Robert Leslie Ellis,William Whewell and Kant: the role of Rev H F C Logan

Reverend H F C Logan is put forward as the formerly unidentified figure to which Robert Leslie Ellis referred in a journal entry of 1840 in which he wrote that it was due to his influence that William Whewell came to uphold particular Kantian views on time and space. The historical evidence of Ellis’s early familiarity with, and later commitment to Kant is noteworthy for at least two reasons. Firstly, it puts into doubt the accepted view of the second generation of reformers of British algebra as non-philosophical, practice-oriented mathematicians. Secondly, in so far as Logan was the correspondent of William Rowan Hamilton, it re-emphasizes that the role of Kantianism in the transition from ‘symbolical’ to ‘abstract’ algebra in nineteenth-century British algebra requires closer scrutiny.     

Mistakes concerning a chance encounter between Francis Galton and John Venn

A chance encounter at Bournemouth between Francis Galton and John Venn has lain in some obscurity because of a slip by Galton himself and a second mistake by Karl Pearson. The contact with Venn provides insight into the development of Galton's perception of statistical dispersion, his disenchantment with the notion of ‘probable error’ and adoption of population variability.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

Using dynamic geometry software for teaching conditional probability with area-proportional Venn diagrams

This classroom note illustrates how dynamic visualization can be used to teach conditional probability and Bayes’ theorem. There are two features of the visualization that make it an ideal pedagogical tool in probability instruction. The first feature is the use of area-proportional Venn diagrams that, along with showing qualitative relationships, describe the quantitative relationship between two sets. The second feature is the slider and animation component of dynamic geometry software enabling students to observe how the change in the base rate of an event influences conditional probability. A hypothetical instructional sequence using a well-known breast cancer example is described.     

Simple, Reducible Venn Diagrams on Five Curves and Hamiltonian Cycles

Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Utilizing these procedures with some new methods introduced here, we determine the number of simple, reducible spherical Venn diagrams of five sets. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum. Among others, we construct a simple, reducible Venn diagram with five congruent ellipses. We show that this diagram is unique on the sphere and produces two different plane diagrams. This corrects some erroneous statements that started with John Venn more than a century ago in 1880 and have been repeated frequently by others ever since.     

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.     

Improving credit scoring by differentiating defaulter behaviour

We present a methodology for improving credit scoring models by distinguishing two forms of rational behaviour of loan defaulters. It is common knowledge among practitioners that there are two types of defaulters, those who do not pay because of cash flow problems (‘Can’t Pay’), and those that do not pay because of lack of willingness to pay (‘Won’t Pay’). This work proposes to differentiate them using a game theory model that describes their behaviour. This separation of behaviours is represented by a set of constraints that form part of a semi-supervised constrained clustering algorithm, constructing a new target variable summarizing relevant future information. Within this approach the results of several supervised models are benchmarked, in which the models deliver the probability of belonging to one of these three new classes (good payers, ‘Can’t Pays’, and ‘Won’t Pays’). The process improves classification accuracy significantly, and delivers strong insights regarding the behaviour of defaulters.     

Intertwined evolution operators

We construct a convolution algebra of admissible homomorphisms defined on a ‘test space’ to demonstrate the fundamental role of convolution in the study of intertwined evolution operators of linear ordinary differential equations in Banach spaces and probability theory. The choice of test space makes the framework we present quite versatile. The applications include semigroups of linear operators, empathy, integrated semigroups and empathies and the convolution semigroups of probability theory.     

Analysis of Venn Diagrams Using Cycles in Graphs

This paper is the last in a series by the authors on the use of graph theory to analyze Venn diagrams on few curves. We complete the construction (and hence the enumeration) of spherical Venn diagrams on five curves, which yields additional results about conjectures of Grünbaum concerning which Venn diagrams are convex, which are exposed, and which can be drawn with congruent ellipses.     

Venn diagrams and planar graphs

Using graph theory, we develop procedures for the construction of Venn diagrams. This allows us to determine the number of Venn diagrams on three sets, and to address further questions on enumeration of Venn diagrams. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum.Supported in part by a Purdue Research Foundation Summer Faculty Grant.     

An application of probability to combinatorics: a proof of Vandermonde identity

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using ‘induction’, ‘polynomial identities’ nor ‘generating functions’, and will give a proof of the ‘Vandermonde Identity’ using elementary notions of probability.     

Baer Invariants and Cohomology of Precrossed Modules

In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Fröhlich and Furtado-Coelho’s general theory on Baer invariants in varieties of Ω-groups and Modi’s theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the last two decades. We show how to use Baer invariants in order to connect these various homology theories. First, we express the low-dimensional Baer invariants of precrossed modules in terms of a new non-abelian tensor product of a precrossed module. This expression is used to analyze the connection between the Baer invariants and the homological invariants of precrossed modules defined by Conduché and Ellis. Specifically we prove that the second homological invariant of Conduché and Ellis is in general a quotient of the first component of the Baer invariant we consider. The definition of classical Baer invariants is generalized using homological methods. These generalized Baer invariants of precrossed modules are applied to the construction of five term exact sequences connecting the generalized Baer invariants with the cohomology theory of crossed modules considered by Carrasco, Cegarra and R.-Grandjeán and the cohomology theory of precrossed modules.     

