Relating Theories via Renormalization

The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behavior separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the "extended singularity theorem" in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters. Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem. Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications are summarized.Comment: This note is partially a summary of a talk given at the workshop "Part and Whole" in Leiden during the period March 22-26, 201

Extending du Bois-Reymond's Infinitesimal and Infinitary Calculus Theory

The discovery of the infinite integer leads to a partition between finite and infinite numbers. Construction of an infinitesimal and infinitary number system, the Gossamer numbers. Du Bois-Reymond's much-greater-than relations and little-o/big-O defined with the Gossamer number system, and the relations algebra is explored. A comparison of function algebra is developed. A transfer principle more general than Non-Standard-Analysis is developed, hence a two-tier system of calculus is described. Non-reversible arithmetic is proved, and found to be the key to this calculus and other theory. Finally sequences are partitioned between finite and infinite intervals.Comment: Resubmission of 6 other submissions. 99 page

Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system

We study the problem of initiation of excitation waves in the FitzHugh-Nagumo model. Our approach follows earlier works and is based on the idea of approximating the boundary between basins of attraction of propagating waves and of the resting state as the stable manifold of a critical solution. Here, we obtain analytical expressions for the essential ingredients of the theory by singular perturbation using two small parameters, the separation of time scales of the activator and inhibitor, and the threshold in the activator's kinetics. This results in a closed analytical expression for the strength-duration curve.Comment: 10 pages, 5 figures, as accepted to Chaos on 2017/06/2

Difference and Necessity: Dispositionalism, Deleuze, and the Finite Genesis of Transfinite Truths

Difference and Necessity: Dispositionalism, Deleuze, and the Finite Genesis of Transfinite Truth

Infinity and the Sublime

In this paper we intend to connect two different strands of research concerning the origin of what I shall loosely call "formal" ideas: firstly, the relation between logic and rhetoric - the theme of the 2006 Cambridge conference to which this paper was a contribution -, and secondly, the impact of religious convictions on the formation of certain twentieth century mathematical concepts, as brought to the attention recently by the work of L. Graham and J.-M. Kantor. In fact, we shall show that the latter question is a special case of the former, and that investigation of the larger question adds to our understanding of the smaller one. Our approach will be primarily historical.Comment: 29 pages and 3 figure

General Relativity and Gravitation: A Centennial Perspective

To commemorate the 100th anniversary of general relativity, the International Society on General Relativity and Gravitation (ISGRG) commissioned a Centennial Volume, edited by the authors of this article. We jointly wrote introductions to the four Parts of the Volume which are collected here. Our goal is to provide a bird's eye view of the advances that have been made especially during the last 35 years, i.e., since the publication of volumes commemorating Einstein's 100th birthday. The article also serves as a brief preview of the 12 invited chapters that contain in-depth reviews of these advances. The volume will be published by Cambridge University Press and released in June 2015 at a Centennial conference sponsored by ISGRG and the Topical Group of Gravitation of the American Physical Society.Comment: 37 page

Paradox lost : the cost of a virtual world

This paper touches on a number of seemingly disparate topics-Artificial Intelligence, Fuzzy Logic, String Theory, the search for extra-terrestrial intelligence, the Cantorian concept of infinite sets-in order to support the thesis that for a large part of the educated public in the Western world, the very concept of reality has been changing over the last few generations, and that the change is being accelerated by our increasing acceptance of the Virtual as a substitute for the traditional Real. This, as I hope to convince you, is a momentous shift in the our world view, and like so many profound but gradual shifts, has gone largely unnoticed. Whether the shift is ultimately a good thing or a bad, it ought not to go unscrutinized; this paper aims to bring it to public attention. (The paradox whose loss is referred to in the title is discussed at the end of the paper.