Archive for the ‘Mathematics’ Category

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The Problem of Space

May 28, 2014

Joshua Eisenthal

Abstract: I define the Problem of Space as the problem of delimiting the range of candidate geometrical descriptions of physical space. I argue that, ever since the development of non-Euclidean geometries, tackling the Problem of Space has become necessary in order to understand thespatial significance of geometrical structures.

I briefly review the nineteenth century approach to this problem, arriving at the so-called “classical solution”. This solution centered around the claim, advanced by Helmholtz and Poincaré, that candidate physical geometries were just those structures which could represent the free mobility of rigid bodies. As noted originally by Riemann, then argued for by Helmholtz and proved rigorously by Lie, congruence relations which could represent such free mobility existed only in geometries of constant curvature. Helmholtz and Poincaré regarded this fact as providing the means of delimiting the range of spatially significant geometrical structures.

However, I then review how this view was fatally undermined by the development of general relativity. I thus turn to explore the twentieth century solution to the Problem of Space advanced by Hermann Weyl. I conclude by reflecting on the significance of this discussion for a relatively recent dispute regarding the status of the metric field in general relativity. I suggest that this dispute has arisen partially due to a failure to properly appreciate the insights made available by the kind of sophisticated analysis of geometrical concepts exemplified by Weyl’s work. More generally, I argue that the nuances of Weyl’s view demonstrate the importance of engaging with the Problem of Space in interpreting general relativity today.

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Axiomatic Quantum Field Theory in the Philosophy of Quantum Field Theory

March 14, 2014

Bihui Li

Abstract: The philosophy of quantum field theory (QFT) in North America is dominated by philosophers working in the algebraic QFT tradition. They justify their choice of this tradition by a methodological requirement that when philosophers of physics study a theory T, the referent of T must be some mathematically well-defined structure. At the same time, philosophers think that their preferred axiomatic versions of QFT are best suited for “foundational”, “fundamental”, or “ontological” inquiries. I argue that the mathematical structures satisfying the methodological requirement are not always the best starting points for foundational, fundamental, or ontological inquiries. Many foundational, fundamental, and ontological questions are best addressed by constructive QFT, which starts with the ill-defined formalisms of Lagrangian QFT and tries to make them well-defined.  Because different interactions in constructive QFT require different mathematical constructions, the mathematical structures required cannot be determined prior to constructing a solution for a particular interaction. In contrast, the advantage of axiomatic QFT is its independence from the specific interactions being modeled, but this independence also means that it cannot provide many kinds of foundational, fundamental, or ontological information.

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When is Harmonics a Science?

November 22, 2013

Marina Baldissera Pacchetti

Abstract: Aristotle bases the principle of harmony on the diesis being the indivisible unit that can be represented by number. This, in a passages of De Anima and De Sensu, sets aesthetic standards for ratios of numbers representing melodious concords (`symphonia’), which are demonstrated by arithmetic. This allows him to define harmonics to be one of the `more physical of the mathematical sciences’ because it shares the principle of indivisible unit number with arithmetic, and the demonstrations that provide knowledge about concords are arithmetical.

Aristoxenus says that this is not an appropriate treatment of harmonics and a proper explanation of what justifies aesthetic perception of consonant sounds. The principles of this science cannot rely on mathematics, but on movement with respect to pitch space. Harmonious sound arises from intervallic movement (to be defined), which is recognized only in terms of its phenomenology. He does not discard the use of arithmetic, but he rather sees it as a useful tool to calculate modes – as Barker says (note 50 GMW II p.135), Aristoxenus does not have a problem in including intervals smaller than a quarter tone (the diesis) in his `science of harmonics’. The inclusion of tones smaller than a quarter tone was problematic in explanations of concords for his predecessors (esp. Pythagoreans) and those who will come afterwards (Ptolemy 2C AD, Boethius 6C AD, Gafurio 15C AD, Zarlino 16C AD). Aristoxenus does not deny that the use of mathematics is not useful in harmonics, but as his `first principles’ are not mathematical, he seems to be able to accommodate standard calculations within his framework without incurring in inconsistencies in his philosophical framework.

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Day-O-WIPs Beta

June 17, 2013

The second installment of the “Day-O-WIPs” series:

“Toward a Philosophy of Synthetic Science” Julia Bursten

“Can Genes be Darwinian Individuals?” Haixin Dang

“Group Theory or No Group Theory: Understanding Atomic Spectra” Joshua Hunt

“Dynamical Models: A Type of Mathematical Explanation in Neuroscience and Medicine” Lauren Ross

“The Wax & the Mechanical Mind: Reexamining Hobbes’s Objections to Descartes’ Meditations” Marcus Adams