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.

On linear order and computation : The expressiveness of interactive computations on linear orders and computations indexed by ordinals

We solve the Dynamic Ehrenfeucht-Fra\"iss\'e Game on linear orders for both players, yielding a normal form for quantifier-rank equivalence classes of linear orders in first-order logic, infinitary logic, and generalized-infinitary logics with linearly ordered clocks. We show that Scott Sentences can be manipulated quickly, classified into local information, and consistency can be decided effectively in the length of the Scott Sentence. We describe a finite set of linked automata moving continuously on a linear order. Running them on ordinals, we compute the ordinal truth predicate and compute truth in the constructible universe of set-theory. Among the corollaries are a study of semi-models as efficient database of both model-theoretic and formulaic information, and a new proof of the atomicity of the Boolean algebra of sentences consistent with the theory of linear order -- i.e., that the finitely axiomatized theories of linear order are dense.

A Possible and Necessary Consistency Proof

After Gödel's incompleteness theorems and the collapse of Hilbert's programme Gerhard Gentzen continued the quest for consistency proofs of Peano arithmetic. He considered a finitistic or constructive proof still possible and necessary for the foundations of mathematics. For a proof to be meaningful, the principles relied on should be considered more reliable than the doubtful elements of the theory concerned. He worked out a total of four proofs between 1934 and 1939. This thesis examines the consistency proofs for arithmetic by Gentzen from different angles. The consistency of Heyting arithmetic is shown both in a sequent calculus notation and in natural deduction. The former proof includes a cut elimination theorem for the calculus and a syntactical study of the purely arithmetical part of the system. The latter consistency proof in standard natural deduction has been an open problem since the publication of Gentzen's proofs. The solution to this problem for an intuitionistic calculus is based on a normalization proof by Howard. The proof is performed in the manner of Gentzen, by giving a reduction procedure for derivations of falsity. In contrast to Gentzen's proof, the procedure contains a vector assignment. The reduction reduces the first component of the vector and this component can be interpreted as an ordinal less than epsilon_0, thus ordering the derivations by complexity and proving termination of the process.

Tyhjyyden reunalla, helvetin kannella : Yhtenäisen subjektin hajoaminen Saul Bellowin Herzogissa

The study shows that the reading paradigm derived from codes Roland Barthes presented in his S/Z exposes the postmodernic nature of Saul Bellow s Herzog (1964), and embodies in which way the novel is organized as a radical rewriting of modernism and its ideology. I explore how solid, compact and modernistic subject breaks down at every level of Herzog s narration. I actually argue that the heteroglossic novel is representative of an early American postmodern movement in literaure, and it should by no means be dissected narrowly as a realistic or naturalistic novel at all. The intertextual code verifies that the interpretation of Herzog remains inadequate if one doesn t take account of the novel s significant intertextual references to other texts. In fact, even the mind of Moses Herzog, the protagonist of the novel, is a mosaic of citations. It emerges from the dissertation that the figure of Don Quixote follows Moses Herzog as an ambiguous shadow while the professor of history struggles with his anxiety and anguish, and travels in a mentally confused state around the U.S. for five days in the early 1960s, encountering the impending atmosphere of transition as the country is on the threshold of a significant cultural and social change. There is a strong necessity for updated interpretation of Herzog partly because its centrality to Bellow s own career as a writer but mainly because it has been previously read trough modernistic lenses. I shall try to proof in my study that American Jewish Saul Bellow s (1915 2005) Herzog escapes any kind of simple, elemental or essentialistic construction or reading and in real terms it doesn t offer any comprehensive, total or coherent solution or system for those philosophical doctrines it criticizes and makes fun of. The philosophical, conceptual and cultural substance of Herzog does not constitute an independent or autonomous theoretical tract which would have a life of its own. Altough the novel reflects the continuity of Bellow s writing it is clearly some kind of conscious or unconscious experiment during his long career as a writer. He hasn t been so radical before or since the publication of Herzog. It is unarguably his most postmodern novel.The reading paradigm based on specific codes demonstrates how deep into the basic questions of his personal life and existence itself he must dive in oder to find his many ways towards authentic or primordial self in fragmented and shattered world which is constanly rewritten and which makes human being a tourist of his own life. In that ongoing process the protagonist has to accept the ultimate plurality of his mind and self. He must confront that the modernistic definition of identity as a solid, monolithic and stable entity has broken down into different, inconsistent and even contradictory possibilities of identification. Moses Herzog embodies obviously Stuart Hall s description of the postmodern self his identity has turn into a movable feast.

