Computable analysis on the space of marked groups

We investigate decision problems for groups described by word problem algorithms. This is equivalent to studying groups described by labelled Cayley graphs. We show that this corresponds to the study of computable analysis on the space of marked groups, and point out several results of computable analysis that can be directly applied to obtain group theoretical results. Those results, used in conjunction with the version of Higman's Embedding Theorem that preserves solvability of the word problem, provide powerful tools to build finitely presented groups with solvable word problem but with various undecidable properties. We also investigate the first levels of an effective Borel hierarchy on the space of marked groups, and show that on many group properties usually considered, this effective hierarchy corresponds sharply to the Borel hierarchy. Finally, we prove that the space of marked groups is a Polish space that is not effectively Polish. Because of this, many of the most important results of computable analysis cannot be applied to the space of marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and a theorem of Moschovakis. The space of marked groups constitutes the first natural example of a Polish space that is not effectively Polish.Comment: 46 pages, Theorem 4.6 was false as stated, it appears now, having been corrected, as Theorem 5.

Computing Measure as a Primitive Operation in Real Number Computation

We study the power of BSS-machines enhanced with abilities such as computing the measure of a BSS-decidable set or computing limits of BSS-computable converging sequences. Our variations coalesce into just two equivalence classes, each of which also can be described as a lower cone in the Weihrauch degrees. We then classify computational tasks such as computing the measure of ???-set of reals, integrating piece-wise continuous functions and recovering a continuous function from an L?([0, 1])-description. All these share the Weihrauch degree lim

The Strength of Truth-Theories

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? It turns out that, in a wide range of cases, we can get some nice answers to this question, but only if we work in a framework that is somewhat different from those usually employed in discussions of axiomatic theories of truth. These results are then used to address a range of philosophical questions connected with truth, such as what Tarski meant by "essential richness" and the so-called conservativeness argument against deflationism. This draft dates from about 2009, with some significant updates having been made around 2011. Around then, however, I decided that the paper was becoming unmanageable and that I was trying to do too many things in it. I have therefore exploded the paper into several pieces, which will be published separately. These include "Disquotationalism and the Compositional Principles", "The Logical Strength of Compositional Principles", "Consistency and the Theory of Truth", and "What Is Essential Richness?" You should probably read those instead, since this draft remains a bit of a mess. Terminology and notation are inconsistent, and some of the proofs aren't quite right. So, caveat lector. I make it public only because it has been cited in a few places now

Totuus, todistettavuus ja gödeliläiset argumentit : Tarskilaisen totuuden puolustus matematiikassa

