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.

Forensic examination of ink by high-performance thin layer chromatography - The United States Secret Service Digital Ink Library

Forensic examinations of ink have been performed since the beginning of the 20th century. Since the 1960s, the International Ink Library, maintained by the United States Secret Service, has supported those analyses. Until 2009, the search and identification of inks were essentially performed manually. This paper describes the results of a project designed to improve ink samples' analytical and search processes. The project focused on the development of improved standardization procedures to ensure the best possible reproducibility between analyses run on different HPTLC plates. The successful implementation of this new calibration method enabled the development of mathematical algorithms and of a software package to complement the existing ink library.

Positional scanning-synthetic peptide library-based analysis of self- and pathogen-derived peptide cross-reactivity with tumor-reactive Melan-A-specific CTL.

Synthetic combinatorial peptide libraries in positional scanning format (PS-SCL) have recently emerged as a useful tool for the analysis of T cell recognition. This includes identification of potentially cross-reactive sequences of self or pathogen origin that could be relevant for the understanding of TCR repertoire selection and maintenance, as well as of the cross-reactive potential of Ag-specific immune responses. In this study, we have analyzed the recognition of sequences retrieved by using a biometric analysis of the data generated by screening a PS-SCL with a tumor-reactive CTL clone specific for an immunodominant peptide from the melanocyte differentiation and tumor-associated Ag Melan-A. We found that 39% of the retrieved peptides were recognized by the CTL clone used for PS-SCL screening. The proportion of peptides recognized was higher among those with both high predicted affinity for the HLA-A2 molecule and high predicted stimulatory score. Interestingly, up to 94% of the retrieved peptides were cross-recognized by other Melan-A-specific CTL. Cross-recognition was at least partially focused, as some peptides were cross-recognized by the majority of CTL. Importantly, stimulation of PBMC from melanoma patients with the most frequently recognized peptides elicited the expansion of heterogeneous CD8(+) T cell populations, one fraction of which cross-recognized Melan-A. Together, these results underline the high predictive value of PS-SCL for the identification of sequences cross-recognized by Ag-specific T cells.

ROBETH: un logiciel pour les procédés statistiques robustes [ROBETH: a library for robust statistical procedures].

In the past 20 years the theory of robust estimation has become an important topic of mathematical statistics. We discuss here some basic concepts of this theory with the help of simple examples. Furthermore we describe a subroutine library for the application of robust statistical procedures, which was developed with the support of the Swiss National Science Foundation.

Automated scoring of AFLPs using RawGeno v 2.0, a free R CRAN library.

Amplified Fragment Length Polymorphisms (AFLPs) are a cheap and efficient protocol for generating large sets of genetic markers. This technique has become increasingly used during the last decade in various fields of biology, including population genomics, phylogeography, and genome mapping. Here, we present RawGeno, an R library dedicated to the automated scoring of AFLPs (i.e., the coding of electropherogram signals into ready-to-use datasets). Our program includes a complete suite of tools for binning, editing, visualizing, and exporting results obtained from AFLP experiments. RawGeno can either be used with command lines and program analysis routines or through a user-friendly graphical user interface. We describe the whole RawGeno pipeline along with recommendations for (a) setting the analysis of electropherograms in combination with PeakScanner, a program freely distributed by Applied Biosystems; (b) performing quality checks; (c) defining bins and proceeding to scoring; (d) filtering nonoptimal bins; and (e) exporting results in different formats.

Combinatorial peptide library-based identification of peptide ligands for tumor-reactive cytolytic T lymphocytes of unknown specificity.

A novel approach for the identification of tumor antigen-derived sequences recognized by CD8(+) cytolytic T lymphocytes (CTL) consists in using synthetic combinatorial peptide libraries. Here we have screened a library composed of 3.1 x 10(11) nonapeptides arranged in a positional scanning format, in a cytotoxicity assay, to search the antigen recognized by melanoma-reactive CTL of unknown specificity. The results of this analysis enabled the identification of several optimal peptide ligands, as most of the individual nonapeptides deduced from the primary screening were efficiently recognized by the CTL. The results of the library screening were also analyzed with a mathematical approach based on a model of independent and additive contribution of individual amino acids to antigen recognition. This biometrical data analysis enabled the retrieval, in public databases, of the native antigenic peptide SSX-2(41-49), whose sequence is highly homologous to the ones deduced from the library screening, among the ones with the highest stimulatory score. These results underline the high predictive value of positional scanning synthetic combinatorial peptide library analysis and encourage its use for the identification of CTL ligands.