The Letter Collections of Anselm of Canterbury

Anselm of Canterbury (1033–1109) was a prolific letter writer. The modern edition of his letter collection comprises more than 600 folio-size pages in print and includes 472 letters, the vast majority of which were sent by him. Our knowledge of Anselm’s letters is derived from collections of his letters, for none of his correspondence survives in its original form of individual letters. There was no one canonical version of the collection, and the extant manuscripts generally differ substantially: the largest medieval manuscript witnesses include over 400 letters, while the smallest contain only a few. We know 38 manuscript witnesses, but no authorial manuscript survives. Certain references in Anselm’s letters reveal, however, that he collected his correspondence on at least two occasions while he was still abbot of Bec, and this study proposes that a third collection was possibly made under his supervision in Christ Church. The third collection also covered Anselm’s Canterbury period. Whether the third collection was authorial or posthumous is unclear. Certain contextual evidence and references in letters would suggest that the collection was authorial. If so, the collection was probably a register book, which was started in c. 1101 at the earliest. There is no positive proof that any of the three surviving minor collections may be authorial. Each of these collections was circulating at a very early stage, however, some probably in Anselm’s lifetime. Moreover, the minor collections seem to have been put together from smaller source units, which possibly originated at Bec. The contents of these units suggest very early and possibly authorial origins: the letters are mainly from Anselm’s years as prior of Bec. The critical edition by F. S. Schmitt represents the current phase in the textual tradition of Anselm’s letter collection. This study demonstrates that the value of the edition is weakened in particular by the way in which Schmitt selected manuscripts for collation, doubtless influenced by the fact that he had not established the structure of the tradition properly. Ultimately it is impossible to undertake systematic research on the letter collection on the basis of Schmitt’s edition.

"Secoituxesta järjettömäin luondocappalden canssa" : Perversiot, oikeuselämä ja kansankulttuuri 1700-luvun Suomessa

Bestiality was in the 18th century a more difficult problem in terms of criminal policy in Sweden and Finland than in any other Christian country in any other period. In the legal history of deviant sexuality, the phenomenon was uniquely widespread by international comparison. The number of court cases per capita in Finland was even higher than in Sweden. The authorities classified bestiality among the most serious crimes and a deadly sin. The Court of Appeal in Turku opted for an independent line and was clearly more lenient than Swedish courts of justice. Death sentences on grounds of bestiality ended in the 1730s, decades earlier than in Sweden. The sources for the present dissertation include judgment books and Court of Appeal decisions in 253 cases, which show that the persecution of those engaging in bestial acts in 18th century Finland was not organised by the centralised power of Stockholm. There is little evidence of local campaigns that would have been led by authorities. The church in its orthodoxy was losing ground and the clergy governed their parishes with more pragmatism than the Old Testament sanctioned. When exposing bestiality, the legal system was compelled to rely on the initiative of the public. In cases of illicit intercourse or adultery the authorities were even more dependent on the activeness of the local community. Bestiality left no tangible evidence, illegitimate children, to betray the crime to the clergy or secular authorities. The moral views of the church and the local community were not on a collision course. It was a common view that bestiality was a heinous act. Yet nowhere near all crimes came to the authorities' knowledge. Because of the heavy burden of proof, the legal position of the informer was difficult. Passiveness in reporting the crime was partly because most Finns felt it was not their place to intervene in their neighbours' private lives, as long as that privacy posed no serious threat to the neighbourhood. Hidden crime was at least as common as crime more easily exposed and proven. A typical Finnish perpetrator of bestiality was a young unmarried man with no criminal background or mental illness. The suspects were not members of ethnic minorities or marginal social groups. In trials, farmhands were more likely to be sentenced than their masters, but a more salient common denominator than social and economical status was the suspects' young age. For most of the defendants bestiality was a deep-rooted habit, which had been adopted in early youth. This form of subculture spread among the youth, and the most susceptible to experiment with the act were shepherds. The difference between man and animal was not clear-cut or self-evident. The difficulty in drawing the line is evident both in legal sources and Finnish folklore. The law that required that the animal partners be slaughtered led to the killing of thousands of cows and mares, and thereby to substantial material losses to their owners. Regarding bestiality as a crime against property motivated people to report it. The belief that the act would produce human-animal mongrels or that it would poison the milk and the meat horrified the public more than the teachings of the church ever could. Among the most significant aspects in the problems regarding the animals is how profoundly different the worldview of 18th century people was from that of today.