Eräs tärkeimmistä kysymyksistä matematiikanfilosofiassa on totuuden ja formaalin todistettavuuden välinen suhde. Kantaa, jonka mukaan nämä kaksi käsitettä ovat yksi ja sama, kutsutaan deflationismiksi, ja vastakkaista näkökulmaa substantialismiksi. Ensimmäisessä epätäydellisyyslauseessaan Kurt Gödel todisti, että kaikki ristiriidattomat ja aritmetiikan sisältävät formaalit systeemit sisältävät lauseita, joita ei voida sen enempää todistaa kuin osoittaa epätosiksi kyseisen systeemin sisällä. Tällaiset Gödel-lauseet voidaan kuitenkin osoittaa tosiksi, jos laajennamme formaalia systeemiä Alfred Tarskin semanttisella totuusteorialla, kuten Stewart Shapiro ja Jeffrey Ketland ovat näyttäneet semanttisissa argumenteissaan substantialismin puolesta. Heidän mukaansa Gödel-lauseet ovat eksplisiittinen tapaus todesta lauseesta, jota ei voida todistaa, ja siten deflationismi on kumottu. Tätä vastaan Neil Tennant on näyttänyt, että tarskilaisen totuuden sijaan voimme laajentaa formaalia systeemiä ns. pätevyysperiaatteella, jonka mukaan kaikki todistettavat lauseet ovat ”väitettävissä”, ja josta seuraa myös Gödel-lauseiden väitettävyys. Relevantti kysymys ei siis ole se pystytäänkö Gödel-lauseiden totuus osoittamaan, vaan se onko tarskilainen totuus hyväksyttävämpi laajennus kuin pätevyysperiaate. Tässä työssä väitän, että tätä ongelmaa on paras lähestyä ajattelemalla matematiikkaa ilmiönä, joka on laajempi kuin pelkästään formaalit systeemit. Kun otamme huomioon esiformaalin matemaattisen ajattelun, huomaamme että tarskilainen totuus ei itse asiassa ole laajennus lainkaan. Väitän, että totuus on esiformaalissa matematiikassa sitä mitä todistettavuus on formaalissa, ja tarskilainen semanttinen totuuskäsitys kuvaa tätä suhdetta tarkasti. Deflationisti voi kuitenkin argumentoida, että vaikka esiformaali matematiikka on olemassa, voi se silti olla filosofisesti merkityksetöntä mikäli se ei viittaa mihinkään objektiiviseen. Tätä vastaan väitän, että kaikki todella deflationistiset teoriat johtavat matematiikan mielivaltaisuuteen. Kaikissa muissa matematiikanfilosofisissa teorioissa on tilaa objektiiviselle viittaukselle, ja laajennus tarskilaiseen totuuteen voidaan tehdä luonnollisesti. Väitän siis, että mikäli matematiikan mielivaltaisuus hylätään, täytyy hyväksyä totuuden substantiaalisuus. Muita tähän liittyviä aiheita, kuten uusfregeläisyyttä, käsitellään myös tässä työssä, eikä niiden todeta poistavan tarvetta tarskilaiselle totuudelle. Ainoa jäljelle jäävä mahdollisuus deflationistille on vaihtaa logiikkaa niin, että formaalit kielet voivat sisältää omat totuuspredikaattinsa. Tarski osoitti tämän mahdottomaksi klassisille ensimmäisen kertaluvun kielille, mutta muilla logiikoilla ei välttämättä olisi lainkaan tarvetta laajentaa formaaleja systeemejä, ja yllä esitetty argumentti ei pätisi. Vaihtoehtoisista tavoista keskityn tässä työssä eniten Jaakko Hintikan ja Gabriel Sandun ”riippumattomuusystävälliseen” IF-logiikkaan. Hintikka on väittänyt, että IF-kieli voi sisältää oman adekvaatin totuuspredikaattinsa. Väitän kuitenkin, että vaikka tämä onkin totta, tätä predikaattia ei voida tunnistaa totuuspredikaatiksi saman IF-kielen sisäisesti, ja siten tarve tarskilaiselle totuudelle säilyy. IF-logiikan lisäksi myös toisen kertaluvun klassinen logiikka ja Saul Kripken käyttämä Kleenen logiikka epäonnistuvat samalla tavalla

