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.

Molecular insight into heart development and congenital heart disease: An update review from the Arab countries.

Congenital heart defect (CHD) has a major influence on affected individuals as well as on the supportive and associated environment such as the immediate family. Unfortunately, CHD is common worldwide with an incidence of approximately 1% and consequently is a major health concern. The Arab population has a high rate of consanguinity, fertility, birth, and annual population growth, in addition to a high incidence of diabetes mellitus and obesity. All these factors may lead to a higher incidence and prevalence of CHD within the Arab population than in the rest of the world, making CHD of even greater concern. Sadly, most Arab countries lack appropriate public health measures directed toward the control and prevention of congenital malformations and so the importance of CHD within the population remains unknown but is thought to be high. In approximately 85% of CHD patients, the multifactorial theory is considered as the pathologic basis. The genetic risk factors for CHD can be attributed to large chromosomal aberrations, copy number variations (CNV) of particular regions in the chromosome, and gene mutations in specific nuclear transcription pathways and in the genes that are involved in cardiac structure and development. The application of modern molecular biology techniques such as high-throughput nucleotide sequencing and chromosomal array and methylation array all have the potential to reveal more genetic defects linked to CHD. Exploring the genetic defects in CHD pathology will improve our knowledge and understanding about the diverse pathways involved and also about the progression of this disease. Ultimately, this will link to more efficient genetic diagnosis and development of novel preventive therapeutic strategies, as well as gene-targeted clinical management. This review summarizes our current understanding of the molecular basis of normal heart development and the pathophysiology of a wide range of CHD. The risk factors that might account for the high prevalence of CHD within the Arab population and the measures required to be undertaken for conducting research into CHD in Arab countries will also be discussed.