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.

Becoming a teacher: emerging teacher identity in mathematics teacher education

This research examines three aspects of becoming a teacher, teacher identity formation in mathematics teacher education: the cognitive and affective aspect, the image of an ideal teacher directing the developmental process, and as an on-going process. The formation of emerging teacher identity was approached in a social psychological framework, in which individual development takes place in social interaction with the context through various experiences. Formation of teacher identity is seen as a dynamic, on-going developmental process, in which an individual intentionally aspires after the ideal image of being a teacher by developing his/her own competence as a teacher. The starting-point was that it is possible to examine formation of teacher identity through conceptualisation of observations that the individual and others have about teacher identity in different situations. The research uses the qualitative case study approach to formation of emerging teacher identity, the individual developmental process and the socially constructed image of an ideal mathematics teacher. Two student cases, John and Mary, and the collective case of teacher educators representing socially shared views of becoming and being a mathematics teacher are presented. The development of each student was examined based on three semi-structured interviews supplemented with written products. The data-gathering took place during the 2005 2006 academic year. The collective case about the ideal image provided during the programme was composed of separate case displays of each teacher educator, which were mainly based on semi-structured interviews in spring term 2006. The intentions and aims set for students were of special interest in the interviews with teacher educators. The interview data was analysed following the modified idea of analytic induction. The formation of teacher identity is elaborated through three themes emerging from theoretical considerations and the cases. First, the profile of one s present state as a teacher may be scrutinised through separate affective and cognitive aspects associated with the teaching profession. The differences between individuals arise through dif-ferent emphasis on these aspects. Similarly, the socially constructed image of an ideal teacher may be profiled through a combination of aspects associated with the teaching profession. Second, the ideal image directing the individual developmental process is the level at which individual and social processes meet. Third, formation of teacher identity is about becoming a teacher both in the eyes of the individual self as well as of others in the context. It is a challenge in academic mathematics teacher education to support the various cognitive and affective aspects associated with being a teacher in a way that being a professional and further development could have a coherent starting-point that an individual can internalise.

Pentitol phosphate dehydrogenases: Discovery, characterization and use in D-arabitol and xylitol production by metabolically engineered Bacillus subtilis

The ultimate goal of this study has been to construct metabolically engineered microbial strains capable of fermenting glucose into pentitols D-arabitol and, especially, xylitol. The path that was chosen to achieve this goal required discovery, isolation and sequencing of at least two pentitol phosphate dehydrogenases of different specificity, followed by cloning and expression of their genes and characterization of recombinant arabitol and xylitol phosphate dehydrogenases. An enzyme of a previously unknown specificity, D-arabitol phosphate dehydrogenase (APDH), was discovered in Enterococcus avium. The enzyme was purified to homogenity from E. avium strain ATCC 33665. SDS/PAGE revealed that the enzyme has a molecular mass of 41 ± 2 kDa, whereas a molecular mass of 160 ± 5 kDa was observed under non-denaturing conditions implying that the APDH may exist as a tetramer with identical subunits. Purified APDH was found to have narrow substrate specificity, converting only D-arabitol 1-phosphate and D-arabitol 5-phosphate into D-xylulose 5-phosphate and D-ribulose 5-phosphate, respectively, in the oxidative reaction. Both NAD+ and NADP+ were accepted as co-factors. Based on the partial protein sequences, the gene encoding APDH was cloned. Homology comparisons place APDH within the medium chain dehydrogenase family. Unlike most members of this family, APDH requires Mn2+ but no Zn2+ for enzymatic activity. The DNA sequence surrounding the gene suggests that it belongs to an operon that also contains several components of phosphotransferase system (PTS). The apparent role of the enzyme is to participate in arabitol catabolism via the arabitol phosphate route similar to the ribitol and xylitol catabolic routes described previously. Xylitol phosphate dehydrogenase (XPDH) was isolated from Lactobacillus rhamnosus strain ATCC 15820. The enzyme was partially sequenced. Amino acid sequences were used to isolate the gene encoding the enzyme. The homology comparisons of the deduced amino acid sequence of L. rhamnosus XPDH revealed several similar enzymes in genomes of various species of Gram-positive bacteria. Two enzymes of Clostridium difficile and an enzyme of Bacillus halodurans were cloned and their substrate specificities together with the substrate specificity of L. rhamnosus XPDH were compared. It was found that one of the XPDH enzymes of C. difficile and the XPDH of L. rhamnosus had the highest selectivity towards D-xylulose 5-phosphate. A known transketolase-deficient and D-ribose-producing mutant of Bacillus subtilis (ATCC 31094) was further modified by disrupting its rpi (D-ribose phosphate isomerase) gene to create D-ribulose- and D-xylulose-producing strain. Expression of APDH of E. avium and XPDH of L. rhamnosus and C. difficile in D-ribulose- and D-xylulose-producing strain of B. subtilis resulted in strains capable of converting D-glucose into D-arabitol and xylitol, respectively. The D-arabitol yield on D-glucose was 38 % (w/w). Xylitol production was accompanied by co-production of ribitol limiting xylitol yield to 23 %.

