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’.     

Early criticism of the symbolical approach to algebra

For a long time, historians have believed that in the 1830s Sir William Rowan Hamilton was a lone critic of symbolical algebra. Using published and unpublished documents, this article shows that symbolical algebra was a considerably controversial subject among British mathematicians of the 1830s and 1840s. Special attention is paid to William Frend's and Osborne Reynolds' criticism of symbolical algebra. The article ends with a brief discussion of reservations concerning symbolical algebra expressed by Augustus De Morgan, William Whewell, and Philip Kelland.     

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’.     

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.     

An algebra for piecewise-linear minimax problems

A piecewise-linear function whose definition involves the operator max and min may be reformulated as a ‘sum-of-partial-fractions’ by use of an algebraic structure $\text{J}$ and so may be ‘rationalized’ to become a ‘quotient-of-polynomials’ in the notation of $\text{J}$ We show that these ‘partial fractions’ and ‘polynomials’ have algebraic properties closely analogous to those of their counterparts in traditional elementary algebra: in particular an analogue of the fundamental theorem of algebra holds. These formal properties lead to straightforward procedures for finding maxima and minima of such functions.     

The art and the science of British algebra: A study in the perception of mathematical truth

This paper investigates the origins of the concept of mathematical truth by focusing on the development of algebra in England in the early 19th century. In particular, it investigates the reasons why the English, despite their attention to the elements of abstract algebra, never produced a system comparable to modern algebra. Special consideration is given to the works of George Peacock, Augustus DeMorgan, William Whewell, and John Herschel. It is argued that what separated the early development of English algebra from modern algebra is a fundamental difference between 19th- and 20th-century views of truth.     

Pupils’ attitudes towards mathematics: a comparative study of Norwegian and English secondary students

Comparing English and Norwegian pupils’ attitude towards mathematics, in this article I develop a deeper understanding of the factors that may shape and influence ‘pupil attitude towards mathematics’, and argue for it as a socio-cultural construct embedded in and shaped by students’ environment and context in which they learn mathematics. The theoretical framework leans on work by Zan and Di Martino (The Montana Mathematics Enthusiast, Monograph 3, pp. 157–168, 2007) to elicit Norwegian and English pupils’ attitude of mathematics as they experience it in their respective environments. Whilst there were differences which could be seen to be accounted for by differently ‘figured’ environments, there are also many similarities. It was interesting to see that, albeit based on a small statistical sample, in both countries students had a positive attitude towards mathematics in year 7/8, which dropped in year 9, and increased again in years 10/11. This result could be explained and compared with other larger scale studies (e.g. Hodgen et al. in Proceedings of the British Society for Research into Learning Mathematics. 29(3), 2009). The analysis of pupils’ qualitative comments (and classroom observations) suggested seven factors that appeared to influence pupil attitude most, and these had ‘superficial’ commonalities, but the perceptions that appeared to underpin these mentions were different, and could be linked to the environments of learning mathematics in their respective classrooms. In summary, it is claimed that it is not enough to identify the factors that may shape and influence pupil attitude, but more importantly, to study how these are ‘lived’ by pupils, what meanings are made in classrooms and in different contexts, and how the factors interrelate and can be understood.     

George Peacock and the British origins of symbolical algebra

This paper studies the background to and content of George Peacock's work on symbolical algebra. It argues that, in response to the problem of the negative numbers, Peacock, an inveterate reformer, elaborated a system of algebra which admitted essentially “arbitrary” symbols, signs, and laws. Although he recognized that the symbolical algebraist was free to assign somewhat arbitrarily the laws of symbolical algebra, Peacock himself did not exercise the freedom of algebra which he proclaimed. The paper ends with a discussion of Sir William Rowan Hamilton's criticism of symbolical algebra.     

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.     

From Lie algebra crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.     

Structural equivalence of individuals in social networks

The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations among relations. From a detailed social network, patterns of global relations can be extracted, within which classes of equivalently positioned individuals are delineated. The global patterns are derived algebraically through a ‘functorial’ mapping of the original pattern. Such a mapping (essentially a generalized homomorphism) allows systematically for concatenation of effects through the network. The notion of functorial mapping is of central importance in the ‘theory of categories,’ a branch of modern algebra with numerous applications to algebra, topology, logic. The paper contains analyses of two social networks, exemplifying this approach.     

A reform in undergraduate mathematics curriculum : more emphasis on social and pedagogical skills

The paper describes the changes that are being made in the mathematics teachers' subject studies in the Department of Mathematics at the University of Joensuu, in order to provide our mathematics students both with a sufficiently deep knowledge of mathematics and science, and with present-day expertise in their profession as teachers. While the formal structure of the mathematics curriculum remains structured and taught as courses with mostly traditional names like algebra, analysis, and linear algebra, there are also totally new ‘professionally oriented’ courses. Some of the old courses—with rather traditional and rigorous contents—have been changed in a more student-driven direction. In these ‘pedagogically oriented’ courses students are encouraged, and even forced, to study co-operatively in social interaction, for example to negotiate how to solve a problem decently, or how to build a formal definition for a concept with certain wanted attributes. As an ultimate example of a pedagogical experiment we describe in more detail an abstract algebra course, where co-operative learning is combined with intensive programming in a mathematically oriented computer environment.     

Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modelling

Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. This note illustrates the concept of dimensional analysis by examining the simple pendulum problem and shows how dimensionless products can lead to the discovery of the connection between the period of the pendulum swing and its length. Dimensional analysis is shown to lead to interesting systems of linear equations to solve, and can point the way to more quantitative analysis, and two student problems are discussed. It is the authors' experience that dimensional analysis broadens a student's viewpoint to include units and dimensions as an integral part of any physical problem. With this approach coupled with a computer algebra systems such as DERIVE, students can concentrate on understanding the model and the modelling process rather than the solution technique. Finally, it has been observed that students find dimensional analysis fun to do.     

The main sources for the Arte Mayor in sixteenth century Spain

One of the main changes in European Renaissance mathematics was the progressive development of algebra from practical arithmetic, in which equations and operations began to be written with abbreviations and symbols, rather than in the rhetorical way found in earlier arithmetical texts. In Spain, the introduction of algebraic procedures was mainly achieved through certain commercial or arithmetical texts, in which a section was devoted to algebra or the ‘Arte Mayor’. This paper deals with the contents of the first arithmetical texts containing sections on algebra. These allow us to determine how algebraic ideas were introduced into Spain and what their main sources were. The first printed arithmetical Spanish text containing algebra was the Libro primero de Arithmetica Algebratica (1552) by Marco Aurel. Therefore, the aim of this paper is to analyse the possible sources of this book and show the major influence of the German text Coss (1525) by Christoff Rudolff, on Aurel's work.     

On Schur’s theorem and its converses for n-Lie algebras

An n-Lie algebra analogue of Schur’s theorem and its converse as well as a Lie algebra analogue of Baer’s theorem and its converse are presented. Also, it is shown that, an n-Lie algebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to some finite dimensional n-Lie algebra.     

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.     

Mickelsson algebras and Zhelobenko operators

We construct a family of automorphisms of Mickelsson algebra, satisfying braid group relations. The construction uses ‘Zhelobenko cocycle’ and includes the dynamical Weyl group action as a particular case.