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.

Mouse alarm pheromone shares structural similarity with predator scents.

Sensing the chemical warnings present in the environment is essential for species survival. In mammals, this form of danger communication occurs via the release of natural predator scents that can involuntarily warn the prey or by the production of alarm pheromones by the stressed prey alerting its conspecifics. Although we previously identified the olfactory Grueneberg ganglion as the sensory organ through which mammalian alarm pheromones signal a threatening situation, the chemical nature of these cues remains elusive. We here identify, through chemical analysis in combination with a series of physiological and behavioral tests, the chemical structure of a mouse alarm pheromone. To successfully recognize the volatile cues that signal danger, we based our selection on their activation of the mouse olfactory Grueneberg ganglion and the concomitant display of innate fear reactions. Interestingly, we found that the chemical structure of the identified mouse alarm pheromone has similar features as the sulfur-containing volatiles that are released by predating carnivores. Our findings thus not only reveal a chemical Leitmotiv that underlies signaling of fear, but also point to a double role for the olfactory Grueneberg ganglion in intraspecies as well as interspecies communication of danger.

Identification of pyridine analogs as new predator-derived kairomones.

In the wild, animals have developed survival strategies relying on their senses. The individual ability to identify threatening situations is crucial and leads to increase in the overall fitness of the species. Rodents, for example have developed in their nasal cavities specialized olfactory neurons implicated in the detection of volatile cues encoding for impending danger such as predator scents or alarm pheromones. In particular, the neurons of the Grueneberg ganglion (GG), an olfactory subsystem, are implicated in the detection of danger cues sharing a similar chemical signature, a heterocyclic sulfur- or nitrogen-containing motif. Here we used a "from the wild to the lab" approach to identify new molecules that are involuntarily emitted by predators and that initiate fear-related responses in the recipient animal, the putative prey. We collected urines from carnivores as sources of predator scents and first verified their impact on the blood pressure of the mice. With this approach, the urine of the mountain lion emerged as the most potent source of chemical stress. We then identified in this biological fluid, new volatile cues with characteristic GG-related fingerprints, in particular the methylated pyridine structures, 2,4-lutidine and its analogs. We finally verified their encoded danger quality and demonstrated their ability to mimic the effects of the predator urine on GG neurons, on mice blood pressure and in behavioral experiments. In summary, we were able to identify here, with the use of an integrative approach, new relevant molecules, the pyridine analogs, implicated in interspecies danger communication.

Population demography and the evolution of helping behaviors.

Limited dispersal may favor the evolution of helping behaviors between relatives as it increases their relatedness, and it may inhibit such evolution as it increases local competition between these relatives. Here, we explore one way out of this dilemma: if the helping behavior allows groups to expand in size, then the kin-competition pressure opposing its evolution can be greatly reduced. We explore the effects of two kinds of stochasticity allowing for such deme expansion. First, we study the evolution of helping under environmental stochasticity that may induce complete patch extinction. Helping evolves if it results in a decrease in the probability of extinction or if it enhances the rate of patch recolonization through propagules formed by fission of nonextinct groups. This mode of dispersal is indeed commonly found in social species. Second, we consider the evolution of helping in the presence of demographic stochasticity. When fecundity is below its value maximizing deme size (undersaturation), helping evolves, but under stringent conditions unless positive density dependence (Allee effect) interferes with demographic stochasticity. When fecundity is above its value maximizing deme size (oversaturation), helping may also evolve, but only if it reduces negative density-dependent competition.