Podstawy matematyki bez aktualnej nieskończoności

Contemporary mathematics significantly uses notions which belong to ideal mathematics (in Hilbert’s sense) – which is expressed in language which essentially uses actual infinity. However, we do not have a meaningful notion of truth for such languages. We can only reduce the notion of truth to finitistic mathematics via axiomatic theories. Nevertheless, justification of truth of axioms themselves exceeds the capabilities of the theory based on these axioms. On the other hand, we can easily decide the truth or falsity of a statement in finite structures. The aim of this dissertation is to identify the fragment of mathematics, which is of the finitistic character. The fragment of mathematics which can be described without actual infinity. This is the part of mathematics which can be described in finite models and for which the truth of its statements can be verified within finite models.We call this fragment of mathematics with a term introduced by Knuth – the concrete mathematics. This part of mathematics is of computational character and it is closer to our empirical base, which makes it more difficult. We consider concrete foundations of mathematics, in particular the concrete model theory and semantics without actual infinity. We base on the notion of FM–representability, introduced by Mostowski, as an explication of expressibility without actual infinity. By the Mostowski’s FM–representability theorem, FM–representable notions are exactly those, which are recursive with the halting problem as an oracle. We show how to express basic concepts of model theory in the language without actual infinity. We investigate feasibility of the classical model– theoretic constructions in the concrete model theory. We present the Concrete Completeness Theorem and the Low Completeness Theorem; the Concrete Omitting Types Theorem; and Preservation Theorems. We identify the constructions which are not admissible in the concrete model theory by showing stages of these constructions which are not allowed in the concrete framework. We show which arguments from the axiomatic model theory fail in the concrete model theory. Moreover, we investigate how to approximate truth for finite models. In particular we study the properties of approximate FM–truth definitions which are expressible in modal logic. We introduce modal logic SL, axioms of which mimic the properties of a specific approximate FM–truth definition. We show that SL is the modal logic of any approximate FM–truth definition. This is done by proving a theorem analogous to Solovay’s completeness theorem for modal logic GL.Współczesna matematyka w znaczącej mierze posługuje się pojęciami, które należą do matematyki idealnej (w sensie Hilberta) -- wyrażona jest w języku istotnie wykorzystującym aktualną nieskończoność. Dla tego typu języków nie posiadamy sensownego kryterium prawdziwości. Jesteśmy w stanie jedynie redukować je do matematyki skończonościowej poprzez teorie aksjomatyczne. Niemniej uzasadnianie prawdziwości samych aksjomatów znajduje się poza zasięgiem teorii na nich opartej. Z drugiej strony w strukturach skończonych jesteśmy w stanie w prosty sposób rozstrzygać prawdziwość i fałszywość twierdzeń. Celem niniejszej rozprawy jest identyfikacja fragmentu matematyki, który ma skończonościowy charakter. Fragmentu matematyki, do którego opisu nie jest niezbędna aktualna nieskończoność, a wystarczy jedynie nieskończoność potencjalna. Jest to ta część matematyki, której pojęcia można wyrazić w modelach skończonych oraz prawdziwość twierdzeń której można w nich zweryfikować. Tę część matematyki, za Knuthem, nazywamy matematyką konkretną. Ma ona obliczeniowy, kombinatoryczny charakter i jest bliższa naszemu doświadczeniu niż matematyka idealna, a co za tym idzie jest trudniejsza. Rozważamy konkretne podstawy matematyki, w szczególności konkretną teorię modeli oraz semantykę bez aktualnej nieskończoności. Opieramy się na wprowadzonym przez Mostowskiego pojęciu FM--reprezentowalności, jako eksplikacji wyrażalności bez aktualnej nieskończoności oraz twierdzeniu o FM--reprezentowalności identyfikującym FM--reprezentowalne pojęcia z tymi, które są obliczalne z problemem stopu jako wyrocznią. Pokazujemy w jaki sposób można zinterpretować podstawowe pojęcia teorii modeli w języku bez aktualnej nieskończoności. Następnie badamy klasyczne konstrukcje teoriomodelowe pod kątem ich wykonalności w obszarze matematyki konkretnej. Prezentujemy twierdzenie o konkretnej pełności oraz twierdzenie o łatwej pełności, twierdzenie o omijaniu typów oraz twierdzenia o zachowaniu. Przedstawiamy konstrukcje, które są niewykonalne dla modeli konkretnych, identyfikując etapy konstrukcji teoriomodelowych, które nie są wykonalne w teorii modeli konkretnych. Identyfikujemy argumenty z aksjomatycznej teorii mnogości, które nie są dopuszczalne w konkretnej teorii modeli. Ponadto, badamy możliwość przybliżania prawdy arytmetycznej w modelach skończonych. W szczególności rozważamy te własności przybliżonych predykatów prawdy dla modeli skończonych, które wyrażalne są w logice modalnej. Wprowadzamy logikę modalną SL, której aksjomaty odzwierciedlają własności przybliżonych predykatów prawdy. Pokazujemy, że logika SL jest logiką przybliżonych predykatów prawdy -- dowodzimy twierdzenia analogicznego do twierdzenia o pełności dla logiki GL udowodnionego przez Solovaya