ADHD-aikuisten psykologinen kuntoutus : Neljän uuden lyhytintervention tutkimus

ADHD (attention deficit hyperactivity disorder) is developmental neurobiological disability. In adults, the prevalence of ADHD has been estimated to be about 4 %. In addition to the difficulties of attention, the problems in executive functioning are typical. The psychiatric comorbidities are common. The most extensively studied treatments are pharmacological. There is also evidence about the usefulness of the cognitive-behavioural therapy (CBT) in the treatment of adults with ADHD. There are some preliminary results about the effectiveness of cognitive training and hypnosis in children, but there is no scientific proof in adults. This dissertation is based on two intervention studies. In the first study, the usefulness of the new group CBT (n = 29) and the maintenance of the symptom reduction in the follow-up of six months were studied. In the second study, the usefulness of short hypnotherapy (n = 9), short individual CBT (n = 10) and computerized cognitive training (n = 9) were examined by comparing groups with each other and to the control group (n = 10). The participation in the group CBT and the participants' satisfaction were good. There were no changes in self-reports during waiting period of three months. After the rehabilitation, the symptoms decreased. Participants having symptom reduction during rehabilitation maintained their benefit through 6-month follow-up period. In a combined ADHD symptom score based on self-reports, seven participants in the hypnotherapy, six in the CBT, two in the cognitive training and two controls improved. Using independent evaluations, improvement was found in six of the hypnotherapy, seven of the CBT, two of the cognitive training and three of the control participants. There was no treatment-related improvement in cognitive performance. Thus, in the hypnotherapy and CBT groups, some encouraging improvement was seen. In the cognitive training group, there was improvement in the trained tasks but no generalization of the improvement. The results support the earlier results from the usefulness of CBT in the treatment of adults with ADHD. Also the hypnotherapy seems a useful rehabilitation. More research is needed to evaluate the usefulness of cognitive training. These promising results warrant further studies with more participants and with longer treatment duration. Also different measures of cognitive functioning and quality of life are needed. It is important in addition to the medication to arrange psychosocial interventions for the ADHD adults.

LKB1 tumor suppression in Peutz-Jeghers Syndrome

Kasvainten, ajatellaan syntyvän yksittäisen solun perimän mutaatioista, jonka seurauksena tuon solun kasvu häiriintyy. Ruoansulatuskanavan polyyppien syntyä käytetään usein mallina siitä, miten nämä epiteelisoluun kerääntyvät mutaatiot aiheuttavat asteittain pahenevan kasvuhäiriön. Peutz–Jeghersin oireyhtymä (PJS) on perinnöllinen polypoosisyndrooma, jossa oireita aiheuttavat erityisesti maha-suolikanavan hamartomatoottiset polyypit. Noin puolella PJS potilaista havaitaan mutaatioita LKB1 kasvunrajoite geenissä. Hiirille joilta toinen Lkb1 alleeli on poistettu (Lkb1+/-) kehittyy PJS-tyypin maha-suolikanavan polyyppeja, joissa on epiteelin liikakasvun lisäksi merkittävä sileälihaskomponentti, aivan kuten PJS polyypeissa. Kuten myös muissa ruoansulatuskanavan polypooseissa, sekä PJS että hiirten polyypeissa Cyclo-oxygenaasi-2:n (COX-2) määrä on usein kohonnut. PJS-polyyppien kehittymisen molekulaarinen mekanismi on kuitenkin selvittämättä. Koska vain osa PJS potilaista kantaa LKB1 mutaatioita, mutaatiot jossakin toisessa lokuksessa saattaisivat selittää osan PJS tapauksista. Jotta PJS:n geneettinen tausta selviäisi, seulottiin kolmen LKB1:n kanssa interaktoivan proteiinin (BRG1, STRADα ja MO25α) geenit PJS potilaista joilla ei ole havaittu LKB1 mutaatioita. Yhdessäkään tutkituista geeneistä ei havaittu tautia aiheuttavia mutaatioita. Näiden kolmen geenin pois sulkeminen, ja uusien menetelmien ansiosta kasvanut havaittujen Lkb1 mutaatioden määrä viittaavat LKB1:n olevan useimpien PJS tapausten taustalla. COX-2:n estäjien käyttö on tehokkaasti vähentänyt polyyppien määrää familiaarisessa adenomatoottisessa polypoosissa. Tästä johtuen COX-2:n eston tehokkuutta tutkittiin PJS polypoosissa. PJS-tyypin polypoosin havaittin pienenevän merkittävästi Lkb1+/- hiirissä, joilta oli lisäksi poistettu toinen tai molemmat COX-2:n alleeleista. Lisäksi farmakologinen COX-2:n esto Celecoxib:lla vähensi polypoosia tehokkaasti. Näin ollen COX-2:n eston tehokkuutta tutkittiin seuraavaksi PJS potilaissa. Kuuden kuukauden Celecoxib hoidon jälkeen polypoosin havaittiin vähentyneen merkittävästi osalla potilaista (2/6). Nämä tulokset osoittavat COX-2:n roolin PJS-polyyppien kehityksessä, ja viittaavat COX-2:n eston vähentävän polypoosia. Kasvunrajoitegeenin klassisen määritelmän mukaan kasvaimen kehitys vaatii perinnöllisen mutaation lisäksi geenin toisenkin alleelin mutaation, mutta PJS-polyyppien häiriintyneestä epiteelistä ei kuitenkaan systemaattisesti löydy toista LKB1:n mutaatiota. Havainto johti tutkimukseen, jossa selvitettiin voisiko LKB1:n kasvun rajoitus välittyäkin epäsuorasti tukikudokseksi ajatelluista sileälihassoluista. Tätä tutkittiin kehittämällä poistogeeninen hiirimalli jossa Lkb1 on mutatoitunut vain sileälihassoluissa. Näille hiirille kehittyi polyyppeja, jotka ovat kaikin tavoin PJS-polyyppien kaltaisia. Lkb1:n menettäneiden solujen havaittiin tuottavan vähemmän transformoivaa kasvutekijä beetaa (TGFß), joka aiheutti solujen välisen viestinnän heikentymisen ja mahdollisesti viereisten epiteelisolujen liikakasvun. Vastaava häiriö havaittiin myös PJS-potilaiden polyypeissa, mikä viittaa siihen, että potilaillakin sileälihassolujen häiriö on polyyppien taustalla. Havainto suuntaa täten hoitokohteiden etsintää ja osoittaa että LKB1 toimii kasvunrajoittajana epätyypillisellä tavalla pitäen naapurisolujen kasvun kurissa.

