324 research outputs found
Anti-Foundational Categorical Structuralism
The aim of this dissertation is to outline and defend the view here dubbed âanti-foundational categorical structuralismâ (henceforth AFCS). The program put forth is intended to provide an answer the question âwhat is mathematics?â. The answer here on offer adopts the structuralist view of mathematics, in that mathematics is taken to be âthe science of structureâ expressed in the language of category theory, which is argued to accurately capture the notion of a âstructural propertyâ. 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 âmathematical objectsâ, or which involves quantification over a fixed domain of such objects. One who wishesâcontrary to the AFCS viewâto inject mathematics with a âstandardâ 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
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
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 âModellâ, âModellerweiterungâ, und âlogischer Folgerungâ. 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 ânonstandardâ 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 âExplizitbegriffenâ 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 Carnapâs 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. Carnapâs 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 Carnapâs 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 Carnapâs 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 Carnapâs conceptualization of certain aspects of the metatheory of axiomatic theories will be evaluated. The first chapter critically discusses Carnapâs attempt to explicate one of the crucial semantic innovations of formal axiomatics, i.e. the definition of classes of structures, via his notion of âExplizitbegriffeâ. The second chapter analyses the impact of Carnapâs 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 Carnapâs specific formalization of minimal axioms
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop âMathematical Logic: Proof Theory, Constructive Mathematicsâ was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
Recommended from our members
A defence of predicativism as a philosophy of mathematics
A specification of a mathematical object is impredicative if it essentially involves quantification over a domain which includes the object being specified (or sets which contain that object, or similar). The basic worry is that we have no non-circular way of
understanding such a specification. Predicativism is the view that mathematics should be limited to the study of objects which can be specified predicatively.
There are two parts to predicativism. One is the criticism of the impredicative aspects of classical mathematics. The other is the
positive project, begun by Weyl in Das Kontinuum (1918), to reconstruct as much as possible of classical mathematics on the basis of a predicatively acceptable set theory, which accepts only countably infinite objects. This is a revisionary project, and certain parts of mathematics will not be saved.
Chapter 2 contains an account of the historical background to the predicativist project. The rigorization of analysis led to Dedekind's and Cantor's theories of the real numbers, which relied on the new notion of abitrary infinite sets; this became a central part of modern classical set theory. Criticism began with Kronecker; continued in the debate about the acceptability of Zermelo's Axiom of Choice; and was somewhat clarified by Poincaré and Russell. In the
light of this, chapter 3 examines the formulation of, and motivations behind the predicativist position.
Chapter 4 begins the critical task by detailing the epistemological problems with the classical account of the continuum. Explanations of classicism which appeal to second-order logic, set theory, and
primitive intuition are examined and are found wanting.
Chapter 5 aims to dispell the worry that predicativism might collapses into mathematical intuitionism. I assess some of the arguments for intuitionism, especially the Dummettian argument from indefinite
extensibility. I argue that the natural numbers are not indefinitely extensible, and that, although the continuum is, we can nonetheless make some sense of classical quantification over it. We need not reject the Law of Excluded Middle.
Chapter 6 begins the positive work by outlining a predicatively acceptable account of mathematical objects which justifies the Vicious Circle Principle. Chapter 7 explores the appropriate shape of formalized predicative mathematics, and the question of just how much mathematics is predicatively acceptable.
My conclusion is that all of the mathematics which we need can be predicativistically justified, and that such mathematics is
particularly transparent to reason. This calls into question one currently prevalent view of the nature of mathematics, on which
mathematics is justified by quasi-empirical means.Supported by the Arts and Humanities Research Council [grant number 111315]
- âŠ