Linnebo and Pettigrew present some objections to categorytheory 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.
Instead of the half-century old foundational feud between set theory and categorytheory, 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 categorytheory 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)
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 categorytheory. In short, the language of elements & distinctions is the conceptual language in which the category of sets is written, and abstract categorytheory gives the abstract arrows version of those definitions. (shrink)
In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness (“n-awareness”). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the (...) mathematical formalism of higher categorytheory. The beauty of higher categorytheory lies in its universality. Pluralism is categorical. In particular, we model awareness using the theories of derived categories and (infinity, 1)-topoi which will give rise to our meta-language. We then posit a “grammar” (“n-declension”) which could express n-awareness, accompanied by a new temporal ontology (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? Another question which could be re-conceptualized in our model is one of soteriology related to this pluralism: what is a self in this context? A new model of “personal identity over time” is thus introduced. (shrink)
Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, categorytheory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic (...) treatment of concrete universals along with a new concept to abstractly model the functions of a brain. (shrink)
Recently Feferman has outlined a program for the development of a foundation for naive categorytheory. 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 categorytheory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive categorytheory currently a ‘cookbook recipe’ is used (...) for constructing categories, and it is explicitly shown with a formalized argument that this “foundationless” naive categorytheory therefore contains a paradox similar to the Russell paradox of naive set theory. (shrink)
This paper shows how the universals of categorytheory in mathematics 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. The paper also shows how the always-self-predicative universals of categorytheory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus (...) that the paradoxes arose from having one theory (e.g., Frege's Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or always the other). (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 categorytheory can facilitate better understanding of Buddhist philosophy, and bring out the broader philosophical import of categorytheory beyond mathematics. (shrink)
In this paper, I explore a line of argument against one form of realism about fictional characters : abstract artifact theory, the view according to which fictional characters like Harry Potter are part of our reality, but, they are abstract objects created by humans, akin to the institution of marriage and the game of soccer. I will defend artifactualism against an objection that Mark Sainsbury considers decisive against it: the category-mistake objection. The objection has it that artifactualism attributes (...) to people who produce and process sentences and thoughts about Harry Potter massive error, indeed, a category mistake about what kind of thing Harry Potter is; for an abstract object isn’t the sort of thing that can wear glasses, ride a double-decker bus, attend school. Given problems with this objection, artifactualism, I shall conclude, remains a tenable contender. (shrink)
I defend a one category ontology: an ontology that denies that we need more than one fundamental category to support the ontological structure of the world. Categorical fundamentality is understood in terms of the metaphysically prior, as that in which everything else in the world consists. One category ontologies are deeply appealing, because their ontological simplicity gives them an unmatched elegance and spareness. I’m a fan of a one category ontology that collapses the distinction between particular (...) and property, replacing it with a single fundamental category of intrinsic characters or qualities. We may describe the qualities as qualitative charactersor as modes, perhaps on the model of Aristotelian qualitative (nonsubstantial) kinds, and I will use the term “properties” interchangeably with “qualities”. The qualities are repeatable and reasonably sparse, although, as I discuss in section 2.6, there are empirical reasons that may suggest, depending on one’s preferred fundamental physical theory, that they include irreducibly intensive qualities. There are no uninstantiated qualities. I also assume that the fundamental qualitative natures are intrinsic, although physics may ultimately suggest that some of them are extrinsic. On my view, matter, concrete objects, abstract objects, and perhaps even spacetime are constructed from mereological fusions of qualities, so the world is simply a vast mixture of qualities, including polyadic properties (i.e., relations). This means that everything there is, including concrete objects like persons or stars, is a quality, a qualitative fusion, or a portion of the extended qualitative fusion that is the worldwhole. I call my view mereological bundle theory. (shrink)