Singular Propositions,Negation and the Square of Opposition

This paper contains two traditions of diagrammatic studies namely one, the Euler–Venn–Peirce diagram and the other, following tradition of Aristotle, the square of oppositions. We put together both the traditions to study representations of singular propositions (through a diagram system Venn-i, involving constants), their negations and the inter relationship between the two. Along with classical negation we have incorporated negation of another kind viz. absence (taking a cue from the notion of ‘abhãva’ existing in ancient Indian knowledge system). We have also considered the changes that take place in the context of open universe.     

The effect of activity-based instruction on conceptual development of seventh grade students in probability

The purpose of this study is to investigate and compare the effects of activity-based and traditional instructions on students’ conceptual development of certain probability concepts. The study was conducted using a pretest–posttest control group design with 80 seventh graders. A developed ‘Conceptual Development Test’ comprising 12 open-ended questions was administered on both groups of students before and after the intervention. The data were analysed using analysis of covariance, with the pretest as covariate. The results revealed that activity-based instruction (ABI) outperformed the traditional counterpart in the development of probability concepts. Furthermore, ABI was found to contribute students’ conceptual development of the concept of ‘Probability of an Event’ the most, whereas to the concept of ‘Sample Space’ the least. As a consequence, it can be deduced that the designed instructional process was effective in the instruction of probability concepts.     

Probability Modelling and Optimal Location of a Travelling Salesman

The ‘random part’ of an operations research model may be less satisfactory than the ‘deterministic part’, and it may thus be desirable to design algorithms that require few probability assumptions and make few calls to a suitable ‘probability oracle’. We consider here the problem of locating a service facility on a tree network so as to minimize the expected length of a travelling salesman tour through a random set of demand nodes.     

In Defence of Pan-Dispositionalism

Pan-Dispositionalism – the view that all properties (and relations) are irreducibly dispositional – currently appears to have no takers amongst major analytic metaphysicians. There are those, such as Mumford, who are open to the idea but remain uncommitted. And there are those, such as Ellis and Molnar, who accept that some properties are irreducibly dispositional but argue that not all are. In this paper, I defend Pan-Dispositionalism against this ‘Moderate’ Dispositionalism.     

Markov chain models for delinquency: Transition matrix estimation and forecasting

A Markov chain is a natural probability model for accounts receivable. For example, accounts that are ‘current’ this month have a probability of moving next month into ‘current’, ‘delinquent’ or ‘paid‐off’ states. If the transition matrix of the Markov chain were known, forecasts could be formed for future months for each state. This paper applies a Markov chain model to subprime loans that appear neither homogeneous nor stationary. Innovative estimation methods for the transition matrix are proposed. Bayes and empirical Bayes estimators are derived where the population is divided into segments or subpopulations whose transition matrices differ in some, but not all entries. Loan‐level models for key transition matrix entries can be constructed where loan‐level covariates capture the non‐stationarity of the transition matrix. Prediction is illustrated on a $7 billion portfolio of subprime fixed first mortgages and the forecasts show good agreement with actual balances in the delinquency states. Copyright © 2010 John Wiley & Sons, Ltd.     

A new model for Boolean algebra

This new model for set theory is a graph. It is similar in many ways to a Venn diagram or Karnaugh map, but it does not pose as a rival, merely as an alternative model which may be useful in some contexts. Defined with reference to the duality of lines and points, the graph is a fitting framework in which to display the rich duality of Boolean algebra.

In the first four sections the graph is developed as a natural embodiment of Boolean theory and it is hoped that it will be seen, not as a more computational device but as helpful for demonstrating Boolean theory. The second half of the article is devoted to practical applications. The graph can be applied (and has been applied in school teaching) extensively in set theory, in logic, in probability, in genetics and in switching circuits, but space does not allow the elaboration of all these in detail. So this article concentrates mainly on one of these applications, switching circuits. The graph is used to simplify and minimize logic circuits with techniques different from Karnaugh's and in some instances more comprehensive.     

The Elizabethan mathematics of everything: John Dee's ‘Mathematicall praeface’ to Euclid's Elements

This article considers John Dee's famous classification and justification of ‘the Sciences, and Artes Mathematicall’ in his Mathematicall praeface to Henry Billingsley's Elements of geometrie of Euclid of Megara (1570), the first English translation of Euclid. It is a revised version of a lecture presented to the British Society for the History of Mathematics Autumn Meeting, October 2010, under the title ‘John Dee and the Elizabethan Mathematics of Everything’.