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.

Systematic assessment of items influencing Crohn's disease patient's preferences in selecting an anti-TNF agent (Choose TNF trial)

Background and Aims: The three anti-TNF agents infliximab (IFX), adalimumab (ADA) andcertolizumab pegol (CZP) have demonstrated similar efficacy in induction and maintenanceof response and remission in Crohn's disease (CD) treatment. Given the comparability ofthese drugs, patient's preferences may influence the choice of the product. However, dataon patient's preferences for choosing anti-TNF agents are lacking. We therefore aimed toassess the CD patient's appraisal to select the drug of his choice and to identify factorsguiding this decision.Methods: A prospective survey among anti-TNF-naive CD patientswas performed. Patients were provided a description of the three anti-TNF agents focusingon indication, application mode (s.c. vs. i.v.), application time intervals, setting of application(hospital vs. private practice vs. patient's home), average time to apply the medication permonth, typical side effects, and the scientific evidence of efficacy and safety available for everydrug. Patients answered a questionnaire consisting of 17 questions, covering demographic,disease-specific, and medication data.Results: Hundred patients (47f/53m, mean age 45±16years) completed the questionnaire. Disease duration was <1year in 7%, 1-5 years in 31%,and >5 years in 62% of patients. Disease location was ileal in 33%, colonic in 40%, andileocolonic in 27%. Disease phenotype was inflammatory in 68%, stenosing in 29%, andinternally fistulizing in 3% of patients. Additionally, 20% had perianal fistulizing disease.Patients were already treated with the following drugs: mesalamines 61%, budesonide 44%,prednisone 97%, thiopurines 78%, methotrexate 16%. In total, 30% had already heardabout IFX, 20% about ADA, and 11% about CZP. Thirty-six percent voted for treatmentwith ADA, 28% for CZP, and 25% for IFX, whereas 11% were undecided. The followingfactors influenced the patient's decision for choosing a specific anti-TNF drug (severalanswers possible): side effects 76%, physician's recommendation 66%, application mode54%, efficacy experience 52%, time to spend for therapy 27%, patient's recommendations21%, interactions with other medications 12%. The single most important factor for choosinga specific anti-TNF was (1 answer): side effect profile 35%, physician's recommendation22%, efficacy experience 21%, application mode 13%, patient's recommendations 5%, timespent for therapy 3%, interaction with other medications 1%.Conclusions: The majority ofpatients preferred anti-TNF syringes to infusions. The safety profile of the drugs and thephysician's recommendation are major factors influencing the patient's choice for a specificanti-TNF drug. Patient's issues about safety and lifestyle habits should be taken into accountwhen prescribing specific anti-TNF formulations.

Development of a pediatric eosinophilic esophagitis activity index (ped-EEsAI): International experts propose dysphagia, whitish exudates on endoscopy , and intraepithelial eosinophili count as major items related to EoE activity

Background and Aims: The international EEsAI study group aims to develop, validate and evaluate the first pediatric EoE activity index (ped-EEsAI). We report on results of phase 1, which aims to generate candidate items. Methods: This study involves 3 phases: (1) item generation, (2) index derivation and testing on a first patient cohort, and (3) validation in a second cohort. In phase 1, item generation, weighting and reduction are achieved through a Delphi process with an international EoE expert group. The experts proposed and ranked candidate items on a 7-point Likert scale (0 = no, 6 = perfect relationship with EoE activity). Results: 23 international EoE experts proposed and ranked 39 items (20 clinical, 6 endoscopic, 8 histologic, 5 laboratory items). Rank order for clinical items: dysphagia related to food consistencies (median 5, range 2-6), severity of dysphagia (5, 3-6), frequency of dysphagia episodes (5, 3-6), regurgitation and vomiting (4, 2-5), response to dietary restrictions (4, 1-6); endoscopic items: whitish exudates (5, 3-6), furrowing (4, 3-6), corrugated rings (4, 2-6), linear shearing (4, 2-6), strictures (3, 2-6); histologic items: intraepithelial eosinophil count (5, 4-6), lamina propria fibrosis (3, 2-6), basal layer enlargement (3, 1-5); laboratory items: % blood eosinophils (3, 0-5). Conclusions: These items will now be reduced in further Delphi rounds, tested on a cohort of 100 pediatric EoE patients and validated in a second independent cohort, resulting in a robust, broadly accepted disease activity index for use in clinical trials and daily care.

Graphical probabilistic analysis of the combination of items of evidence

Unlike the evaluation of single items of scientific evidence, the formal study and analysis of the jointevaluation of several distinct items of forensic evidence has to date received some punctual, ratherthan systematic, attention. Questions about the (i) relationships among a set of (usually unobservable)propositions and a set of (observable) items of scientific evidence, (ii) the joint probative valueof a collection of distinct items of evidence as well as (iii) the contribution of each individual itemwithin a given group of pieces of evidence still represent fundamental areas of research. To somedegree, this is remarkable since both, forensic science theory and practice, yet many daily inferencetasks, require the consideration of multiple items if not masses of evidence. A recurrent and particularcomplication that arises in such settings is that the application of probability theory, i.e. the referencemethod for reasoning under uncertainty, becomes increasingly demanding. The present paper takesthis as a starting point and discusses graphical probability models, i.e. Bayesian networks, as frameworkwithin which the joint evaluation of scientific evidence can be approached in some viable way.Based on a review of existing main contributions in this area, the article here aims at presentinginstances of real case studies from the author's institution in order to point out the usefulness andcapacities of Bayesian networks for the probabilistic assessment of the probative value of multipleand interrelated items of evidence. A main emphasis is placed on underlying general patterns of inference,their representation as well as their graphical probabilistic analysis. Attention is also drawnto inferential interactions, such as redundancy, synergy and directional change. These distinguish thejoint evaluation of evidence from assessments of isolated items of evidence. Together, these topicspresent aspects of interest to both, domain experts and recipients of expert information, because theyhave bearing on how multiple items of evidence are meaningfully and appropriately set into context.

Development of an activity index for adult eosinophilic esophagitis: international experts propose dysphagia, withish exudates on endoscopy and peripheral eosinophilia as items with closest association to EoE activity

Background and Aims: Eosinophilic Esophagitis (EoE) is reported with increasing frequency over the last two decades. However, it is still unknown whether this reflects a true increase in incidence or just an increased awareness by gastroenterologists. Therefore, we evaluated the incidence and cumulative prevalence of EoE in Olten county over the last 20 years. Methods: Olten county is an area of approximately 91,000 inhabitants without pronounced demographic changes in the last two decades. EoE evaluation is based upon two gastroenterology centers and one pathology center. No public programs for increased EoE awareness were implemented in this region. All EoE patients diagnosed from 1989 to 2009 were entered prospectively into the Olten county database. Results: Fourty-six patients (76% males, mean age 41±16 yrs) were diagnosed with EoE from 1989 to 2009. Ninety-four percent presented with dysphagia. In 70% of the patients concomitant allergies were found. The number of upper endoscopies per year was stable during the entire observation period. An average annual incidence rate of 2/100,000 was found (range 0-8) with a marked increase in the period from 2001 to 2009. A current cumulative EoE prevalence of 43/100,000 inhabitants was calculated. The mean diagnostic delay (time from first symptoms to diagnosis) was 4.3 years from 1989 to 1998 and 4.8 years from 1999 to 2009. Conclusions: Over the last 20 years, a significant increase in EoE incidence was found in a stable indicator region of Switzerland. The constant rate of upper endoscopies, the constant diagnostic delay, as well as the lack of EoE awareness programs in Olten county indicate a true increase in EoE incidence.