An Algorithmic Interpretation of Quantum Probability

The Everett (or relative-state, or many-worlds) interpretation of quantum mechanics has come under fire for inadequately dealing with the Born rule (the formula for calculating quantum probabilities). Numerous attempts have been made to derive this rule from the perspective of observers within the quantum wavefunction. These are not really analytic proofs, but are rather attempts to derive the Born rule as a synthetic a priori necessity, given the nature of human observers (a fact not fully appreciated even by all of those who have attempted such proofs). I show why existing attempts are unsuccessful or only partly successful, and postulate that Solomonoff's algorithmic approach to the interpretation of probability theory could clarify the problems with these approaches. The Sleeping Beauty probability puzzle is used as a springboard from which to deduce an objectivist, yet synthetic a priori framework for quantum probabilities, that properly frames the role of self-location and self-selection (anthropic) principles in probability theory. I call this framework "algorithmic synthetic unity" (or ASU). I offer no new formal proof of the Born rule, largely because I feel that existing proofs (particularly that of Gleason) are already adequate, and as close to being a formal proof as one should expect or want. Gleason's one unjustified assumption--known as noncontextuality--is, I will argue, completely benign when considered within the algorithmic framework that I propose. I will also argue that, to the extent the Born rule can be derived within ASU, there is no reason to suppose that we could not also derive all the other fundamental postulates of quantum theory, as well. There is nothing special here about the Born rule, and I suggest that a completely successful Born rule proof might only be possible once all the other postulates become part of the derivation. As a start towards this end, I show how we can already derive the essential content of the fundamental postulates of quantum mechanics, at least in outline, and especially if we allow some educated and well-motivated guesswork along the way. The result is some steps towards a coherent and consistent algorithmic interpretation of quantum mechanics

Wittgenstein on the Foundations of Mathematics

In Part I, an attempt is made to survey the original source material on which any detailed assessment of Wittgenstein's remarks on the foundations of mathematics from his middle and later periods ought to be based. This survey is presented within the context of a sketch of Wittgenstein's biography, which also mentions some of the major developments in his thinking. In addition, certain main themes are emphasized; these have to do primarily with the Kantian aspects of Wittgenstein's thought and with his mysticism or the 'religious point of view'. In Part II, Kreisel's critique of Wittgenstein's remarks on the foundations of mathematics, which has been developed since 1958 in a series of published articles, receives close examination, and, in connection with this, different approaches to the philosophical investigation of mathematics are considered which represent genuine alternatives to Wittgenstein's approach. There are separate sections on Lakatos's Proofs and Refutations and Bourbaki's 'L'Architecture des Mathématiques'. Finally, besides a bibliography which surveys the reception of Wittgenstein's views on the foundations of mathematics, there are two substantial appendices, which are supplemental to Part I. The first of these gives the manuscript sources for typescripts 221 and 222-4, and the correspondences in both directions between these typescripts. The second appendix is part of a chronological version of von Wright's catalogue of Wittgenstein's papers, beginning in 1929