The precondition of any feminist politics – a usable category of ‘woman’ – has proved to be difﬁcult to construct, even proposed to be impossible, given the ‘problem of exclusion’. This is the inevitable exclusion of at least some women, as their lives or experiences do not ﬁt into the necessary and sufﬁcient condition(s) that denotes group membership. In this paper, I propose that the problem of exclusion arises not because of inappropriate category membership criteria, but because of (...) the presumption that categories can only be organised by identity relations or shared properties among their members. This criterion of sameness as well as the characterisation of this exclusion as essentialism attests to a metaphysics that is not conducive to resistance and liberatory projects. Following a strain of hybrid thinking in feminist and post-colonial theory, I outline an alternative pluralist logic that confronts oppressive binaries that impede theory work in gender, sexuality, and race theory, and limit political action and resistance. The problem of exclusion is neither irresolvable nor is it essentialism. Instead it is a denial of subjectivity due to pseudodualistic self/Other dichotomies that can be resisted by adopting a new categorial logic. While this paper focuses on the speciﬁc problem of formulating a category of ‘woman’, it has implications for other areas of gender, critical race, and postcolonial theory. Rather than working toward an inclusive category founded on sameness, theorists need to develop independent and positive categories grounded in difference. Our current categorial logic does not permit such a project, and therefore a new metaphysics must be adopted. (shrink)
Thomas Polger and Laurence Shapiro argue that Carl Gillett's much publicized dimensioned theory of realization is incoherent, being subject to a reductio. Their argument turns on the fact that Gillett's definition of realization makes property instances the exclusive relata of the realization relation, while his belief in multiple realization implies its denial, namely, that properties are the relata of the realization relation on occasions of multiple realization. Others like Sydney Shoemaker have also expressed their view of realization in terms (...) of property instances, yet they too have accepted the multiple realizability of properties. Thus I am interested in the more general issue raised by Polger and Shapiro's argument. Specifically, I show how to supplement a theory of realization with a category-inclusive auxiliary assumption, which avoids the stated reductio. I then offer a few reasons to justify the proposed category-inclusive view of realization, making some comparisons to supervenience and causation along the way. (shrink)
We introduce the category of mereotopology Mtop as an alternative category to that of topology Top, stating ontological consequences throughout. We consider entities such as boundaries utilizing Brentano’s thesis and holes utilizing homotopy theory with a rigorous proof of Hausdorff Spaces satisfying [GEM]TC axioms. Lastly, we mention further areas of study in this category.
In this paper, I present a categorial theory of meaning which asserts that the meaning of a sentence is the function from the actualization of some potentiality or the potentiality of some actuality to the truth of the sentence. I argue that it builds on the virtues of David Lewis’s Possible World Semantics but advances beyond problems that Lewis’s theory faces with its distinctly Aristotelian turn toward actuality and potentiality.
John Hick’s theory concerning plurality of religions is an ontologic pluralism according to which all religions are authentic ways for man to attain the "real an sich". Gods of religions are real as perceived and veridical hallucinations; while the “real an sich” has ineffable substantial and trans-categorical properties. Hick’s view suffers from several problems. As a second order analysis of religions, Hick’s view is not a correct one. To reject naturalism, it falls into an epistemological circle, where distinction between (...) formal and substantial properties fades away. It seems that Hick is captured by a category mistake in the presentation of his own theory concerning authenticity of all religions to attain the "real an sich". (shrink)
Since its formal definition over sixty years ago, categorytheory 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)
Kripke’s main argument against descriptivism is rooted in a category error that confuses statements about the world with statements about models of the world. It is only because of the ambiguity introduced by the fact that a single sentence can frame two different propositions, one necessary and the other a posteriori, that one reaches the mistaken conclusion that there can be necessary a posteriori truths. This ambiguity from language was carried over into modal logic by Kripke. However, we must (...) consider the two different propositions (1) and (2) separately. Doing so reveals that a given proposition is either necessary and a priori or contingent and a posteriori. It cannot be both. (shrink)
Is it possible to get by with just one ontological category? We evaluate L.A. Paul's attempt to do so: the mereological bundle theory. The upshot is that Paul's attempt to construct a one category ontology may be challenged with some of her own arguments. In the positive part of the paper we outline a two category ontology with property universals and kind universals. We will also examine Paul's arguments against a version of universal bundle theory (...) that takes spatiotemporal co-location instead of compresence or coinstantiation as the feature by which we can identify genuine bundles. We compare this novel theory, bundle theory with kinds, and Paul's mereological bundle theory and apply them to a case study concerning entangled fermions and co-located bosons. (shrink)
