Online research in philosophy

Giandomenico Sica’s volume is a collection of eleven papers on category theory by philosophers, mathematicians, and mathematical physicists. In addition to papers of direct interest to philosophers of mathematics, the volume contains some introductory expositions of category theory along with a valuable discussion of the relationship between category theory and physics by Bob Coecke. While there are several technically difficult papers, the volume as a whole is reasonably accessible to those with some familiarity with the basics of category theory. The importance of the volume lies in the possibility that it will encourage broader interest in category theory among philosophers.

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 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.

The Unified Medical Language System and the Gene Ontology are among the most widely used terminology resources in the biomedical domain. However, when we evaluate them in the light of simple principles for wellconstructed ontologies we find a number of characteristic inadequacies. Employing the theory of granular partitions, a new approach to the understanding of ontologies and of the relationships ontologies bear to instances in reality, we provide an application of this theory in relation to an example drawn from the context of the pathophysiology of hypertension. This exercise is designed to demonstrate how, by taking ontological principles into account we can create more realistic biomedical ontologies which will also bring advantages in terms of efficiency and robustness of associated software applications.

Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S should allow us to establish the existence of the usual basic mathematical structures and carry out the usual set-theoretical operations; and (R4) S should be shown to be consistent relative to currently accepted systems of set theory. This paper explains how all but parts of (R3) can be met using a system S extending NFU enriched by a stratified pairing operation; to meet more of (R3) a stronger system S∗ is introduced, but there are still some real obstacles to meeting this requirement in full. For (R4) it is sketched how both S and S∗ are shown to be consistent.

This paper, accessible for a general philosophical audience having only some fleeting acquaintance with set-theory and category-theory, concerns the philosophy of mathematics, specifically the bearing of category-theory on the foundations of mathematics. We argue for six claims. (I) A founding theory for category-theory based on the primitive concept of a set or a class is worthwile to pursue. (II) The extant set-theoretical founding theories for category-theory are conceptually flawed. (III) The conceptual distinction between a set and a class can be seen to be formally codified in Ackermann's axiomatisation of set-theory. (IV) A slight but significant deductive extension of Ackermann's theory of sets and classes founds Cantorian set-theory as well as category-theory, and therefore can pass as a founding theory of the whole of mathematics. (V) The extended theory does not suffer from the conceptual flaws of the extant set-theoretical founding theories. (VI) The extended theory is not only conceptually but also logically superior to the competing set-theories because its consistency can be proved on the basis of weaker assumptions than the consistency of the competition.

A systems theoretical thinking on the categorial object and morphism is developed, leading to a reflection on the philosophical and mathematical foundations of category theory, which allows for the introduction of a formal language for category theory and of a categorial calculus as a morphic web-based logical calculus. A formal system, built from such calculus, is proposed and the logical semantics is addressed. Both syntax and semantics are independent from set theory.

Formal Axiomatic method as exemplified in Hilbert’s Grundlagen der Geometrie is based on a structuralist vision of mathematics and science according to which theories and objects of these theories are to be construed “up to isomorphism”. This structuralist approach is tightly linked with the idea of making Set theory into foundations of mathematics. Category theory suggests a generalisation of Formal Axiomatic method, which amounts to construing objects and theories “up to general morphism” rather than up to isomorphism. It is shown that this category-theoretic method of theorybuilding better fits mathematical and scientific practice. Moreover so since the requirement of being determined up to isomorphism (i.e. categoricity in the usual model-theoretic sense) turns to be unrealistic in many important cases. The category-theoretic approach advocated in this paper suggests an essential revision of the structuralist philosophy of mathematics and science. It is argued that a category should be viewed as a far-reaching generalisation of the notion of structure rather than a particular kind of structure. Finally, I compare formalisation and categorification as two alternative epistemic strategies.

We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal. We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence.

In this paper I argue that category theory ought to be seen as providing the language for mathematical discourse. Against foundational approaches, I argue that there is no need to reduce either the content or structure of mathematical concepts and theories to the constituents of either the universe of sets or the category of categories. I assign category theory the role of organizing what we say about the content and structure of both mathematical concepts and theories. Insofar, then, as the structuralist sees mathematics as talking about structures and their morphology, I contend that category theory furnishes a framework for mathematical structuralism.