Millikan’s theory of content purports to rely heavily on the existence of isomorphisms between a system of representations and the things in the world which they represent — “the mapping requirement for being intentional signs” (Millikan 2004, p. 106). This paper asks whether those isomorphisms are doing any substantive explanatory work. Millikan’s isomorphism requirement is deployed for two main purposes. First, she claims that the existence of an isomorphism is the basic representing relation, with teleology playing a subsidiary (...) role — to account for misrepresentation (the possibility of error). Second, Millikan relies on an isomorphism requirement in order to guarantee that a system of representations displays a kind of productivity. This seemingly strong reliance on isomorphism has prompted the objection that isomorphism is too liberal to be the basic representing relation: there are isomorphisms between any system of putative representations and any set, of the same cardinality, of items putatively represented. This paper argues that all the work in fixing content is in fact done by the teleology. Deploying Millikan’s teleology-based conditions to ascribe contents will ensure that there is an isomorphism between representations and the things they represent, but the isomorphism ‘requirement’ is playing no substantive explanatory role in Millikan’s account of content determination. So an objection to her theory based on the liberality of isomorphism is misplaced. The second role for isomorphism is to account for productivity. If some kind of productivity is indeed necessary for representation, then functional isomorphism will again be too liberal a constraint to account for that feature. The paper suggests an alternative way of specifying the relation between a system of representations and that which they represent which is capable of playing an explanatory role in accounting for Millikan’s type of productivity. In short, the liberality of isomorphism is no objection to Millikan’s teleosemantics, since the isomorphism ‘requirement’ need play no independent substantive role in Millikan’s account of representation. (shrink)

The relevance of Wolfgang K hler's psychoneural isomorphism principle to contemporary cognitive neuroscience is explored. K hler's approach to the mind—body problem is interpreted as a response to the foundational crisis of psychology at the beginning of the twentieth century. Some aspects of his isomorphism doctrine are discussed, with a view to reaching an interpretation that is both historically accurate and pertinent to issues currently debated in the philosophy of psychology. The principle was meant to be empirically verifiable. (...) Accordingly, some similarities between K hler's approach and current neural network modeling are pointed out, and it is shown that some recent trends in the neurosciences are broadly compatible with K hler's views on cortical functioning. Isomorphism is interpreted as a form of neuroreductionism constrained by bridging laws relating mental phenomena to macrosocopic parameters of neural function. While isomorphism is probably valid for perceptual phenomena, its applicability to higher mental processes remains doubtful. (shrink)

A serious crisis is identified in theories of neurocomputation, marked by a persistent disparity between the phenomenological or experiential account of visual perception and the neurophysiological level of description of the visual system. In particular, conventional concepts of neural processing offer no explanation for the holistic global aspects of perception identified by Gestalt theory. The problem is paradigmatic and can be traced to contemporary concepts of the functional role of the neural cell, known as the Neuron Doctrine. In the (...) absence of an alternative neurophysiologically plausible model, I propose a perceptual modeling approach, to model the percept as experienced subjectively, rather than modeling the objective neurophysiological state of the visual system that supposedly subserves that experience. A Gestalt Bubble model is presented to demonstrate how the elusive Gestalt principles of emergence, reification, and invariance can be expressed in a quantitative model of the subjective experience of visual consciousness. That model in turn reveals a unique computational strategy underlying visual processing, which is unlike any algorithm devised by man, and certainly unlike the atomistic feed-forward model of neurocomputation offered by the Neuron Doctrine paradigm. The perceptual modeling approach reveals the primary function of perception as that of generating a fully spatial virtual-reality replica of the external world in an internal representation. The common objections to this picture-in-the-head concept of perceptual representation are shown to be ill founded. Key Words: brain-anchored; Cartesian theatre; consciousness; emergence; extrinsic constraints; filling-in; Gestalt; homunculus; indirect realism; intrinsic constraints; invariance; isomorphism; multistability; objective phenomenology; perceptual modeling; perspective; phenomenology; psychophysical parallelism; psychophysical postulate; qualia; reification; representationalism; structural coherence. (shrink)