The concept of instantiation is realized differently across a variety of metaphysical theories. A certain realization of the concept in a given theory depends on what roles are specified and associated with the concept and its corresponding term as well as what entities are suited to fill those roles. In this paper, the classic realization of the concept of instantiation in a one-category ontology of abstract particulars or tropes is articulated in a novel way and defended against unaddressed (...) objections. (shrink)
There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by categorytheory. 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 (...)categorytheory and is focused on the interpretation and application of the mathematical concepts. (shrink)
This sketch of a perhaps future 'Elementary Theory of the Category of Mereological Sums (including Mereological Wholes and Parts)' relates to my previous papers "The Topos of Emergence" and "Intelligible Gunk". I assert that for successfully categorizing Mereology one has to start with a specific setting of gunk. In this paper we will give a sketch of a categorically version of particular mereological structures. I.e. we will follow the example of F.W.Lawvere’s “An elementary theory of the (...) class='Hi'>category of sets” -/- where he introduced a categorised version of set theory. We refer to sirenians mainly for the reason that they are the closest living relatives of elephants. -/- But they are also rather underestimated creatures - at the first moment rather dull and clumsy, but moving gracefully and elegant in their natural habitat.And some sailors already reckognized their bewitching attraction before we did.Yet we still think this sirenia is a modest creature compared with the elephant in the room. (shrink)
The aim of this paper is make a contribution to the ongoing search for an adequate concept of the a priori element in scientific knowledge. The point of departure is C.I. Lewis’s account of a pragmatic a priori put forward in his "Mind and the World Order" (1929). Recently, Hasok Chang in "Contingent Transcendental Arguments for Metaphysical Principles" (2008) reconsidered Lewis’s pragmatic a priori and proposed to conceive it as the basic ingredient of the dynamics of an embodied scientific reason. (...) The present paper intends to further elaborate Chang’s account by relating it with some conceptual tools from cognitive semantics and certain ideas that first emerged in the context of the category-theoretical foundations of mathematics. (shrink)
A qualitative study was conducted to examine the philosophy of technology of K-12 technology leaders, and explore the influence of their thinking on technology decision making. The research design aligned with CORBIN and STRAUSS grounded theory methods, and I proceeded from a research paradigm of critical realism. The subjects were school technology directors and instructional technology specialists, and data collection consisted of interviews and a written questionnaire. Data analysis involved the use of grounded theory methods including memo writing, (...) open and axial coding, constant comparison, the use of purposive and theoretical sampling, and theoretical saturation of categories. Three broad philosophy of technology views were widely held by participants: an instrumental view of technology, technological optimism, and a technological determinist perspective that saw technological change as inevitable. Technology leaders were guided by two main approaches to technology decision making, represented by the categories Educational goals and curriculum should drive technology, and Keep up with Technology (or be left behind). The core category and central phenomenon that emerged was that technology leaders approached technology leadership by placing greater emphasis on keeping up with technology, being influenced by an ideological orientation to technological change, and being concerned about preparing students for a technological future. (shrink)
In the contemporary biomedical literature, every disease is considered genetic. This extension of the concept of genetic disease is usually interpreted either in a trivial or genocentrist sense, but it is never taken seriously as the expression of a genetic theory of disease. However, a group of French researchers defend the idea of a genetic theory of infectious diseases. By identifying four common genetic mechanisms (Mendelian predisposition to multiple infections, Mendelian predisposition to one infection, and major gene and (...) polygenic predispositions), they attempt to unify infectious diseases from a genetic point of view. In this article, I analyze this explicit example of a genetic theory, which relies on mechanisms and is applied only to a specific category of diseases, what we call “a regional genetic theory.” I have three aims: to prove that a genetic theory of disease can be devoid of genocentrism, to consider the possibility of a genetic theory applied to every disease, and to introduce two hypotheses about the form that such a genetic theory could take by distinguishing between a genetic theory of diseases and a genetic theory of Disease. Finally, I suggest that network medicine could be an interesting framework for a genetic theory of Disease. (shrink)
