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.

Rare genomic structural variants in complex disease: lessons from the replication of associations with obesity.

The limited ability of common variants to account for the genetic contribution to complex disease has prompted searches for rare variants of large effect, to partly explain the 'missing heritability'. Analyses of genome-wide genotyping data have identified genomic structural variants (GSVs) as a source of such rare causal variants. Recent studies have reported multiple GSV loci associated with risk of obesity. We attempted to replicate these associations by similar analysis of two familial-obesity case-control cohorts and a population cohort, and detected GSVs at 11 out of 18 loci, at frequencies similar to those previously reported. Based on their reported frequencies and effect sizes (OR≥25), we had sufficient statistical power to detect the large majority (80%) of genuine associations at these loci. However, only one obesity association was replicated. Deletion of a 220 kb region on chromosome 16p11.2 has a carrier population frequency of 2×10(-4) (95% confidence interval [9.6×10(-5)-3.1×10(-4)]); accounts overall for 0.5% [0.19%-0.82%] of severe childhood obesity cases (P = 3.8×10(-10); odds ratio = 25.0 [9.9-60.6]); and results in a mean body mass index (BMI) increase of 5.8 kg.m(-2) [1.8-10.3] in adults from the general population. We also attempted replication using BMI as a quantitative trait in our population cohort; associations with BMI at or near nominal significance were detected at two further loci near KIF2B and within FOXP2, but these did not survive correction for multiple testing. These findings emphasise several issues of importance when conducting rare GSV association, including the need for careful cohort selection and replication strategy, accurate GSV identification, and appropriate correction for multiple testing and/or control of false discovery rate. Moreover, they highlight the potential difficulty in replicating rare CNV associations across different populations. Nevertheless, we show that such studies are potentially valuable for the identification of variants making an appreciable contribution to complex disease.

Adult native septic arthritis: a review of 10 years of experience and lessons for empirical antibiotic therapy.

Objectives To review the epidemiology of native septic arthritis to establish local guidelines for empirical antibiotic therapy as part of an antibiotic stewardship programme. Methods We conducted a 10 year retrospective study based on positive synovial fluid cultures and discharge diagnosis of septic arthritis in adult patients. Microbiology results and medical records were reviewed. Results Between 1999 and 2008, we identified 233 episodes of septic arthritis. The predominant causative pathogens were methicillin-susceptible Staphylococcus aureus (MSSA) and streptococci (respectively, 44.6% and 14.2% of cases). Only 11 cases (4.7%) of methicillin-resistant S. aureus (MRSA) arthritis were diagnosed, among which 5 (45.5%) occurred in known carriers. For large-joint infections, amoxicillin/clavulanate or cefuroxime would have been appropriate in 84.5% of cases. MRSA and Mycobacterium tuberculosis would have been the most frequent pathogens that would not have been covered. In contrast, amoxicillin/clavulanate would have been appropriate for only 75.3% of small-joint infections (82.6% if diabetics are excluded). MRSA and Pseudomonas aeruginosa would have been the main pathogens not covered. Piperacillin/tazobactam would have been appropriate in 93.8% of cases (P < 0.01 versus amoxicillin/clavulanate). This statistically significant advantage is lost after exclusion of diabetics (P = 0.19). Conclusions Amoxicillin/clavulanate or cefuroxime would be adequate for empirical coverage of large-joint septic arthritis in our area. A broad-spectrum antibiotic would be significantly superior for small-joint infections in diabetics. Systematic coverage of MRSA is not justified, but should be considered for known carriers. These recommendations are applicable to our local setting. They might also apply to hospitals sharing the same epidemiology.