In this paper I provide a metatheoretical analysis of speech perception research. I argue that the central turning point in the history of speech perception research has not been well understood. While it is widely thought to mark a decisive break with what I call "the alphabetic conception of speech," I argue that it instead marks the entrenchment of this conception of speech. In addition, I argue that the alphabetic conception of speech continues to underwrite speech perception research today and (...) moreover that it functions as a dogma which ought to be rejected. (shrink)

Associated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional languages are recursively isomorphic (...) where a language is undecidable iff it has at least one operation or relation symbol of two or more places or at least two unary operation symbols, and a language is functional iff it has exactly one unary operation symbol and all other symbols are for unary relations, constants, or propositions. (shrink)

Let L be a computable vocabulary, let X_L be the space of L-structures with universe ω and let f:2^\omega \rightarrow X_L be a hyperarithmetic function such that for all x,y \in 2^\omega, if x \equiv _h y then f(x) \cong f(y). One of the following two properties must hold. (1) The Scott rank of f(0) is \omega _1^{CK} + 1. (2) For all x \in 2^\omega, f(x) \cong f(0).

If reports given on experimental awakenings validly represent mental activity that was underway before the awakening, REM sleep is neither necessary nor sufficient for dreaming. Another intuitively attractive hypothesis for its function – that REM consolidates or otherwise modifies memory traces acquired while awake – is not supported by the preponderant evidence. There is growing acceptance of the possibility that REM functions to support sleep rather than waking brain processes. [Hobson et al.; Nielsen; Solms; Vertes & Eastman].

This paper critiques the view that consciousness is likely something extra which accompanies or is produced by neural states, something beyond the functional cognitive processes realized in the brain. Such a view creates the `explanatory gap'between function and nomenology which many suppose cannot be filled by functionalist theories of mind. Given methodological considerations of simplicity, ontological parsimony, and theoretical conservatism, an alternative hypothesis is recommended, that subjective qualitative experience is identical to certain information-bearing, behaviour-controlling functions, not something which emerges (...) from them. This hypothesis explains the isomorphism between the structure of experience and neural organization, while providing a naturalistic account of qualiait's relational properties of informational states, not a sparate ontology of phenomenal essences. On this functionalist view, the hard, empirical problem of consciousness is to discover precisely which neural functions constitute subjective experience. (shrink)

The significance of the concept of a mathematical transformation is explained. In particular, it is explained how to construct true statements concerning n-dimensional spaces, for arbitrary n, on the basis of true statements concerning two-dimensional spaces.

In this article I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss’s ε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely mathematical definition of observational (...) equivalence are unpromising. Despite this critique, I suggest that Werndl’s proposals are valuable because they clarify the distinction between provable and unprovable elements in arguments for observational equivalence. (shrink)

Recent work in the philosophy of science has generated an apparent conflict between theories attempting to explicate the nature of scientific representation. On one side, there are what one might call 'informational' views, which emphasize objective relations (such as similarity, isomorphism, and homomorphism) between representations (theories, models, simulations, diagrams, etc.) and their target systems. On the other side, there are what one might call 'functional' views, which emphasize cognitive activities performed in connection with these targets, such as interpretation (...) and inference. The main sources of the impression of conflict here are arguments by some functionalists to the effect that informational theories are flawed: it is suggested that relations typically championed by informational theories are neither necessary nor sufficient for scientific representation, and that any theory excluding functions is inadequate. In this paper I critically examine these arguments, and contend that, as it turns out, informational and functional theories are importantly complementary. (shrink)

In this paper I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss's ε-congruenceε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely mathematical definition of observational (...) equivalence are unpromising. Despite this critique, I suggest that Werndl's proposals are valuable because they clarify the distinction between provable and unprovable elements in arguments for observational equivalence. (shrink)

Journal Name: Cognitive Linguistics Issue: Ahead of print.

