On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

Airway surface liquid volume regulation determines different airway phenotypes in liddle compared with betaENaC-overexpressing mice.

Studies in cystic fibrosis patients and mice overexpressing the epithelial Na(+) channel beta-subunit (betaENaC-Tg) suggest that raised airway Na(+) transport and airway surface liquid (ASL) depletion are central to the pathogenesis of cystic fibrosis lung disease. However, patients or mice with Liddle gain-of-function betaENaC mutations exhibit hypertension but no lung disease. To investigate this apparent paradox, we compared the airway phenotype (nasal versus tracheal) of Liddle with CFTR-null, betaENaC-Tg, and double mutant mice. In mouse nasal epithelium, the region that functionally mimics human airways, high levels of CFTR expression inhibited Liddle epithelial Nat channel (ENaC) hyperfunction. Conversely, in mouse trachea, low levels of CFTR failed to suppress Liddle ENaC hyperfunction. Indeed, Na(+) transport measured in Ussing chambers ("flooded" conditions) was raised in both Liddle and betaENaC-Tg mice. Because enhanced Na(+) transport did not correlate with lung disease in these mutant mice, measurements in tracheal cultures under physiologic "thin film" conditions and in vivo were performed. Regulation of ASL volume and ENaC-mediated Na(+) absorption were intact in Liddle but defective in betaENaC-Tg mice. We conclude that the capacity to regulate Na(+) transport and ASL volume, not absolute Na(+) transport rates in Ussing chambers, is the key physiologic function protecting airways from dehydration-induced lung disease.

A Playful Glance at Hierarchical Questions for Two-Way Alternating Automata

Two-way alternating automata were introduced by Vardi in order to study the satisfiability problem for the modal μ-calculus extended with backwards modalities. In this paper, we present a very simple proof by way of Wadge games of the strictness of the hierarchy of Motowski indices of two-way alternating automata over trees.