Online research in philosophy

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be algebraic structuralists all the way down.

In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents.

The paper proposes to amend structuralism in mathematics by saying what places in a structure and thus mathematical objects are. They are the objects of the canonical system realizing a categorical structure, where that canonical system is a minimal system in a specific essentialistic sense. It would thus be a basic ontological axiom that such a canonical system always exists. This way of conceiving mathematical objects is underscored by a defense of an essentialistic version of Leibniz’ principle according to which each object is uniquely characterized by its proper and possibly relational essence (where “proper” means “not referring to identity").

In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of its complementary dimensions: paradigmatic generalization (domain extension, descriptive definitions, creative role of the symbolism...) and thematic reflexivity of concepts (promotion of operations to objects of a higher type).

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics.

In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a “nature” than that confers on them.

We are used to talking about the “structure” posited by a given theory of physics. We say that relativity is a theory about spacetime structure. Special relativity posits one spacetime structure; different models of general relativity posit different spacetime structures. We also talk of the “existence” of these structures. Special relativity says the world’s spacetime structure is Minkowskian: it posits that this spacetime structure exists. Understanding structure in this sense seems important for understanding what physics is telling us about the world. But it is not immediately obvious just what this structure is, or what we mean by the existence of one structure, rather than another. The idea of mathematical structure is relatively straightforward. There is geometric structure, topological structure, algebraic structure, and so forth. Mathematical structure tells us how abstract mathematical objects t together to form different types of mathematical spaces. Insofar as we understand mathematical objects, we can understand mathematical structure. Of course, what to say about the nature of mathematical objects isn’t easy. But there seems to be no further problem for understanding mathematical structure. Modern theories of physics are formulated in terms of these mathematical structures. In order to understand “structure” as used in physics, then, it seems we must simply look at the structure of the mathematics that is used to state the physics. But it is not that simple. Physics is supposed to be telling us about the nature of the world. If our physical theories are formulated in mathematical language, using mathematical objects, then this mathematics is somehow telling us about the physical make-up of the world. What is..

Moral theology is given force by punishment and reward, which is, in turn, comprehensible only in the presence of free will. Yet free will has been bedevilled with philosophical difficulties, not least among them the tension between omniscience and autonomy. The paper, building on a theory of temptation and sin published in /Mind/, gives a partial resolution to that tension using a mathematical argument.

In this paper, we defend and extend a (simple) mathematical model of /akrasia/. The probability is due to Professor Peter Milne.