324 research outputs found

    Anti-Foundational Categorical Structuralism

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    The aim of this dissertation is to outline and defend the view here dubbed “anti-foundational categorical structuralism” (henceforth AFCS). The program put forth is intended to provide an answer the question “what is mathematics?”. The answer here on offer adopts the structuralist view of mathematics, in that mathematics is taken to be “the science of structure” expressed in the language of category theory, which is argued to accurately capture the notion of a “structural property”. In characterizing mathematical theorems as both conditional and schematic in form, the program is forced to give up claims to securing the truth of its theorems, as well as give up a semantics which involves reference to special, distinguished “mathematical objects”, or which involves quantification over a fixed domain of such objects. One who wishes—contrary to the AFCS view—to inject mathematics with a “standard” semantics, and to provide a secure epistemic foundation for the theorems of mathematics, in short, one who wishes for a foundation for mathematics, will surely find this view lacking. However, I argue that a satisfactory development of the structuralist view, couched in the language of category theory, accurately represents our best understanding of the content of mathematical theorems and thereby obviates the need for any foundational program

    On the relationship between plane and solid geometry

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    Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area

    Carnap's early semantics

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    In jĂŒngerer Zeit hat sich ein verstĂ€rktes Interesse an den historischen und technischen Details von Carnaps Philosophie der Logik und Mathematik entwickelt. Meine Dissertation knĂŒpft an diese Entwicklung an und untersucht dessen frĂŒhe und formative BeitrĂ€ge aus den spĂ€ten 1920er Jahren zu einer Theorie der formalen Semantik. Carnaps zu Lebzeiten unveröffentlichtes Manuskript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) beinhaltet ein Reihe von erstmals formal entwickelten Definitionen der Begriffe ‚Modell’, ‚Modellerweiterung’, und ‚logischer Folgerung’. Die vorliegende Dissertation entwickelt eine logische und philosophische Analyse dieser semantischen Begriffsbildungen. DarĂŒber hinaus wird Carnaps frĂŒhe Semantik in ihrem historisch-intellektuellen Entwicklungskontext diskutiert. Der Fokus der Arbeit liegt in der Thematisierung einiger interpretatorischer Fragen zu dessen implizit gehaltenen Annahmen bezĂŒglich der VariabilitĂ€t des Diskursuniversums von Modellen sowie zur Interpretation seiner typen-theoretischen logischen Sprache. Mit Bezug auf eine Reihe von historischen Dokumenten aus Carnaps Nachlass, insbesondere zu dem geplanten zweiten Teil der Untersuchungen wird erstens gezeigt, dass dessen VerstĂ€ndnis von Modellen in wesentlichen Punkten heterodox gegenĂŒber dem modernen BegriffsverstĂ€ndnis ist. Zweitens, dass Carnap von einer ‚nonstandard’ Interpretation der logischen Hintergrundtheorie fĂŒr seine Axiomatik ausgeht. Die Konsequenzen dieser semantischen Annahmen fĂŒr dessen Konzeptualisierung von metatheoretischen Begriffen werden nĂ€her diskutiert. Das erste Kapitel entwickelt eine kritische Analyse von Carnaps Versuch, die axiomatische Definition von Klassen von mathematischen Strukturen mittels des Begriffs von ‚Explizitbegriffen’ formal zu rekonstruieren. Im zweiten Kapitel werden die Implikationen von Carnaps frĂŒhem Modellbegriff fĂŒr seine Theorie von Extremalaxiomen nĂ€her beleuchtet. Das letzte Kapitel bildet eine Diskussion der konkreten historischen EinflĂŒsse, insbesondere durch den Mengentheoretiker Abraham Fraenkel, auf Carnaps formale Theorie von Minimalaxiomen.In recent years one was able to witness an intensified interest in the technical and historical details of Carnap’s philosophy of logic and mathematics. In my thesis I will take up this line and focus on his early, formative contributions to a theory of semantics around 1928. Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) includes some of the first formal definitions of the genuinely semantic concepts of a model, model extensions, and logical consequence. In the dissertation, I provide a detailed conceptual analysis of their technical details and contextualize Carnap’s results in their historic and intellectual environment. Certain interpretative issues related to his tacit assumptions concerning the domain of a model and the semantics of type theory will be addressed. By referring to unpublished material from Carnap’s Nachlass I will present archival evidence as well as more systematic arguments to the view that Carnap holds a heterodox conception of models and a nonstandard semantics for his type-theoretic logic. Given these semantic background assumptions, their impact on Carnap’s conceptualization of certain aspects of the metatheory of axiomatic theories will be evaluated. The first chapter critically discusses Carnap’s attempt to explicate one of the crucial semantic innovations of formal axiomatics, i.e. the definition of classes of structures, via his notion of ‘Explizitbegriffe’. The second chapter analyses the impact of Carnap’s early theory of model for his theory of extremal axioms. The final chapter reviews the mathematical influences, most importantly by the set theoretician Abraham Fraenkel on Carnap’s specific formalization of minimal axioms

    Mathematical Logic: Proof theory, Constructive Mathematics

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    The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
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