Recently Feferman has outlined a program for the development of a foundation for naive category theory. While Ernst has shown that the resulting axiomatic system is still inconsistent, the purpose of this note is to show that nevertheless some foundation has to be developed before naive category theory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive category theory currently a ‘cookbook (...) recipe’ is used for constructing categories, and it is explicitly shown with a formalized argument that this “foundationless” naive category theory therefore contains a paradox similar to the Russell paradox of naive set theory. (shrink)

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the (...) mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. (shrink)

Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited” or “naive” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes (...) care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution. (shrink)

Recent work on children’s inferences concerning biological and chemical categories has suggested that children (and perhaps adults) are essentialists— a view known as psychological essentialism. I distinguish three varieties of psychological essentialism and investigate the ways in which essentialism explains the inferences for which it is supposed to account. Essentialism succeeds in explaining the inferences, I argue, because it attributes to the child belief in causal laws connecting category membership and the possession of certain characteristic appearances and behavior. This (...) suggests that the data will be equally well explained by a non-essentialist hypothesis that attributes belief in the appropriate causal laws to the child, but makes no claim as to whether or not the child represents essences. I provide several reasons to think that this non-essentialist hypothesis is in fact superior to any version of the essentialist hypothesis. (shrink)

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.

In their recent paper, “Practical Knowledge without Luminosity,” Bob Beddor and Carlotta Pavese (2022) claim that the doctrine of essentially intentional actions, or “essentialism,” is false. Essentialism states that some actions are essentially intentional, such that, “whenever they are performed, they are performed intentionally” (2022, p. 926). Beddor and Pavese work to reject essentialism, which figures as a key premise in Juan Piñeros Glasscock’s anti-luminosity argument against the knowledge condition for intentional action (Piñeros Glasscock, p. 1240). Historically, essentialism has received (...) little attention from philosophers since its inception in Elizabeth Anscombe’s Intention (2000, §47). However, I believe that essentially intentional actions can play an important role in an ontology of action. In my paper, I develop and argue for a variety of essentialism in the context of naive action theory, which I call naive essentialism. Naive essentialism is a two-fold thesis, which claims that (1) essentially intentional actions exist, and (2) that essentially intentional actions ground accidentally intentional actions. My paper has four parts. In the first part, I distinguish between essentially and accidentally intentional actions, and unpack the relevant principles of naive action theory. Second, I present the grounding thesis that accidentally intentional actions are grounded in essentially intentional actions. Next, I provide an argument for the existence of essentially intentional actions. Lastly, I briefly respond to a possible objection to my argument. The upshot of my arguments is that essentially intentional actions form the metaphysical and explanatory bedrock in a naive ontology of action, and that there are good reasons for accepting a key premise in Piñeros Glasscock’s anti-luminosity argument. (shrink)

The epistemological literature on conspiracy theory has established that strict generalism about conspiracy theories is untenable. This chapter argues, however, that this does not license a move to naive or strict particularism. Rather, any consideration of specific conspiracy claims needs to address conspiracy theory not simply as a formal category of explanation, but as a distinctive social practice, with a history and explanatory repertoire that can give us important, if defeasible, reasons for rejecting at least some (...) such types of claim. These reasons are not merely epistemic. The moral costs of participating in such practices, and of levelling conspiracy accusations in general, generate corresponding ethical requirements for reticence about entertaining conspiracy theories. This doesn’t mean we can retreat to generalism; but it does mean particularism must be governed by a reluctance that, in practice, will rule out a great many conspiracy theory claims as candidates for serious investigation. (shrink)

An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences. Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. (...) Information is inherently dynamic; the same ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines. Written in an engaging and straightforward style, and assuming little background in mathematics, the book is rigorous but accessible to non-mathematicians. Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs—categories in disguise. After explaining the “big three” concepts of category theory—categories, functors, and natural transformations—the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions. Category Theory for the Sciences is intended to create a bridge between the vast array of mathematical concepts used by mathematicians and the models and frameworks of such scientific disciplines as computation, neuroscience, and physics. (shrink)

