Type Classes for Mathematics in Type Theory

The introduction of first-class type classes in the Coq system calls for re-examination of the basic interfaces used for mathematical formalization in type theory. We present a new set of type classes for mathematics and take full advantage of their unique features to make practical a particularly flexible approach formerly thought infeasible. Thus, we address both traditional proof engineering challenges as well as new ones resulting from our ambition to build upon this development a library of constructive analysis in which abstraction penalties inhibiting efficient computation are reduced to a minimum. The base of our development consists of type classes representing a standard algebraic hierarchy, as well as portions of category theory and universal algebra. On this foundation we build a set of mathematically sound abstract interfaces for different kinds of numbers, succinctly expressed using categorical language and universal algebra constructions. Strategic use of type classes lets us support these high-level theory-friendly definitions while still enabling efficient implementations unhindered by gratuitous indirection, conversion or projection. Algebra thrives on the interplay between syntax and semantics. The Prolog-like abilities of type class instance resolution allow us to conveniently define a quote function, thus facilitating the use of reflective techniques

Formalising Mathematics in Simple Type Theory

Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL. Harrison's technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic type classes. However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism

Classes, why and how

This paper presents a new approach to the class-theoretic paradoxes. In the first part of the paper, I will distinguish classes from sets, describe the function of class talk, and present several reasons for postulating type- free classes. This involves applications to the problem of unrestricted quantification, reduction of properties, natural language semantics, and the epistemology of mathematics. In the second part of the paper, I will present some axioms for type-free classes. My approach is loosely based on the Gödel-Russell idea of limited ranges of significance. It is shown how to derive the second-order Dedekind-Peano axioms within that theory. I conclude by discussing whether the theory can be used as a solution to the problem of unrestricted quantification. In an appendix, I prove the consistency of the class theory relative to Zermelo-Fraenkel set theory

Mathematical specialized knowledge of a mathematics teacher educator for teaching divisibility

The knowledge of Mathematics teachers has been a very prominent focus of attention in the last decades. However, it leaves aside one of the dimensions involved in the development of this type of knowledge, specifically the knowledge of Mathematics teacher educators. In this paper, we discuss a mathematics teacher educator’s knowledge in the context of classes on Euclid’s division algorithm theorem in a Number Theory course for prospective secondary teachers. Some indicators of this specialized knowledge of mathematics teacher educators are presented and discussed

Church's thesis and related axioms in Coq's type theory

"Church's thesis" ($\mathsf{CT}$) as an axiom in constructive logic states that every total function of type $\mathbb{N} \to \mathbb{N}$ is computable, i.e. definable in a model of computation. $\mathsf{CT}$ is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak K\"onig's Lemma and the fan theorem, respectively. Recently, $\mathsf{CT}$ was proved consistent for (univalent) constructive type theory. Since neither Weak K\"onig's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that $\mathsf{CT}$ is inconsistent only with a combination of classical logic and choice axioms. We study consequences of $\mathsf{CT}$ and its relation to several classes of axioms in Coq's type theory, a constructive type theory with a universe of propositions which does neither prove classical logical axioms nor strong choice axioms. We thereby provide a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant

Differential Teachers’ Attention to Boys and Girls in Mathematics Whole Class Interaction Sequences

The relationship of gender to the quality and type of interaction in the mathematics classroom is a question that has concerned researchers for some time. Past studies have indicated that patterns of interaction can be gender-specific, despite individual teachers regarding themselves as teaching in a gender-neutral manner. Drawing on interaction theory and socialization theory, this quantitative observational study tracked the number and type of interactions in 38 high school Mathematics classes across three schools, involving 899 students and 26 teachers (12 male, 14 female). The study investigated whether there were significant gender differences in the treatment of students during their engagement in different types of questioning behavior. It found that there were significant gender differences at student level that favored boys, but that questioning behaviors amongst individual teachers were highly variable. Teachers’ gender did not explain the differences. It is recommended that gender awareness be incorporated into teacher training

An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space

2010 Mathematics Subject Classification: 53A07, 53A35, 53A10.The present article is a survey of some of our recent results on the theory of two-dimensional surfaces in the four-dimensional Euclidean or Minkowski space. We present our approach to the theory of surfaces in Euclidean or Minkowski 4-space, which is based on the introduction of an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This invariant map allows us to introduce principal lines and an invariant moving frame field at each point of the surface. Writing derivative formulas of Frenet-type for this frame field, we obtain a system of invariant functions, which determine the surface up to a motion. We formulate the fundamental theorems for the general classes of surfaces in Euclidean or Minkowski 4-space in terms of the invariant functions. We show that the basic geometric classes of surfaces, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix and the normal curvature ellipse. We apply our theory to some special classes of surfaces in Euclidean or Minkowski 4-space

Lectures on Duflo isomorphisms in Lie algebra and complex geometry

International audienceDuflo isomorphism first appeared in Lie theory and representation theory. It is an isomorphism between invariant polynomials of a Lie algebra and the center of its universal enveloping algebra, generalizing the pioneering work of Harish-Chandra on semi-simple Lie algebras. Later on, Duflo’s result was refound by Kontsevich in the framework of deformation quantization, who also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. The present book, which arose from a series of lectures by the first author at ETH, derives these two isomorphisms from a Duflo-type result for Q-manifolds.All notions mentioned above are introduced and explained in the book, the only prerequisites being basic linear algebra and differential geometry. In addition to standard notions such as Lie (super)algebras, complex manifolds, Hochschild and Chevalley–Eilenberg cohomologies, spectral sequences, Atiyah and Todd classes, the graphical calculus introduced by Kontsevich in his seminal work on deformation quantization is addressed in details.The book is well-suited for graduate students in mathematics and mathematical physics as well as for researchers working in Lie theory, algebraic geometry and deformation theory

Chern class identities from tadpole matching in type IIB and F-theory

In light of Sen's weak coupling limit of F-theory as a type IIB orientifold, the compatibility of the tadpole conditions leads to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. We present the physical argument leading to the identity, and a mathematical derivation of a Chern class identity which confirms it, after taking into account singularities of the relevant loci. This identity of Chern classes holds in arbitrary dimension, and for varieties that are not necessarily Calabi-Yau. Singularities are essential in both the physics and the mathematics arguments: the tadpole relation may be interpreted as an identity involving stringy invariants of a singular hypersurface, and corrections for the presence of pinch-points. The mathematical discussion is streamlined by the use of Chern-Schwartz-MacPherson classes of singular varieties. We also show how the main identity may be obtained by applying `Verdier specialization' to suitable constructible functions.Comment: 26 pages, 1 figure, references added, typos correcte