A contemporary analysis of the six “Theories of Mathematics Education” theses of Hans-Georg Steiner

In this contribution we discuss the six theses presented by Hans-Georg Steiner (1987), which were instrumental in the community becoming interested in theories and philosophies of mathematics education. We discuss overlooked aspects of this seminal paper particularly in light of recent developments in the field of mathematics education. Nearly 20 years later, we reflect on the development of Steiner’s program for theory development and examine if any progress has been made at all on the open questions that Steiner (1987) posed to the community.     

Hans-Georg Steiner: a life dedicated to the development of didactics of mathematics as a scientific discipline

In our introductory paper to this special issue we follow two goals. First of all, we take on the challenge to give an account of more than 40 years of academic work by one of the leading members of our discipline by looking at Hans Georg Steiner’s contributions to the development of didactics of mathematics as a scientific discipline in Germany as well as internationally. Therefore, we try to highlight major research interests, publications and conferences during his early years in Münster, Karlsruhe and Bayreuth as well as during the 20 years at the IDM in Bielefeld. Closely linked to these periods of this life and work are specific research interests, professional contacts and friendships. Hence, the second goal of our paper is to emphasise Hans-Georg Steiner’s relationships with national and international colleagues (many of whom became friends) and their shared interests and collaborations in the development of mathematics education through a selection of invited papers that address different stages and professional foci in the life of Hans-Georg Steiner. These papers are organised in four sections: (1) Revisiting the New Math reform, (2) Developing specific research domains in didactics of mathematics, (3) Discussing theories of mathematics education (TME), and (4) Reflecting on goals and results of mathematics education.     

Towards theoretical foundations of mathematics education

Although research in mathematics education developed in the last decades as a vivid scientific field, the nature and the place of theories in the field are still under discussion. H.-G. Steiner contributed to this debate in several ways. He not only intervened in the scientific debate but also played an active role in organizing the discipline and the scientific discussion at the international level. This paper attempts to give a synthetic view of the evolution of mathematics education with respect to theory by focusing on the French situation of research in mathematics education and paying a particular attention to the role and contribution of H.-G. Steiner.     

Scalar products of dirichlet series and the distribution of integer points on toric varieties

The scalar product of Hecke series with Größencharakteren, introduced by Yu. V. Linnik, and its generalizations are studied. Analytic properties of the functions under consideration are closely related to the distribution of integer points on toric varieties. Bibliography: 22 titles.     

Frege on Identity as a Relation of Names

This essay offers a detailed philosophical criticism of Frege’s popular thesis that identity is a relation of names. I consider Frege’s position as articulated both in ‘On Sense and Reference’, and in the Grundgesetze, where he appears to take an objectual view of identity, arguing that in both cases Frege is clearly committed to the proposition that identity is a relation holding between names, on the grounds that two different things can never be identical. A counterexample to Frege’s thesis is considered, and a positive thesis is developed according to which, in contradistinction to the Fregean position, identity is a reflexive, symmetric, and transitive relation holding only between a thing and itself which can be expressed as a relation between names.     

A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann

This article provides a detailed discussion of Otto Hölder's ideas on the foundations of mathematics. It focuses on a paper he published in 1892 written in reaction to a book published in 1891 by Robert Graßmann, which Hölder saw as an attempt to axiomatize arithmetic. Hölder's paper is important for at least three reasons: First, it represents what might be called Hölder's research manifesto on the foundations of mathematics, containing a wealth of ideas which Hölder gradually developed in a variety of publications until the end of his life. Second, Hölder's analysis of R. Graßmann's foundational ideas provides an important assessment of the contribution of Hermann and Robert Graßmann to the axiomatization of arithmetic, a contribution which, though often mentioned, is itself still not widely acknowledged and not fully understood. Third, the effort of exposing the weak spots in R. Graßmann's ideas led Hölder to formulate the main problems confronting formal axiomatics: independence of the axioms, consistency, completeness, and the issue of the relationship between pure mathematics and its applications. The first part of this paper presents R. Graßmann's ideas on the foundations of mathematics as outlined in two closely related works published in 1872 and 1891. The second part focuses on Hölder's analysis of these ideas.     

Frege,Dedekind, and the Modern Epistemology of Arithmetic

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.     

A generalization of the Pólya–Szegö inequality for one-dimensional functionals

We consider the Pólya–Szegö type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner’s symmetrization.     

Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Hölder–McCarthy’s inequality for positive operators and Ky Fan’s inequality for real numbers are given as well.     

A note on Gröbner bases

In this paper we introduce the new notion of co-polynomials as polynomials arising from the Graßmann–Plücker polynomials. Pairs of co-polynomials are shown to be critical in the computation of a Gröbner basis for the chirotope ideal.     

Human Rationality Challenges Universal Logic

Tarski’s conceptual analysis of the notion of logical consequence is one of the pinnacles of the process of defining the metamathematical foundations of mathematics in the tradition of his predecessors Euclid, Frege, Russell and Hilbert, and his contemporaries Carnap, Gödel, Gentzen and Turing. However, he also notes that in defining the concept of consequence “efforts were made to adhere to the common usage of the language of every day life.” This paper addresses the issue of what relationship Tarski’s analysis, and Béziau’s further generalization of it in universal logic, have to reasoning in the everyday lives of ordinary people from the cognitive processes of children through to those of specialists in the empirical and deductive sciences. It surveys a selection of relevant research in a range of disciplines providing theoretical and empirical studies of human reasoning, discusses the value of adopting a universal logic perspective, answers the questions posed in the call for this special issue, and suggests some specific research challenges.     

