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.

Fum i miralls: trobant l'ordre en un món caòtic

El calidoscopi és una joguina que converteix fragments de vidre acolorit en patrons infinitament variats mitjançant l'ús de miralls. Hi ha quelcom satisfactori en els patrons –els necessitem per crear significat, i trobem bellesa en la seva simetria. No obstant, els sistemes complexos desobeeixen les fórmules simples. Per exemple, és impossible descriure una cosa tan prosaica com el comportament del fum mentre ascendeix fent servir les matemàtiques clàssiques. Agafant en préstec idees de diferents disciplines, podem considerar l'alfabetització informacional com una invitació per als nostres estudiants per jugar amb la imprevisible varietat de l'experiència humana i descobrir patrons que els resultin plens de significat.

Symmetry for a Dirichlet-Neumann problem arising in water waves

"Vegeu el resum a l'inici del document del fitxer adjunt."

Bilateral symmetry of radial pulse in high-level tennis players: implications for the validity of central aortic pulse wave analysis.

BACKGROUND: Reconstruction of the central aortic pressure wave from the noninvasive recording of the radial pulse with applanation tonometry has become a standard tool in the field of hypertension. It is not presently known whether recording the radial pulse on the dominant or the nondominant side has any effect on such reconstruction. METHOD: We carried out radial applanation tonometry on both forearms in young, healthy, male volunteers, who were either sedentary (n = 11) or high-level tennis players (n = 10). The purpose of including tennis players was to investigate individuals with extreme asymmetry between the dominant and nondominant upper limb. RESULTS: In the sedentary individuals, forearm circumference and handgrip strength were slightly larger on the dominant (mean +/- SD respectively 27.9 +/- 1.5 cm and 53.8 +/- 10 kg) than on nondominant side (27.3 +/- 1.6 cm, P < 0.001 vs. dominant, and 52.1 +/- 11 kg, P = NS). In the tennis players, differences between sides were more conspicuous (forearm circumference: dominant 28.0 +/- 1.7 cm nondominant 26.4 +/- 1.5 cm, P < 0.001; handgrip strength 61.4 +/- 10.8 vs. 53.4 +/- 9.7 kg, P < 0.001). We found that in both sedentary individuals and tennis players, the radial pulse had identical shape on both sides and, consequently, the reconstructed central aortic pressure waveforms, as well as derived indices of central pulsatility, were not dependent on the side where applanation tonometry was carried out. CONCLUSION: Evidence from individuals with maximal asymmetry of dominant vs. nondominant upper limb indicates that laterality of measurement is not a methodological issue for central pulse wave analysis carried out with radial applanation tonometry.

Broken symmetry.

Review of the book : Lucifer's legacy, the meaning of asymmetry, by F. Close, Oxford University Press, Oxford, UK

Evaluation of the 3D symmetry of the knees when walking using na ambulatory inertia system: results 14 months after ACL Plasty.

Purpose of the study: Reconstruction of the anterior cruciate ligament (ACL) controls laxity but does not enable restoration of strictly normal 3D kinematics. The purpose of this study was to compare the kinematics of the pathological knee with that of the healthy knee after ACL plasty. This study applied a new ambulatory system using miniature captors. Material and method: Five patients with an isolated injury of the ACL participated in this study. The patients were assessed after injury (T1), at five months (T2), and at 14 months (T3) after surgery. The assessment included laxity (KT-1000), the IKDC score and the Lysholm score. The 3D angles of the knees were measured when walking 30 m on flat ground using a system composed of to small inertia units (3D accelerometer and 3D gyroscope) and a portable recorder. Functional settings were optimised and validating to ensure easy precise measurement of the 3D angles. Symmetry of the two knees was quantified using a symmetry index (SI) (difference in amplitude normalised in relation to mean amplitude) and the correlation coefficient CC. Results: Clinical indicators improved during the follow-up (IKDC T1: 3C, 2C; T2: 5B; T3: 2A, 3B; subjective IKD: 53-95; Lysholm 67-96). Mean laxity improved from 8.6m to 2.5 mm. The gait analysis showed increased symmetry in terms of amplitude for flexion-extension (SI: −17% at T1, −1% at T2, 1% at T3), and an increase in symmetry in terms of the rotation signature (CC: 0.16 at T1, 0.99 at T2, 0.99 at T3). There was no trend to varus-valgus. Discussion: This study demonstrates the clinical application of the new ambulatory system for measuring 3D angles of the knee joint. Joint symmetry increased after ACL plasty but still showed some perturbation at 14 months. The results observed here are in agreement with the literature. Other patients and other types of gait are being analysed. Conclusion: This portable system allows gait analysis outside the laboratory, before and after ACL injury. It is very useful for follow-up after surgery.

