Brouwer and Euclid

We explore the relationship between Brouwer’s intuitionistic mathematics and Euclidean geometry. Brouwer wrote a paper in 1949 called The contradictority of elementary geometry. In that paper, he showed that a certain classical consequence of the parallel postulate implies Markov’s principle, which he found intuitionistically unacceptable. But Euclid’s geometry, having served as a beacon of clear and correct reasoning for two millennia, is not so easily discarded.Brouwer started from a “theorem” that is not in Euclid, and requires Markov’s principle for its proof. That means that Brouwer’s paper did not address the question whether Euclid’s Elements really requires Markov’s principle. In this paper we show that there is a coherent theory of “non-Markovian Euclidean geometry”. We show in some detail that our theory is an adequate formal rendering of (at least) Euclid’s Book I, and suffices to define geometric arithmetic, thus refining the author’s previous investigations (which include Markov’s principle as an axiom).Philosophically, Brouwer’s proof that his version of the parallel postulate implies Markov’s principle could be read just as well as geometric evidence for the truth of Markov’s principle, if one thinks the geometrical “intersection theorem” with which Brouwer started is geometrically evident.     

Euclid and the fundamental theorem of arithmetic

Slight changes or benevolent interpretations of certain theorems and proofs in Euclid's Elements make his demonstration of the fundamental theorem of arithmetic satisfactory for square-free numbers, but Euclid's methods cannot be adapted to prove the uniqueness for numbers containing square factors.     

The debate between Peletier and Clavius on superposition

This article considers the sixteenth-century debate between Jacques Peletier du Mans and Christoph Clavius over the admissibility of superposition as a means to demonstrate the equality of figures in Euclidean geometry. It notably aims to determine, in the first part, which understanding of superposition motivated its rejection by Peletier, especially whether and to which extent his critical position towards this method was related to its kinematic implications. In the second part, the article presents the critical response Clavius addressed to Peletier in order to defend the legitimacy of superposition in Euclid's Elements.     

Johann Heinrich Lambert,mathematician and scientist, 1728 – 1777

1977 is the two hundredth anniversary of the death of Johann Heinrich Lambert, a little known but nonetheless intriguing figure in 18th century science. In the general histories of science and mathematics Lambert's contributions are often described piecemeal, with each discovery and invention usually divorced both from the method by which he arrived at it and from the totality of his intellectual endeavour. To the student of optics he is remembered for his cosine law in photometry, to the astronomer for his work on comets, to the meteorologist for his design of a gut hygrometer, and to the mathematician for his work on non-Euclidean geometry and his demonstration of the irrationality of π and e. There is no doubt that each of these contributions had a definite importance of its own; but it is not the aim of the present article to enumerate in this way the high points of Lambert's scientific and mathematical work, rather to describe it for once as a unified whole, and to relate it to the contemporary intellectual outlook.     

Felix Klein's “Erlanger Antrittsrede”: A Transcription with English Translation and Commentary

One of Felix Klein's leading interests was the role of mathematics education not only in the German universities but in the secondary schools as well. Klein played a leading role in the educational reform movements that flourished during the twenty-year period prior to World War I, and in 1908 he was elected President of the International Mathematics Instruction Commission. The “Erlanger Antrittsrede” of 1872, presented herein, gives a clear expression of Klein's views on mathematics education at the very beginning of his career. While previous writers, including Klein himself, have stressed the continuity between the Antrittsrede and his later views on mathematics education, the following commentary presents an analysis of the text together with external evidence supporting exactly the opposite conclusion.     

Torsten Brodén's work on the foundations of Euclidean geometry

The Swedish mathematician Torsten Brodén (1857–1931) wrote two articles on the foundations of Euclidean geometry. The first was published in 1890, almost a decade before Hilbert's first attempt, and the second was published in 1912. Brodén's philosophical view of the nature of geometry is discussed and his thoughts on axiomatic systems are described. His axiomatic system for Euclidean geometry from 1890 is considered in detail and compared with his later work on the foundations of geometry. The two continuity axioms given are compared to and proved to imply Hilbert's two continuity axioms of 1903.     

An outline of the life and work of E. B. Christoffel (1829–1900)

This article begins with an account of the life of Christoffel and of the intellectual milieu in which he worked, namely, the Institutes of Technology in Zürich and Berlin, and the University of Strasbourg. A survey of Christoffel's work follows. His major contributions lie in diverse areas: (i) geometry, including the foundations of tensor analysis; (ii) function theory, including conformal mappings, abelian integrals, and theta functions; (iii) numerical analysis; (iv) orthogonal polynomials and continued fractions; (v) ordinary and partial differential equations, potential theory; (vi) theory of shock waves; and (vii) dispersion of light. Finally, an attempt is made to assess Christoffel's work and its subsequent influence upon the development of modern mathematics, physics, and mechanics.     

On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying.     

Albertus magnus and mathematics: A translation with annotations of those portions of the commentary on Euclid's Elements published by Bernhard Geyer

This article consists of an English translation, with mathematical and philosophical notes, of three sections of the commentary by Albertus Magnus on Euclid's Elements: (1) the Prologue, (2) the question “Is an angle a quantity?” and (3) Book I, Proposition 11. The critical apparatus which Bernhard Geyer provided for (1) and (2) is also translated into English and updated by references to the new Cologne edition of Albert's Opera Omnia.     

A thirteenth century ‘proof’ of the parallel postulate

Proposition I.14 of Witelo's Perspectiva purports to provide a proof of the claim contained in Euclid's fifth postulate. The Latin text of the proposition is presented and translated into English; a commentary on the nature of the ‘proof’ is also provided.     

Cayley's anticipation of a generalised Cayley-Hamilton theorem

The principal result of Cayley's famouus memoir on matrices of 1858 is his contribution to what is now known as ‘the Cayley-Hamilton theorem’. We discuss this theorem and show that prior to its publication Cayley was aware of a more general theorem, a result that he left unpublished. This theorem is associated with the binary algebraic form det (μP ? λQ) analogous to the standard characteristic polynomial det (A ? λI).     

Combinatorial analysis of proofs in projective and affine geometry

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates (i.e., solves the word problem). This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry.     

Personae at play. ‘Men of Mathematics’ in commentary

Mathematical writers, above all, Euclid, tend to present their theorems as decontextualized, abstract propositions, which has become the standard modus of textual presentation in theoretical mathematics. Mathematical commentators, however, provide their readers with personal names and historical facts in order to elucidate problems, provide contexts of discovery, or construct doxographies, among other things. Modern readers have used such information for the construction of histories of science. When we look at these passages, however, we see that personal names and information about mathematicians of the past can serve quite a range of different objectives, such as the strategic self-positioning of the commentator vis-à-vis the past or present of mathematics, the education of the reader, mathematical or moral, the construction of the history of the field, etc. Not only does the commentator present a persona of himself to the reader, he can also turn colleagues and predecessors into personae. This paper attempts to elucidate the practice, by offering four examples of such plays of and with personae, in Pappus, Eutocius, al-Nayrīzī, and Proclus.     

Some generalized type functional dependencies formalized as equality set on matrices

In this paper we deal with functional (f) dependencies and their generalizations, the dual, strong (d, s) and weak (w) dependencies. We give new axioms instead of Armstrong's and Czédli's and axiomatize the full w-families. Our axioms are based on a characterization of equality-sets of matrices. We demonstrate an essential difference between the weak dependency and the rest. We give an estimation for the number of rows which is needed for some antichain of an n-element set to represent it as the set of candidate keys in a relation. Finally, we give a combinatorial characterization of the generating sets with minimal cardinality of full f-families.