Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (...) (essentially a formulation of universal mapping properties using hets) can then be combined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action (as opposed to the passive reception of sense-data or the reflex generation of behavior). (shrink)

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory (...) and is focused on the interpretation and application of the mathematical concepts. (shrink)

This work gives an extended presentation of the treatment of variable-binding operators adumbrated in [3:1993d]. Illustrative examples include elementary languages with quantifiers and lambda-equipped categorial languages. Some remarks are also offered to illustrate the philosophical import of the resulting picture. Particularly, a certain conception of logic emerges from the account: the view that logics are true theories in the model-theoretic sense, i.e. the result of selecting a certain class of models as the only “admissible” interpretation structures (for a given language).

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called “determination through universals” based on universal mapping properties. A recently developed “heteromorphic” theory about (...) adjoints suggests a conceptual structure, albeit abstract and atemporal, for how new relatively autonomous behavior can emerge within a system obeying certain laws. The focus here is on applications in the life sciences (e.g., selectionist mechanisms) and human sciences (e.g., the generative grammar view of language). (shrink)

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, (...) a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)

In Writing the Book of the World, Ted Sider argues that David Lewis’s distinction between those predicates which are ‘perfectly natural’ and those which are not can be extended so that it applies to words of all semantic types. Just as there are perfectly natural predicates, there may be perfectly natural connectives, operators, singular terms and so on. According to Sider, one of our goals as metaphysicians should be to identify the perfectly natural words. Sider claims that there is a (...) perfectly natural first-order quantifier. I argue that this claim is not justified. Quine has shown that we can dispense with first-order quantifiers, by using a family of ‘predicate functors’ instead. I argue that we have no reason to think that it is the first-order quantifiers, rather than Quine’s predicate functors, which are perfectly natural. The discussion of quantification is used to provide some motivation for a general scepticism about Sider’s project. Shamik Dasgupta’s ‘generalism’ and Jason Turner’s critique of ‘ontological nihilism’ are also discussed. (shrink)

Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects in (...) different categories), then he, in effect, isolated the concept of a left representation solving a universal mapping problem. When dualized to obtain the concept of a right representation, the two halves only need to be united to obtain an adjunction. Thus Samuel was only a now-simple dualization away for formulating adjoints in 1948. Apparently, Bodo Pareigis' 1970 text was the first and perhaps only text to give the heterodox "new characterization" (i.e., heteromorphic treatment) of adjoints. Orthodox category theory uses various relatively artificial devices to avoid formally recognizing hets--even though hets are routinely used by the working mathematician. Finally we consider a "philosophical" question as to whether the most important concept in category theory is the notion of an adjunction or the notion of a representation giving a universal mapping property (where adjunctions arise as the special case of a bi-representation of dual universal mapping problems). (shrink)

The paper proposes a way for adherents of Fregean, structured propositions to designate propositions and other complex senses/concepts using a special kind of functor. I consider some formulations from Peacocke's works and highlight certain problems that arise as we try to quantify over propositional constituents while referring to propositions using "that"-clauses. With the functor notation, by contrast, we can quantify over senses/concepts with objectual, first-order quantifiers and speak without further ado about their involvement in propositions. The functor (...) notation also turns out to come with an important kind of expressive strengthening, and is shown to be neutral on several controversial issues. (shrink)

One well known problem regarding quantifiers, in particular the 1st order quantifiers, is connected with their syntactic categories and denotations.The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated (...) by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functor-function defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