End-User Attitudes Towards EDM Use in Construction Project Work: Case Study

This article reports on a cross-sectional case study of a large construction project in which Electronic document management (EDM) was used. Attitudes towards EDM from the perspective of individual end users were investigated. Responses from a survey were combined with data from system usage log files to obtain an overview of attitudes prevalent in different user segments of the total population of 334 users. The survey was followed by semi-structured interviews with representative users. A strong majority of users from all segments of the project group considered EDM as a valuable aid in their work processes, despite certain functional limitations of the system used and the complexity of the information mass. Based on the study a model describing the key factors affecting end user EDM adoption is proposed. The model draws on insight from earlier studies of EDM enabled projects and theoretical frameworks on technology acceptance and success of information systems, as well as the insights gained from the case study.

Language Influence in Services: Perceived Importance of Native Language Use in Service Encounters

A defining characteristic of most service encounters is that they are strongly influenced by interactions in which both the consumer and the service personnel are playing integral roles. Such is the importance of this interaction that it has even been argued that for the consumer, these encounters are in fact the service. Given this, it is not surprising that interactions involving communication and customer participation in the service encounters have received considerable attention within the field of services marketing. Much of the research on interactions and communication in services, however, appear to have assumed that the consumer and the service personnel by definition are perfectly able to interact and communicate effortlessly with each other. Such communication would require a common language, and in order to be able to take this for granted the market would need to be fairly homogenous. The homogenous country, however, and with it the homogenous market, would appear to be gone. It is estimated that more than half the consumers in the world are already speaking more than one language. For a company entering a new market, language can be a major barrier that firms may underestimate, and understanding language influence across different markets is important for international companies. The service literature has taken a common language between companies and consumers for granted but this is not matched by the realities on the ground in many markets. Owing to the communicational and interaction-oriented nature of services, the lack of a common language between the consumer and the service provider is a situation that could cause problems. A gap exists in the service theory, consisting of a lack of knowledge concerning how language influences consumers in service encounters. By addressing this gap, the thesis contributes to an increased understanding of service theory and provides a better practical understanding for service companies of the importance of native language use for consumers. The thesis consists of four essays. Essay one is conceptual and addresses how sociolinguistic research can be beneficial for understanding consumer language preferences. Essay two empirically shows how the influence of language varies depending on the nature of the service, essay three shows that there is a significant difference in language preferences between female and male consumers while essay four empirically compares consumer language preferences in Canada and Finland, finding strong similarities but also indications of difference in the motives for preferring native language use. The introduction of the thesis outlines the existence of a research gap within the service literature, a gap consisting of the lack of research into how native language use may influence consumers in service encounters. In addition, it is described why this gap is of importance to services and why its importance is growing. Building on this situation, the purpose of the thesis is to establish the existence of language influence in service encounters and to extend the knowledge of how language influences consumers on multilingual markets.