Constructive Perspectives on Inductive Logic

Constructive (intuitionist, anti-realist) semantics has thus far been lacking an adequate concept of truth in infinity concerning factual (i.e., empirical, non-mathematical) sentences. One consequence of this problem is the difficulty of incorporating inductive reasoning in constructive semantics. It is not possible to formulate a notion for probable truth in infinity if there is no adequate notion of what truth in infinity is. One needs a notion of a constructive possible world based on sensory experience. Moreover, a constructive probability measure must be defined over these constructively possible empirical worlds. This study defines a particular kind of approach to the concept of truth in infinity for Rudolf Carnap's inductive logic. The new approach is based on truth in the consecutive finite domains of individuals. This concept will be given a constructive interpretation. What can be verifiably said about an empirical statement with respect to this concept of truth, will be explained, for which purpose a constructive notion of epistemic probability will be introduced. The aim of this study is also to improve Carnap's inductive logic. The study addresses the problem of justifying the use of an "inductivist" method in Carnap's lambda-continuum. A correction rule for adjusting the inductive method itself in the course of obtaining evidence will be introduced. Together with the constructive interpretation of probability, the correction rule yields positive prior probabilities for universal generalizations in infinite domains.

Mathematical models on the impact of noise and dyadic molecular structures on the properties of a cardiac myocyte

In cardiac myocytes (heart muscle cells), coupling of electric signal known as the action potential to contraction of the heart depends crucially on calcium-induced calcium release (CICR) in a microdomain known as the dyad. During CICR, the peak number of free calcium ions (Ca) present in the dyad is small, typically estimated to be within range 1-100. Since the free Ca ions mediate CICR, noise in Ca signaling due to the small number of free calcium ions influences Excitation-Contraction (EC) coupling gain. Noise in Ca signaling is only one noise type influencing cardiac myocytes, e.g., ion channels playing a central role in action potential propagation are stochastic machines, each of which gates more or less randomly, which produces gating noise present in membrane currents. How various noise sources influence macroscopic properties of a myocyte, how noise is attenuated and taken advantage of are largely open questions. In this thesis, the impact of noise on CICR, EC coupling and, more generally, macroscopic properties of a cardiac myocyte is investigated at multiple levels of detail using mathematical models. Complementarily to the investigation of the impact of noise on CICR, computationally-efficient yet spatially-detailed models of CICR are developed. The results of this thesis show that (1) gating noise due to the high-activity mode of L-type calcium channels playing a major role in CICR may induce early after-depolarizations associated with polymorphic tachycardia, which is a frequent precursor to sudden cardiac death in heart failure patients; (2) an increased level of voltage noise typically increases action potential duration and it skews distribution of action potential durations toward long durations in cardiac myocytes; and that (3) while a small number of Ca ions mediate CICR, Excitation-Contraction coupling is robust against this noise source, partly due to the shape of ryanodine receptor protein structures present in the cardiac dyad.

Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

Using Mathematical Morphology for Geometric Music Retrieval

The usual task in music information retrieval (MIR) is to find occurrences of a monophonic query pattern within a music database, which can contain both monophonic and polyphonic content. The so-called query-by-humming systems are a famous instance of content-based MIR. In such a system, the user's hummed query is converted into symbolic form to perform search operations in a similarly encoded database. The symbolic representation (e.g., textual, MIDI or vector data) is typically a quantized and simplified version of the sampled audio data, yielding to faster search algorithms and space requirements that can be met in real-life situations. In this thesis, we investigate geometric approaches to MIR. We first study some musicological properties often needed in MIR algorithms, and then give a literature review on traditional (e.g., string-matching-based) MIR algorithms and novel techniques based on geometry. We also introduce some concepts from digital image processing, namely the mathematical morphology, which we will use to develop and implement four algorithms for geometric music retrieval. The symbolic representation in the case of our algorithms is a binary 2-D image. We use various morphological pre- and post-processing operations on the query and the database images to perform template matching / pattern recognition for the images. The algorithms are basically extensions to classic image correlation and hit-or-miss transformation techniques used widely in template matching applications. They aim to be a future extension to the retrieval engine of C-BRAHMS, which is a research project of the Department of Computer Science at University of Helsinki.