Parallel algorithms for Gröbner-basis construction

The problem of the Gröbner-basis construction is important both from the theoretical and applied points of view. As examples of applications of Gröbner bases, one can mention the consistency problem for systems of nonlinear algebraic equations and the determination of the number of solutions to a system of nonlinear algebraic equations. The Gröbner bases are actively used in the constructive theory of polynomial ideals and at the preliminary stage of numerical solution of systems of nonlinear algebraic equations. Unfortunately, many real examples cannot be processed due to the high computational complexity of known algorithms for computing the Gröbner bases. However, the efficiency of the standard basis construction can be significantly increased in practice. In this paper, we analyze the known algorithms for constructing the standard bases and consider some methods for increasing their efficiency. We describe a technique for estimating the efficiency of paralleling the algorithms and present some estimates.     

Optimising Gröbner Bases on Bivium

Bivium is a reduced version of the stream cipher Trivium. In this paper we investigate how fast a key recovery attack on Bivium using Gröbner bases is. First we explain the attack scenario and the cryptographic background. Then we identify the factors that have impact on the computation time and show how to optimise them. As a side effect these experiments benchmark several Gröbner basis implementations. The optimised version of the Gröbner attack has an expected running time of 2^39.12 s, beating the attack time of our previous SAT solver attack by a factor of more than 330. Furthermore this approach is faster than an attack based on BDDs, an exhaustive key search, a generic time-memory trade-off attack and a guess-and-determine strategy.     

Kronecker’s density theorem and irrational numbers in constructive reverse mathematics

To prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an irrational number as a real number that is bounded away from each rational number. In fact, once one understands “irrational” merely as “not rational”, then the theorem becomes equivalent to Markov’s principle. To see this we undertake a systematic classification, in the vein of constructive reverse mathematics, of logical combinations of “rational” and “irrational” as predicates of real numbers.     

Jacobi's Criticism of Lagrange: The Changing Role of Mathematics in the Foundations of Classical Mechanics

C. G. J. Jacobi in hisLectures on Analytical Mechanics(Berlin, 1847–1848) gives a detailed and critical discussion of Lagrange's mechanics. Lagrange's view that mechanics could be pursued as an axiomatic-deductive science forms the center of Jacobi's criticism and is rejected on mathematical and philosophical grounds. In this paper, Jacobi's arguments are presented and analyzed. It is shown that Jacobi's criticism is motivated by a changed evaluation of the role of mathematics in the empirical sciences. This change is interpreted as a process of dissolution of Euclideanism (in the sense of Lakatos) that dominated theoretical mechanics up to Jacobi as the leading ideal of science.Copyright 1998 Academic Press.In seinenVorlesungen über analytische Mechanik(Berlin, 1847–1848) setzt sich C. G. J. Jacobi ausführlich und kritisch mit Lagranges Mechanik auseinander. Im Mittelpunkt der Kritik steht dabei Lagranges Auffassung, Mechanik könne als eine axiomatisch-deduktive Wissenschaft betrieben werden. Diese Auffassung wird von Jacobi aus mathematischen und philosophischen Gründen zurückgewiesen. Im vorliegenden Aufsatz werden Jacobis Argumente dargestellt und analysiert. Es wird gezeigt, daß seine Kritik auf einer veränderten Beurteilung der Rolle der Mathematik in den empirischen Wissenschaften beruht. Diese Veränderung wird als Prozeß der Auflösung des Euklidianismus (im Sinne von Lakatos) als leitendem Wissenschaftsideal der theoretischen Mechanik bis hin zu Jacobi interpretiert.Copyright 1998 Academic Press.Dans sonCours de Mécanique analytique(Berlin, 1847–1848), C. G. J. Jacobi présente une discussion détaillée et critique de la mécanique de Lagrange. La critique de Jacobi concerne avant tout la conception de la mécanique comme science axiomatique et déductive défendue par Lagrange. Il rejette cette conception pour des raisons mathématiques et philosophiques. Dans cet article seront présentés et analysés les différents arguments de Jacobi, démontrant que sa critique est fondée sur un changement d'appréciation de la fonction de la mathématique dans les sciences empiriques. La position nouvelle de Jacobi sera interprétée comme conséquence de l'élimination de l'Euclidianisme (dans le sens de Lakatos), le prototype scientifique de la mécanique rationnelle jusqu'à Jacobi.Copyright 1998 Academic Press.MSC 1991 subject classifications; 01A50, 01A55, 70–03, 70A05, 70D10     

Gröbner Bases for Ideals of σ-PBW Extensions

In this article, we introduce the σ-PWB extensions and construct the theory of Gröbner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincaré–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Gröbner bases of left ideals.     

Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

On monotonicity of some functionals under rearrangements

We consider the Pólya–Szegö type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner’s symmetrization. In particular we fill the gap in the paper of Brock (Calc Var PDEs 8:15–25, 1999) for 1D case.     

Parametrization of extremals of Grötzsch’s problem

We give parametric expressions for extremals of the Grötzsch’s problem.