The comments above (see below for who made them and when) represent one type of thinking about infinity. There are other types, as we will see, and they all create difficulties for students, philosophers and even mathematicians. The purpose of this article is to show how a particular theory about how people come to understand mathematics, APOS Theory, can be helpful in understanding the thinkin...

These notes discuss sets, mappings between sets, and the notion of countably infinite. This material supplements the lecture on countably and uncountably infinite sets in the EE 503 probability class. The material is useful because probability theory is defined over abstract sets, probabilities are defined as measures on subsets, and random variables are defined by mappings from an abstract set...

We generalize Penrose’s notion of conformal infinity of spacetime, to situations with anisotropic scaling. This is relevant not only for Lifshitz-type anisotropic gravity models, but also in standard general relativity and string theory, for spacetimes exhibiting a natural asymptotic anisotropy. Examples include the Lifshitz and Schrödinger spaces (proposed as AdS/CFT duals of nonrelativistic f...

In this article I concentrate on three issues. First, Graham Oppy’s treatment of the relationship between the concept of infinity and Zeno’s paradoxes lay bare several porblems that must be dealt with if the concept of infinity is to do any intellectual work in philosophy of religion. Here I will expand on some insightful remarks by Oppy in an effort ot adequately respond to these problems. Sec...

Within the context of the standard model of rationality within economic modelling we show the existence of a utility function that rationalises a demand correspondence, hence completely characterizes the associated preference structure, by taking a dense demand sample. This resolves the problem of revealed preferences under some very mild assumptions on the demand correspondence which are close...

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α > 0. The high-frequency (or:

Probability inequalities are. obtained for the supremum of a weighted empirical process indexed. by a Vapnik-Cervonenkis class C of sets. These inequalities are particularly useful under the assumption P(ufC E: C : P(C) < t D -? 0 as t -? o. They are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity, to find an asymptotic sample modulu...

We study the problem of classification of d-dimensional vectors into two classes (one of which is 'pure noise') based on a training sample of size m. The main specific feature is that the dimension d can be very large. We suppose that the difference between the distribution of the population and that of the noise is only in a shift, which is a sparse vector. For Gaussian noise, fixed sample siz...

We apply a spin-coherent states formalism to study the simplest version of a central-spin model, the so-called Mermin model, in which a given (central) spin 1/2 interacts symmetrically with a "bath" composed of N spins 1/2, characterized by a common frequency. The symmetric interaction allows one to restrict the problem to the fully symmetric sector of the Hilbert space associated with these N ...