Knowledge through Discussion : An Interpretation of Epistemology in Plato's Meno

According to Meno s paradox we cannot inquire into what we do not know because we do not know what we are inquiring into. There are many ways to interpret the paradox but the central issue about our ability to reach truth is a profound one. In the dialogue Meno, Plato presents the paradox and an outline of a solution which enables us to reach knowledge (epistēmē) through philosophical discussion. During the last century Meno has often been considered transitional between Socratic thinking and Plato s own philosophy, and thus the dialogue has not been adequately interpreted as an integrated whole. Therefore the distinctive epistemology of the dialogue has not gained due notice. In this thesis the dialogue is analysed as an integrated whole and the philosophical interpretation also takes into account its dramatic features. The thesis emphasises the role of language and definitions in acquiring knowledge. Among the results concerning these subjects is a new interpretation of Socrates s defintion of shape (schēma). The theory of anamnēsis all learning is recollection in the Meno is argued to answer the paradox philosophically although Plato s presentation also contains playful and ironic elements. The background of the way Plato presents his case is that he appreciated the fact that no argument can plausibly demonstrate that argumentation is able to reach truth. In the Meno, Plato makes the earliest explicit distinction between knowledge and true belief in the history of Western philosophy. He also gives a definition of knowledge which is the basis of the so called classical definition of knowledge as justified true belief. In the Meno, true beliefs become knowledge when someone ties them down by reasoning about the explanation. The analysis of the epistemology of the dialogue from this perspective gives an interpretation which integrates the central concepts of the epistemology in the dialogue elenchos, anamnēsis and hypothetical inquiry into a unified whole which contains a plausible argument according to which the ignorant can reach knowledge through discussion. The conception that emerges by such an analysis is interesting both from the point of view of current interests and that of the history of philosophy. The method of knowledge acquisition in the Meno can, for example, be seen as a predecessor of modern scientific methods. The Meno is the earliest Greek mathematical text that has survived in its original form. The analysis presented in the thesis of the geometric passages in the dialogue provides new results both concerning Socrates s geometry lesson with the slave and the example presenting the hypothetical method. Concerning the latter, a new interpretation is presented. Keywords: anamnēsis, epistēmē, knowledge, Meno s paradox, Plato

Models of Dispersal and Diversification

Ecology and evolutionary biology is the study of life on this planet. One of the many methods applied to answering the great diversity of questions regarding the lives and characteristics of individual organisms, is the utilization of mathematical models. Such models are used in a wide variety of ways. Some help us to reason, functioning as aids to, or substitutes for, our own fallible logic, thus making argumentation and thinking clearer. Models which help our reasoning can lead to conceptual clarification; by expressing ideas in algebraic terms, the relationship between different concepts become clearer. Other mathematical models are used to better understand yet more complicated models, or to develop mathematical tools for their analysis. Though helping us to reason and being used as tools in the craftmanship of science, many models do not tell us much about the real biological phenomena we are, at least initially, interested in. The main reason for this is that any mathematical model is a simplification of the real world, reducing the complexity and variety of interactions and idiosynchracies of individual organisms. What such models can tell us, however, both is and has been very valuable throughout the history of ecology and evolution. Minimally, a model simplifying the complex world can tell us that in principle, the patterns produced in a model could also be produced in the real world. We can never know how different a simplified mathematical representation is from the real world, but the similarity models do strive for, gives us confidence that their results could apply. This thesis deals with a variety of different models, used for different purposes. One model deals with how one can measure and analyse invasions; the expanding phase of invasive species. Earlier analyses claims to have shown that such invasions can be a regulated phenomena, that higher invasion speeds at a given point in time will lead to a reduction in speed. Two simple mathematical models show that analysis on this particular measure of invasion speed need not be evidence of regulation. In the context of dispersal evolution, two models acting as proof-of-principle are presented. Parent-offspring conflict emerges when there are different evolutionary optima for adaptive behavior for parents and offspring. We show that the evolution of dispersal distances can entail such a conflict, and that under parental control of dispersal (as, for example, in higher plants) wider dispersal kernels are optimal. We also show that dispersal homeostasis can be optimal; in a setting where dispersal decisions (to leave or stay in a natal patch) are made, strategies that divide their seeds or eggs into fractions that disperse or not, as opposed to randomized for each seed, can prevail. We also present a model of the evolution of bet-hedging strategies; evolutionary adaptations that occur despite their fitness, on average, being lower than a competing strategy. Such strategies can win in the long run because they have a reduced variance in fitness coupled with a reduction in mean fitness, and fitness is of a multiplicative nature across generations, and therefore sensitive to variability. This model is used for conceptual clarification; by developing a population genetical model with uncertain fitness and expressing genotypic variance in fitness as a product between individual level variance and correlations between individuals of a genotype. We arrive at expressions that intuitively reflect two of the main categorizations of bet-hedging strategies; conservative vs diversifying and within- vs between-generation bet hedging. In addition, this model shows that these divisions in fact are false dichotomies.

