Introducing Formalism in Economics: The Growth Model of John von Neumann

The objective is to interpret John von Neumann's growth model as a decisive step of the forthcoming formalist revolution of the 1950s in economics. This model gave rise to an impressive variety of comments about its classical or neoclassical underpinnings. We go beyond this traditional criterion and interpret rather this model as the manifestation of von Neumann's involvement in the formalist programme of mathematician David Hilbert. We discuss the impact of Kurt Gödel’s discoveries on this programme. We show that the growth model reflects the pragmatic turn of the formalist programme after Gödel and proposes the extension of modern axiomatisation to economics..Von Neumann, Growth model, Formalist revolution, Mathematical formalism, Axiomatics

Axiomatic method and the law

Whether an axiomatic approach to law is possible and useful today has to be perceived as unanswered. Perception of the axiomatic method among lawyers, however, is clouded by misunderstanding. Clarifying them may generate new discussion about the axiomatization of legal theories

Axiomatics for the external numbers of nonstandard analysis

Neutrices are additive subgroups of a nonstandard model of the real numbers. An external number is the algebraic sum of a nonstandard real number and a neutrix. Due to the stability by some shifts, external numbers may be seen as mathematical models for orders of magnitude. The algebraic properties of external numbers gave rise to the so-called solids, which are extensions of ordered fields, having a restricted distributivity law. However, necessary and sufficient conditions can be given for distributivity to hold. In this article we develop an axiomatics for the external numbers. The axioms are similar to, but mostly somewhat weaker than the axioms for the real numbers and deal with algebraic rules, Dedekind completeness and the Archimedean property. A structure satisfying these axioms is called a complete arithmetical solid. We show that the external numbers form a complete arithmetical solid, implying the consistency of the axioms presented. We also show that the set of precise elements (elements with minimal magnitude) has a built-in nonstandard model of the rationals. Indeed the set of precise elements is situated between the nonstandard rationals and the nonstandard reals whereas the set of non-precise numbers is completely determined

Electric/magnetic reciprocity in premetric electrodynamics with and without magnetic charge, and the complex electromagnetic field

We extend an axiomatic approach to classical electrodynamics, which we developed recently, to the case of non-vanishing magnetic charge. Then two axioms, namely those of the existence of the Lorentz force (Axiom 2) and of magnetic flux conservation (Axiom 3) have to be generalized. Electric/magnetic reciprocity constitutes a guiding principle for this undertaking. The extension of the axioms can be implemented at a premetric stage, i.e., when metric and connection of spacetime don't play a role. Complex Riemann-Silberstein fields of the form $(E\pm i {\cal H},{\cal D}\pm i B)$ have a natural place in the theory, independent of the Hodge duality mapping defined by any particular metric.Comment: 13 pages in latex, 3 references added, text slightly revise

Locus configurations and ∨\vee-systems

We present a new family of the locus configurations which is not related to $\vee$-systems thus giving the answer to one of the questions raised in \cite{V1} about the relation between the generalised quantum Calogero-Moser systems and special solutions of the generalised WDVV equations. As a by-product we have new examples of the hyperbolic equations satisfying the Huygens' principle in the narrow Hadamard's sense. Another result is new multiparameter families of $\vee$-systems which gives new solutions of the generalised WDVV equation.Comment: 12 page

A Characterisation of the Weylian Structure of Space-Time by Means of Low Velocity Tests

The compatibility axiom in Ehlers, Pirani and Schild's (EPS) constructive axiomatics of the space-time geometry that uses light rays and freely falling particles with high velocity, is replaced by several constructions with low velocity particles only. For that purpose we describe in a space-time with a conformal structure and an arbitrary path structure the radial acceleration, a Coriolis acceleration and the zig-zag construction. Each of these quantities give effects whose requirement to vanish can be taken as alternative version of the compatibility axiom of EPS. The procedural advantage lies in the fact, that one can make null-experiments and that one only needs low velocity particles to test the compatibility axiom. We show in addition that Perlick's standard clock can exist in a Weyl space only.Comment: to appear in Gen.Rel.Gra