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.

Inheritance of ductile and brittle structures in the development of large rock slope instabilities: Examples from western Norway

The high density of slope failures in western Norway is due to the steep relief and to the concentration of various structures that followed protracted ductile and brittle tectonics. On the 72 investigated rock slope instabilities, 13 were developed in soft weathered mafic and phyllitic allochthons. Only the intrinsic weakness of such rocks increases the susceptibility to gravitational deformation. In contrast, the gravitational structures in the hard gneisses reactivate prominent ductile or/and brittle fabrics. At 30 rockslides along cataclinal slopes, weak mafic layers of foliation are reactivated as basal planes. Slope-parallel steep foliation forms back-cracks of unstable columns. Folds are specifically present in the Storfjord area, together with a clustering of potential slope failures. Folding increases the probability of having favourably orientated planes with respect to the gravitational forces and the slope. High water pressure is believed to seasonally build up along the shallow-dipping Caledonian detachments and may contribute to destabilization of the rock slope upwards. Regional cataclastic faults localized the gravitational structures at 45 sites. The volume of the slope instabilities tends to increase with the amount of reactivated prominent structures and the spacing of the latter controls the size of instabilities.

The cells and peripheral representation of sodium taste in mice.

Salt taste in mammals can trigger two divergent behavioural responses. In general, concentrated saline solutions elicit robust behavioural aversion, whereas low concentrations of NaCl are typically attractive, particularly after sodium depletion. Notably, the attractive salt pathway is selectively responsive to sodium and inhibited by amiloride, whereas the aversive one functions as a non-selective detector for a wide range of salts. Because amiloride is a potent inhibitor of the epithelial sodium channel (ENaC), ENaC has been proposed to function as a component of the salt-taste-receptor system. Previously, we showed that four of the five basic taste qualities-sweet, sour, bitter and umami-are mediated by separate taste-receptor cells (TRCs) each tuned to a single taste modality, and wired to elicit stereotypical behavioural responses. Here we show that sodium sensing is also mediated by a dedicated population of TRCs. These taste cells express the epithelial sodium channel ENaC, and mediate behavioural attraction to NaCl. We genetically engineered mice lacking ENaCalpha in TRCs, and produced animals exhibiting a complete loss of salt attraction and sodium taste responses. Together, these studies substantiate independent cellular substrates for all five basic taste qualities, and validate the essential role of ENaC for sodium taste in mice.

PHYTOCHROME KINASE SUBSTRATE1 regulates root phototropism and gravitropism.

Light promotes the expression of PHYTOCHROME KINASE SUBSTRATE1 (PKS1) in the root of Arabidopsis thaliana, but the function of PKS1 in this organ is unknown. Unilateral blue light induced a negative root phototropic response mediated by phototropin 1 in wild-type seedlings. This response was absent in pks1 mutants. In the wild type, unilateral blue light enhanced PKS1 expression in the subapical region of the root several hours before bending was detectable. The negative phototropism and the enhanced PKS1 expression in response to blue light required phytochrome A (phyA). In addition, the pks1 mutation enhanced the root gravitropic response when vertically oriented seedlings were placed horizontally. The negative regulation of gravitropism by PKS1 occurred even in dark-grown seedlings and did not require phyA. Blue light also failed to induce negative phototropism in pks1 under reduced gravitational stimulation, indicating that the effect of pks1 on phototropism is not simply the consequence of the counteracting effect of enhanced gravitropism. We propose a model where the background level of PKS1 reduces gravitropism. After a phyA-dependent increase in its expression, PKS1 positively affects root phototropism and both effects contribute to negative curvature in response to unilateral blue light.