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.

The use of mannan antigen and anti-mannan antibodies in the diagnosis of invasive candidiasis: recommendations from the Third European Conference on Infections in Leukemia.

INTRODUCTION: Timely diagnosis of invasive candidiasis (IC) remains difficult as the clinical presentation is not specific and blood cultures lack sensitivity and need a long incubation time. Thus, non-culture-based methods for diagnosing IC have been developed. Mannan antigen (Mn) and anti-mannan antibodies (A-Mn) are present in patients with IC. On behalf of the Third European Conference on Infections in Leukemia, the performance of these tests was analysed and reviewed. METHODS: The literature was searched for studies using the commercially available sandwich enzyme-linked immunosorbent assays (Platelia™, Bio-Rad Laboratories, Marnes-la-Coquette, France) for detecting Mn and A-Mn in serum. The target condition of this review was IC defined according to 2008 European Organization for Research and Treatment of Cancer/Mycoses Study Group criteria. Sensitivity, specificity and diagnostic odds ratios (DOR) were calculated for Mn, A-Mn and combined Mn/A-Mn testing. RESULTS: Overall, 14 studies that comprised 453 patients and 767 controls were reviewed. The patient populations included in the studies were mainly haematological and cancer cases in seven studies and mainly intensive care unit and surgery cases in the other seven studies. All studies but one were retrospective in design. Mn sensitivity was 58% (95% confidence interval [CI], 53-62); specificity, 93% (95% CI, 91-94) and DOR, 18 (95% CI 12-28). A-Mn sensitivity was 59% (95% CI, 54-65); specificity, 83% (95% CI, 79-97) and DOR, 12 (95% CI 7-21). Combined Mn/A-Mn sensitivity was 83% (95% CI, 79-87); specificity, 86% (95% CI, 82-90) and DOR, 58 (95% CI 27-122). Significant heterogeneity of the studies was detected. The sensitivity of both Mn and A-Mn varied for different Candida species, and it was the highest for C. albicans, followed by C. glabrata and C. tropicalis. In 73% of 45 patients with candidemia, at least one of the serological tests was positive before the culture results, with mean time advantage being 6 days for Mn and 7 days for A-Mn. In 21 patients with hepatosplenic IC, 18 (86%) had Mn or A-Mn positive test results at a median of 16 days before radiological detection of liver or spleen lesions. CONCLUSIONS: Mn and A-Mn are useful for diagnosis of IC. The performance of combined Mn/A-Mn testing is superior to either Mn or A-Mn testing.

Setting boundaries: brain dynamics of modal and amodal illusory shape completion in humans.

Normal visual perception requires differentiating foreground from background objects. Differences in physical attributes sometimes determine this relationship. Often such differences must instead be inferred, as when two objects or their parts have the same luminance. Modal completion refers to such perceptual "filling-in" of object borders that are accompanied by concurrent brightness enhancement, in turn termed illusory contours (ICs). Amodal completion is filling-in without concurrent brightness enhancement. Presently there are controversies regarding whether both completion processes use a common neural mechanism and whether perceptual filling-in is a bottom-up, feedforward process initiating at the lowest levels of the cortical visual pathway or commences at higher-tier regions. We previously examined modal completion (Murray et al., 2002) and provided evidence that the earliest modal IC sensitivity occurs within higher-tier object recognition areas of the lateral occipital complex (LOC). We further proposed that previous observations of IC sensitivity in lower-tier regions likely reflect feedback modulation from the LOC. The present study tested these proposals, examining the commonality between modal and amodal completion mechanisms with high-density electrical mapping, spatiotemporal topographic analyses, and the local autoregressive average distributed linear inverse source estimation. A common initial mechanism for both types of completion processes (140 msec) that manifested as a modulation in response strength within higher-tier visual areas, including the LOC and parietal structures, is demonstrated, whereas differential mechanisms were evident only at a subsequent time period (240 msec), with amodal completion relying on continued strong responses in these structures.

Systematic evaluation of risk factors for diagnostic delay in inflammatory bowel disease.

The diagnosis of inflammatory bowel disease (IBD), comprising Crohn's disease (CD) and ulcerative colitis (UC), continues to present difficulties due to unspecific symptoms and limited test accuracies. We aimed to determine the diagnostic delay (time from first symptoms to IBD diagnosis) and to identify associated risk factors. A total of 1591 IBD patients (932 CD, 625 UC, 34 indeterminate colitis) from the Swiss IBD cohort study (SIBDCS) were evaluated. The SIBDCS collects data on a large sample of IBD patients from hospitals and private practice across Switzerland through physician and patient questionnaires. The primary outcome measure was diagnostic delay. Diagnostic delay in CD patients was significantly longer compared to UC patients (median 9 versus 4 months, P < 0.001). Seventy-five percent of CD patients were diagnosed within 24 months compared to 12 months for UC and 6 months for IC patients. Multivariate logistic regression identified age <40 years at diagnosis (odds ratio [OR] 2.15, P = 0.010) and ileal disease (OR 1.69, P = 0.025) as independent risk factors for long diagnostic delay in CD (>24 months). In UC patients, nonsteroidal antiinflammatory drug (NSAID intake (OR 1.75, P = 0.093) and male gender (OR 0.59, P = 0.079) were associated with long diagnostic delay (>12 months). Whereas the median delay for diagnosing CD, UC, and IC seems to be acceptable, there exists a long delay in a considerable proportion of CD patients. More public awareness work needs to be done in order to reduce patient and doctor delays in this target population.