Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.

Aritmetiikasta algebraan : Muutoksia osaamisessa peruskoulun päättöluokalla 20 vuoden aikana

From Arithmetic to Algebra. Changes in the skills in comprehensive school over 20 years. In recent decades we have emphasized the understanding of calculation in mathematics teaching. Many studies have found that better understanding helps to apply skills in new conditions and that the ability to think on an abstract level increases the transfer to new contexts. In my research I take into consideration competence as a matrix where content is in a horizontal line and levels of thinking are in a vertical line. The know-how is intellectual and strategic flexibility and understanding. The resources and limitations of memory have their effects on learning in different ways in different phases. Therefore both flexible conceptual thinking and automatization must be considered in learning. The research questions that I examine are what kind of changes have occurred in mathematical skills in comprehensive school over the last 20 years and what kind of conceptual thinking is demonstrated by students in this decade. The study consists of two parts. The first part is a statistical analysis of the mathematical skills and their changes over the last 20 years in comprehensive school. In the test the pupils did not use calculators. The second part is a qualitative analysis of the conceptual thinking of pupils in comprehensive school in this decade. The study shows significant differences in algebra and in some parts of arithmetic. The largest differences were detected in the calculation skills of fractions. In the 1980s two out of three pupils were able to complete tasks with fractions, but in the 2000s only one out of three pupils were able to do the same tasks. Also remarkable is that out of the students who could complete the tasks with fractions, only one out of three pupils was on the conceptual level in his/her thinking. This means that about 10% of pupils are able to understand the algebraic expression, which has the same isomorphic structure as the arithmetical expression. This finding is important because the ability to think innovatively is created when learning the basic concepts. Keywords: arithmetic, algebra, competence

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.

Holistic Spirituality in the Thinking of Ellen White

Spiritualiteetti viittaa syvälliseen, inhimilliseen ulottuvuuteen ja ominaisuuteen, jonka tarkka määritteleminen on haasteellista, ellei mahdotonta. Sitä vastaa yhtäältä uskonnollisuuden kautta toteutuva, elämän tarkoitukseen ja syvemmän olemuksen etsintään liittyvä hengellisyys, mutta toisaalta myös kaikkea muuta hengen viljelyä ja mielekkään olemisen tavoittelua tarkoittava henkisyys. John Swintonin mukaan hengen ulottuvuus on se inhimilliseen olemukseen kuuluva, dynaaminen elinvoima, joka virkistää ja elävöittää ihmistä ja motivoi häntä etsimään Jumalaa, arvoja, merkitystä, tarkoitusta ja toivoa. Tämä tutkimus nostaa tarkastelun kohteeksi kokonaisvaltaisen hengellisyyden, jolloin huomio kiinnitetään niihin sidoksiin, joiden kautta hengen ulottuvuus liittyy muihin inhimillisen elämän olennaisiin toimintoihin ja näkökulmiin. Tällaisia ovat 1) ajattelu 2) teot ja käytännön toiminta 3) suhteet ja vuorovaikutusverkostot 4) tunteet ja kanssakäymistä ohjaavat asenteet 5) olemassaolon ja olemisen ulottuvuudet. Kokemusten merkitys, arvo ja mielekkyys hahmottuvat juuri hengen alueella, toisin sanoen sisäisesti, hengellisenä ja henkisenä asiana. Tutkimusmateriaalina tässä tutkimuksessa on amerikkalaisen vuosina 1827 1915 eläneen Ellen Whiten kuusi myöhäiskauden teosta vuosilta 1892 1905 ja tutkimusmenetelmänä on käytetty systemaattista analyysiä. Olennaista Whiten tavassa käsitellä uskonnon harjoitukseen liittyviä aiheita on hänen käytännöllinen ja elämän arkeen kiinteästi niveltyvä otteensa. Tutkimus paljastaa, että Martti Lutherin käsitykset ovat merkittävästi vaikuttaneet Whiten ajatteluun. Lähteistä paljastuu samankaltaisuutta hänen näkemystensä ja uusimman suomalaisen Luther-tutkimuksen Martti Lutherin tuotannosta esiin nostaman ajattelutavan välillä. Vaikka teologisen oppineisuuden kannalta White ja Luther ovat eri tasoilla, kummankin käsitys ihmisen ja Jumalan välisen suhteen perusolemuksesta on samankaltainen: Lähtökohtana sille on Jumalan rakkaus ja hänen armostaan lähtenyt toiminta. Toiseksi, ihmisen ja Kristuksen välinen, olemuksellinen yhteys, unio , on perustana sille, että Jumala hyväksyy ihmisen ja huolehtii hänestä nyt ja ikuisesti. Kolmanneksi, tämä ihmisen ja Kristuksen liittoutuminen ja yhdistyminen ilmenee yhteistoimintana ja kumppanuutena yhteisten tavoitteiden saavuttamiseksi maailmassa. White korostaa ihmisen ja Kristuksen välisen hengellisen suhteen vuorovaikutteista ja toiminnallista luonnetta, joka tulee ilmi epäitsekkyytenä, toisten ihmisten ja heidän tarpeittensa huomioimisena sekä myötätuntona ja kykynä asettua toisen asemaan. Terveellistä elämäntapaa ja kasvatusta koskevat ajatuksensa White liittää siihen laaja-alaiseen näkemykseen hengellisyydestä, jonka tavoitteena on ihmisen kokonaisvaltainen hyvinvointi. Hän ei näe spiritualiteettia elämän arjesta irrallisena tai erillisenä saarekkeena, vaan ihmistä kaikessa ohjaavana, voimaannuttavana ja mielekkyyttä tuottavana, ensisijaisena ulottuvuutena. Tutkimuksen kuluessa myös Whiten usein käyttämät Jumalalle antautumisen ja luonteen käsitteet nousevat tarkastelun kohteiksi. Hänen mukaansa ihminen ei tahdonponnistuksillaan yksin pysty tavoittamaan Jumalaa vaan hänen on lakattava Jumalan rakastavan kutsun edessä itse tahtomasta ja suostuttava liittymään Jumalan tahtoon ja tarkoitukseen. Tämä liittyy siihen sisäiseen muutokseen, jota White kuvaa luonteen käsitteen avulla. Jumalan armon vaikuttama tahdon uudelleen suuntaaminen muuttaa ihmisen olemusta, arvoja, asennoitumisen tapaa ja myötätuntoisen vuorovaikutuksen kykyä niin ettei ihminen ole enää aivan sama kuin ennen. Kysymys on toisaalta yhtäkkisestä ja kertakaikkisesta olemuksellisesta muuttumisesta, mutta samalla myös hiljaisesta, elämänmittaisesta kasvusta ja kypsymisestä. Juuri luonteen käsitteen avulla White kuvaa hengellisyyttä ja siihen kuuluvaa sisästä matkaa. Tässä tutkimuksessa spiritualiteettia lähestytään yleisinhimillisenä piirteenä ja ominaisuutena, jolloin huomio ei ole ensisijaisesti yksittäisissä opillisissa käsityksissä tai uskonnollisuuden harjoittamisen muodoissa. Tarkoituksena on luoda kokoava rakenne, jonka puitteissa holistinen spiritualiteetti voidaan selkeämmin hahmottaa ja yksilöidymmin ymmärtää.