The Elementary Process Theory (EPT) is a collection of seven elementary process-physical principles that describe the individual processes by which interactions have to take place for repulsive gravity to exist. One of the two main problems of the EPT is that there is no proof that the four fundamental interactions (gravitational, electromagnetic, strong, and weak) as we know them can take place in the elementary processes described by the EPT. This paper sets forth the method by which it can (...) be proven that the EPT agrees with the knowledge that derives from the successful predictions of a modern interaction theory T. This determines a fundamentally new research program in theoretical physics. (shrink)
People construe reality by using words as basic units of meaningful categorization. The present theory-driven study applied the method of a free association task to explore how people express the concepts of the world and the self in words. The respondents were asked to recall any five words relating with the word world. Afterwards they were asked to recall any five words relating with the word self. The method of free association provided the respondents with absolute freedom to choose (...) any words they wanted. Such free recall task is suggested as being a relatively direct approach to the respondents’ self- and world-related conceptual categories, without enormous rational processing. The results provide us, first, with associative ranges for constructs of the world and the self, where some associative dimensions are defined by semantic polarities in the meanings of peripheral categories (e.g., Nature vs. Culture). Second, our analysis showed that some groups of verbal categories that were associated with the words world and self are central, while others are peripheral with respect to the central position. Third, the analysis of category networks revealed that some categories play the role of a transmitter, mediating the pathway between other categories in the network. (shrink)
This article maintains that the so-called theory-practice divide in legal education is not only factually false but semantically impossible. -/- As to the divide's falsity, practitioners have of course performed excellent scholarship and academics have excelled in practice. As to the divide's semantic impossibility, this article examines, among other things: -/- (1) the essential role of experience in meaning, -/- (2) the resulting inseparability of theory and practice in the world of experience, -/- (3) problems the divide shares (...) in common with debunked Cartesian dualism, and -/- (4) modern cognitive psychology’s notions of embodied meaning which further underscore the semantic impossibility of separating theory from practice in the world of experience. -/- Using insights from such examinations, this article also explores implications of a debunked theory-practice divide for, among other things, law school curriculums and law school faculty hiring standards. -/- Keywords: legal education, legal writing, semantics, theory, practice, experience, Charles Sanders Peirce, embodied meaning, cognitive psychology, Cartesian dualism, affordance knowledge, metaphor, George Lakoff, category, humanities, Langdell, pragmatism, semiotics, philosophy. (shrink)
I explore the possibility of a structuralist interpretation of homotopy type theory (HoTT) as a foundation for mathematics. There are two main aspects to HoTT's structuralist credentials. First, it builds on categorical set theory (CST), of which the best-known variant is Lawvere's ETCS. I argue that CST has merit as a structuralist foundation, in that it ascribes only structural properties to typical mathematical objects. However, I also argue that this success depends on the adoption of a strict typing (...) system which undermines the metaphysical seriousness of this structuralism. Homotopy type theory adds to CST a distinctive theory of identity between sets, which arguably allows its objects to be seen as ante rem structures. I examine the prospects for such a view, and address many other interpretive problems as they arise. (shrink)
The aim of this paper is to describe and analyze the epistemological justification of a proposal initially made by the biomathematician Robert Rosen in 1958. In this theoretical proposal, Rosen suggests using the mathematical concept of “category” and the correlative concept of “natural equivalence” in mathematical modeling applied to living beings. Our questions are the following: According to Rosen, to what extent does the mathematical notion of category give access to more “natural” formalisms in the modeling of living (...) beings? Is the so -called “naturalness” of some kinds of equivalences (which the mathematical notion of category makes it possible to generalize and to put at the forefront) analogous to the naturalness of living systems? Rosen appears to answer “yes” and to ground this transfer of the concept of “natural equivalence” in biology on such an analogy. But this hypothesis, although fertile, remains debatable. Finally, this paper makes a brief account of the later evolution of Rosen’s arguments about this topic. In particular, it sheds light on the new role played by the notion of “category” in his more recent objections to the computational models that have pervaded almost every domain of biology since the 1990s. (shrink)