Novel conditional mouse models for targeted kidney research : Emphasis on the analysis of the biological function of nephrin

Nephrin is a transmembrane protein belonging to the immunoglobulin superfamily and is expressed primarily in the podocytes, which are highly differentiated epithelial cells needed for primary urine formation in the kidney. Mutations leading to nephrin loss abrogate podocyte morphology, and result in massive protein loss into urine and consequent early death in humans carrying specific mutations in this gene. The disease phenotype is closely replicated in respective mouse models. The purpose of this thesis was to generate novel inducible mouse-lines, which allow targeted gene deletion in a time and tissue-specific manner. A proof of principle model for succesful gene therapy for this disease was generated, which allowed podocyte specific transgene replacement to rescue gene deficient mice from perinatal lethality. Furthermore, the phenotypic consequences of nephrin restoration in the kidney and nephrin deficiency in the testis, brain and pancreas in rescued mice were investigated. A novel podocyte-specific construct was achieved by using standard cloning techniques to provide an inducible tool for in vitro and in vivo gene targeting. Using modified constructs and microinjection procedures two novel transgenic mouse-lines were generated. First, a mouse-line with doxycycline inducible expression of Cre recombinase that allows podocyte-specific gene deletion was generated. Second, a mouse-line with doxycycline inducible expression of rat nephrin, which allows podocyte-specific nephrin over-expression was made. Furthermore, it was possible to rescue nephrin deficient mice from perinatal lethality by cross-breeding them with a mouse-line with inducible rat nephrin expression that restored the missing endogenous nephrin only in the kidney after doxycycline treatment. The rescued mice were smaller, infertile, showed genital malformations and developed distinct histological abnormalities in the kidney with an altered molecular composition of the podocytes. Histological changes were also found in the testis, cerebellum and pancreas. The expression of another molecule with limited tissue expression, densin, was localized to the plasma membranes of Sertoli cells in the testis by immunofluorescence staining. Densin may be an essential adherens junction protein between Sertoli cells and developing germ cells and these junctions share similar protein assembly with kidney podocytes. This single, binary conditional construct serves as a cost- and time-efficient tool to increase the understanding of podocyte-specific key proteins in health and disease. The results verified a tightly controlled inducible podocyte-specific transgene expression in vitro and in vivo as expected. These novel mouse-lines with doxycycline inducible Cre recombinase and with rat nephrin expression will be useful for conditional gene targeting of essential podocyte proteins and to study in detail their functions in the adult mice. This is important for future diagnostic and pharmacologic development platforms.