The Restless Love of Thinking : The Concept Liebe in Hegel's Philosophy

The German philosopher G.W.F.Hegel (1770–1831) is best known for his idealistic system philosophy, his concept of spirit [Geist] and for his dictum that the existing and the rational overlap. This thesis offers a new perspective: it examines the working of the concept ‘love’ in Hegel’s philosophy by looking at the contexts and function he puts it to, from his earliest writings to the very last lectures he gave. The starting point of the inquiry is that he applied the concept Liebe to different contexts for different purposes, but each time to provide an answer to a specific philosophical problem. His formulation, reformulation and use of ‘love’ give possible solutions to problems the solving of which was crucial to the development of his thought as a whole. The study is divided into three parts, each analysing the different problems and solutions to which Hegel applied the concept of love. The first part, "Love, morality and ethical life", examines these interconnected themes in Hegel’s early work. The main questions he addressed during this period concerned how to unite Kant’s philosophy and the Greek ideal of the good life. In this context, the concept ‘love’ did three things. First, it served to formulate his grounding idea of the relation between unity and difference, or the manifold. Secondly, it was the key to his attempt to base an ideal folk religion on Christianity interpreted as a religion of love. Finally, it provided the means to criticise Kant’s moral philosophy. The question of the moral value of love helped Hegel to break away from Kant’s thought and develop his own theory about love and ethical life. The second part of the study, "Love and the political realm", considers the way 'Liebe' functions in connection with questions concerning the community and political life in Hegel’s work. In addition to questioning the universal applicability of the concept of recognition as a key to his theory of social relations, the chapters focus on gender politics and the way he conceptualised the gender category ‘woman’ through the concept ‘love’. Another line of inquiry is the way the figure of Antigone was used to conceptualise the differentiated spheres of action for men and women, and the part ‘love’ played in Hegel’s description of Antigone’s motives. Thirdly, Hegel’s analogy of the family and the state and the way ‘love’ functions in an attempt to promote understanding of the relation between citizens and the state are examined. The third and final part of the study, "Love as absolute spirit", focuses on ‘love’ within Hegel’s systemic thought and the way he continued to characterise Geist through the language of Liebe up until and including his very last works. It is shown how Liebe functions in his hierarchical organisation of the domains of art, religion and philosophy, and how both art and religion end up in similar structural positions with regard to philosophy. One recurrent theme in the third part is Hegel’s complex relation to Romantic thought. Another line of investigation is how he reconstructed Christianity as a religion of love in his mature work. In striking contrast to his early thought, in his last works Hegel introduced a new concept of love that incorporated negativity, and that could also function as the root of political action.

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.