In Cybernetics (1961 Edition), Professor Norbert Wiener noted that “The role of information and the technique of measuring and transmitting information constitute a whole discipline for the engineer, for the neuroscientist, for the psychologist, and for the sociologist”. Sociology aside, the neuroscientists and the psychologists inferred “information transmitted” using the discrete summations from Shannon Information Theory. The present author has since scrutinized the psychologists’ approach in depth, and found it wrong. The neuroscientists’ approach is highly related, but remains unexamined. (...) Neuroscientists quantified “the ability of [physiological sensory] receptors (or other signal-processing elements) to transmit information about stimulus parameters”. Such parameters could vary along a single continuum (e.g., intensity), or along multiple dimensions that altogether provide a Gestalt – such as a face. Here, unprecedented scrutiny is given to how 23 neuroscience papers computed “information transmitted” in terms of stimulus parameters and the evoked neuronal spikes. The computations relied upon Shannon’s “confusion matrix”, which quantifies the fidelity of a “general communication system”. Shannon’s matrix is square, with the same labels for columns and for rows. Nonetheless, neuroscientists labelled the columns by “stimulus category” and the rows by “spike-count category”. The resulting “information transmitted” is spurious, unless the evoked spike-counts are worked backwards to infer the hypothetical evoking stimuli. The latter task is probabilistic and, regardless, requires that the confusion matrix be square. Was it? For these 23 significant papers, the answer is No. (shrink)
Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a (...) class='Hi'>theory that fruitfully interrelates a `structural' perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)
The aim of this article is to critically examine what I call Action-Centric Theories of Representation (ACToRs). I include in this category theories of representation that (1) reject construing representation in terms of a relation that holds between representation itself (the representational vehicle) and what is represented, and instead (2) try to bring the function that representations play for cognitive systems to the center stage. Roughly speaking, according to proponents of ACToRs, what makes a representation (that is, what is (...) constitutive of it being a representation) is its being functionally involved in preselecting or guiding the actions of cognitive systems. I intend to argue that while definitely valuable, ACToRs are underconstrained and thus not entirely satisfying, since there exist structures that would count as representations according to ACToRs, but which do not play functional roles that could be nontrivially or in an explanatorily valuable way classified as representing something for a cognitive system. I outline a remedy for this theoretical situation by postulating that a fully satisfying theory of representation in cognitive science should have two factors; i.e., it should combine the pragmatic, action-oriented aspect present in ACToRs with an element that emphasizes the importance of the relation holding between a representational vehicle and what is represented. (shrink)
Categorytheory 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)
This paper has two parts: In the first part, I give a general survey of the various reasons 17th and 18th century life scientists and metaphysicians endorsed the theory of pre-existence according to which God created all living beings at the creation of the universe, and no living beings are ever naturally generated anew. These reasons generally fall into three categories. The first category is theological. For example, many had the desire to account for how all humans are (...) stained by original sin (we were all there). As another example of a theological motivation, some take the organism as an obvious starting point for a teleological argument for God’s existence, and this staring point is sometimes developed into a full-blown theory of pre-existence. The second category could be thought of as non-theological metaphysical, and paramount here is the desire to deal with the metaphysical problem of individuation. So, for example, Leibniz embraces a version of hylomorphism in order to overcome difficulties with Descartes’ theory of material substance, including the difficulty of how to account for enduring material individuals, and Leibniz’s hylomorphism is closely linked with his embrace of pre-existence. The third category might be termed “biological”, and one example of such a concern is how to explain the organic unity of living beings where the whole seems to ontologically precede the parts. This is frequently translated into a temporal priority of whole to parts, and thus pre-existence is posited. Of course, many natural philosophers of the early modern period embrace pre-existence for more than one reason, but in general, these are the three classes of motivations one might have for embracing the theory. In the second part of the paper I examine in detail one argument that appears in the work of Malebranche. On the face of it, this argument seems to be a biological one, specifically the biological or organic holism argument mentioned above. But upon closer examination, I shall argue, Malebranche’s reasons for endorsing pre-existence bring together several of the arguments discussed in the first part of the paper. I conclude with some considerations about what we can learn about Malebranche as a natural philosopher from his motivations for holding the pre-existence doctrine of generation. (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, categorytheory 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)
