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.

Visuaalis-spatiaalisten työmuistivalmiuksien yhteys (esi)matemaattisiin taitoihin ja merkitys osana matemaattisilta taidoiltaan heikkojen lasten ja nuorten kognitiivista profiilia

Tämän itsenäisistä osatutkimuksista koostuvan tutkimussarjan tavoitteena oli pyrkiä täydentämään kuvaa matemaattisilta taidoiltaan heikkojen lasten ja nuorten tiedonkäsittelyvalmiuksista selvittämällä, ovatko visuaalis-spatiaaliset työmuistivalmiudet yhteydessä matemaattiseen suoriutumiseen. Teoreettinen viitekehys rakentui Baddeleyn (1986, 1997) kolmikomponenttimallin ympärille. Työmuistikäsitys oli kuitenkin esikuvaansa laajempi sisällyttäen visuaalis-spatiaaliseen työmuistiin Cornoldin ja Vecchin (2003) termein sekä passiiviset varastotoiminnot että aktiiviset prosessointitoiminnot. Yhteyksiä työmuistin ja matemaattisten taitojen välillä tarkasteltiin viiden eri osatutkimuksen avulla. Kaksi ensimmäistä keskittyivät alle kouluikäisten lukukäsitteen hallinnan ja visuaalis-spatiaalisten työmuistivalmiuksen tutkimiseen ja kolme jälkimmäistä peruskoulun yhdeksäsluokkalaisten matemaattisten taitojen ja visuaalis-spatiaalisten työmuistitaitojen välisten yhteyksien selvittämiseen. Tutkimussarjan avulla pyrittiin selvittämään, ovatko visuaalis-spatiaaliset työmuistivalmiudet yhteydessä matemaattiseen suoriutumiseen sekä esi- että yläkouluiässä (osatutkimukset I, II, III, IV, V), onko yhteys spesifi rajoittuen tiettyjen visuaalis-spatiaalisten valmiuksien ja matemaattisen suoriutumisen välille vai onko se yleinen koskien matemaattisia taitoja ja koko visuaalis-spatiaalista työmuistia (osatutkimukset I, II, III, IV, V) tai työmuistia laajemmin (osatutkimukset II, III) sekä onko yhteys työmuistispesifi vai selitettävissä älykkyyden kaltaisella yleisellä päättelykapasiteetilla (osatutkimukset I, II, IV). Tutkimussarjan tulokset osoittavat, että kyky säilyttää ja käsitellä hetkellisesti visuaalis-spatiaalista informaatiota on yhteydessä matemaattiseen suoriutumiseen eikä yhteyttä voida selittää yksinomaan joustavalla älykkyydellä. Suoriutuminen visuaalis-spatiaalista työmuistia mittaavissa tehtävissä on yhteydessä sekä alle kouluikäisten esimatemaattisten taitojen hallintaan että peruskoulun yhdeksäsluokkalaisten matematiikan taitoihin. Matemaattisilta taidoiltaan heikkojen lasten ja nuorten visuaalis-spatiaalisten työmuistiresurssien heikkoudet vaikuttavat kuitenkin olevan sangen spesifejä rajoittuen tietyntyyppisissä muistitehtävissä vaadittaviin valmiuksiin; kaikissa visuaalis-spatiaalisen työmuistin valmiuksia mittaavissa tehtävissä suoriutuminen ei ole yhteydessä matemaattisiin taitoihin. Työmuistivalmiuksissa ilmenevät erot sekä alle kouluikäisten että kouluikäisten matemaattisilta taidoiltaan heikkojen ja normaalisuoriutujien välillä näyttävät olevan kuitenkin jossain määrin yhteydessä kielellisiin taitoihin viitaten vaikeuksien tietynlaiseen kasautumiseen; niillä matemaattisesti heikoilla, joilla on myös kielellisiä vaikeuksia, on keskimäärin laajemmat työmuistiheikkoudet. Osalla matematiikassa heikosti suoriutuvista on näin ollen selvästi keskimääräistä heikommat visuaalis-spatiaaliset työmuistivalmiudet, ja tämä heikkous saattaa olla yksi mahdollinen syy tai vaikeuksia lisäävä tekijä heikon matemaattisen suoriutumisen taustalla. Visuaalis-spatiaalisen työmuistin heikkous merkitsee konkreettisesti vähemmän mentaalista prosessointitilaa, joka rajoittaa oppimista ja suoritustilanteita. Tiedonkäsittelyvalmiuksien heikkous liittyy nimenomaan oppimisnopeuteen, ei asioiden opittavuuteen sinänsä. Mikäli oppimisympäristö ottaa huomioon valmiuksien rajallisuuden, työmuistiheikkoudet eivät todennäköisesti estä asioiden oppimista sinänsä. Avainsanat: Työmuisti, visuaalis-spatiaalinen työmuisti, matemaattiset taidot, lukukäsite, matematiikan oppimisvaikeudet

