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.

Identifying the neural correlates of executive functions in early cerebral microangiopathy: a combined VBM and DTI study.

Cerebral microangiopathy (CMA) has been associated with executive dysfunction and fronto-parietal neural network disruption. Advances in magnetic resonance imaging allow more detailed analyses of gray (e.g., voxel-based morphometry-VBM) and white matter (e.g., diffusion tensor imaging-DTI) than traditional visual rating scales. The current study investigated patients with early CMA and healthy control subjects with all three approaches. Neuropsychological assessment focused on executive functions, the cognitive domain most discussed in CMA. The DTI and age-related white matter changes rating scales revealed convergent results showing widespread white matter changes in early CMA. Correlations were found in frontal and parietal areas exclusively with speeded, but not with speed-corrected executive measures. The VBM analyses showed reduced gray matter in frontal areas. All three approaches confirmed the hypothesized fronto-parietal network disruption in early CMA. Innovative methods (DTI) converged with results from conventional methods (visual rating) while allowing greater spatial and tissue accuracy. They are thus valid additions to the analysis of neural correlates of cognitive dysfunction. We found a clear distinction between speeded and nonspeeded executive measures in relationship to imaging parameters. Cognitive slowing is related to disease severity in early CMA and therefore important for early diagnostics.

Targetting of the ventro-intermediate nucleus using ultra-high field (7T) MRI for gamma knife surgery purposes: A pilot in vivo study on healthy subjects.

INTRODUCTION: Gamma Knife surgery (GKS) is a non-invasive neurosurgical stereotactic procedure, increasingly used as an alternative to open functional procedures. This includes targeting of the ventro-intermediate nucleus of the thalamus (e.g. Vim) for tremor. We currently perform an indirect targeting, as the Vim is not visible on current 3Tesla MRI acquisitions. Our objective was to enhance anatomic imaging (aiming at refining the precision of anatomic target selection by direct visualisation) in patients treated for tremor with Vim GKS, by using high field 7T MRI. MATERIALS AND METHODSH: Five young healthy subjects were scanned on 3 (T1-w and diffusion tensor imaging) and 7T (high-resolution susceptibility weighted images (SWI)) MRI in Lausanne. All images were further integrated for the first time into the Gamma Plan Software(®) (Elekta Instruments, AB, Sweden) and co-registered (with T1 was a reference). A simulation of targeting of the Vim was done using various methods on the 3T images. Furthermore, a correlation with the position of the found target with the 7T SWI was performed. The atlas of Morel et al. (Zurich, CH) was used to confirm the findings on a detailed analysis inside/outside the Gamma Plan. RESULTS: The use of SWI provided us with a superior resolution and an improved image contrast within the basal ganglia. This allowed visualization and direct delineation of some subgroups of thalamic nuclei in vivo, including the Vim. The position of the target, as assessed on 3T, perfectly matched with the supposed one of the Vim on the SWI. Furthermore, a 3-dimensional model of the Vim-target area was created on the basis of the obtained images. CONCLUSION: This is the first report of the integration of SWI high field MRI into the LGP, aiming at the improvement of targeting validation of the Vim in tremor. The anatomical correlation between the direct visualization on 7T and the current targeting methods on 3T (e.g. quadrilatere of Guyot, histological atlases) seems to show a very good anatomical matching. Further studies are needed to validate this technique, both by improving the accuracy of the targeting of the Vim (potentially also other thalamic nuclei) and to perform clinical assessment.

Mutism and Amnesia following High-Voltage Electrical Injury: Psychogenic Symptomatology Triggered by Organic Dysfunction?

Background: Mutism and dense retrograde amnesia are found both in organic and dissociative contexts. Moreover, dissociative symptoms may be modulated by right prefrontal activity. A single case, M.R., developed left hemiparesis, mutism and retrograde amnesia after a high-voltage electric shock without evidence of lasting brain lesions. M.R. suddenly recovered from his mutism following a mild brain trauma 2 years later. Methods: M.R.'s neuropsychological pattern and anatomoclinical correlations were studied through (i) language and memory assessment to characterize his deficits, (ii) functional neuroimaging during a standard language paradigm, and (iii) assessment of frontal and left insular connectivity through diffusion tractography imaging and transcranial magnetic stimulation. A control evaluation was repeated after recovery. Findings: M.R. recovered from the left hemiparesis within 90 days of the accident, which indicated a transient right brain impairment. One year later, neurobehavioral, language and memory evaluations strongly suggested a dissociative component in the mutism and retrograde amnesia. Investigations (including MRI, fMRI, diffusion tensor imaging, EEG and r-TMS) were normal. Twenty-seven months after the electrical injury, M.R. had a very mild head injury which was followed by a rapid recovery of speech. However, the retrograde amnesia persisted. Discussion: This case indicates an interaction of both organic and dissociative mechanisms in order to explain the patient's symptoms. The study also illustrates dissociation in the time course of the two different dissociative symptoms in the same patient.