Motivated by Scholze and Fargues' geometrization of the local Langlands correspondence using perfectoid diamonds and Clausen and Scholze's work on the K-theory of adic spaces using condensed mathematics, we introduce the Efimov K-theory of diamonds. We propose a pro-diamond, a large stable (infinity,1)-category of diamonds D^{diamond}, a diamond spectra and chromatic tower, and a localization sequence for diamond spectra. Commensurate with the localization sequence, we detail three potential applications of the Efimov K-theory of D^{diamond}: to emergent (...) time as a pro-emergence (v-stack time) in quantum gravity in a diamond holographic principle using Scholze's six operations in the 'etale cohomology of diamonds; to D^{diamond}-cryptography; and to nonlocality in perfectoid quantum physics. (shrink)
Shannon’s information theory has been a popular component of first-order cybernetics. It quantifies information transmitted in terms of the number of times a sent symbol is received as itself, or as another possible symbol. Sent symbols were events and received symbols were outcomes. Garner and Hake reinterpreted Shannon, describing events and outcomes as categories of a stimulus attribute, so as to quantify the information transmitted in the psychologist’s category (or absolute judgment) experiment. There, categories are represented by specific (...) stimuli, and the human subject must assign those stimuli, singly and in random order, to the categories that they represent. Hundreds of computations ensued of information transmitted and its alleged asymptote, the sensory channel capacity. The present paper critically re-examines those estimates. It also reviews estimates of memory capacity from memory experiments. It concludes that absolute judgment is memory-limited and that channel capacities are actually memory capacities. In particular, there are factors that affect absolute judgment that are not explainable within Shannon’s theory, factors such as feedback, practice, motivation, and stimulus range, as well as the anchor effect, sequential dependences, the rise in information transmitted with the increase in number of stimulus dimensions, and the phenomena of masking and stimulus duration dependence. It is recommended that absolute judgments be abandoned, because there are already many direct estimates of memory capacity. (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 categorytheory 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 categorytheory 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)
Propounded in relation to a peculiar mode in the view of an oscillating or cyclic universe, the concept of Return of Power, or of ontic recurrence as further increase in ontic Power signifies the determination of the existing entity according to its own selective recurrence as dialectically exceeding a previous status. Based thus upon the assumption that the actual ontological existence of the entity lies in its own potentiated recurrence (for it is maintained that only what is able to return (...) to itself as potentiated signifies actual ontological existence) and that an ontic eternal return can only occur in terms of further Power, this notion embodies the culminating phase within change as an ultimate overcoming of a previous ontic status or situation respecting the category Power as a primary criterion within a constantly increasing Real (as Power, precisely). An ontic selective ultimate event respecting becoming as a general increment of the Real, this concept exceeds both the ontic dispersion in nothingness of the linear time and the identical ontic repeatability of the Eternal Return of the Same, the latter insofar as asserting the perennial return of man's own status of powerlessness, and as rendering accordingly impossible any actual (dialectical) transition to a well-defined superior ontic phase. (shrink)
Alvin Plantinga’s theory of knowledge, as developed in his Warrant trilogy, has shaped the debates surrounding many areas in epistemology in profound ways. Plantinga has received his share of criticism, however, particularly in his treatment of belief in God as being “properly basic”. There has also been much confusion surrounding his notions of warrant and proper function, to which Plantinga has responded numerous times. Many critics remain unsatisfied, while others have developed alternative understandings of warrant in order to rescue (...) Plantinga’s theory from certain objections. The most promising of such attempts fall under the broad category of “virtue epistemology” or a “virtue-theoretic” approach. The work being done in virtue epistemology is still in its early stages and a consensus on what actually constitutes virtue epistemology has yet to be reached. While some have attempted to structure an entire theory of knowledge based on the virtues possessed by the knower, others have focused more on the role of epistemic virtues as an attempt to supplement existing theories, including Plantinga’s. In this paper, I will offer an analysis of what such an attempt might look like and evaluate the potential success of broadening Plantinga’s original model. -/- My proposal is that certain features of a virtue-theoretic approach (also referred to as “agent-reliabilism”) could improve Plantinga’s model in significant ways. Not only would such a broadened approach be better equipped to handle common objections, but it would also be better suited to contribute an enhanced understanding of the task of epistemology, one that seeks to discover multiple epistemic goods other than what has been traditionally confined to the realm of knowledge. I conclude by applying this approach to Plantinga’s treatment of theistic belief in Warranted Christian Belief and by articulating a few of the ways in which epistemic virtues can increase the degree of warrant enjoyed by such belief. (shrink)
