Aunt Alice in Wonderland

George and Mary Everest Boole’s daughter Alicia learnt Hinton’s methods for visualizing hypercubes during her teens and developed an incredible facility for four-dimensional geometry. As a housewife and without any formal training in mathematics she discovered all six regular four-dimensional polytopes (the four-dimensional Platonic solids) and made models of their three-dimensional sections. The regular polytopes had previously been discovered by Ludwig Schläfli and William Irving Stringham, but their work had received little attention and was probably unknown to Alicia Stott Boole. Alicia later worked for a time with the Dutch mathematician Pieter Hendrik Schoute. When she was in her 70s, her nephew, later Professor Geoffrey Ingram Taylor, introduced her to a young Cambridge student called Donald Coxeter. Coxeter developed her ideas further, and in 1948 published the first edition of his book Regular Polytopes. Coxeter is now recognized as the greatest classical geometer of the twentieth century.

Algol Genes

In 1969 a “Report on the Algorithmic Language ALGOL 68” was published in the journal Numerische Mathematik. The authors of the report were also its designers, all academic computer scientists, Adriaan van Wijngaarden and C. H. A. Koster from the Netherlands and Barry Mailloux and John Peck from Canada. The Algol 68 project was, by then, 4 years old. The International Federation for Information Processing (IFIP) had under its umbrella a number of technical committees devoted to various specialties; each technical committee in turn had, under its jurisdiction, several working groups given to subspecialties. One such committee was the technical committee TC2, on programming; and in 1965 one of its constituent working groups WG2.1 (programming languages) mandated the development of a new international language as a successor to Algol 60. The latter, developed by an international committee of computer scientists between 1958 and 1963, had had considerable theoretical and practical impact in the first age of computer science. The Dutch mathematician-turned-computer scientist Adriaan van Wijngaarden, one of the codesigners of Algol 60 was entrusted with heading this task. The goal for Algol 68 was that it was to be a successor of Algol 60 and that it would have to be accepted and approved by IFIP as the “official” international programming language. Prior to its publication in 1969, the language went through a thorough process of review, first within the ranks of WG2.1, then by its umbrella body TC2, and finally by the IFIP General Assembly before being officially recommended for publication. The words review and recommendation mask the fact that the Algol 68 project manifested some of the features of the legislative process with its attendant politics. Thus, at a meeting of WG2.1 in Munich in December 1968— described by one of the Algol 68 codesigners John Peck as “dramatic”— where the Algol 68 report was to be approved by the working group, the designers presented their language proposal much as a lawmaker presents a bill to a legislative body; and just as the latter debates over the bill, oftentimes acrimoniously, before putting the bill to a vote, so also the Algol 68 proposal was debated over by members of WG2.1 and was finally voted on.

Plerosis and Atomic Gestalts

SummaryFranz Brentano, 1838–1917, introduced the intriguing concept of “plerosis” in order to account for aspects of the continuum that were “explained” by formal mathematics in ways that he considered absurd from the perspective of intuition, especially visual awareness and imagery. In doing this, he pointed in directions later developed by the Dutch mathematician Luitzen Brouwer. Brentano’s notion of plerosis involves distinct though coincident points, which one might call “atomic entities with parts”. This notion fits the modern concepts of “receptive field” in neurophysiology, “perceptive field” in psychology and “differential operator” in the formal theory of scale space. We identify Brentano’s boundary points as the primordial atomic Gestalts of visual imagery. The concept deserves to play a key role in Gestalt theory.

Huygens' headache

Background Christiaan Huygens (1629–1695) was a Dutch mathematician, physicist, and astronomer. He became well-known as inventor of the pendulum clock and described light as a wave phenomenon. He became Fellow of the Royal Society (London) and member of the Académie des Sciences (Paris). From the correspondence with family members and famous scientists, we learn that he suffered from frequent headaches. Aim To study Huygens' 22-volume Oeuvres Complètes (1888–1950) to find letters in which his headaches are mentioned and translate pertinent sections into English. Conclusions Although a posthumous diagnosis of Huygens' headaches is somewhat hazardous, the recurrent episodes with incapacitating headache and family history over two generations are suggestive for migraine. It becomes clear that it impeded his writing, reading, and research. From the letters we get an impression of the impact of the headache upon his life and the treatments that were applied in the 17th century.

Intuitionism vs. Classicism

In the early twentieth century, the Dutch mathematician L.E.J. Brouwer launched a powerful attack on the prevailing mathematical methods and theories. He developed a new kind of constructive mathematics, called intuitionism, which seems to allow for a rigorous refutation of widely accepted mathematical assumptions including fundamental principles of classical logic. Following an intense mathematical debate esp. in the 1920s, Brouwer's revolutionary criticism became a central philosophical concern in the 1970s, when Michael Dummett tried to substantiate it with meaning-theoretic considerations. Since that time, the debate between intuitionists and classicists has remained a central philosophical dispute with far-reaching implications for mathematics, logic, epistemology, and semantics. In this book, Nick Haverkamp presents a detailed analysis of the intuitionistic criticism of classical logic and mathematics. The common assumption that intuitionism and classicism are equally legitimate enterprises corresponding to different understandings of logical or mathematical expressions is investigated and rejected, and the major intuitionistic arguments against classical logic are scrutinised and repudiated. Haverkamp argues that the disagreement between intuitionism and classicism is a fundamental logical and mathematical dispute which cannot be resolved by means of meta-mathematical, epistemological, or semantic considerations.

The gaps between sums of two squares

Problems concerning the setof numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identitywhich shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.