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"Infinite History."
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Infinity
This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy and theology offer a rich intellectual exchange among various current viewpoints, rather than displaying a static picture of accepted views on infinity. The book starts with a historical examination of the transformation of infinity from a philosophical and theological study to one dominated by mathematics. It then offers technical discussions on the understanding of mathematical infinity. Following this, the book considers the perspectives of physics and cosmology: can infinity be found in the real universe? Finally, the book returns to questions of philosophical and theological aspects of infinity.
eBook
Infinity : new research frontiers
\"'The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.' David Hilbert (1862-1943). This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy, and theology offer a rich intellectual exchange among various current viewpoints, rather than a static picture of accepted views on infinity. The book starts with a historical examination of the transformation of infinity from a philosophical and theological study to one dominated by mathematics. It then offers technical discussions on the understanding of mathematical infinity. Following this, the book considers the perspectives of physics and cosmology: Can infinity be found in the real universe? Finally, the book returns to questions of philosophical and theological aspects of infinity\"--Provided by publisher.
Book
Infinite-dimensional Gaussian change of variables’ formula
In this paper, we study the Banach space ℓ∞ of the bounded real sequences, and a measure N(a,Γ) over R∞,B∞ analogous to the finite-dimensional Gaussian law. The main result of our paper is a change of variables’ formula for the integration, with respect to N(a,Γ), of the measurable real functions on E∞,B∞E∞, where E∞ is the separable Banach space of the convergent real sequences. This change of variables is given by some m,σ functions, defined over a subset of E∞, with values on E∞, with properties that generalize the analogous ones of the finite-dimensional diffeomorphisms.
Journal Article
The structure of asymptotic idealization
Robert Batterman and others have argued that certain idealizing explanations have an asymptotic form: they account for a state of affairs or behavior by showing that it emerges \"in the limit\". Asymptotic idealizations are interesting in many ways, but is there anything special about them as idealizations? To understand their role in science, must we augment our philosophical theories of idealization? This paper uses simple examples of asymptotic idealization in population genetics to argue for an affirmative answer and proposes a general schema for asymptotic idealization, drawing on insights from Batterman's treatment and from John Norton's subsequent critique.
Journal Article
Combining finite and infinite elements: Why do we use infinite idealizations in engineering?
This contribution sheds light on the role of infinite idealization in structural analysis, by exploring how infinite elements and finite element methods are combined in civil engineering models. This combination, I claim, should be read in terms of a 'complementarity function' through which the representational ideal of completeness is reached in engineering model-building. Taking a cue from Weisberg's definition of multiple-model idealization, I highlight how infinite idealizations are primarily meant to contribute to the prediction of structural behavior in Multiphysics approaches.
Journal Article
Berkeley's philosophy of mathematics
In this first modern, critical assessment of the place of mathematics in Berkeley's philosophy and Berkeley's place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley's work. Jesseph challenges the prevailing view that Berkeley's mathematical writings are peripheral to his philosophy and argues that mathematics is in fact central to his thought, developing out of his critique of abstraction. Jesseph's argument situates Berkeley's ideas within the larger historical and intellectual context of the Scientific Revolution. Jesseph begins with Berkeley's radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a \"science of abstractions.\" Since this view seriously conflicted with Berkeley's critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley's unique treatments of geometry and arithmetic and his famous critique of the calculus in The Analyst. By putting Berkeley's mathematical writings in the perspective of his larger philosophical project and examining their impact on eighteenth-century British mathematics, Jesseph makes a major contribution to philosophy and to the history and philosophy of science.
eBook
On the paradox of reversible processes in thermodynamics
This paper discusses an argument by Norton (in: European Philosophy of Science—Philosophy of Science in Europe and the Viennese Heritage, Vienna Circle Institute Yearbook, vol 17, Springer, Dordrecht, pp 197-210, 2014, 2016) to the effect that reversible processes in thermodynamics have paradoxical character, due to the infinite-time limit. For Norton, one can \"dispel the fog of paradox\" by adopting a distinction between idealizations and approximations, which he himself puts forward. Accordingly, reversible processes ought to be regarded as approximations, rather than idealizations. Here, we critically assess his proposal. In doing so, we offer a resolution of his alleged paradox based on the original work by Tatiana Ehrenfest-Afanassjeva on the foundations of thermodynamics.
Journal Article
Interpreting the Infinitesimal Mathematics of Leibniz and Euler
We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass's ghost behind some of the received historiography on Euler's infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a \"primary point of reference for understanding the eighteenth-century theories.\" Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler's own. Euler's use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler's principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro's assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler's work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.
Journal Article
Effective theories and infinite idealizations
Williams and J. Fraser have recently argued that effective field theory methods enable scientific realists to make more reliable ontological commitments in quantum field theory (QFT) than those commonly made. In this paper, I show that the interpretative relevance of these methods extends beyond the specific context of QFT by identifying common structural features shared by effective theories across physics. In particular, I argue that effective theories are best characterized by the fact that they contain intrinsic empirical limitations, and I extract from their structure one central interpretative constraint for making more reliable ontological commitments in different subfields of physics. While this is in principle good news, this constraint still raises a challenge for scientific realists in some contexts, and I bring the point home by focusing on Williams’s and J. Fraser’s defense of selective realism in QFT.
Journal Article