By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically categorytheory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the (...) rôle he envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised. (shrink)
The subject of this paper is Charles Morris’ semiotic theory that has as one of its major projects the unification of all sciences of signs. However, since the above project has proven to be unsuccessful, we will try to examine here the reasons that led to this. Accordingly, we will argue that to transcend the particularities of individual disciplines that he wanted to unify, Morris had to make certain ontological assumptions, instead of theoretical and methodological ones, that they could (...) share. However, because the 'sign' as an ontological category could in our view only be established if we follow the principles of the pragmatic philosophical tradition, we will try to show that the reasons for this failure should be primarily sought in different effects that consistent application of the pragmatic principles has in each of them (primarily in linguistics and the philosophy of language). On the other hand, this should enable us to draw several important conclusions regarding Morris’ project: namely, that his failure does not have to mean giving up semiotics as a potentially key discipline in approaching some fundamental philosophical problems, but also that it would demand return to the original semiotics developed in Peirce’s works. (shrink)
Evil acts are not merely wrong; they belong to a different moral category. For example, telling a minor lie might be wrong but it is not evil, whereas the worst act of gratuitous torture that you can imagine is evil and not merely wrong. But how do wrongs and evils differ? A theory or conception of evil should, among other things, answer that question. But once a theory of evil has been developed, how do we defend or (...) refute it? The most commonly used method for doing this in the literature has been to, respectively, provide pro-examples or counter-examples. While this method might be sufficient for establishing that a theory is at least a prima facie plausible theory of evil, it is often insufficient for making fine-grained distinctions between otherwise plausible theories of evil. To supplement this insufficiency I propose that we also focus on five theoretical virtues that a theory of evil should have. These virtues are: 1) meshing well with important theories of moral wrongdoing; 2) being based on a plausible moral psychology; 3) explaining the basis of our judgments about evil; 4) being able to alter, revise and expand our judgments about evil; and 5) being pitched at the right level of generality. The main result of this paper will be to show that these five theoretical virtues provide a useful analytical tool for interrogating plausible theories of evil. The secondary result will be to show that my theory of evil has these five virtues. (shrink)
Information flow in a system is a core cybernetics concept. It has been used frequently in Sensory Psychology since 1951. There, Shannon Information Theory was used to calculate "information transmitted" in "absolute identification" experiments involving human subjects. Originally, in Shannon's "system", any symbol received ("outcome") is among the symbols sent ("events"). Not all symbols are received as transmitted, hence an indirect noise measure is calculated, "information transmitted", which requires knowing the confusion matrix, its columns labeled by "event" and its (...) rows labeled by "outcome". Each matrix entry is dependent upon the frequency with which a particular outcome corresponds to a particular event. However, for the sensory psychologist, stimulus intensities are "events"; the experimenter partitions the intensity continuum into ranges called "stimulus categories" and "response categories", such that each confusion-matrix entry represents the frequency with which a stimulus from a stimulus category falls within a particular response category. Of course, a stimulus evokes a sensation, and the subject's immediate memory of it is compared to the memories of sensations learned during practice, to make a categorization. Categorizing thus introduces "false noise", which is only removed if categorizations can be converted back to their hypothetical evoking stimuli. But sensations and categorizations are both statistically distributed, and the stimulus that corresponds to a given mean categorization cannot be known from only the latter; the relation of intensity to mean sensation, and of mean sensation to mean categorization, are needed. Neither, however, are presently knowable. This is a quandary, which arose because sensory psychologists ignored an ubiquitous component of Shannon's "system", the uninvolved observer, who calculates "information transmitted". Human sensory systems, however, are within de facto observers, making "false noise" inevitable. (shrink)