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

Tekstiilityötä ja matematiikkaa yhdistävän oppimateriaalin käyttökelpoisuus perusopetuksen yläluokilla

In this study the researcher wanted to show the observed connection of mathematics and textile work. To carry this out the researcher designed a textbook by herself for the upper secondary school in Tietoteollisuuden Naiset TiNA project at Helsinki University of Technology (URL:http://tina.tkk.fi/). The assignments were designed as additional teaching material to enhance and reinforce female students confidence in mathematics and in the management of their textile work. The research strategy applied action research, out of which two cycles two have been carried out. The first cycle consists of establishing the textbook and in the second cycle its usability is investigated. The third cycle is not included in this report. In the second cycle of the action research the data was collected from 15 teachers, five textile teachers, four mathematics teachers and six teachers of both subjects. They all got familiar with the textbook assignments and answered a questionnaire on the basis of their own teaching experience. The questionnaire was established by applying the theories of usability and teaching material assessment study. The data consisted of qualitative and quantitative information, which was analysed by content analysis with computer assisted table program to either qualitative or statistical description. According to the research results, the textbook assignments seamed to be applied better to mathematics lessons than textile work. The assignments pointed out, however, the clear interconnectedness of textile work and mathematics. Most of the assignments could be applied as such or as applications in the upper secondary school textile work and mathematics lessons. The textbook assignments were also applicable in different stages of the teaching process, e.g. as introduction, repetition or to support individual work or as group projects. In principle the textbook assignments were in well placed and designed in the correct level of difficulty. Negative findings concerned some too difficult assignments, lack of pupil motivation and unfamiliar form of task for the teacher. More clarity for some assignments was wished for and there was especially expressed a need for easy tasks and assignments in geometry. Assignments leading to the independent thinking of the pupil were additionally asked for. Two important improvements concerning the textbook attainability would be to get the assignments in html format over the Internet and to add a handicraft reference book.

Distortion of Dimension under Quasiconformal Mappings

Quasiconformal mappings are natural generalizations of conformal mappings. They are homeomorphisms with 'bounded distortion' of which there exist several approaches. In this work we study dimension distortion properties of quasiconformal mappings both in the plane and in higher dimensional Euclidean setting. The thesis consists of a summary and three research articles. A basic property of quasiconformal mappings is the local Hölder continuity. It has long been conjectured that this regularity holds at the Sobolev level (Gehring's higher integrabilty conjecture). Optimal regularity would also provide sharp bounds for the distortion of Hausdorff dimension. The higher integrability conjecture was solved in the plane by Astala in 1994 and it is still open in higher dimensions. Thus in the plane we have a precise description how Hausdorff dimension changes under quasiconformal deformations for general sets. The first two articles contribute to two remaining issues in the planar theory. The first one concerns distortion of more special sets, for rectifiable sets we expect improved bounds to hold. The second issue consists of understanding distortion of dimension on a finer level, namely on the level of Hausdorff measures. In the third article we study flatness properties of quasiconformal images of spheres in a quantitative way. These also lead to nontrivial bounds for their Hausdorff dimension even in the n-dimensional case.

Locality and order-invariant logics

Malli on logiikassa käytetty abstraktio monille matemaattisille objekteille. Esimerkiksi verkot, ryhmät ja metriset avaruudet ovat malleja. Äärellisten mallien teoria on logiikan osa-alue, jossa tarkastellaan logiikkojen, formaalien kielten, ilmaisuvoimaa malleissa, joiden alkioiden lukumäärä on äärellinen. Rajoittuminen äärellisiin malleihin mahdollistaa tulosten soveltamisen teoreettisessa tietojenkäsittelytieteessä, jonka näkökulmasta logiikan kaavoja voidaan ajatella ohjelmina ja äärellisiä malleja niiden syötteinä. Lokaalisuus tarkoittaa logiikan kyvyttömyyttä erottaa toisistaan malleja, joiden paikalliset piirteet vastaavat toisiaan. Väitöskirjassa tarkastellaan useita lokaalisuuden muotoja ja niiden säilymistä logiikkoja yhdistellessä. Kehitettyjä työkaluja apuna käyttäen osoitetaan, että Gaifman- ja Hanf-lokaalisuudeksi kutsuttujen varianttien välissä on lokaalisuuskäsitteiden hierarkia, jonka eri tasot voidaan erottaa toisistaan kasvavaa dimensiota olevissa hiloissa. Toisaalta osoitetaan, että lokaalisuuskäsitteet eivät eroa toisistaan, kun rajoitutaan tarkastelemaan äärellisiä puita. Järjestysinvariantit logiikat ovat kieliä, joissa on käytössä sisäänrakennettu järjestysrelaatio, mutta sitä on käytettävä siten, etteivät kaavojen ilmaisemat asiat riipu valitusta järjestyksestä. Määritelmää voi motivoida tietojenkäsittelyn näkökulmasta: vaikka ohjelman syötteen tietojen järjestyksellä ei olisi odotetun tuloksen kannalta merkitystä, on syöte tietokoneen muistissa aina jossakin järjestyksessä, jota ohjelma voi laskennassaan hyödyntää. Väitöskirjassa tutkitaan minkälaisia lokaalisuuden muotoja järjestysinvariantit ensimmäisen kertaluvun predikaattilogiikan laajennukset yksipaikkaisilla kvanttoreilla voivat toteuttaa. Tuloksia sovelletaan tarkastelemalla, milloin sisäänrakennettu järjestys lisää logiikan ilmaisuvoimaa äärellisissä puissa.