In a recent series of papers, I and others have advanced new logical approaches to familiar paradoxes. The key to these approaches is to accept full classical logic, and to accept the principles that cause paradox, while preventing trouble by allowing a certain sort ofnontransitivity. Earlier papers have treated paradoxes of truth and vagueness. The present paper will begin to extend the approach to deal with the familiar paradoxes arising in naive set theory, pointing out some of the (...) promises and pitfalls of such an approach. (shrink)

Naive set theory, as found in Frege and Russell, is almost universally believed to have been shown to be false by the set-theoretic paradoxes. The standard response has been to rank sets into one or other hierarchy. However it is extremely difficult to characterise the nature of any such hierarchy without falling into antinomies as severe as the set-theoretic paradoxes themselves. Various attempts to surmount this problem are examined and criticised. It is argued that the rejection of (...) class='Hi'>naive set theory inevitably leads one into a severe scepticism with regard to the feasibility of giving a systematic semantics for set theory. It is further argued that this is not just a problem for philosophers of mathematics. Semantic scepticism in set theory will almost inevitably spill over into total pessimism regarding the prospects for an explanatory theory of language and meaning in general. The conclusion is that those who wish to avoid such intellectual defeatism need to look seriously at the possibility that it is the logic used in the derivation of the paradoxes, and not the naive set theory itself, which is at fault. (shrink)

The question "Why?" that is deployed in these exchanges evidently bears the "special sense" Elizabeth Anscombe has linked to the concepts of intention and of a reason for action; it is the sort of question "Why?" that asks for what Donald Davidson later called a "rationalization".2 The special character of what is given, in each response, as formulating a reason ── a description, namely, of the agent as actually doing something, and, moreover, as..

A Naive Realist Theory of Colour defends the view that colours are mind-independent properties of things in the environment, that are distinct from properties identified by the physical sciences. This view stands in contrast to the long-standing and wide-spread view amongst philosophers and scientists that colours don't really exist - or at any rate, that if they do exist, then they are radically different from the way that they appear. It is argued that a naive realist (...) class='Hi'>theory of colour best explains how colours appear to perceiving subjects, and that this view is not undermined either by reflecting on variations in colour perception between perceivers and across perceptual conditions, or by our modern scientific understanding of the world. A Naive Realist Theory of Colour also illustrates how our understanding of what colours are has far-reaching implications for wider questions about the nature of perceptual experience, the relationship between mind and world, the problem of consciousness, the apparent tension between common sense and scientific representations of the world, and even the very nature and possibility of philosophical inquiry. (shrink)

Two distinct and apparently "dual" traditions of non-classical logic, three-valued logic and paraconsistent logic, are considered here and a unified presentation of "easy-to-handle" versions of these logics is given, in which full naive set theory, i.e. Frege's comprehension principle + extensionality, is not absurd.

Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.

Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as (...) is customary in other branches of contemporary mathematics. If such a specification suffices, then a category-theoretical approach will be highly appropriate. But if sets have a richer `nature' than is preserved under isomorphism, then such an approach will be inadequate. (shrink)

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set (...) theory can be carried out relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and (...) non-substantial ontology in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science. (shrink)

A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems, as well as numerous examples and exercises.

Linnebo and Pettigrew present some objections to category theory as an autonomous foundation. They do a commendable job making clear several distinct senses of ‘autonomous’ as it occurs in the phrase ‘autonomous foundation’. Unfortunately, their paper seems to treat the ‘categorist’ perspective rather unfairly. Several infelicities of this sort were addressed by McLarty. In this note I address yet another apparent infelicity.

Social categorization is an early emerging and robust component of social cognition, yet the role that social categories play in children's understanding of the social world has remained unclear. The present studies examined children's explanations of social behavior to provide a window into their intuitive theories of how social categories constrain human action. Children systematically referenced category memberships and social relationships as causal-explanatory factors for specific types of social interactions: harm among members of different categories more than harm among (...) members of the same category. In contrast, they systematically referred to agents' mental states to explain the reverse patterns of behaviors: harm among members of the same category more than harm among members of different categories. These data suggest that children view social category memberships as playing a causal-explanatory role in constraining social interactions. (shrink)