This paper concerns an important and exciting book by Marek Dobrzeniecki Ukrytość i Wcielenie. Teistyczna odpowiedź na argument Johna L. Schellenberga za nieistnieniem Boga [Hiddenness and the Incarnation: A Theistic Response to John L. Schellenberg’s Argument for Divine Nonexistence]. After a brief discussion of the content of the book’s chapters, critical remarks are presented. They concern the adopted method and approach to Schellenberg’s philosophy in general and the argument from hiddenness in particular. The conceptual framework serving as a typologization of (...) the propositional dimension of the phenomenon of theistic belief/faith is dubious and requires in-depth research. It is worth considering the possibility of asymmetry of the attitudes of belief/faith and unbelief, which requires consideration in the reasoning whose legitimacy would assume the transfer of features from one to the other and the negation functor. Regarding the voluntarism of belief in God, the paper suggests that extreme, purely intellectual, and purely volitional positions should be considered, as well as moderate and mixed ones. It is argued extensively that Dobrzeniecki’s criticism against the arguments of Rea and Mourad is not conclusive. Just the possibility of voluntariness of disbelief undermines the validity of Schellenberg’s argument and should therefore be ruled out as much as possible. The paper suggests supplementing Dobrzeniecki’s approach with a ‘trans-paradigmatic’ approach consisting of external criticism, which, in the light of Schellenberg’s arguments being probably incompatible with the described phenomenon of religious belief in God, leads to a modification of the theoretical apparatus of Schellenberg’s paradigm. Such a modification deprives the latter’s argument of justification. --- Niniejszy artykuł dotyczy ważnej i interesującej książki Marka Dobrzenieckiego Ukrytość i Wcielenie. Teistyczna odpowiedź na argument Johna L. Schellenberga za nieistnieniem Boga. Po krótkim omówieniu zawartości poszczególnych rozdziałów książki prezentuję swoje uwagi krytyczne. Dotyczą one przyjętej metody i podejścia do filozofii Schellenberga w ogólności, a argumentu z ukrytości w szczególności. Wątpliwa i wymagająca pogłębionego badania jest siatka pojęciowa pełniąca funkcję typologizacji propozycjonalnego wymiaru fenomenu wiary teistycznej. Warta rozważania jest możliwość niesymetryczności postaw wiary i niewiary, co wymaga uwzględnienia w rozumowaniach, których zasadność zakładałaby przenoszenie cech z jednej na drugą oraz funktora negacji. W sprawie woluntaryzmu wiary w Boga sugeruję, że należy rozpatrywać nie tylko stanowiska skrajne, czysto intelektualistyczne i czysto wolicjonalne, ale także u miarkowane i mieszane. Argumentuję szeroko, że krytyka Dobrzenieckiego skierowana przeciwko argumentom Rea i Mourada nie jest konkluzywna. Argumentuję, że samo możliwość woluntarności niewiary podważa zasadność argumentu Schellenberga i dlatego powinna ona być wykluczona w sposób maksymalnie możliwie uzasadniony. Sugeruję uzupełnienie podejścia Dobrzenieckiego o podejście ‘transparadygmatyczne’ polegające na rodzaju krytyki zewnętrznej, która w świetle nieprzystawania argumentacji Schellenberga do opisywanego fenomenu prowadzi do modyfikacji aparatu teoretycznego paradygmatu Schellenberga. Taka modyfikacja odbiera argumentowi tego ostatniego moc uzasadniającą. (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing (...) a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (shrink)

Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set of (...) theorems. The concept of an accessible domain comes from Wilfried Sieg's analysis of proof-theoretic practices, starting with his dissertation. In particular, he noticed the special epistemological character of elements of an accessible domain: they can always be uniquely identified with their build-up. Generally, the unique build-up of elements justifies the principles of induction and recursion. -/- I use category theory to give an abstract characterization of accessible domains. I claim that accessible domains are all instances of initial algebras for endofunctors. Grounded in the historical roots of Sieg's discussions, this dissertation shows how the properties of initial algebras for endofunctors and accessible domains coincide in a satisfying and natural way. -/- Filling out this characterization, I show how important examples of accessible domains fit into this broad characterization. I first characterize some accessible domains by relatively simple functors (e.g. finite, polynomial, those that preserve certain colimits) where we can see how iterating the functor can produce the accessible domain. Then I describe accessible domains that result from more involved specifications (e.g. ordinals and segments of the cumulative hierarchies associated with CZF, IZF, and ZF) by relying heavily on algebraic set theory. I end with a discussion of some of the methodological features of category theory in particular that helped characterize accessible domains. (shrink)

By pure calculus of names we mean a quantifier-free theory, based on the classical propositional calculus, which defines predicates known from Aristotle’s syllogistic and Leśniewski’s Ontology. For a large fragment of the theory decision procedures, defined by a combination of simple syntactic operations and models in two-membered domains, can be used. We compare the system which employs `ε’ as the only specific term with the system enriched with functors of Syllogistic. In the former, we do not need an empty name (...) in the model, so we are able to construct a 3-valued matrix, while for the latter, for which an empty name is necessary, the respective matrices are 4-valued. (shrink)