Legal Reasoning in Context

This study in EU law analyses the reasoning of the Court of Justice (the Court of Justice of the European Union) in a set of its preliminary rulings. Preliminary rulings are answers to national courts questions on the interpretation (and validity) of EU law called preliminary references. These questions concern specific legal issues that have arisen in legal disputes before the national courts. The Court of Justice alone has the ultimate authority to interpret EU law. The preliminary rulings bind the national courts in the cases giving rise to the preliminary reference, and the interpretations of EU law offered in the preliminary rulings are considered generally binding on all instances applying EU law. EU law is often described as a dynamic legal order and the Court of Justice as at the vanguard of developing it. It is generally assumed that the Court of Justice is striving to realise the EU s meta-level purpose (telos): integration. Against this backdrop one can understand the criticism the Court of Justice is often faced with in certain fields of EU law that can be described as developing. This criticism concerns the Court s (negatively) activist way of not just stating the law but developing or even making law. It is difficult to analyse or prove wrong this accusation as it is not in methodological terms clearly established what constitutes judicial activism, or more exactly where the threshold of negative activism lies. Moreover, one popular approach to assessing the role of the Court of Justice described as integration through law has become fairly political, neglecting to take into consideration the special nature of law as both facilitating and constraining action, not merely a medium for furthering integration. This study offers a legal reasoning approach of a more legalist nature, in order to balance the existing mix of approaches to explaining what the Court of Justice does and how. Reliance on legal reasoning is found to offer a working framework for analysis, whereas the tools for an analysis based on activism are found lacking. The legal reasoning approach enables one to assess whether or not the Court of Justice is pertaining to its own established criteria of interpretation of EU law, and if it is not, one should look more in detail at how the interpretation fits with earlier case-law and doctrines of EU law. This study examines the reasoning of the Court of Justice in a set of objectively chosen cases. The emphasis of the study is on analysing how the Court of Justice applies the established criteria of interpretation it has assumed for itself. Moreover, the judgments are assessed not only in terms of reasoning but also for meaningful silences they contain. The analysis is furthermore contextualised by taking into consideration how the cases were commented by legal scholars, their substantive EU law context, and also their larger politico-historical context. In this study, the analysis largely shows that the Court of Justice is interpreting EU law in accordance with its previous practice. Its reasoning retains connection with the linguistic or semiotic criteria of interpretation, while emphasis lies on systemic reasoning. Moreover, although there are a few judgments where the Court of Justice offers clearly dynamic reasoning or what can be considered as substantive reasoning stemming from, for example, common sense or reasonableness, such reasons are most often given in addition to systemic ones. In this sense and even when considered in its broader context, the case-law analysed in this study does not portray a specifically activist image of the Court of Justice. The legal reasoning approach is a valid alternative for explaining how and why the Court of Justice interprets EU law as it does.

Mathematical modelling of groundwater pollution in a small heterogeneous aquifer at Kärkölä, southern Finland

Yhteenveto: Kärkölän likaantuneen pohjavesialueen matemaattinen mallinnus