Prevalence and Forms of Bullying among Business Professionals: A Comparison of Two Different Strategies for Measuring Bullying

The aim of this study was to analyse the prevalence and forms of workplace bullying among business professionals holding predominantly managerial or expert positions. A cross-sectional survey study was conducted among Finnish professionals with a university degree in business studies. In the study two different strategies for measuring bullying were used and compared. When provided a definition of bullying, 8.8% of the respondents reported that they had at least occasionally been bullied during the past 12 months. However, when using a slightly modified version of the Negative Acts Questionnaire (Einarsen & Hoel 2001), containing a list of 32 predefined negative and potentially harassing acts, as many as 24.1% of the respondents reported that they had been subjected to at least one of the negative acts on a weekly basis. The respondents had experienced predominantly work-related negative acts, e.g. that their opinions and views were ignored, that they were given unreasonable deadlines or that information was withheld. Although the prevalence rates reported with the two strategies varied considerably, there was still consistency between the two strategies in the sense that those who had classified themselves as bullied also reported higher exposure rates to almost all of the negative acts included.

Asukkaan Helsinki? : Tapaustutkimus vuorovaikutteisuudesta kaupunkisuunnittelussa

This study examines the interaction between inhabitants and urban planning. In addition to traditional methods of participation there can be seen an increasing need to find new channels and means to influence on one´s environment. Hearing of inhabitants during planning is hoped to decrease the amount of claims and this way to speed up the planning process. Discussion that emphasizes competitiveness and innovativeness in planning has brought out the role of inhabitants as selective consumers and end users. This extension of civil perspective completes the thought of participation in city planning and adds the pressure on developing the interaction and user orientation. The aim of the study is to point out the present situation of inhabitant´s participation and influencing in Helsinki. Helsinki City Planning Department opened a new information and exhibiotion hall called Laituri in 2008. Laituri provides the latest information about planning projects in Helsinki and temporary exhibitions as well as it operates as communication channel and information point for the department. In this study Laituri is examined as a case study of interaction between citizen and planner. The study is divided into two principal themes. The aim is to research action and interactivity at Laituri from the inhabitants´and planners´point of view. The qualitative study has inquiries, interview surveys and observation as research methods. Empiricial data of the study consists of three parts which complement each other: Laituri operational reseach, inquiry directed to the members of Laituri team and interviews of three experts. The aim is to find answers to questions like, does Laituri reach the citizens and will the opportunities to participate improve along Laituri. The study examines also how the local knowledge of inhabitants will come across to planners and further to planning. The study combines discussions of inhabitants´ influencing in Finnish society and science community. Cornerstones of the study are inhabitant participation, interaction and local knowledge in urban planning. The theory behind the study is communicative planning theory. In addition the theory consists of key concepts. The study introduces a concept of Inhabitant´s Helsinki, which reflects the inhabitant as customer-citizen who is an active product developer in participative urban planning. According to the research results the experts of Laituri and majority of inhabitants in Helsinki experience that the inhabitant´s possibilities to participate will improve along Laituri. However half of the citizens in Helsinki believe that local knowledge and ideas will only have minor impact on the final plan. According to city dwellers the present practise used by Helsinki City Planning Department supports only partially adequate interaction. The experts of Laituri experience that the role of Laituri is first of all forum of communication and discussion channel instead of effective local data collector. Based on the results the study introduces a model of inhabitant´s participation field. According to the model Laituri can be seen as phenomenom in Helsinki urban planning which has the elements of network municipality. The planner is more like diplomatic trend-setter and visionary. The inhabitant of Helsinki is an expert in city living and participative producer of local knowledge. Participation methods are increasingly segmented and tailored in every plan and project. The study argues that Inhabitant´s Helsinki is a pluralistic milieu in constant pressure for change. Therefore reaching the everyday life experiencies of inhabintants should be at higher degree in Helsinki City Planning Department´s operations.