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.

Studies on Stone Age Housepits in Fennoscandia (4000 - 2000 cal BC) : Changes in ground plan, site location and degree of sedentism

Housepits have a remarkably short research history as compared to Fennoscandian archaeological research on the Stone Age in general. The current understanding of the numbers and the distribution of Stone Age housepits in the Nordic countries has, for the most part, been shaped by archaeological studies carried out over the last twenty to thirty years. The main subjects of this research are Neolithic housepits, which are archaeological remains of semi-subterranean pithouses. This dissertation consists of five peer-reviewed articles and a synthesis paper. The articles deal with the development of housepits as seen in the data gathered from Finland (the Lake Saimaa area and south-eastern Finland) and Russia (the Karelian Isthmus). This synthesis expands the discussion of the changes observed in the Papers to include Fennoscandian housepit research as a whole. Certain changes in the size, shape, environmental location, and clustering of housepits extended into various cultures and ecological zones in northern Fennoscandia. Previously, the evolution of housepits has been interpreted to have been caused by the adaptation of Neolithic societies to prevailing environmental circumstances or to re-organization following contacts with the agrarian Corded Ware/Battle Axe Cultures spreading to North. This dissertation argues for two waves of change in the pithouse building tradition. Both waves brought with them certain changes in the pithouses themselves and in the practices of locating the dwellings in the environment/landscape. The changes in housepits do not go hand in hand with other changes in material culture, nor are the changes restricted to certain ecological environments. Based on current information, it appears that the changes relate primarily to the spread of new concepts of housing and possibly to new technology, as opposed to representing merely a local response to environmental factors. This development commenced already before the birth of the Corded Ware/Battle Axe Cultures. Therefore, the changes are argued to have resulted from the spreading of new ideas through the same networks that actively distributed commodities, exotic goods, and raw materials over vast areas between the southern Baltic Sea, the north-west Russian forest zone, and Fennoscandia.

Embedded Storytelling and Gender Negotiations in Githa Hariharan's First Three Novels

This thesis treats Githa Hariharan s first three novels The Thousand Faces of Night (1992), The Ghosts of Vasu Master (1994) and When Dreams Travel (1999) as a trilogy, in which Hariharan studies the effects of storytelling from different perspectives. The thesis examines the relationship between embedded storytelling and the primary narrative level, the impact of character-bound storytelling on personal development and interpersonal relationships. Thus, I assume that an analysis of the instabilities between characters, and tensions between sets of values introduced through storytelling displays the development of the central characters in the novels. My focus is on the tensions between different sets of values and knowledge systems and their impact on the gender negotiations. The tensions are articulated through a resistance and/or adherence to a cultural narrative, that is, a formula, which underlies specific narratives. Conveyed or disputed by embedded storytelling, the cultural narrative circumscribes what it means to be gendered in Hariharan s novels. The analysis centres on how the narratee in The Thousand Faces of Night and the storyteller in The Ghosts of Vasu Master relate to and are affected by the storytelling, and how they perceive their gendered positions. The analysis of the third novel When Dreams Travel focuses on storytelling, and its relationship to power and representation. I argue that Hariharan's use of embedded storytelling is a way to renegotiate and even reconceptualise gender. Hariharan s primary concern in all three novels is the tensions between tradition or repetition, and change, and how they affect gender. Although the novels feature ancient mythical heroes, mice and flies, or are set in a fantasy world of jinnis and ghouls, they are, I argue, deeply concerned with contemporary issues. Whereas the first novel questions the demands set by family and society on the individual, the second strives to articulate an ethical relationship between the self and the other. The third novel examines the relationship between representation and power. These issues could not be more contemporary, and they loom large over both the regional and global arenas.

Constructions and situations : a constructivist reading of Hegel's System

The object of this work is Hegel's Logic, which comprises the first third of his philosophical System that also includes the Philosophy of Nature and the Philosophy of Spirit. The work is divided into two parts, where the first part investigates Hegel s Logic in itself or without an explicit reference to rest of Hegel's System. It is argued in the first part that Hegel's Logic contains a methodology for constructing examples of basic ontological categories. The starting point on which this construction is based is a structure Hegel calls Nothing, which I argue to be identical with an empty situation, that is, a situation with no objects in it. Examples of further categories are constructed, firstly, by making previous structures objects of new situations. This rule makes it possible for Hegel to introduce examples of ontological structures that contain objects as constituents. Secondly, Hegel takes also the very constructions he uses as constituents of further structures: thus, he is able to exemplify ontological categories involving causal relations. The final result of Hegel's Logic should then be a model of Hegel s Logic itself, or at least of its basic methods. The second part of the work focuses on the relation of Hegel's Logic to the other parts of Hegel's System. My interpretation tries to avoid, firstly, the extreme of taking Hegel's System as a grand metaphysical attempt to deduce what exists through abstract thinking, and secondly, the extreme of seeing Hegel's System as mere diluted Kantianism or a second-order investigation of theories concerning objects instead of actual objects. I suggest a third manner of reading Hegel's System, based on extending the constructivism of Hegel's Logic to the whole of his philosophical System. According to this interpretation, transitions between parts of Hegel's System should not be understood as proofs of any sort, but as constructions of one structure or its model from another structure. Hence, these transitions involve at least, and especially within the Philosophy of Nature, modelling of one type of object or phenomenon through characteristics of an object or phenomenon of another type, and in the best case, and especially within the Philosophy of Spirit, transformations of an object or phenomenon of one type into an object or phenomenon of another type. Thus, the transitions and descriptions within Hegel's System concern actual objects and not mere theories, but they still involve no fallacious deductions.

The Art of Adaptation : A study on the Europeanization of Finland's foreign and security policy

This study examines how Finnish foreign and security policy has been influenced by the European Union and its Common Foreign and Security Policy. It points to a growing interplay and misfit between the external expectations originating from the European level and the domestic expectations and traditional ways-of-doing-things. It is concluded that the deepening European integration in the sphere of foreign, security and defence policy has played a significant role in a number of transformations in the Finnish policies since 1995. New, more European, meanings have been attached to the key concepts of Finnish foreign and security policy. Neutrality and traditional peacekeeping have been replaced by a minimalist reading of military non-alignment and participation in crisis management operations and EU battle groups. Traditional small state identity has been recast more and more as small member stateness . At the same time Finland has entered an era of post-consensus in national foreign and security policy. A key theoretical argument in the background of the study is that collective understandings attached to European policies, when not resonating well with domestic understandings, cause adaptation pressures on domestic-level processes and may lead to changes in the way interests and identities are constructed. This means that Europeanization is principally seen as identity reconstruction. Consequently, the theoretical framework of the study builds on the Europeanization research literature and constructivist IR theory on state identity. Foreign and security policy is defined as the practice in which state identity is reproduced, and the key foreign and security policy concepts are seen as the vehicles of identity production. It is concluded that for Finland, participation in the EU s foreign, security and defence policies represents not only a tool for responding to the changes in the international security environment but also a new means of self-identification. Concerning the Finnish attempts of projecting national interests on the European security policy agenda, it is concluded that they mainly relate to the compatibility of the potential development of EU s defence dimension with the Finnish military non-alignment. Although neutrality was cast aside in the official security policy when Finland joined the EU, the analysis shows that its impact has continued in the domestic political debate and in the mind-set of the decision-makers. The primary research material includes official Finnish foreign and security policy documentation and the related parliamentary debates from 1994 to 2007. This study serves also as a comprehensive empirical overview on Finland s reactions and contributions to the EU Common Foreign and Security Policy.

A Priori and the A Posteriori

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.