Programmation et indécidabilités dans les systèmes complexes

N/AUn système complexe est un système constitué d'un ensemble d'entités quiinteragissent localement, engendrant des comportements globaux, émergeant dusystème, qu'on ne sait pas expliquer à partir du comportement local, connu, desentités qui le constituent. Nos travaux ont pour objet de mieux cerner lesliens entre certaines propriétés des systèmes complexes et le calcul. Parcalcul, il faut entendre l'objet d'étude de l'informatique, c'est-à-dire ledéplacement et la combinaison d'informations. À l'aide d'outils issus del'informatique, l'algorithmique et la programmation dans les systèmes complexessont abordées selon trois points de vue. Une première forme de programmation,dite externe, consiste à développer l'algorithmique qui permet de simuler lessystèmes étudiés. Une seconde forme de programmation, dite interne, consiste àdévelopper l'algorithmique propre à ces systèmes, qui permet de construire desreprésentants de ces systèmes qui exhibent des comportements programmés. Enfin,une troisième forme de programmation, de réduction, consiste à plonger despropriétés calculatoires complexes dans les représentants de ces systèmes pourétablir des résultats d'indécidabilité -- indice d'une grande complexitécalculatoire qui participe à l'explication de la complexité émergente. Afin demener à bien cette étude, les systèmes complexes sont modélisés par desautomates cellulaires. Le modèle des automates cellulaires offre une dualitépertinente pour établir des liens entre complexité des propriétés globales etcalcul. En effet, un automate cellulaire peut être décrit à la fois comme unréseau d'automates, offrant un point de vue familier de l'informatique, etcomme un système dynamique discret, une fonction définie sur un espacetopologique, offrant un point de vue familier de l'étude des systèmesdynamiques discrets.Une première partie de nos travaux concerne l'étude de l'objet automatecellulaire proprement dit. L'observation expérimentale des automatescellulaires distingue, dans la littérature, deux formes de dynamiques complexesdominantes. Certains automates cellulaires présentent une dynamique danslaquelle émergent des structures simples, sortes de particules qui évoluentdans un domaine régulier, se rencontrent lors de brèves collisions, avant degénérer d'autres particules. Cette forme de complexité, dans laquelletransparaît une notion de quanta d'information localisée en interaction, estl'objet de nos études. Un premier champ de nos investigations est d'établir uneclassification algébrique, le groupage, qui tend à rendre compte de ce type decomportement. Cette classification met à jour un type d'automate cellulaireparticulier : les automates cellulaires intrinsèquement universels. Un automatecellulaire intrinsèquement universel est capable de simuler le comportement detout automate cellulaire. C'est l'objet de notre second champ d'investigation.Nous caractérisons cette propriété et démontrons son indécidabilité. Enfin, untroisième champ d'investigation concerne l'algorithmique des automatescellulaires à particules et collisions. Étant donné un ensemble de particuleset de collisions d'un tel automate cellulaire, nous étudions l'ensemble desinteractions possibles et proposons des outils pour une meilleure programmationinterne à l'aide de ces collisions.Une seconde partie de nos travaux concerne la programmation par réduction. Afinde démontrer l'indécidabilité de propriétés dynamiques des automatescellulaires, nous étudions d'une part les problèmes de pavage du plan par desjeux de tuiles finis et d'autre part les problèmes de mortalité et depériodicité dans les systèmes dynamiques discrets à fonction partielle. Cetteétude nous amène à considérer des objets qui possèdent la même dualité entredescription combinatoire et topologique que les automates cellulaires. Unenotion d'apériodicité joue un rôle central dans l'indécidabilité des propriétésde ces objets

Geometry In Architecture: An Approach Developed On Architectural And Philosophical Discussions