Many philosophers and psychologists have attempted to elucidate the nature of mental representation by appealing to notions like isomorphism or abstract structural resemblance. The ‘structural representations’ that these theorists champion are said to count as representations by virtue of functioning as internal models of distal systems. In his 2007 book, Representation Reconsidered, William Ramsey endorses the structural conception of mental representation, but uses it to develop a novel argument against representationalism, the widespread view that cognition essentially involves the manipulation (...) of mental representations. Ramsey argues that although theories within the ‘classical’ tradition of cognitive science once posited structural representations, these theories are being superseded by newer theories, within the tradition of connectionism and cognitive neuroscience, which rarely if ever appeal to structural representations. Instead, these theories seem to be explaining cognition by invoking so-called ‘receptor representations’, which, Ramsey claims, aren’t genuine representations at all—despite being called representations, these mechanisms function more as triggers or causal relays than as genuine stand-ins for distal systems. I argue that when the notions of structural and receptor representation are properly explicated, there turns out to be no distinction between them. There only appears to be a distinction between receptor and structural representations because the latter are tacitly conflated with the ‘mental models’ ostensibly involved in offline cognitive processes such as episodic memory and mental imagery. While structural representations might count as genuine representations, they aren’t distinctively mental representations, for they can be found in all sorts of non-intentional systems such as plants. Thus to explain the kinds of offline cognitive capacities that have motivated talk of mental models, we must develop richer conceptions of mental representation than those provided by the notions of structural and receptor representation. (shrink)

In our contribution we will observe phenomenal architecture of a mind and operational architectonics of the brain and will show their intimate connectedness within a single integrated metastable continuum. The notion of operation of different complexity is the fundamental and central one in bridging the gap between brain and mind: it is precisely by means of this notion that it is possible to identify what at the same time belongs to the phenomenal conscious level and to the neurophysiological level of (...) brain activity organization, and what mediates between them. Implications for linguistic semantics, self-organized distributed computing algorithms, artificial machine consciousness, and diagnosis of dynamic brain diseases will be discussed briefly. (shrink)

The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.

A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its effective power over a cohesive set of natural numbers. A cohesive set is an infinite set of natural numbers that is indecomposable with respect to computably enumerable sets. It plays the role of an ultrafilter, and the elements of a cohesive power are the equivalence classes of certain partial computable functions determined by the cohesive (...) set. Thus, unlike many classical ultrapowers, a cohesive power is a countable structure. In this paper we focus on cohesive powers of graphs, equivalence structures, and computable structures with a single unary function satisfying various properties, which can also be viewed as directed graphs. For these computable structures, we investigate the isomorphism types of their cohesive powers, as well as the properties of cohesive powers when they are not isomorphic to the original structure. (shrink)

This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal (...) notations, the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties). (shrink)

Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a version of Parigot’s free deduction. The elimination rules are “general,” but can be systematically simplified. Cut-elimination and normalization hold. Restriction to a single formula in the succedent yields intuitionistic versions of these systems. The rules also yield generalized lambda calculi providing proof terms for (...) natural deduction proofs as in the Curry-Howard isomorphism. Addition of an indirect proof rule yields classical single-conclusion versions of these systems. Gentzen’s standard systems arise as special cases. (shrink)

One goal of recent philosophy of mind has been to ‘naturalize’ intentionality by showing how a purely physical system could have states that represent or are about items (objects, properties, facts) in the world. The project is reductionist in spirit, the aim being to explain intentional relations—to say what they really are—and to do so in terms that do not themselves utilize intentional or semantic concepts. In this vein there are attempts to explain intentional relations in terms of causal relations, (...) informational relations, teleological or functional relations, relations involving abstract similarity or isomorphism, and various combinations thereof. What makes these accounts naturalistic is the presumed objectivity and scientific respectability of the properties appelated to in the explanans. What makes them all reductive is their shared presumption that intentionality can be explained in terms that have a wider application to intentional systems as well as to systems that have no mental properties at all. (shrink)

Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family is limitwise monotonic (l.m.) if every set is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice. The semilattices exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of ‐computable families. We show that every (...) Rogers semilattice of a ‐computable family is isomorphic to some semilattice. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices. In particular, there is an l.m. family S such that is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all ‐computable numberings. We prove that inside this class, the index set of l.m. numberings is ‐complete. (shrink)

is the second-order theory of the part-whole relation. It can express such hypotheses about the size of Reality as that there are inaccessibly many atoms. Take a non-empty class to have exactly its non-empty subclasses as parts; hence, its singleton subclasses as atomic parts. Then standard set theory becomes the theory of the member-singleton function—better, the theory of all singleton functions—within the framework of megethology. Given inaccessibly many atoms and a specification of which atoms are urelements, a singleton function exists, (...) unique up to isomorphism. (This article is partly abridged from my Parts of Classes, partly a sequel.). (shrink)

The drive to describe cultural history as an evolutionary process has two sources. One from within social theory is part of the impetus to convert social studies into "social sciences" providing them with the status accorded to the natural sciences. The other comes from within biology and biological anthropology in the belief that the theory of evolution must be universal in its application to all functions of all living organisms. The social scientific theory of cultural evolution is pre-Darwinian, employing a (...) developmental model of unfolding characterized by intrinsic directionality, by definable stages that succeed each other and by some criterion of progress. It is arbitrary in its definitions of progress and has had the political problem that a diachronic claim of cultural progress implies a synchronic differential valuation of present-day cultures. The biological scheme creates an isomorphism between the Darwinian mechanism of evolution and cultural history, postulating rules of cultural "mutation", cultural inheritance and some mechanism of natural selection among cultural alternatives. It uses simplistic ad hoc notions of individual acculturation and of the differential survival and reproduction of cultural elements. It is unclear what useful work is done by substituting the metaphor of evolution for history. (shrink)

Realization can be roughly understood as a kind of role-playing, a relationship between a property that plays a role and a property characterized by that role. This rough sketch previously received only moderate elaboration; recently, however, several substantive theories of realization have been proposed. But are there any general constraints on a theory of realization? What is a theory of realization supposed to accomplish? I first argue that a view of realization is viable, in part, to the extent that physical (...) realization under that view explains or accounts for why instances of realized properties are necessitated by how things are physically in a modally strong sense. In this sense, I claim that physical realization should explain physical supervenience. I then call into question two alternative desiderata and raise a challenge for attempts to explicate realization in terms of isomorphism or analogy. Finally, I explain how a causal-functional account of realization, as well as less demanding accounts, can meet the preferred desideratum. (shrink)

Grounded in institutional theory, this study provides an overview of the corporate social responsibility initiatives of Turkey's 30 largest corporations through a thematic content analysis. The study focuses on the G-20 member Turkey and investigates the influence of isomorphism mechanisms on the adoption of CSR initiatives in a developing country context. The aim of this study is to integrate Carroll's CSR dimensions, the type of CSR engagement and coercive, mimetic and normative isomorphism mechanisms proposed by institutional theory. Through (...) this integration the study makes a unique contribution to the literature by providing a different perspective. Findings reveal industry characteristics do not influence the selection of CSR initiatives. While business-to-business companies focus on CSR activities linked to their core business functions, business-to-consumer companies focus on CSR initiatives that are more discretionary, varied and philanthropic. In addition, findings show that multinational corporations implement CSR initiatives at the global level rather than focusing on local needs. (shrink)

The paper deals with the semantics of mathematical notation. In arithmetic, for example, the syntactic shape of a formula represents a particular way of specifying, arriving at, or constructing an arithmetical object (that is, a number, a function, or a truth value). A general definition of this sense of "construction" is proposed and compared with related notions, in particular with Frege's concept of "function" and Carnap's concept of "intensional isomorphism." It is argued that constructions constitute the proper subject matter (...) of both logic and mathematics, and that a coherent semantic account of mathematical formulas cannot be given without assuming that they serve as names of constructions. (shrink)