The book is a collection of papers addressing the role of logic in forming and developing philosophy. In particular, on the ground of modern development of logic, it is shown that philosophy can be established (and, in fact, to a large extent is established) as a modern science. The following problems are addressed: general relationship between philosophy and science (especially from a logical viewpoint); the use of logic in ordinary language; names and descriptions; Quine's pragmatic extensional Platonism and predicate-functor (...) logic; philosophical and logical aspects of Gödel's onto-theological system. In the last two chapters, an early critique and emendement of algebraic logic by Croatian philosophers Franjo Marković and Mate Meršić, respectively, is examined. (shrink)

Our conscious experiences are qualitative and unitary. The qualitative universals given in particular experiences, i.e. qualia, combine into the seamless unity of our conscious experience. The problematics of quality and cohesion are not unique to consciousness studies. In mathematics, the study of qualities (e.g., shape) resulting from quantitative variations in cohesive spaces led to the axiomatization of cohesion and quality. Using the mathematical definition of quality, herein we model qualia space as a categorical product of qualities. Thus modeled qualia space (...) is a codomain space wherein composite qualities (e.g. shape AND color) of conscious experiences can be valued. As part of characterizing the qualia space, we provide a detailed exemplification of the mathematics of quality and cohesion in terms of the categories of idempotents and reflexive graphs. More specifically, with qualities as commutative triangles formed of cohesion-preserving functors, first we calculate the product of commutative triangles. Next, we explicitly show that the category of idempotents is a quality type. Lastly, as part of showing that the category of reflexive graphs is cohesive, we characterize the adjointness between functors relating cohesive graphs to discrete sets. In conclusion, our category theoretic construction of qualia space is a formalization of the binding of qualitative features (colors and shapes) into the cohesive objects (colored-shapes) of conscious experiences. Compared to the feature-vector accounts of conscious experiences, our product-of-qualities account of consciousness is a substantial theoretical advance. (shrink)

Cognition involves physical stimulation, neural coding, mental conception, and conscious perception. Beyond the neural coding of physical stimuli, it is not clear how exactly these component processes constitute cognition. Within mathematical sciences, category theory provides tools such as category, functor, and adjointness, which are indispensable in the explication of the mathematical calculations involved in acquiring mathematical knowledge. More speci cally, functorial semantics, in showing that theories and models can be construed as categories and functors, respectively, and in establishing the (...) adjointness between abstraction (of theories) and interpretation (to obtain models), mathematically accounts for knowing-within-mathematics. Here we show that mathematical knowing recapitulates--in an elementary form--ordinary cognition. The process of going from particulars (physical stimuli) to their concrete models (conscious percepts) via abstract theories (mental concepts) and measured properties (neural coding) is common to both mathematical knowing and ordinary cognition. Our investigation of the similarity between knowing-within-mathematics and knowing-in-general leads us to make a case for the development of the basic science of cognition in terms of the functorial semantics of mathematical knowing. (shrink)

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. -/- It is well known from universal algebra that, for every signature Sigma, there exist algebras over Sigma which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of Sigma-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of Sigma-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor U, from the category of Sigma-multialgebras to Set, does not have a left adjoint. -/- In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that U does not have a left adjoint. (shrink)

We develop an approach to temporal logic that replaces the traditional objective, agent- and event-independent notion of time with a constructive, event-dependent notion of time. We show how to make this event-dependent time entropic and hence well-defined. We use sheaf-theoretic techniques to render event-dependent time functorial and to construct memories as sequences of observed and constructed events with well-defined limits that maximize the consistency of categorizations assigned to objects appearing in memories. We then develop a condensed formalism that represents memories (...) as pure constructs from single events. We formulate an empirical hypothesis that human episodic memory implements a particular constructive functor. (shrink)

The following text is divided in four parts. The first presents the inner relation between the phenomenological concept of intentionality and space in a general mathematical sense. Following this train of though the second part brie_ly characterizes the use of the geometrical concept of manifold (Mannigfaltigkeit) in Husserl’s work. In the third part we present some examples of the use of the concept in Husserl’s analyses of space, time and intersubjectivity, pointing out some dif_iculties in his endeavor. In the fourth (...) and _inal part we offer some points of coincidence between phenomenology and category theory suggesting that the latter can work as a formal frame for ontology in the former. Our thesis is that intentionality operates in different levels as a morphism, functor and natural transformation. (shrink)

The following text is divided in four parts. The first presents the inner relation between the phenomenological concept of intentionality and space in a general mathematical sense. Following this train of though the second part brie_ly characterizes the use of the geometrical concept of manifold (Mannigfaltigkeit) in Husserl’s work. In the third part we present some examples of the use of the concept in Husserl’s analyses of space, time and intersubjectivity, pointing out some dif_iculties in his endeavor. In the fourth (...) and _inal part we offer some points of coincidence between phenomenology and category theory suggesting that the latter can work as a formal frame for ontology in the former. Our thesis is that intentionality operates in different levels as a morphism, functor and natural transformation. (shrink)

