508 research outputs found
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 Gödelâs 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
Ethics and economics in Karl Menger: how did social sciences cope with Hilbertism
This paper deals with the contributions made to the social sciences by the mathematician Karl Menger (1902-1985), the son of the more famous economist, Carl Menger. Mathematician and a logician, he focused on whether it was possible to explain the social order in formal terms.1 He stressed the need to find the appropriate means with which to treat them, avoiding recourse to historical descriptions, which are unable to yield social laws. He applied Hilbertism to economics and ethics in order to build an axiomatic and formalized model of the individual behavior and the dynamics of social groups.
Mathematics as the role model for neoclassical economics (Blanqui Lecture)
Born out of the conscious effort to imitate mechanical physics, neoclassical economics ended up in the mid 20th century embracing a purely mathematical notion of rigor as embodied by the axiomatic method. This lecture tries to explain how this could happen, or, why and when the economistsâ role model became the mathematician rather than the physicist. According to the standard interpretation, the triumph of axiomatics in modern neoclassical economics can be explained in terms of the disciplineâs increasing awareness of its lack of good experimental and observational data, and thus of its intrinsic inability to fully abide by the paradigm of mechanics. Yet this story fails to properly account for the transformation that the word ârigorâ itself underwent first and foremost in mathematics as well as for the existence of a specific motivation behind the economistsâ decision to pursue the axiomatic route. While the full argument is developed in Giocoli 2003, these pages offer a taste of a (partially) alternative story which begins with the so-called formalist revolution in mathematics, then crosses the economistsâ almost innate urge to bring their discipline to the highest possible level of generality and conceptual integrity, and ends with the advent and consolidation of that very core set of methods, tools and ideas that constitute the contemporary image of economics.Axiomatic method, formalism, rationality, neoclassical economics
Geometric Approach to Digital Quantum Information
We present geometric methods for uniformly discretizing the continuous
N-qubit Hilbert space. When considered as the vertices of a geometrical figure,
the resulting states form the equivalent of a Platonic solid. The
discretization technique inherently describes a class of pi/2 rotations that
connect neighboring states in the set, i.e. that leave the geometrical figures
invariant. These rotations are shown to generate the Clifford group, a general
group of discrete transformations on N qubits. Discretizing the N-qubit Hilbert
space allows us to define its digital quantum information content, and we show
that this information content grows as N^2. While we believe the discrete sets
are interesting because they allow extra-classical behavior--such as quantum
entanglement and quantum parallelism--to be explored while circumventing the
continuity of Hilbert space, we also show how they may be a useful tool for
problems in traditional quantum computation. We describe in detail the discrete
sets for one and two qubits.Comment: Introduction rewritten; 'Sample Application' section added. To appear
in J. of Quantum Information Processin
A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media
In this work, we are interested in the spectrum of the diffusively excited
granular gases equation, in a space inhomogeneous setting, linearized around an
homogeneous equilibrium.
We perform a study which generalizes to a non-hilbertian setting and to the
inelastic case the seminal work of Ellis and Pinsky about the spectrum of the
linearized Boltzmann operator. We first give a precise localization of the
spectrum, which consists in an essential part lying on the left of the
imaginary axis and a discrete spectrum, which is also of nonnegative real part
for small values of the inelasticity parameter. We then give the so-called
inelastic "dispersion relations", and compute an expansion of the branches of
eigenvalues of the linear operator, for small Fourier (in space) frequencies
and small inelasticity.
One of the main novelty in this work, apart from the study of the inelastic
case, is that we consider an exponentially weighted Banach
setting instead of the classical Hilbertian
case, endorsed with Gaussian weights. We prove in particular that the results
of Ellis and Pinsky holds also in this space.Comment: 30 pages, 2 figure
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