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2011Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2011Üniversitesi : İstanbul Teknik Üniversitesi Enstitüsü : Fen Bilimleri Anabilim Dalı : Mimarlık Programı : Mimari Tasarım Tez Danışmanı : Prof. Dr. Yurdanur DÜLGEROĞLU YÜKSEL Tez Türü ve Tarihi : Doktora –Haziran 2011 ÖZET MİMARLIKTA GEOMETRİ: FELSEFİ VE MİMARİ TARTIŞMALAR ÜZERİNE GELİŞTİRİLMİŞ BİR YAKLAŞIM Ayşe SIKIÇAKAR YÜCEL Hem mimarlıkta hem felsefede yer alan bir tartışmanın iki farklı ayağını içeren bu tez, geometrinin modernitede ve moderniteye yönelik eleştirilerde oynadığı rol üzerine odaklanmıştır. Mimari temsil, görsel ağırlıklı sunuşlar, algı ve metaforu içermesi nedeniyle, tasarım sürecinin sezilebilen ve sezilemeyen tüm adımlarını ilgilendirmektedir. Modern düşüncenin yaşayan dünyadaki çeşitliliği görsel biçim kaynaklı anlayışlarla analiz etme çabası, görselleştirmenin baskın etkisi, bu amaçla geometriye dayalı uygulamalardaki yoğunluk, mimarlıkta mekanın tasarım ve ifade süreçleri üzerine bu konulara dayalı olarak yeniden eğilmeyi önemli kılmıştır. Bu tez, kendinden önce bu konuları birbirinden ayrı olarak ele alan pek çok yaklaşımın ötesinde; geometrik, şematik yapıya sahip ve metaforik doğası olan düşünce ve sunumların değerlendirmesini bir araya getirerek yapar. Bu çalışmada Kant ve Heidegger, Modern ve karşıtı örnekleri oluşturmak üzere, mekan ve geometriye bakışları açısından karşılaştırılmak istenmektedir. Kant’ın birey ve Heidegger’in varlık anlayışları düşünüldüğünde farklı uçları oluşturdukları görülmektedir. Kant’ın ilgi alanları geniştir ve bireyin tanımında hemen her konuya kendince hatasız bir şekilde cevap veren, lineer bir şematik kurguya sahip bir sistem tarif eder. Heidegger’in de ilgi alanları çeşitlilik gösterir, ilk bakışta karmaşık görünse de, mimarlığa epeyce tartışma konusu katmaktadır. Tezin bütününde, mimari mekanın algı ve sunum sistemleri, şematik kurguya dayalı algı modelleri karşısında duran, tüm bedenin dahil olduğu algı anlayışları ve unların çağdaşları olan sunum sistemleri bu alanlarla ilgili tartışmaların kapsamında yer almaktadır. Anahtar Kelimeler: Geometri, Şema, Algı, Temsil, Kant, Heidegger, Metafor, Mimari Mekan Bilim Dalı Sayısal Kodu: 601.01.02University : Istanbul Technical University Institute : Institute of Science and Technology Science Programme : Architecture Programme : Architectural Design Supervisor : Prof. Dr. Yurdanur DÜLGEROĞLU YÜKSEL Degree Awarded and Date : PhD – June 2011 ABSTRACT GEOMETRY IN ARCHITECTURE: AN APPROACH DEVELOPED ON THE ARCHITECTURAL AND PHILOSOPHICAL DISCUSSIONS Ayşe SIKIÇAKAR YÜCEL Main topic in this dissertation is based on comparison of two different ends of a major argument existing with some similar points in philosophy and architecture. The argument is on the role of geometry in modernity and its criticisms, and involves in the conflict between architectural representation, visual demonstrations and metaphor, predictable and unpredictable aspects of design process. Investigating the connections between the major changes of geometry used by science and architecture is necessarily a matter of concern. This dissertation combines the evaluation of arguments over geometric, schematic and metaphoric representations, unlike some previous approaches which deal with them individually. Considering the arguments in the philosophical area, this dissertation aims at examining Kant s and Heidegger’s ways of dealing with space and geometry, as well as art and to some extent architecture. Considering one’s understanding of subject and other’s Dasein, Kant and Heidegger constitutes two far ends of a line. Kant is versatile, makes himself difficult to be understood but he describes such a linear structured system that covers as many questions as possible and for him without any failure. Heidegger is also wide in his interests, confusing at first glance and above all lending quite a few topics to architecture. Comments on systems of perception and representation of architectural space, schematic perception models vs. bodily involvements and representation systems of corresponding periods constituted related areas to these arguments. Keywords: Geometry, Schematizm, Perception, Representation, Kant, Heidegger, Metaphor, Architectural Space Science Code: 601.01.02DoktoraPh