If episodic memory is constructive, experienced time is also a construct. We develop an event-based formalism that replaces the traditional objective, agent-independent notion of time with a constructive, agent-dependent notion of time. We show how to make this agent-dependent time entropic and hence well-defined. We use sheaf-theoretic techniques to render agent-dependent time functorial and to construct episodic memories as sequences of observed and constructed events with well-defined limits that maximize the consistency of categorizations assigned to objects appearing in memories. We (...) then develop a condensed formalism that represents episodic memories as pure constructs from single events. We formulate an empirical hypothesis that human episodic memory implements a particular time-symmetric constructive functor, and discuss possible experimental tests. (shrink)

We define consciousness as the category of all conscious experiences. This immediately raises the question: What is the essence in which every conscious experience in the category of conscious experiences partakes? We consider various abstract essences of conscious experiences as theories of consciousness. They are: (i) conscious experience is an action of memory on sensation, (ii) conscious experience is experiencing a particular as an exemplar of a general, (iii) conscious experience is an interpretation of sensation, (iv) conscious experience is referring (...) sensation to an object as its cause, and (v) conscious experience is a model of stimulus. Corresponding to each one of these theories we obtain a category of models of conscious experiences: (i) category of actions, (ii) category of idempotents, (iii) category of two sequential maps, (iv) category of brain-generalized figures, and (v) functor categories with intuition as base and conceptual repertoire as exponent, respectively. For each theory of consciousness we also calculate its truth value object and characterize the objective logic intrinsic to the corresponding category of models of consciousness experiences. (shrink)

Notions such as Sunyata, Catuskoti, and Indra's Net, which figure prominently in Buddhist philosophy, are difficult to readily accommodate within our ordinary thinking about everyday objects. Famous Buddhist scholar Nagarjuna considered two levels of reality: one called conventional reality and the other ultimate reality. Within this framework, Sunyata refers to the claim that at the ultimate level objects are devoid of essence or "intrinsic properties", but are interdependent by virtue of their relations to other objects. Catuskoti refers to the claim (...) that four truth values, along with contradiction, are admissible in reasoning. Indra's Net refers to the claim that every part of a whole is reflective of the whole. Here we present category theoretic constructions which are reminiscent of these Buddhist concepts. The universal mapping property definition of mathematical objects, wherein objects of a universe of discourse are defined not in terms of their content, but in terms of their relations to all objects of the universe is reminiscent of Sunyata. The objective logic of perception, with perception modeled as [a category of] two sequential processes (sensation followed by interpretation), and with its truth value object of four truth values, is reminiscent of the Buddhist logic of Catuskoti. The category of categories, wherein every category has a subcategory of sets with zero structure within which every category can be modeled, is reminiscent of Indra's Net. Our thorough elaboration of the parallels between Buddhist philosophy and category theory can facilitate better understanding of Buddhist philosophy, and bring out the broader philosophical import of category theory beyond mathematics. (shrink)

That the state-of-affairs of cognitive science is not good is brought into figural salience in "What happened to cognitive science?" (Núñez et al., 2019). We extend their objective description of 'what's wrong' to a prescription of 'how to correct'. Cognitive science, in its quest to elucidate 'how we know', embraces a long list of subjects, while ignoring Mathematics (Fig. 1a, Núñez et al., 2019). Mathematics is known for making the unknown to be known (cf. solving for unknowns). This acknowledgement naturally (...) raises the question: does mathematical knowing inform knowing in general? Here we show that drawing parallels to mathematical knowing can facilitate the advancement of cognitive science (Lawvere, 1994). (shrink)

The philosopher and professor Tudor Cătineanu proposes to "seekers of metaphysical reason" a volume of studies on Rhetoric and Metaphysics. Entitled Rhetoric and Metaphysics: In Search of Metaphysical Reason (Romanian Academy Publishing House, 2019, 571 p.), the book is the result of elaborate studies, therefore pre- and post-December 1989. The work, as the author points out, “is organized by combining two major perspectives, historical and systematic” (p. 7), and the predominant method of work used is the „interdisciplinary methods” called “matrix (...) of levels”. But the researcher uses, also, “the matrix of functors” and "the matrix of stylistic". (shrink)