Mathematics

We provide a faithful translation of Hans Richter's important 1948 paper "Das isotrope Elastizitätsgesetz" from its original German version into English. Our introduction summarizes Richter's achievements.

We give an uncountability proof of the reals which relies on their order completeness instead of their sequential completeness. We use neither a form of the axiom of choice nor the law of excluded middle, therefore the proof applies to the MacNeille reals in any flavor of constructive mathematics. The proof leans heavily on Levy's unusual proof of the uncountability of the reals.

We correct a common (but mistaken) attribution of the evaluation of the probability integral, usually attributed to Poisson, Gauss, or Laplace.

The International Congress of Mathematicians (ICM), inaugurated in 1897, is the greatest effort of the mathematical community to strengthen international communication and connections across all mathematical fields. Meetings of the ICM have historically hosted some of the most prominent mathematicians of their time. Receiving an invitation to present a talk at an ICM signals the high international reputation of the recipient, and is akin to entering a `hall of fame for mathematics'. Women mathematicians attended the ICMs from the start. With the invitation of Laura Pisati to present a lecture in 1908 in Rome and the plenary talk of Emmy Noether in 1932 in Zurich, they entered the grand international stage of their field. At the congress in 2014 in Seoul, Maryam Mirzakhani became the first woman to be awarded the Fields Medal, the most prestigious award in mathematics. In this article, we dive into assorted data sources to follow the footprints of women among the ICM invited speakers, analyzing their demographics and topic distributions, and providing glimpses into their diverse biographies.

In La Géométrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation. Descartes' geometric foundational approach (analysis without infinitary objects and synthesis with diagrammatic constructions) has been extended beyond the limits of algebraic polynomials in two different periods: by late 17th century tractional motion and by early 20th century differential algebra. This paper proves that, adopting these extensions, it is possible to define a new convergence of machines (analog computation), algebra (symbolic manipulations) and a well determined class of mathematical objects that gives scope for a constructive foundation of (a part of) infinitesimal calculus without the conceptual need of infinity. To establish this balance, a clear definition of the constructive limits of tractional motion is provided by a differential universality theorem.

Taking up the challenge McConnell laid down at the end of his proof of the law of cosines, we give a completely visual dissection proof of this theorem, which applies to any triangle. In order to avoid the trigonometric expressions of Cuoco-McConnell's proof, we replaced the equal-area rectangles with congruent triangles. As a matter of fact, trigonometric expressions are implicitly based on the similarity of two right triangles with a common non-right angle. So they are conceptually less simple than our congruent triangles which are, moreover, easy to visualize. This makes our proof the only dissection proof and the simplest proof of its family, and thus one of the best options for a course of geometry.

We examine the preparation and context of the paper "The Crisis in Contemporary Mathematics" by Errett Bishop, published 1975 in Historia Mathematica. Bishop tried to moderate the differences between Hilbert and Brouwer with respect to the interpretation of logical connectives and quantifiers. He also commented on Robinson's Non-standard Analysis, fearing that it might lead to what he referred to as 'a debasement of meaning.' The 'debasement' comment can already be found in a draft version of Bishop's lecture, but not in the audio file of the actual lecture of 1974. We elucidate the context of the 'debasement' comment and its relation to Bishop's position vis-a-vis the Law of Excluded Middle. Keywords: Constructive mathematics; Robinson's framework; infinitesimal analysis.

We generalize Pappus chain theorem and give an analogue to this theorem.

As a generalization of planar Fibonacci spirals that are based on the recurrence relation F n = F n−1 + F n−2 , we draw assembled spirals stemming from analytic solutions of the recurrence relation G n =a G n−1 +b G n−2 +cd n , with positive real initial values G 0 and G 1 and coefficients a , b , c , and d . The principal coordinates given in closed-form correspond to finite sums of alternating even- or alternating odd-indexed terms G n . For rectangular spirals made of straight line segments (a.k.a. spirangles), the even-indexed and the odd-indexed directional corner points asymptotically lie on mutually orthogonal oblique lines. We calculate the points of intersection and show them in the case of inwinding spirals to coincide with the point of convergence. In the case of outwinding spirals, an n -dependent quadruple of points of intersection may form. For arched spirals, interpolation between principal coordinates is performed by means of arcs of quarter-ellipses. A three-dimensional representation is exhibited, too. The continuation of the discrete sequence { G n } to the complex-valued function G(t) with real argument t ∈ R , exhibiting spiral graphs and oscillating curves in the Gaussian plane, subsumes the values G n for t ∈ N 0 as the zeros. Besides, we provide a matrix representation of G n in terms of transformed Horadam numbers, retrieve the Shannon product difference identity as applied to G n , and suggest a substitution method for finding a variety of other identities and summations related to G n .

We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by Möbius, using hyperbolic geometry.