Backward Error Analysis as a Model of Computation for Numerical Methods

This thesis delineates a generally applicable perspective on numerical meth­ ods for scientific computation called residual-based a posteriori backward er­ ror analysis, based on the concepts of condition, backward error, and residual, pioneered by Turing and Wilkinson. The basic underpinning of this perspec­ tive, that a numerical method’s errors should be analyzable in the same terms as physical and modelling errors, is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily in­ terpretable in the broader context of mathematical modelling. It is applied in this thesis mainly to numerical solution of differential equations. We examine the condition of initial-value problems for ODEs and present a residual-based error control strategy for methods such as Euler’s method, Taylor series meth­ ods, and Runge-Kutta methods. We also briefly discuss solutions of continuous chaotic problems and stiff problems

The Reasonable Effectiveness of Mathematics in the Natural Sciences

One of the most unsettling problems in the history of philosophy examines how mathematics can be used to adequately represent the world. An influential thesis, stated by Eugene Wigner in his paper entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences, claims that the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. Contrary to this view, this thesis delineates and implements a strategy to show that the applicability of mathematics is very reasonable indeed. I distinguish three forms of the problem of the applicability of mathematics, and focus on one I call the problem of uncanny accuracy: Given that the construction and manipulation of mathematical representations is pervaded by uncertainty, error, approximation, and idealization, how can their apparently uncanny accuracy be explained? I argue that this question has found no satisfactory answer because our rational reconstruction of scientific practice has not involved tools rich enough to capture the logic of mathematical modelling. Thus, I characterize a general schema of mathematical analysis of real systems, focusing on the selection of modelling assumptions, on the construction of model equations, and on the extraction of information, in order to address contextually determinate questions on some behaviour of interest. A concept of selective accuracy is developed to explain the way in which qualitative and quantitative solutions should be utilized to understand systems. The qualitative methods rely on asymptotic methods and on sensitivity analysis, whereas the quantitative methods are best understood using backward error analysis. The basic underpinning of this perspective is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. In addition, this perspective is used to discuss the nature of theories, the role of scaling, and the epistemological and semantic aspects of experimentation. In conclusion, we argue for a method of local and global conceptual analysis that goes beyond the reach of the tools standardly used to capture the logic of science; on their basis, the applicability of mathematics finds itself demystified

Les enjeux de la controverse Frege-Hilbert sur les fondements de la géométrie : une étude philosophique sur la logique et les mathématiques

L’auteur entreprend dans ce mémoire de faire une présentation des débats axiologiques de philosophie de la logique sous-jacents à la controverse opposant Frege et Hilbert sur les fondements de la géométrie. Contre le parti pris philosophique selon lequel la logique est une discipline achevée, l’auteur entreprend une mise en contexte des positions de Frege et Hilbert afin de montrer que dans leur conception de la logique se trouvent des paradigmes incommensurables, résultant de l’influence de traditions philosophiques et scientifiques diverses. Dans cette perspective, Frege est le défenseur de la vision traditionnelle de la logique comme medium universel de la science, tel qu’incarnée dans la géométrie euclidienne. La logique symbolique de Frege est ainsi vue comme la mise en oeuvre de moyens raffinés pour lutter contre la (( perversion des sciences )) ayant lieu au 19ième siècle et pour la défense de la vision traditionnelle de la science. à l’opposé, l’approche métathéorique de Hilbert représente la conception moderne dite algébrique de la logique telle que développée au 19ième sous l’influence des métamathématiques, et certains rapprochements avec les conceptions (( model-theoretic )) et catégorielles de la logique viennent appuyer cette thèse.This memoir presents some axiological debates of philosophy of logic underlying the Frege-Hilbert controversy on the foundations of geometry. Against the philosophical bias according to which logic is an achieved discipline, a contextualized presentation of the respective positions of Frege and Hilbert is done in order to show that incommensurable paradigms are found in their view of logic, that is due to the influence of various philosophical and scientific traditions. From this standpoint, Frege is the defender of the traditionalist view of logic as the universal medium of science, as embodied in Euclidean geometry. In this perspective, Frege’s symbolic logic is seen as the achievement of a refined means to counter the 19th-century perversion of science with the purpose of defending the traditional conception of the role of science. On the other hand, Hilbert’s metatheoretical approach represents the so-called algebraic modern conception of logic as developed in the 19th century under the influence of metamathematics. Following this, parallels between Hilbert’s approach and the model-theoretical and categorical conceptions of logic are drawn to show their proximity