Imitation, deliberation, and mindreading are characteristically human sociocognitive skills. Research on imitation and its role in social cognition is flourishing across various disciplines. Imitation is surveyed in this target article under headings of behavior, subpersonal mechanisms, and functions of imitation. A model is then advanced within which many of the developments surveyed can be located and explained. The shared circuits model (SCM) explains how imitation, deliberation, and mindreading can be enabled by subpersonal mechanisms of control, mirroring, and simulation. It is (...) cast at a middle, functional level of description, that is, between the level of neural implementation and the level of conscious perceptions and intentional actions. The SCM connects shared informational dynamics for perception and action with shared informational dynamics for self and other, while also showing how the action/perception, self/other, and actual/possible distinctions can be overlaid on these shared informational dynamics. It avoids the common conception of perception and action as separate and peripheral to central cognition. Rather, it contributes to the situated cognition movement by showing how mechanisms for perceiving action can be built on those for active perception.;>;>The SCM is developed heuristically, in five layers that can be combined in various ways to frame specific ontogenetic or phylogenetic hypotheses. The starting point is dynamic online motor control, whereby an organism is closely attuned to its embedding environment through sensorimotor feedback. Onto this are layered functions of prediction and simulation of feedback, mirroring, simulation of mirroring, monitored inhibition of motor output, and monitored simulation of input. Finally, monitored simulation of input specifying possible actions plus inhibited mirroring of such possible actions can generate information about the possible as opposed to actual instrumental actions of others, and the possible causes and effects of such possible actions, thereby enabling strategic social deliberation. Multiple instances of such shared circuits structures could be linked into a network permitting decomposition and recombination of elements, enabling flexible control, imitative learning, understanding of other agents, and instrumental and strategic deliberation. While more advanced forms of social cognition, which require tracking multiple others and their multiple possible actions, may depend on interpretative theorizing or language, the SCM shows how layered mechanisms of control, mirroring, and simulation can enable distinctively human cognitive capacities for imitation, deliberation, and mindreading. (shrink)

We show that every model of IΔ0 has an end extension to a model of a theory where log-space computable function are formalizable. We also show the existence of an isomorphism between models of IΔ0 and models of linear arithmetic LA.

Modus ponens provides the central theme. There are laws, of the form A→C. A logic L collects such laws. Any datum A provides input to the laws of L. The central ternary relation R relates theories L,T and U, where U consists of all of the outputs C got by applying modus ponens to major premises from L and minor premises from T. Underlying this relation is a modus ponens product operation on theories L and T, whence RLTU iff LTU. (...) These ideas have been expressed, especially with Routley, as worlds semantics for relevant and other substructural logics.Worlds are best demythologized as theories, subject to truth-functional and other constraints. The chief constraint is that theories are taken as closed under logical entailment, which clearly begs the question if we are using the semantics to determine which theory L is Logic itself. Instead we draw the modal logicians’ conclusion—there are many substructural logics, each with its appropriate ternary relational postulates.Each logic L gives rise to a Calculus of L-theories, on which particular candidate logical axioms have the combinatorial properties expected from the well-known Curry–Howard isomorphism . We apply their bubbling lemma, updating the Fools Model of Combinatory Logic at the pure → level for the system BT. We make fusion an explicit connective, proving a combinator correspondence theorem. Having taken relevant → as a left residual for , we explore its right residual mate →r. Finally we concentrate on and prove a finite model property for the classical minimal relevant logic CB, a conservative extension of the minimal positive relevant logic B+. (shrink)

In a well-known paper, Timothy Williamson claimed to prove with a coin-flipping example that infinitesimal-valued probabilities cannot save the principle of Regularity, because on pain of inconsistency the event ‘all tosses land heads’ must be assigned probability 0, whether the probability function is hyperreal-valued or not. A premise of Williamson’s argument is that two infinitary events in that example must be assigned the same probability because they are isomorphic. It was argued by Howson that the claim of isomorphism fails, (...) but a more radical objection to Williamson’s argument is that it had been, in effect, refuted long before it was published. (shrink)

