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.

Novel ADAM9 homozygous mutation in a consanguineous Egyptian family with severe cone-rod dystrophy and cataract.

OBJECTIVE: To genetically and phenotypically describe a new ADAM9 homozygous mutation in a consanguineous family from Egypt with autosomal recessive cone-rod dystrophy (arCRD), anterior polar and posterior subcapsular cataract. DESIGN, SETTING AND PARTICIPANTS: The parents and their six children were included. They underwent a complete ophthalmic examination with fundus photography and optical coherence tomography (OCT). INTERVENTION: DNA was extracted from peripheral blood from all family members. Screening for mutations in genes known to be implicated in retinal disorders was done with the IROme, an in-solution enrichment array, followed by high-throughput sequencing. Validation of the results was done by bidirectional Sanger sequencing of ADAM9 exon 14, including exon-intron junctions. Screening of normal controls was done by denaturing high-performance liquid chromatography. RESULTS: arCRD was diagnosed in the mother and two of her children. Bilateral anterior polar and posterior subcapsular cataract was observed in the mother and bilateral dot cataract was diagnosed in three of the four children not affected with arCRD, one of whom also had glaucoma. The characteristics of the arCRD were childhood-onset visual impairment, reorganisation of the retinal pigment epithelium with mid-periphery greyish-white discolouration, attenuated retinal vasculatur and optic disc pallor. A coloboma-like macular lesion was observed in one of the arCRD-affected children. IROme analysis identified a c.1396-2A>G homozygous mutation in the splice acceptor site of intron 13 of ADAM9. This mutation was homozygous in the two children affected by arCRD and in their affected mother. This mutation was heterozygous in the unaffected father and the four unaffected children. CONCLUSIONS AND RELEVANCE: We identified a novel autosomal recessive ADAM9 mutation causing arCRD in a consanguineous Egyptian family. The percentage of arCRD cases caused by mutation in ADAM9 remains to be determined. Few families are reported in the literature to date; hence extensive clinical descriptions of families with ADAM9 mutations are of significant importance.

PAX6 mutations in Egyptian families with aniridia

Methods Ten patients with aniridia from 3 families of Egyptian origin underwent full ophthalmologic, general and neurological examination, and blood drawing. Cerebral MRI was performed in the index case of each family. Genomic DNA was prepared from venous leukocytes and direct sequencing of all the exons and intron-exon junctions of the PAX6 gene was performed after PCR amplification. Results Common features observed in the three families included absence of iris tissue, corneal pannus with different degrees of severity and foveal hypoplasia with severely reduced visual acuity. In families 2 and 3, additional findings such as lens dislocation, lens opacities or polar cataract and glaucoma were observed. We identified two novel (c.170-174delTGGGC [p.L57fs17] and c.475delC [p.R159fs47]) and one known (c.718C>T) PAX6 mutations in the affected members of the 3 families. Systemic and neurological examination was normal in all ten affected patients. Cerebral MRI showed absence of the pineal gland in all three index patients. Severe hypoplasia of the brain anterior commissure was associated to the p.L57fs17mutation, absence of the posterior commissure to both p.R159fs47 and p.R240X, and optic chiasma atrophy and almost complete agenesis of the corpus callosum to p.R240X. Conclusions We identified two novel PAX6 mutations in families with severe aniridia from Northern Egypt, an ethnic group which is not well studied. In addition to common phenotype of aniridia and despite normal neurological examination, absence of the pineal gland was observed in all 3 index patients. The heterogeneity of brain anomalies related to PAX6 mutations is underexplored and is highlighted in this study.

Compound Heterozygous VSX2 Mutation Causing Bilateral Anophthalmia in a Consanguineous Egyptian Family

Purpose: To report the clinical and genetic study of a child with bilateral anophthalmia. Methods: A 14-year-old Egyptian boy, born from consanguineous parents, underwent a general and a full ophthalmological examination. Mutation screen of the A/M genes with recessive inheritance was done stepwise and DNA was analyzed by Sanger sequencing. Results: Bilateral anophthalmia, arachnodactyly of the feet and high arched palate were observed on general examination. The parents were first cousins and healthy. Sequencing analysis revealed a novel compound heterozygous mutation in one of the copy of exon 2 of VSX2 and a possible deletion of at least exon 2 on the other allele. Conclusions: A compound heterozygous VSX2 mutation associated with anophthalmia was identified in a patient from an Egyptian consanguineous family. This report brings the number of VSX2 mutation in anophthalmia/microphthalmia (A/M) to 13. Functional consequences of the reported changes still need to be characterized, as well as the percentage of A/M caused by mutations in the VSX2 gene. This family also shows that despite consanguinity, heterozygous mutations can also happen and one should not restrict the molecular analysis to homozygous mutations.