Semantic Layering and the Success of Mathematical Sciences

What are the pillars on which the success of modern science rest? Although philosophers have much discussed what is behind science's success, this paper argues that much of the discussion is misdirected. The extant literature rightly regards the semantic and inferential tools of formal logic and probability theory as pillars of scientific rationality, in the sense that they reveal the justificatory structure of important aspects of scientific practice. As key elements of our rational reconstruction toolbox, they make a fundamental contribution to our understanding of the success of science. At the same time, any science, however exact, is dominated by approximation, error, and uncertainty, a fact that makes one wonder how science can be so successful. This paper articulates and illustrates general themes---e.g., that truth-preserving arguments often fail to preserve approximate truth---that highlight the need for additional semantic resources. Thus, our proposal is that persistent failures to unravel the reasons behind the success of science in the face of pervasive error and uncertainty should be attributed to an insufficiently rich way of rationally reconstructing scientific and mathematical knowledge. What is missing? This paper claims that there is a third formal method of reasoning that constitutes a distinct pillar on which rests the success of science, namely, perturbation theory. The paper outlines how the representational and inferential tools of perturbation theory differ from those of logic and probability theory, and how they enable us to understand the apparently elusive aspects of the success of science. However, compared to its peers, perturbative reasoning has not received the attention it deserves. As the paper explains, this partly results from the circumstances in which perturbation theory is taught, and partly from the fact that perturbation theory first appears to be a vaguely related collection of methods offering no systematic semantic insight. In an attempt to show that this first impression is wrong, this paper presents its contribution to the semantic dimension of scientific representation and inference in terms of what I call ``semantic layering.'

Idealization in Scientific Explanation

Many phenomena pose interesting “fundamental” questions for both physics and philosophy of science. Understanding and explanation often seem to require non-Galilean, essential idealizations. But idealizations are false. This fact suggests that we need to give up on the view that truth is a necessary condition for explanation

Sociologie de la santé et du monde médical

Nicolas Dodier, directeur d’étudesJanine Barbot, chargée de recherche à l’INSERMEmannuelle Fillion, doctorante Nous sommes partis d’une question nouvelle : comment sont traitées les défaillances de la médecine ? Cette question a été tout d’abord examinée en prenant appui sur le cadre d’analyse des transformations des institutions médicales, élaboré au cours des années précédentes. Nous avons ainsi montré comment, dans chacune des grandes formes politiques du monde médical (la tradition cliniq..

Pushing the limits of surface-enhanced raman spectroscopy (SERS) with deep learning : identification of multiple species with closely related molecular structures

Raman spectroscopy is a non-destructive and label-free molecular identification technique capable of producing highly specific spectra with various bands correlated to molecular structure. Moreover, the enhanced detection sensitivity offered by Surface-Enhanced Raman spectroscopy (SERS) allows analyzing mixtures of related chemical species in a relatively short measurement time. Combining SERS with deep learning algorithms allows in some cases to increase detection and classification capabilities even further. The present study evaluates the potential of applying deep learning algorithms to SERS spectroscopy to differentiate and classify different species of bile acids, a large family of molecules with low Raman cross sections and molecular structures that often differ by a single hydroxyl group. Moreover, the study of these molecules is of interest for the medical community since they have distinct pathological roles and are currently viewed as potential markers of gut microbiome imbalances. A Convolutional Neural Network (CNN) model was developed and used to classify SERS spectra from five bile acid species. The model succeeded in identifying the five analytes despite very similar molecular structures and was found to be reliable even at low analyte concentrations

Oral health-related quality of life after prosthodontic treatment for patients with partial edentulism: A systematic review and meta-analysis

Statement of problem. Clinicians are currently unable to quantify the psychosocial, functional, and esthetic effects of prosthetic interventions to replace teeth. Understanding the effects of treatment to replace teeth on oral health-related quality of life (OHRQoL) is important for informed consent. A systematic review of the evidence of OHRQoL improvements with prosthodontic tooth replacement and a comparison of outcomes between treatment modalities is therefore indicated. Purpose. The purpose of this systematic review was to examine the OHRQoL of patients with partial edentulism after different dental prosthetic treatments. Material and methods. Electronic database and manual searches were conducted to identify cohort studies and clinical trials reporting on the OHRQoL of individuals receiving implantsupported crowns (ISCs), implant-supported fixed dental prostheses (IFDPs), implant-supported removable dental prostheses (IRDPs), tooth-supported fixed dental prostheses (TFDPs), and removable partial dentures (RPDs). Two reviewers independently conducted article selection, data extraction, and quality assessment. Random-effects models were used to compare OHRQoL change scores (standardised mean change [SMC], 95% confidence intervals [CI]). Results. Of the 2147 identified studies, 2 randomized controlled trials and 21 cohort studies met the inclusion criteria. Overall, studies were of low or moderate risk of bias. Pooled mean OHRQoL change ≤9 months was 15.3 for TFDP, 11.9 for RPD, and 14.9 for IFDP. Pooled SMC OHRQoL change >9 months was 13.2 for TFDP and 15.8 for IFDP. Direct comparisons ≤9 months between TFDP against IFDP and RPD against IFDP significantly favored IFDP in both cases. Conclusions. TFDP and IFDP had short- and long-term positive effects on OHRQoL. RPDs positively affected OHRQoL in the short-term. IFDP showed greater short-term improvement in OHRQoL than RPD and TFDP. CLINICAL IMPLICATIONS Clinicians may advise patients with multiple missing teeth that implant-supported FDPs make greater improvements to OHRQoL than tooth-supported FDPs or RPDs. They should also inform patients that, while RPDs improve OHRQoL in the short term, these effects may be less noticeable in the medium term