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.

Potential of education-based insulin therapy for achievement of good metabolic control: a real-life experience.

Diabet. Med. 28, 539-542 (2011) ABSTRACT: Aims  Achievement of good metabolic control in Type 1 diabetes is a difficult task in routine diabetes care. Education-based flexible intensified insulin therapy has the potential to meet the therapeutic targets while limiting the risk for severe hypoglycaemia. We evaluated the metabolic control and the rate of severe hypoglycaemia in real-life clinical practice in a centre using flexible intensified insulin therapy as standard of care since 1990. Methods  Patients followed for Type 1 diabetes (n = 206) or those with other causes of absolute insulin deficiency (n = 17) in our outpatient clinic were analysed in a cross-sectional study. Mean age (± standard deviation) was 48.9 ± 15.7 years, with diabetes duration of 21.4 ± 14.4 years. Outcome measures were HbA(1c) and frequency of severe hypoglycaemia. Results  Median HbA(1c) was 7.1% (54 mmol/mol) [interquartile range 6.6-7.8 (51-62 mmol/mol)]; a good or acceptable metabolic control with HbA(1c) < 7.0% (53 mmol/mol) or 7.5% (58 mmol/mol) was reached in 43.5 and 64.6% of the patients, respectively. The frequency of severe hypoglycaemic episodes was 15 per 100 patient years: 72.3% of the patients did not experience any such episodes during the past 5 years. Conclusions  Good or acceptable metabolic control is achievable in the majority of patients with Type 1 diabetes or other causes of absolute insulin deficiency in routine diabetes care while limiting the risk for severe hypoglycaemia.

Hyperactivity and attention problems in a Swiss sample of school-aged children: effects of school achievement, child gender, and informants.

OBJECTIVE: The sensitivity and tolerance regarding ADHD symptoms obviously differ from one culture to another and according to the informants (parents, teachers, or children). This stimulates the comparison of data across informants and countries. METHOD: Parents and teachers of more than 1,000 school-aged Swiss children (5 to 17 years old) fill in Conners's questionnaires on ADHD. Children who are older than 10 years old also fill in a self-report questionnaire. Results are compared to data from a North American sample. RESULTS: Swiss parents and teachers tend to report more ADHD symptoms than American parents and teachers as far as the oldest groups of children are concerned. Interactions are evidenced between school achievement, child gender, and informants. A relatively low rate of agreement between informants is found. CONCLUSION: These results strengthen the importance to take into account all informants in the pediatric and the child psychiatry clinic, as well as in the epidemiological studies.

When performance-approach goals predict academic achievement and when they do not: A social value approach

Research on achievement goal promotion at University has shown that performance-approach goals are perceived as a means to succeed at University (high social utility) but are not appreciated (low social desirability). We argue that such a paradox could explain why research has detected that performance-approach goals consistently predict academic grades. First-year psychology students answered a performance-approach goal scale with standard, social desirability and social utility instructions. Participants' grades were recorded at the end of the semester. Results showed that the relationship between performance-approach goals and grades was inhibited by the increase of these goals' social desirability and facilitated by the increase of their social utility, revealing that the predictive validity of performance-approach goals depend on social value.