The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

Dependence Logic: Investigations into Higher-Order Semantics Defined on Teams

This thesis is a study of a rather new logic called dependence logic and its closure under classical negation, team logic. In this thesis, dependence logic is investigated from several aspects. Some rules are presented for quantifier swapping in dependence logic and team logic. Such rules are among the basic tools one must be familiar with in order to gain the required intuition for using the logic for practical purposes. The thesis compares Ehrenfeucht-Fraïssé (EF) games of first order logic and dependence logic and defines a third EF game that characterises a mixed case where first order formulas are measured in the formula rank of dependence logic. The thesis contains detailed proofs of several translations between dependence logic, team logic, second order logic and its existential fragment. Translations are useful for showing relationships between the expressive powers of logics. Also, by inspecting the form of the translated formulas, one can see how an aspect of one logic can be expressed in the other logic. The thesis makes preliminary investigations into proof theory of dependence logic. Attempts focus on finding a complete proof system for a modest yet nontrivial fragment of dependence logic. A key problem is identified and addressed in adapting a known proof system of classical propositional logic to become a proof system for the fragment, namely that the rule of contraction is needed but is unsound in its unrestricted form. A proof system is suggested for the fragment and its completeness conjectured. Finally, the thesis investigates the very foundation of dependence logic. An alternative semantics called 1-semantics is suggested for the syntax of dependence logic. There are several key differences between 1-semantics and other semantics of dependence logic. 1-semantics is derived from first order semantics by a natural type shift. Therefore 1-semantics reflects an established semantics in a coherent manner. Negation in 1-semantics is a semantic operation and satisfies the law of excluded middle. A translation is provided from unrestricted formulas of existential second order logic into 1-semantics. Also game theoretic semantics are considerd in the light of 1-semantics.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Numerical computation of the module of a quadrilateral

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.

On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

The Catholic Internet discussion on cloning : A study in cognitive rhetoric

This work combines the cognitive theory of folk-theoretical thought with the classical Aristotelian theory of artistic proof in rhetoric. The first half of the work discusses the common ground shared by the elements of artistic proof (logos, pathos, ethos) and the elements of folk-theoretical thought (naïve physics, folk biology, folk psychology, naïve sociology). Combining rhetoric with the cognitive theory of folk-theoretical thought creates a new point of view for argumentation analysis. The logos of an argument can be understood as the inferential relations established between the different parts of an argument. Consequently, within this study the analysis of logos is to be viewed as the analysis of the inferential folk-theoretical elements that make the suggested factual states-of-things appear plausible within given argumentative structures. The pathos of an argumentative structure can be understood as determining the quality of the argumentation in question in the sense that emotive elements play a great part in what can be called a distinction between good and deceptive rhetoric. In the context of this study the analysis of pathos is to be viewed as the analysis of the emotive content of argumentative structures and of whether they aim at facilitating surface- or deep cognitive elaboration of the suggested matters. The ethos of an argumentative structure means both the speaker-presentation and audience-construct that can be discerned within a body of argumentation. In the context of this study, the analysis of ethos is to be understood as the analysis of mutually manifest cognitive environments in the context of argumentation. The theory is used to analyse Catholic Internet discussion concerning cloning. The discussion is divided into six themes: Human Dignity, Sacred Family, Exploitation / Dehumanisation, Playing God, Monsters and Horror Scenarios and Ensoulment. Each theme is analysed for both the rhetorical and the cognitive elements that can be seen creating persuasive force within the argumentative structures presented. It is apparent that the Catholic voices on the Internet extensively oppose cloning. The voices utilise rhetoric that is aggressive and pejorative more often than not. Furthermore, deceptive rhetoric (in the sense presented above) plays a great part in argumentative structures of the Catholic voices. The theory of folk-theoretical thought can be seen as a useful tool for analysing the possible reasons why the Catholic speakers think about cloning and choose to present cloning in their argumentation as they do. The logos utilized in the argumentative structures presented can usually be viewed as based on folk-theoretical inference concerning biology and psychology. The structures of pathos utilized generally appear to aim at generating fear appeal in the assumed audiences, often incorporating counter-intuitive elements. The ethos utilised in the arguments generally revolves around Christian mythology and issues of social responsibility. These structures can also be viewed from the point of view of folk psychology and naïve sociological assumptions.