Letters to Joel

A professor and his students answer his brother’s question: Is mathematics art

The Deconstruction of Mathematics: A criticism of Reuben Hersh\u27s What Is Mathematics, Really? and the Humanist Philosophy of Mathematics

Mathematics, as an academic discipline, has stood for many years as the last bastion against a growing tide of intellectual relativism that has become all but ubiquitous. More recently, however, efforts have been made to humanize mathematics by advocating a social-constructivist approach to the philosophy of mathematics, both in practice and education. This paper is intended to serve as a critical response to one advocate of this approach, Reuben Hersh (What Is Mathematics, Really?, 1997), and in the process a defense of Platonism

Mathematics As Worship

This paper treats worship in a broad, inclusive sense as the primary and necessary response of human beings to their creator. It provides a brief overview of the relationship between mathematics and theology from the Pythagoreans to the present. It argues that all knowledge is contingent upon faith and thus that contemplation of mathematical insights can lead to worship of God

The Hidden Injuries Of Overloading \u27ADT

The most commonly stated definition of abstract data type (ADT) is that it is a domain of values and the operations over that domain. So, for example, a language\u27s built-in types, like int are seen to be ADTs. It is our opinion that a pure interpretation of this definition yields a semantics in which using an ADT is the same as using built-in types: the operations are side effect free and there is no concern over alias, shallow copy or synchronization problems. Unfortunately, the term abstract data type has over time been associated with at least three distinct meanings, and those incompatible definitions have often been conflated, causing confusion to students and textbook authors alike. We believe that this has resulted in a loss of appreciation for the value-based semantics of ADTs

Reconciling Shannon and Scott with a Lattice of Computable Information

This paper proposes a reconciliation of two different theories of information. The first, originally proposed in a lesser-known work by Claude Shannon (some five years after the publication of his celebrated quantitative theory of communication), describes how the information content of channels can be described qualitatively, but still abstractly, in terms of information elements, where information elements can be viewed as equivalence relations over the data source domain. Shannon showed that these elements have a partial ordering, expressing when one information element is more informative than another, and that these partially ordered information elements form a complete lattice. In the context of security and information flow this structure has been independently rediscovered several times, and used as a foundation for understanding and reasoning about information flow. The second theory of information is Dana Scott\u27s domain theory, a mathematical framework for giving meaning to programs as continuous functions over a particular topology. Scott\u27s partial ordering also represents when one element is more informative than another, but in the sense of computational progress - i.e. when one element is a more defined or evolved version of another. To give a satisfactory account of information flow in computer programs it is necessary to consider both theories together, in order to understand not only what information is conveyed by a program (viewed as a channel, \ue0 la Shannon) but also how the precision with which that information can be observed is determined by the definedness of its encoding (\ue0 la Scott). To this end we show how these theories can be fruitfully combined, by defining the Lattice of Computable Information (LoCI), a lattice of preorders rather than equivalence relations. LoCI retains the rich lattice structure of Shannon\u27s theory, filters out elements that do not make computational sense, and refines the remaining information elements to reflect how Scott\u27s ordering captures possible varieties in the way that information is presented. We show how the new theory facilitates the first general definition of termination-insensitive information flow properties, a weakened form of information flow property commonly targeted by static program analyses

Book Review

A book review of Fashionable Nonsense: Postmodern Intellectuals\u27 Abuse of Science by Alan Sokal and Jean Bricmont

The Deconstruction of Mathematics

This paper is a criticism of Reuben Hersh\u27s What is Mathematics, Really? and the humanist philosophy of mathematics

Design Early Considered Harmful: Graduated Exposure to Complexity and Structure Based on Levels of Cognitive Development

We have recognized that the natural tendency to teach according to the structure of one’s own understanding runs contrary to established models of cognitive development. Bloom’s Taxonomy has provided a basis for establishing a more efficacious pedagogy. Emphasizing a hierarchical progression of skill sets and gradual learning through example, our approach advocates teaching software development from the inside/out rather than beginning with either console apps or monolithic designs

Magnetic Evolution and Temperature Variation in a Coronal Hole

We have explored the magnetic flux evolution and temperature variation in a coronal-hole region, using Big Bear Solar Observatory (BBSO) deep magnetograms and {\it SOHO}/EIT images observed from 2005 October 10 to 14. For comparison, we also investigated a neighboring quiet region of the Sun. The coronal hole evolved from its mature stage to its disappearance during the observing period. We have obtained the following results: (1) When the coronal hole was well developed on October 10, about 60 % of the magnetic flux was positive. The EUV brightness was 420 counts pixel$^{-1}$, and the coronal temperature, estimated from the line ratio of the EIT 195 {\AA} and 171 {\AA} images, was 1.07 MK. (2) On October 14, when the coronal hole had almost disappeared, 51 % of the magnetic flux was positive, the EUV radiance was 530 counts pixel$^{-1}$, and the temperature was 1.10 MK. (3) In the neighboring quiet region, the fraction of positive flux varied between 0.49 and 0.47. The EUV brightness displayed an irregular variation, with a mean value of 870 counts pixel$^{-1}$. The temperature was almost constant at 1.11 MK during the five-day observation. Our results demonstrate that in a coronal hole less imbalance of the magnetic flux in opposite polarities leads to stronger EUV brightness and higher coronal temperatures

Mycobacterium tuberculosis lineage 4 comprises globally distributed and geographically restricted sublineages

Generalist and specialist species differ in the breadth of their ecological niches. Little is known about the niche width of obligate human pathogens. Here we analyzed a global collection of Mycobacterium tuberculosis lineage 4 clinical isolates, the most geographically widespread cause of human tuberculosis. We show that lineage 4 comprises globally distributed and geographically restricted sublineages, suggesting a distinction between generalists and specialists. Population genomic analyses showed that, whereas the majority of human T cell epitopes were conserved in all sublineages, the proportion of variable epitopes was higher in generalists. Our data further support a European origin for the most common generalist sublineage. Hence, the global success of lineage 4 reflects distinct strategies adopted by different sublineages and the influence of human migration