A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let be a computable model and let R be an extra relation on the domain of . That is, R is not named in the language of . We define to be the set of Turing degrees of the images f under all isomorphisms f from to computable models. We investigate conditions on and R which are sufficient and necessary for to (...) contain every Turing degree. These conditions imply that if every Turing degree 0″ can be realized in via an isomorphism of the same Turing degree as its image of R, then contains every Turing degree. We also discuss an example of and R whose coincides with the Turing degrees which are 0′. (shrink)

Abstract: Kant follows a substantial tradition by defining judgment so that it must involve a relation of concepts, which raises the question of why he thinks that single-term existential judgments should still qualify as judgments. There is a ready explanation if Kant is somehow anticipating a Fregean second-order account of existence, an interpretation that is already widely held for separate reasons. This paper examines Kant's early (1763) critique of Wolffian accounts of existence, finding that it provides the key idea in (...) his mature model of existential judgment, which is in fact sharply opposed to the Fregean strategy. By relating this to Kant's theory of judgment in general—in particular, to his claim for an isomorphism between the assertoric function of judgment and the category of existence—a preliminary case is made that absolute positing, far from being a marginal special case, accomplishes the primary function of judgment. This argument shows the importance of distinguishing between contexts in which Kant is treating judgment as a vehicle for inference (e.g. pure general logic) and contexts in which he is treating it, more robustly, as the cognition of an object. (shrink)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

Freedom is first apprehended as the pursuit of an activity which implies the choice to defend a thesis among other possible ones. This translation of the problem of freedom in an articulate language presupposes a complex nervous system and sensory apparatuses which we take for granted. In this study, I try to explore the undergrounds of the problem of freedom along with the suggestion that the notion of coding could enable one to bridge nature and the mind. When organisms invent, (...) are they doing it in a spontaneous manner, inscribing in their hereditary and mnemonic instructions a stochastic contrivance of random accidents, or are they attempting to select among a limited number of schemes endowed with some optimality of functioning? If we consider them as submitted to physical forces, it is to the extent that we make them part of a strategy to extend the "laws" of a nature understood to respond passively. I suggest in this study that the epistemological understanding must regionalize itself and admit a hierarchy of dispositions in relation to the phenomenon of selection. I end by suggesting the pursuit of affirmation instead of negation, as this alone contains the requirement of integration to the knowing subject as well as the form in its act of understanding, without giving it a spontaneist position. (shrink)

This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the same fact. (...) The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations. (shrink)

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced (...) by a computable permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non‐periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non‐commutative and non‐Boolean c.e. ring. (shrink)

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank (...) at most n can be naturally identified with the set S of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class Sof rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ of x is defined to be the function. (shrink)

We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω (...) in which the functions and relations have degree at most c. We show that every degree d ≥ 0 (ω) is the ωth jump degree of a Boolean algebra, but that for $n no Boolean algebra has nth-jump degree $\mathbf{d} > 0^{(n)}$ . The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties. (shrink)