Desenvolupament de codis numèrics per a la resolució de fluxos turbulents en ordinadors paral·lels emprant discretitzacions tipus symmetry-preserving

Projecte de recerca elaborat a partir d’una estada a la University of Groningen, Holanda, entre 2007 i 2009. La simulació directa de la turbulència (DNS) és una eina clau dins de la mecànica de fluids computacional. Per una banda permet conèixer millor la física de la turbulència i per l'altra els resultats obtinguts són claus per el desenvolupament dels models de turbulència. No obstant, el DNS no és una tècnica vàlida per a la gran majoria d'aplicacions industrials degut al elevats costos computacionals. Per tant, és necessari cert grau de modelització de la turbulència. En aquest context, s'han introduïts importants millores basades en la modelització del terme convectiu (no lineal) emprant symmetry-preserving regularizations. En tracta de modificar adequadament el terme convectiu a fi de reduir la producció d'escales més i més petites (vortex-stretching) tot mantenint tots els invariants de les equacions originals. Fins ara, aquest models s'han emprat amb èxit per nombres de Rayleigh (Ra) relativament elevats. En aquest punt, disposar de resultats DNS per a configuracions més complexes i nombres de Ra més elevats és clau. En aquest contexte, s'han dut a terme simulacions DNS en el supercomputador MareNostrum d'una Differentially Heated Cavity amb Ra=1e11 i Pr=0.71 durant el primer any dels dos que consta el projecte. A més a més, s'ha adaptat el codi a fi de poder simular el fluxe al voltant d'un cub sobre una pared amb Re=10000. Aquestes simulacions DNS són les més grans fetes fins ara per aquestes configuracions i la seva correcta modelització és un gran repte degut la complexitat dels fluxes. Aquestes noves simulacions DNS estan aportant nous coneixements a la física de la turbulència i aportant resultats indispensables per al progrés de les modelitzacións tipus symmetry-preserving regularization.

Symmetry-breaking in cumulative measures of shapes of polymer models.

Using numerical simulations we investigate shapes of random equilateral open and closed chains, one of the simplest models of freely fluctuating polymers in a solution. We are interested in the 3D density distribution of the modeled polymers where the polymers have been aligned with respect to their three principal axes of inertia. This type of approach was pioneered by Theodorou and Suter in 1985. While individual configurations of the modeled polymers are almost always nonsymmetric, the approach of Theodorou and Suter results in cumulative shapes that are highly symmetric. By taking advantage of asymmetries within the individual configurations, we modify the procedure of aligning independent configurations in a way that shows their asymmetry. This approach reveals, for example, that the 3D density distribution for linear polymers has a bean shape predicted theoretically by Kuhn. The symmetry-breaking approach reveals complementary information to the traditional, symmetrical, 3D density distributions originally introduced by Theodorou and Suter.

Reduction of symmetry in grandite solid solution

The results of a crystal structure refinement of an anisotropic grandite garnet specimen with composition Gro36-4 And63-6 are given. The structure obtained has orthorrombic symmetry (space group Fddd) and is compared with similar results obtained by other authors. In all cases the reduction of symmetry is due to the ordering of Fe3+ and Al in octahedral sites. Non cubic structures of grandites are discussed in connection with optical, morphological an grou-th features of these minerals.

Density dependence of the symmetry free energy of hot nuclei

The density and excitation energy dependence of symmetry energy and symmetry free energy for finite nuclei are calculated microscopically in a microcanonical framework, taking into account thermal and expansion effects. A finite-range momentum and density-dependent two-body effective interaction is employed for this purpose. The role of mass, isospin, and equation of state (EOS) on these quantities is also investigated; our calculated results are in consonance with the available experimental data.