Making up Numbers

"Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research. The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis. Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms. Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject.

Some Extensions to Classic Lotka-Volterra Modeling For Predator Prey Applications

In this paper we present some specific cases of the classic Nonlinear Lotka-Volterra (NLV) approach to modeling predator-prey dynamic systems [1,5], and propose to implement them using "mathematical" (Matlab) approach as well as "ad-hoc" approach using Agent Based Modeling (implemented using NetLogo modeling environment), [6].   Examples of various scenarios are introduced in a gradual way, from simpler to more complex ones. The emphasis is given to gaining insight into predator-prey relationship, as well as some structural results [2,3] as applied to classic complex systems modeling and control, as well as understanding stability in multispecies communities. The paper sets the scene for further research using NLV (mathematical) and ABM (ad-hoc) models. With this "parallel" approach we hope to address some classic problems such as Gause's Law and Paradox of the Plankton,  Paradox of Enrichment (system level instability), Oksanen's description and trophic level numbers,  and other current Complex Systems paradigms such as adaptivity, emergence, etc.

Making up Numbers

"Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research. The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis. Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms. Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject.

Jewish Revenge: Haredi Action in the Zionist Sphere

This is a pre-copyedited version of an article accepted for publication in (journal title, volume and issue numbers, and year) following peer review. The definitive publisher-authenticated version is available from Wayne State University Press.Jewish ultra-Orthodox (Haredi) cinema in Israel has become increasingly prominent in recent years. Emerging as a highly controversial, secluded, and gender-segregated form of “amateur cinema,” it is currently seeing gradual professionalization. This article discusses Haredi cinema in the context of the Haredi community’s relationship with the Israeli state and the doctrine of Zionism. Appropriating generic conventions of mainstream Hollywood cinema, yet keeping within the secluded Haredi space, this form of minority cinema functions as an alternative (virtual) sphere in which a complex set of negotiations occurs between Jewish ultra-Orthodox ideals and those of the surrounding Israeli society and Zionism. It is reflective of and engaged in the production of recent social and discursive transformations within the Haredi community in Israel. We examine this phenomenon through a focused analysis of the male action genre, specifically the popular series Jewish Revenge (Yehuda Grovais, 2000–2010). As we demonstrate, the mode of representation and the narratives of these films bring models of masculinities and notions of heroism under scrutiny. The Zionist narrative, the national body, and the (imaginary) place of the Haredi within it are being reconfigured through the prism of body politics and fantasies of transgression

The co-evolution of number concepts and counting words

Humans possess a number concept that differs from its predecessors in animal cognition in two crucial respects: (1) it is based on a numerical sequence whose elements are not confined to quantitative contexts, but can indicate cardinal/quantitative as well as ordinal and even nominal properties of empirical objects (e.g. ‘five buses’: cardinal; ‘the fifth bus’: ordinal; ‘the #5 bus’: nominal), and (2) it can involve recursion and, via recursion, discrete infinity. In contrast to that, the predecessors of numerical cognition that we find in animals and human infants rely on finite and iconic representations that are limited to cardinality and do not support a unified concept of number. In this paper, I argue that the way such a unified number concept could evolve in humans is via verbal sequences that are employed as numerical tools, that is, sequences of words whose elements are associated with empirical objects in number assignments. In particular, I show that a certain kind of number words, namely the counting sequences of natural languages, can be characterised as a central instance of verbal numerical tools. I describe a possible scenario for the emergence of such verbal numerical tools in human history that starts from iconic roots and that suggests that in a process of co-evolution, the gradual emergence of counting sequences and the development of an increasingly comprehensive number concept supported each other. On this account, it is language that opened the way for numerical cognition, suggesting that it is no accident that the same species that possesses the language faculty as a unique trait, should also be the one that developed a systematic concept of number