A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint

Reciprocity as a foundation of financial economics

This paper argues that the subsistence of the fundamental theorem of contemporary financial mathematics is the ethical concept ‘reciprocity’. The argument is based on identifying an equivalence between the contemporary, and ostensibly ‘value neutral’, Fundamental Theory of Asset Pricing with theories of mathematical probability that emerged in the seventeenth century in the context of the ethical assessment of commercial contracts in a framework of Aristotelian ethics. This observation, the main claim of the paper, is justified on the basis of results from the Ultimatum Game and is analysed within a framework of Pragmatic philosophy. The analysis leads to the explanatory hypothesis that markets are centres of communicative action with reciprocity as a rule of discourse. The purpose of the paper is to reorientate financial economics to emphasise the objectives of cooperation and social cohesion and to this end, we offer specific policy advice

Pascal and Fermat: Religion, Probability, and Other Mathematical Discoveries

This final project primarily discusses how Blaise Pascal and Pierre de Fermat, two French seventeenth century mathematicians, founded the field of mathematical Probability and how this area continued to evolve after their contributions. Also included in this project is an analysis of how Pascal and Fermat were affected, or not, in their mathematical work by the widespread impact that the Catholic Church had on life in France during this time period. I further discuss two other central discoveries by these theorists: Pascal’s Triangle and Fermat’s Last Theorem. Lastly, the project analyzes how all of these aspects: the influence of the contemporary religion of the period on science and mathematics, Pascal’s discoveries, and Fermat’s different method of approach, impacted the trajectory of mathematical history

There are no multiply-perfect Fibonacci numbers

Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors

If Archimedes would have known functions

These are notes and slides from a Pecha-Kucha talk given on March 6, 2013. The presentation tinkered with the question whether calculus on graphs could have emerged by the time of Archimedes, if the concept of a function would have been available 2300 years ago. The text first attempts to boil down discrete single and multivariable calculus to one page each, then presents the slides with additional remarks and finally includes 40 "calculus problems" in a discrete or so-called 'quantum calculus' setting. We also added some sample Mathematica code, gave a short overview over the emergence of the function concept in calculus and included comments on the development of calculus textbooks over time.Comment: 31 pages, 36 figure

The dual process account of reasoning: historical roots, problems and perspectives.

Despite the great effort that has been dedicated to the attempt to redefine expected utility theory on the grounds of new assumptions, modifying or moderating some axioms, none of the alternative theories propounded so far had a statistical confirmation over the full domain of applicability. Moreover, the discrepancy between prescriptions and behaviors is not limited to expected utility theory. In two other fundamental fields, probability and logic, substantial evidence shows that human activities deviate from the prescriptions of the theoretical models. The paper suggests that the discrepancy cannot be ascribed to an imperfect axiomatic description of human choice, but to some more general features of human reasoning and assumes the “dual-process account of reasoning” as a promising explanatory key. This line of thought is based on the distinction between the process of deliberate reasoning and that of intuition; where in a first approximation, “intuition” denotes a mental activity largely automatized and inaccessible from conscious mental activity. The analysis of the interactions between these two processes provides the basis for explaining the persistence of the gap between normative and behavioral patterns. This view will be explored in the following pages: central consideration will be given to the problem of the interactions between rationality and intuition, and the correlated “modularity” of the thought.