This paper reveals errors within Norwich et al.’s Entropy Theory of Perception, errors that have broad implications for our understanding of perception. What Norwich and coauthors dubbed their “informational theory of neural coding” is based on cybernetics, that is, control and communication in man and machine. The Entropy Theory uses information theory to interpret human performance in absolute judgments. There, the continuum of the intensity of a sensory stimulus is cut into categories and the subject is (...) shown exemplar stimuli of each category. The subject must then identify individual exemplars by category. The identifications are recorded in the Garner-Hake version of the Shannon “confusion matrix”. The matrix yields “H”, the entropy (degree of uncertainty) about what stimulus was presented. Hypothetically, uncertainty drops as a stimulus lengthens, i.e. a plot of H vs. stimulus duration should fall monotonically. Such “adaptation” is known for both sensation and firing rate. Hence, because “the physiological adaptation curve has the same general shape as the psychophysical adaptation curve”, Norwich et al. assumed that both have the same time course; sensation and firing rate were thus both declared proportional to H. However, a closer look reveals insurmountable contradictions. First, the peripheral neuron hypothetically cannot fire in response to a stimulus of a given intensity until after somehow computing H from its responses to stimuli of various intensities. Thus no sensation occurs until firing rate adapts, i.e. attains its spontaneous rate. But hypothetically, once adaptation is complete, certainty is reached and perception ends. Altogether, then, perception cannot occur until perception is over. Secondly, sensations, firing rates, and H’s are empirically not synchronous, contrary to assumption. In sum, the core concept of the cybernetics-based Entropy Theory of Perception, that is, that uncertainty reduction is the basis for perception, is irrational. (shrink)
In this paper, we propose a mathematical model of subjective experience in terms of classes of hierarchical geometries of representations (“n-awareness”). We first outline a general framework by recalling concepts from higher categorytheory, homotopy theory, and the theory of (infinity,1)-topoi. We then state three conjectures that enrich this framework. We first propose that the (infinity,1)-category of a geometric structure known as perfectoid diamond is an (infinity,1)-topos. In order to construct a topology on the (infinity,1)- (...) class='Hi'>category of diamonds we then propose that topological localization, in the sense of Grothendieck-Rezk-Lurie (infinity,1)-topoi, extends to the (infinity,1)-category of diamonds. We provide a small-scale model using triangulated categories. Finally, our meta-model takes the form of Efimov K-theory of the (infinity,1)-category of perfectoid diamonds, which illustrates structural equivalences between the category of diamonds and subjective experience (i.e.its privacy, self-containedness, and self-reflexivity). Based on this, we investigate implications of the model. We posit a grammar (“n-declension”) for a novel language to express n-awareness, accompanied by a new temporal scheme (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? We also examine the notion of “self” within our framework. A new model of personal identity is introduced which resembles a categorical version of the “bundle theory”: selves are not substances in which properties inhere but (weakly) persistent moduli spaces in the K-theory of perfectoid diamonds. (shrink)
This article, written in Bengali ('Gonit Dorshon' means `philosophy of mathematics' ), briefly reviews a few of the major points of view toward mathematics and the world of mathematical entities, and interprets the philosophy of mathematics as an interaction between these. The existence of these different points of view is indicative that mathematics, in spite of being of universal validity, can nevertheless accommodate alternatives. In particular, I review the alternative viewpoints of Platonism and Intuitionism and present the case that in (...) spite of their great differences, they are not mutually exclusive - that both can be accommodated within the infinite edifice of mathematics. This, in turn, is argued to be consistent with the viewpoint of CategoryTheory that holds the promise of an entirely new interpretation of the world of mathematics and the relation of that world to the world of our concepts and ideas: mathematics is a human enterprise and mathematical logic is a reflection of how our ideas and concepts are formed and combined with one another. I venture that this, perhaps, is the view of mathematics that Ludwig Wittgenstein would espouse. (shrink)
Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key (...) element here is the systematic nature of these maps in a categorical framework and I suggest that, from that point of view, one can see an architectonic of mathematics emerging clearly. Moreover, they force us to reconsider the nature of mathematical knowledge itself. Thus, to understand certain fundamental aspects of mathematics, categorytheory is necessary (at least, in the present state of mathematics). (shrink)
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