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.

Genotype 3 is associated with rapid histological progression in chronic HCV infection

While several risk factors for the histological progression of chronic hepatitis C have been identified, the contribution of HCV genotypes to liver fibrosis evolution remains controversial. The aim of the present study was to assess independent predictors for fibrosis progression. Methods: We identified 1540 patients from the Swiss Hepatitis C Cohort database with at least one liver biopsy prior to antiviral treatment. Factors associated with fibrosis stage, steatosis and histological activity were assessed in univariate and multivariate regression models. Fibrosis progression rate per year was calculated in a subgroup of 1263 patients, in whom risk factors were assessed by cumulative incidence curves, logistic and linear regression models. Results: Independent risk factors for rapid fibrosis progression included male sex (OR = 1.66, 95% CI 1.25-2.21, P <0.001), age at infection (OR = 1.08, 95% CI 1.06-1.10, P <0.001), histological activity (OR = 2.14, 95% CI 1.61-2.85, P <0.001) and genotype 3 (OR = 1.97, 95% CI 1.43-2.72, P <0.001). Genotype 2 was associated with slow progression (OR = 0.51, 95% CI 0.30-0.89, P = 0.02), but this observation may be due to the decreased prevalence of genotype 2 over the last decades, leading to an overrepresentation of subjects with genotype 2 with a slow progression rate. Conclusion: This study shows a significant association of genotype 3 with accelerated fibrosis. While assessing risk factors for fibrosis progression, the changing epidemiology of HCV genotypes over time needs to be taken into account.

Renoncement aux soins: comment appréhender cette réalité en médecine de premier recours [Patients forgoing health care for economic reasons: how to identify this in a primary care setting?].

Although the performance of the Swiss health system is high, one out of ten patients in general practitioner's (GP) office declares having foregone care in the previous twelve months for economic reasons. Reasons for foregoing care are several and include a lack of knowledge of existing social aids in getting health insurance, unavailability of GPs and long waiting lists for various types of care. Although long term knowledge of patients or a psychosocial history of deprivation or poverty may help identify individuals at risk of foregoing care, many may remain undetected. We propose then a few instruments to help GPs to identify, in a simple and structured approach, patients at risk of forgoing care for economic reasons; these patients are frequently deprived and sometimes poor.

Repeat Gamma Knife surgery for recurrent trigeminal neuralgia: long-term outcomes and systematic review.

Object The purpose of this study was to establish the safety and efficacy of repeat Gamma Knife surgery (GKS) for recurrent trigeminal neuralgia (TN). Methods Using the prospective database of TN patients treated with GKS in Timone University Hospital (Marseille, France), data were analyzed for 737 patients undergoing GKS for TN Type 1 from July 1992 to November 2010. Among the 497 patients with initial pain cessation, 34.4% (157/456 with ≥ 1-year follow-up) experienced at least 1 recurrence. Thirteen patients (1.8%) were considered for a second GKS, proposed only if the patients had good and prolonged initial pain cessation after the first GKS, with no other treatment alternative at the moment of recurrence. As for the first GKS, a single 4-mm isocenter was positioned in the cisternal portion of the trigeminal nerve at a median distance of 7.6 mm (range 4-14 mm) anterior to the emergence of the nerve (retrogasserian target). A median maximum dose of 90 Gy (range 70-90 Gy) was delivered. Data for 9 patients with at least 1-year followup were analyzed. A systematic review of literature was also performed, and results are compared with those of the Marseille study. Results The median time to retreatment in the Marseille study was 72 months (range 12-125 months) and in the literature it was 17 months (range 3-146 months). In the Marseille study, the median follow-up period was 33.9 months (range 12-96 months), and 8 of 9 patients (88.9%) had initial pain cessation with a median of 6.5 days (range 1-180 days). The actuarial rate for new hypesthesia was 33.3% at 6 months and 50% at 1 year, which remained stable for 7 years. The actuarial probabilities of maintaining pain relief without medication at 6 months and 1 year were 100% and 75%, respectively, and remained stable for 7 years. The systematic review analyzed 20 peer-reviewed studies reporting outcomes for repeat GKS for recurrent TN, with a total of 626 patients. Both the selection of the cases for retreatment and the way of reporting outcomes vary widely among studies, with a median rate for initial pain cessation of 88% (range 60%-100%) and for new hypesthesia of 33% (range 11%-80%). Conclusions Results from the Marseille study raise the question of surgical alternatives after failed GKS for TN. The rates of initial pain cessation and recurrence seem comparable to, or even better than, those of the first GKS, according to different studies, but toxicity is much higher, both in the Marseille study and in the published data. Neither the Marseille study data nor literature data answer the 3 cardinal questions regarding repeat radiosurgery in recurrent TN: which patients to retreat, which target is optimal, and which dose to use.