Mathematical existence results for the Doi-Edwards polymer model

In this paper, we present some mathematical results on the Doi-Edwards model describing the dynamics of flexible polymers in melts and concentrated solutions. This model, developed in the late 1970s, has been used and tested extensively in modeling and simulation of polymer flows. From a mathematical point of view, the Doi-Edwards model consists in a strong coupling between the Navier-Stokes equations and a highly nonlinear constitutive law. The aim of this article is to provide a rigorous proof of the well-posedness of the Doi-Edwards model, namely it has a unique regular solution. We also prove, which is generally much more difficult for flows of viscoelastic type, that the solution is global in time in the two dimensional case, without any restriction on the smallness of the data.Comment: 48 page

Aristotle on Mathematical Existence

Do mathematical objects exist in some realm inaccessible to our senses? It may be tempting to deny this. For one thing, how we could come to know mathematical truths, if such knowledge must arise from causal interaction with non-empirical objects? However, denying that mathematical objects exist altogether has unsettling consequences. If you deny the existence of mathematical objects, then you must reject all claims that commit you to such objects, which means rejecting much of mathematics as it is standardly understood. For, as David Papineau (1990) vividly puts it, it is doublethink to deny that mathematical objects exist but to continue to believe, for example, that there are two prime numbers between ten and fifteen. Two current responses to this problem are literalism and fictionalism. Both literalists and fictionalists deny the existence of a world of mathematical objects distinct from the empirical world. But they differ markedly in this denial. Literalists argue that mathematical objects simply exist in the empirical world; on this account, mathematical assertions assert true beliefs about perceivable objects. Fictionalists, on the other hand, hold that, strictly speaking, mathematical objects do not exist at all, and so exist in neither the empirical world nor in some realm distinct from the empirical world. They argue that mathematical objects are not actual objects but rather harmless fictions; on this account, mathematical assertions do not assert true beliefs about the world but merely fictional attitudes. Although these two positions are apparently quite opposed to one another, they nonetheless have been both ascribed to Aristotle. Indeed, as I’ll argue, Aristotle’s philosophy of mathematics exhibits some of the features characteristic of literalism and some of the features characteristic of fictionalism. However, Aristotle’s position also exhibits features interestingly different from both literalism and fictionalism. The paper comes in three parts. In the first part, I’ll quickly survey the variety of descriptions which Aristotle uses to characterize the relation between mathematical objects and the perceivable world. This will help to explain how apparently opposed positions have been ascribed to Aristotle. In the second part, I’ll discuss literalism in contemporary philosophy of mathematics, the ascription of literalism to Aristotle and the points of agreement and disagreement between Aristotle and literalists. In the third and final part of the paper, I’ll discuss fictionalism in contemporary philosophy of mathematics, the ascription of fictionalism to Aristotle and the points of agreement and disagreement between Aristotle and fictionalists

The non-unique Universe

The purpose of this paper is to elucidate, by means of concepts and theorems drawn from mathematical logic, the conditions under which the existence of a multiverse is a logical necessity in mathematical physics, and the implications of Godel's incompleteness theorem for theories of everything. Three conclusions are obtained in the final section: (i) the theory of the structure of our universe might be an undecidable theory, and this constitutes a potential epistemological limit for mathematical physics, but because such a theory must be complete, there is no ontological barrier to the existence of a final theory of everything; (ii) in terms of mathematical logic, there are two different types of multiverse: classes of non-isomorphic but elementarily equivalent models, and classes of model which are both non-isomorphic and elementarily inequivalent; (iii) for a hypothetical theory of everything to have only one possible model, and to thereby negate the possible existence of a multiverse, that theory must be such that it admits only a finite model

Probability of immortality and God’s existence. A mathematical perspective

What are the probabilities that this universe is repeated exactly the same with you in it again? Is God invented by human imagination or is the result of human intuition? The intuition that the same laws/mechanisms (evolution, stability winning probability) that have created something like the human being capable of self-awareness and controlling its surroundings, could create a being capable of controlling all what it exists? Will be the characteristics of the next universes random or tend to something? All these ques-tions that with different shapes (but the same essence) have been asked by human be-ings from the beginning of times will be developed in this paper

Hilbert, logicism, and mathematical existence

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom)

The Relativity of Existence

Despite the success of modern physics in formulating mathematical theories that can predict the outcome of experiments, we have made remarkably little progress towards answering the most fundamental question of: why is there a universe at all, as opposed to nothingness? In this paper, it is shown that this seemingly mind-boggling question has a simple logical answer if we accept that existence in the universe is nothing more than mathematical existence relative to the axioms of our universe. This premise is not baseless; it is shown here that there are indeed several independent strong logical arguments for why we should believe that mathematical existence is the only kind of existence. Moreover, it is shown that, under this premise, the answers to many other puzzling questions about our universe come almost immediately. Among these questions are: why is the universe apparently fine-tuned to be able to support life? Why are the laws of physics so elegant? Why do we have three dimensions of space and one of time, with approximate locality and causality at macroscopic scales? How can the universe be non-local and non-causal at the quantum scale? How can the laws of quantum mechanics rely on true randomness