear a relation atemporally. I show that standard presentations of the theories considered are indeterminate as to which of these two understandings is the intended one. I claim that even if both understandings are admissible, one of them is more attractive and has more to recommend than the other.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Possibly v. actually the case: Davidson’s omniscient interpreter at twenty

The publication of Davidson 2001, anthologizing articles from the 1980s and 1990s, encourages reconsidering arguments contained in them. One such argument is Davidson’s omniscient-interpreter argument (‘OIA’) in Davidson 1983. The OIA allegedly establishes that it is necessary that most beliefs are true. Thus the omniscient interpreter, revived in 2001 and now 20 years old, was born to answer the skeptic. In Part I of this paper, I consider charges that the OIA establishes only that it is possible that most beliefs are true; if correct, then it is also possibly the case that most beliefs are false—the skeptic’s very position. Next, I consider two responses on Davidson’s behalf, showing that each fails. In Part II, I show that the OIA establishes neither that it is necessarily merely possibly but actually the case that most beliefs are true. I then conclude that this is enough to answer the skeptic.     

Polar and singular value decomposition of 3×3 magic squares

In the literature there have been reported a number of studies of probabilities involved in Monopoly, the worldwide popular game designed by Charles B. Darrow in the 1930s. Simple assumptions often lead to inaccurate results. In this paper, having distinguished between ‘In Jail’ and ‘Just Visiting’, both theoretical computations and computer simulations are conducted to determine the probability of landing on ‘JAIL’. Theoretical and simulation results are consistent. The sum of the probabilities of landing on ‘In Jail’ and ‘Just Visiting’ is found to be 6.40%, both in theory and in simulation, compared with Ian Stewart's 5.89%. The long-term probabilities of landing on the other 39 spaces are computed and it is concluded that the best properties to buy are the orange ones.     

Russell on Spinoza’s Substance Monism

Russell??s critique of substance monism is an ideal starting point from which to understand some main concepts in Spinoza??s difficult metaphysics. This paper provides an in-depth examination of Spinoza??s proof that only one substance exists. On this basis, it rejects Russell??s interpretation of Spinoza??s theory of reality as founded upon the logical doctrine that all propositions consist of a predicate and a subject. An alternative interpretation is offered: Spinoza??s substance is not a bearer of properties, as Russell implied, but an eternally active, self-actualizing creative power. Eventually, Spinoza the Monist and Russell the Pluralist are at one in holding that process and activity rather than enduring things are the most fundamental realities.     

Remarks on the idealist and empiricist interpretation of frequentism: Robert Leslie Ellis versus John Venn

The goal of this paper is to correct a widespread misconception about the work of Robert Leslie Ellis and John Venn, namely that it can be considered as the ‘British empiricist’ reaction against the traditional theory of probability. It is argued, instead, that there was no unified ‘British school’ of frequentism during the nineteenth century. Where Ellis arrived at frequentism from a metaphysical idealist transformation of probability theory’s mathematical calculations, Venn did so on the basis of an empiricist critique of its ‘inverse application’.     

Dynamic modelling of life table data

In this paper we formulate a dynamic model expressing the human life table data by using the first-passage-time theory for a stochastic process. The model is derived analytically and then is applied to the mortality data in Belgium and France. A stochastic simulation is also performed for the ‘health state function’ proposed and the related stochastic paths. Furthermore the implications of the proposed model and the results derived for pension funds and option theory are discussed.     

错用罗素悖论-康托在集合论中的两个逻辑性错误

分析了罗素悖论与康托的实数集合不可数证明及康托定理S〈P（S）证明之间的本质性联系，发现康托的这两个非构造性证明与罗素悖论有完全相同的思路，但是康托犯了两个逻辑性错误而使他误用了这个悖论思路。得到明确的结论：康托在集合论中如上两个证明里的核心部分实际上是罗素悖论的翻版，这两个证明中的思路与做法是错误的，这样的证明结果没有科学性。     

Operational Research for the Real World: big questions from a small island

The address explores three questions that are important for operational research: ‘Where do we come from?’, ‘What are we?’ and ‘Where are we going?’ First, it looks briefly at the history of O.R. from Babbage to Blackett and the importance of a ‘union of theory and practice’ in our work. Then it looks at O.R.'s identity and image and argues that we do have a reasonably clear idea of our identity, although we do not give sufficient prominence to our contribution to improving system design. It also argues that the ‘brand image’ of O.R. has become diluted and needs to be simple, distinctive, relevant and compelling. In considering how we might achieve that, by addressing the third question, it proposes that we should grow our capability and activity in accommodating ‘analytics’, in building in behavioural science, in coping with complex systems, in developing design thinking and in encouraging evaluation. It also suggests strengthening bridges between academics and practitioners, between those working in different specialisms and in diverse sectors and between O.R. and the public. Finally, it argues that such developments would assist O.R. to reinforce our bridge between science and the humanities and enable us to better meet the challenges of the ‘real world’.     

From Optimizing to Learning: A Development of Systems Thinking for the 1990s

In any subject concerned with rational intervention in human affairs, theory must lead to practice; but practice is the source of theory: neither theory nor practice is prime. We can examine this ‘groundless’ relation by asking what intellectual framework F is applied in what methodology M to what area of application A? If we do this for O.R., systems analysis, systems engineering etc., we see that F and M have changed dramatically between the 1950s and the 1980s, yielding the ‘hard’ and ‘soft’ traditions of systems thinking. The ‘hard’ tradition, based on goal seeking, is examined in the work of Simon and contrasted with the ‘soft’ tradition, based on learning, as exemplified in the work of Vickers and the development of soft systems methodology. The two are complementary, but the relation between them is that the ‘hard’ is a special case of ‘soft’ systems thinking. This analysis makes sense of the recent history of management science and helps to prepare us for the 1990s.     

Leibniz’s Logic and the “Cube of Opposition”

After giving a short summary of the traditional theory of the syllogism, it is shown how the square of opposition reappears in the much more powerful concept logic of Leibniz (1646–1716). Within Leibniz’s algebra of concepts (which may be regarded as an “intensional” counterpart of the extensional Boolean algebra of sets), the categorical forms are formalized straightforwardly by means of the relation of concept-containment plus the operator of concept-negation as ‘S contains P’ and ‘S contains Not-P’, ‘S doesn’t contain P’ and ‘S doesn’t contain Not-P’, respectively. Next we consider Leibniz’s version of the so-called Quantification of the Predicate which consists in the introduction of four additional forms ‘Every S is every P’, ‘Some S is every P’, ‘Every S isn’t some P’, and ‘Some S isn’t some P’. Given the logical interpretation suggested by Leibniz, these unorthodox propositions also form a Square of Opposition which, when added to the traditional Square, yields a “Cube of Opposition”. Finally it is shown that besides the categorical forms, also the non-categorical forms can be formalized within an extension of Leibniz’s logic where “indefinite concepts” X, Y, Z\({\ldots}\) function as quantifiers and where individual concepts are introduced as maximally consistent concepts.     

A tale of mathematical myth-making: E T Bell and the ‘arithmetization of algebra’

Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamous penchant for mathematical ‘myth-making’.