For centuries, the processes of social differentiation associated with Modernity have often been thought to intensify the need for site-specific forms of role training and knowledge production, threatening the university’s survival either through fragmentation or through failure to adapt. Other lines of argument emphasize the extent to which the Modern system creates and relies on an integrated knowledge system, but most of the literature stresses functional differentiation and putative threats to the university. And yet over this period the university (...) has flourished. In our view, this seeming paradox is explained by the fact that modern society rests as much on universalistic cosmological bases as it does on differentiation. The university expands over recent centuries because – as it has from its religious origins – it casts cultural and human materials in universalistic terms. Our view helps explain empirical phenomena that confound standard accounts: the university’s extraordinary expansion and global diffusion, its curricular and structural isomorphism, and its relatively unified structure. All of this holds increasingly true after World War II, as national state societies made up of citizens are increasingly embedded in a world society constituted of empowered individuals. The redefinition of society in global and individual terms reduces nationally bounded models of nature and culture, extends the pool of university beneficiaries and investigators, and empowers the human persons who are understood to root it all. The changes intensify universalization and the university’s rate of worldwide growth. For the university’s knowledge and “knowers,” and for the pedagogy that joins them together, the implications are many. The emerging societal context intensifies longstanding processes of cultural rationalization and ontological elaboration, yielding great expansions in what can and should be known, and in who can and should know. These changes in turn alter the menu of approved techniques for joining knowledge and knower as one. The “knowledge society” that results is distinguished by the extraordinary degree to which the university is linked to society. But it is also distinguished by the degree to which society is organized around the university’s abstracted and universalized understandings of the world and its degree-certified graduates. (shrink)

This work addresses a broad range of questions which belong to four fields: computation theory, general philosophy of science, philosophy of cognitive science, and philosophy of mind. Dynamical system theory provides the framework for a unified treatment of these questions. ;The main goal of this dissertation is to propose a new view of the aims and methods of cognitive science--the dynamical approach . According to this view, the object of cognitive science is a particular set of dynamical systems, which I (...) call "cognitive systems". The goal of a cognitive study is to specify a dynamical model of a cognitive system, and then use this model to produce a detailed account of the specific cognitive abilities of that system. The dynamical approach does not limit a-priori the form of the dynamical models which cognitive science may consider. In particular, this approach is compatible with both computational and connectionist modeling, for both computational systems and connectionist networks are special types of dynamical systems. ;To substantiate these methodological claims about cognitive science, I deal first with two questions in two different fields: What is a computational system? What is a dynamical explanation of a deterministic process? ;Intuitively, a computational system is a deterministic system which evolves in discrete time steps, and which can be described in an effective way. In chapter 1, I give a formal definition of this concept which employs the notions of isomorphism between dynamical systems, and of Turing computable function. In chapter 2, I propose a more comprehensive analysis which is based on a natural generalization of the concept of Turing machine. ;The goal of chapter 3 is to develop a theory of the dynamical explanation of a deterministic process. By a "dynamical explanation" I mean the specification of a dynamical model of the system or process which we want to explain. I start from the analysis of a specific type of explanandum--dynamical phenomena--and I then use this analysis to shed light on the general form of a dynamical explanation. Finally, I analyze the structure of those theories which generate explanations of this form, namely dynamical theories. (shrink)

Properties of mental representation are related to findings in cognitive science and ideas in shamanism. A selective review of research in cognitive science suggests visual images and spatial memory preserve important functional information regarding physical principles and the behavior of objects in the natural world, and notions of second‐order isomorphism and the perceptual cycle developed to account for such findings are related to shamanic experience. Possible roles of implicit processes in shamanic cognition, and the idea that shamanic experience (...) may involve normally unconscious information becoming temporarily available to consciousness, are considered. The existence of a cognitive module dedicated to processing information relevant to social knowledge and social interaction is consistent with cognitive science and with shamanism, and may help account for the extension of intentionality and meaning that characterize shamanic practice. Overall, findings from cognitive science and ideas from shamanism exhibit a number of correspondences and similarities regarding basic properties of cognition, and this suggests that shamanic and nonshamanic cognition may not be fundamentally different. (shrink)

Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If T satisfies $(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory (...) in L κ +, ω of cardinality κ at most, and φ (x, y) ∈ L λ +, ω }. This concept is interesting because of THEOREM 1. Let $T \subseteq L_{\kappa^+,\omega}$ of cardinality ≤ κ, and φ (x̄, ȳ) ∈ L λ +, ω . If $(\forall\mu then $(\forall_\chi > \kappa) I(\chi, T) = 2^\chi$ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ). Many years ago the second author proved that $\mu^\ast (\lambda, \kappa) \leq \beth_{(2^\lambda)^+}$ . Here we continue that work by proving. THEOREM 2. $\mu^\ast (\lambda, \aleph_0) = \beth_{\lambda^+}$ . THEOREM 3. For every κ ≤ λ we have $\mu^\ast (\lambda, \kappa) \leq \beth)_{(\lambda^\kappa)}^+$ . For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem. THEOREM 4. For every $\kappa \leq \lambda, T \subseteq L_{\kappa^+,\omega}$ , and any set of formulas $\Lambda \subseteq L_{\lambda^+,\omega}$ such that $\Lambda \subseteq L_{\kappa^+,\omega}$ , if T is (Λ,μ)-unstable for μ satisfying μ μ * (λ, κ) = μ then T is Λ-unstable (i.e. for every χ ≥ λ, T is (Λ, χ)-unstable). Moreover, T is L κ +, ω -unstable. In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers. (shrink)

It is shown that by realizing the isomorphism features of the frequency and geometric interpretations of probability, Reichenbach comes very close to the idea of identifying mathematical probability theory with measure theory in his 1949 work on foundations of probability. Some general features of Reichenbach’s axiomatization of probability theory are pointed out as likely obstacles that prevented him making this conceptual move. The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of (...) Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. The functioning of the Maxim of Probabilism is illustrated by the example of conditioning via conditional expectations. (shrink)

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary “*” operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of (...) Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism. (shrink)

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. (...) From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough. (shrink)

Let $\to $ be a continuous $\protect \operatorname {\mathrm {[0,1]}}$ -valued function defined on the unit square $\protect \operatorname {\mathrm {[0,1]}}^2$, having the following properties: $x\to = y\to $ and $x\to y=1 $ iff $x\leq y$. Let $\neg x=x\to 0$. Then the algebra $W=$ satisfies the time-honored Łukasiewicz axioms of his infinite-valued calculus. Let $x\to _{\text {\tiny \L }}y=\min $ and $\neg _{\text {\tiny \L }}x=x\to _{\text {\tiny \L }} 0 =1-x.$ Then there is precisely one isomorphism $\phi $ (...) of W onto the standard Wajsberg algebra $W_{\text {\tiny \L }}= $. Thus $x\to y= \phi ^{-1}+\phi ))$. (shrink)

According to the Thomistic tradition, the Principle of Totality (TPoT) articulates a secondary principle of natural law which guides the exercise of human ownership or dominium over creation. In its general signification, TPoT is a principle of distributive justice determining the right ordering of wholes to their parts. In the medical field it is traditionally understood as entailing an absolute prohibition of bodily mutilation as irrational and immoral, and an imperfect obligation to use the parts of one’s body for the (...) perfection of the bodily whole. TPoT is thus a key element of the system of principles within which an individual exercises her right to life: it helps specify the nature, scope and limits of those actions by which an agent permissibly acts in order to preserve her life. While the Thomistic tradition and the Catholic Church have drawn clear conclusions from the principle regarding, e.g., direct sterilization and non-therapeutic experimentation on human subjects, less attention has been given to the implications of TPoT for non-therapeutic procedures that may positively impact biological functioning or supra-biological goals–that is, for human “enhancement.” We will explore the degree to which TPoT non-univocally guides our use of both artifacts and bodies. We will argue that a careful analysis of these distinct kinds of totalities suggests that the application of TPoT to artifacts and bodies is strongly isomorphic, which is what tempts advocates of the Principle of Autonomy to invalidly infer the absolute dominium of the individual over her body. The inference is invalid because this isomorphism also includes a principle of intrinsic value whose function is to resist the instrumentalization of both artifacts and bodies in some contexts; we are not even related to artifacts as advocates of absolute autonomy believe we are, let alone to our bodies. Rather, the limits of human dominium are determined by the nature and finalities, inherent or acquired, of the objects in question, and it will be argued that articulating these limits raises important, understudied, and fascinating questions about the permissibility of various kinds of human enhancement. (shrink)