As a metaphysical theory, radical ontic structural realism is characterised mainly in terms of the ontological primacy it places on relations and structures, as opposed to the individual relata and objects that inhabit these relations/structures. The most popular criticism of ROSR is that its central thesis is incoherent. Bain attempts to address this criticism by arguing that the mathematical language of category theory allows for a coherent articulation of ROSR’s key thesis. Subsequently, Wüthrich and Lam and Lal (...) and Teh have criticised Bain’s arguments and claimed that category theory fares no better than set theory in coherently articulating the main ideas of ROSR. In this paper, we defend Bain’s main arguments against these critiques, and attempt to elaborate on the sense in which category theory can be seen as providing a coherent articulation of ROSR. We also consider the relationship between ROSR and Categorical Quantum Mechanics. (shrink)

The article surveys some past and present debates within mathematics over the meaning of category theory. It argues that such conceptual analyses, applied to a field still under active development, must be in large part either predictions of, or calls for, certain programs of further work.

The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided (...) into 5 sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory. (shrink)

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. (shrink)

The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can (...) be abstracted in purely arrow-theoretic way for abstract category theory. In short, the language of elements & distinctions is the conceptual language in which the category of sets is written, and abstract category theory gives the abstract arrows version of those definitions. (shrink)

economists. According to Rosenberg, Milton Friedman's positive methodology is being supplanted by Lakatos's methodology of scientific research programs (MSRP). At any rate, the Kuhnian wave of the seventies is being swallowed up by the Lakatosian program. (Redman 142) There have been a number of attempts to comprehend mainstream (bourgeois) economics as a Lakatosian research program, or as a set of competing research programs. (Latsis, ed. passim; de Marchi and Blaug, eds.)i In contrast, the extent to which the Marxian study of (...) capitalism can be interpreted from this perspective has hardly been explored.ii In the following sections some provisional steps in this direction will be taken. I shall first introduce a reading of Marx's Capital that emphasizes the systematic dialectic of economic categories unifying that work. I shall then ask whether we can arrive at a better understanding of Marx's systematic dialectic through considering how it contributes to a scientific research 1 program in Lakatos's sense of the term. Two other themes will arise along the way. Certain shortcomings in Lakatos's framework will be discussed. Also, a number of reasons for considering the Marxian research program superior to that of neoclassical economics will be mentioned. It should be stressed that the main topic of the present paper is the role of systematic dialectics in the Marxian research program. The other themes go far beyond the scope of single paper. Lakatos's work can be understood as a response to the dead-end of naive falsificationism. According to this simplistic methodology, theories are to be tested by deducing predictions from them and then determining whether the events predicted occur. If the predicted events do take place at some later time, this does not verify the validity of the theory. That would be to commit the fallacy of affirming the consequent; there may be many other theories that also lead to the prediction in question. In contrast, if the predicted events do not take place, that is sufficient to falsify the proposed theory.. (shrink)

Dialetheism is the metaphysical claim that there are true contradictions. And based on this view, Graham Priest and his collaborators have been suggesting solutions to a number of paradoxes. Those paradoxes include Russell’s paradox in naive set theory. For the purpose of dealing with this paradox, Priest is known to have argued against the presence of classical negation in the underlying logic of naive set theory. The aim of the present paper is to challenge this view (...) by showing that there is a way to handle classical negation. (shrink)

This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.

Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We also (...) address the relation of our axiom systems to alternative proposals from the literature, including an axiom set proposed by Freyd and Scedrov for which we reveal a technical issue (when encoded in free logic where free variables range over defined and undefined objects): either all operations, e.g. morphism composition, are total or their axiom system is inconsistent. The repair for this problem is quite straightforward, however. (shrink)

Category theory doesn't support Mathematical Structuralism but suggests a new philosophical view on mathematics, which differs both from Structuralism and from traditional Substantialism about mathematical objects. While Structuralism implies thinking of mathematical objects up to isomorphism the new categorical view implies thinking up to general morphism.

This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal (...) profile of $\Omega$-logical validity, and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility. (shrink)

Reasonning in naive set theory (with unlimited comprehension), we derive a paradox (a formal contradiction) which can be seen as a variant of the Burali-Forti paradox.

Rocco Gangle addresses the methodological questions raised by a commitment to immanence in terms of how diagrams may be used both as tools and as objects of philosophical investigation. Gangle integrates insights from Spinoza, Pierce and Deleuze in conjunction with the formal operations of category theory.