Biogenic approaches investigate cognition from the standpoint of evolutionary function, asking what cognition does for a living system and then looking for common principles and exhibitions of cognitive strategies in a vast array of living systems—non-neural to neural. One worry which arises for the biogenic approach is that it is overly permissive in terms of what it construes as cognition. In this paper I critically engage with a recent instance of this way of criticising biogenic approaches in order to clarify (...) their theoretical commitments and prospects. In his critique of the biogenic approach, Fred Adams uses the presence of intentional states with conceptual content as a criterion to demarcate cognition-driven behaviour from mere sensory response. In this paper I agree with Adams that intentionality is the mark of the cognitive, but simultaneously reject his overly restrictive conception of intentionality. I argue that understanding intentionality simpliciter as the mark of the mental is compatible with endorsing the biogenic approach. I argue that because cognitive science is not exclusively interested in behaviour driven by intentional states with the kind of content Adams demands, the biogenic approach’s status as an approach to cognition is not called into question. I then go on to propose a novel view of intentionality whereby it is seen to exist along a continuum which increases in the degree of representational complexity: how far into the future representational content can be directed and drive anticipatory behaviour. Understanding intentionality as existing along a continuum allows biogenic approaches and anthropogenic approaches to investigate the same overarching capacity of cognition as expressed in its different forms positioned along the continuum of intentionality. Even if all organisms engage in some behaviour that is driven by weak intentional dynamics, this does not suggest that every behaviour of all organisms is so driven. As such, the worry that the biogenic approach is overly permissive can be avoided. (shrink)

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly (...) pick out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

We often speak of 'Eastern' and 'Western' philosophy, yet it is not always easy to distinguish the key factors that justify this distinction. This essay explores the two very different conceptions of the continuum that underlie these traditions of thought and knowledge. The views of Hermann Weyl are given and it is proposed they are correct. Attention is drawn to the mutually-exclusive visions of the continuum that separate the philosophies of East and West, for they offer us a (...) way of pinning down a definition of these vague geographical terms so as to give them, in at least one respect, a clear philosophical and scientific meaning. (shrink)

It is often argued that the indispensability of continuum models comes from their empirical adequacy despite their decoupling from the microscopic details of the modelled physical system. There is thus a commonly held misconception that temperature varying across a region of space or time can always be accurately represented as a continuous function. We discuss three inter-related cases of temperature modelling — in phase transitions, thermal boundary resistance and slip flows — and show that the continuum view is (...) fallacious on the ground that the microscopic details of a physical system are not necessarily decoupled from continuum models. We show how temperature discontinuities are present in both data (experiments and simulations) and phenomena (theory and models) and how discontinuum models of temperature variation may have greater empirical adequacy and explanatory power. The conclusions of our paper are: a) continuum idealisations are not indispensable to modelling physical phenomena and both continuous and discontinuous representations of phenomena work depending on the context; b) temperature is not necessarily a continuously defined function in our best scientific representations of the world; and c) that its continuity, where applicable, is a contingent matter. We also raise a question as to whether discontinuous representations should be considered truly de-idealised descriptions of physical phenomena. (shrink)

Here we will go beyond the variety of violence to show its unity, common points and continuities. For although there are multiple forms of violence, they are interrelated: they define a continuum from trivial to extreme violence. Violence against oneself, things, living things such as plants and animals, other nations, the other, one’s fellow human beings, therefore the violence of society against its members, which returns to self-violence. Another continuum is its spiral development, with violence generating violence and (...) pushing it to grow. Violence can also be learned, we progress ever further in violence: in gangs, in armies, in society... Everyone is capable of violence, sometimes to a good advantage as in self-defence. Here it resides in necessity, that of survival, but in general it is impunity that allows and encourages it. In closed, totalitarian, universes: family, work, hospital, army, state... The proximity and distance between the perpetrator and the victim defines yet another continuity. We will discuss the various elements that contribute to its development. And paradoxically, to see that violence also serves to avoid violence. In this way, evacuating it, refusing it, is in fact feeding it, which makes today’s violence, which is an evolution of previous violence and which prepares and defines tomorrow’s violence. (shrink)

Although Peirce frequently insisted that continuity was a core component of his philosophical thought, his conception of it evolved considerably during his lifetime, culminating in a theory grounded primarily in topical geometry. Two manuscripts, one of which has never before been published, reveal that his formulation of this approach was both earlier and more thorough than most scholars seem to have realized. Combining these and other relevant texts with the better-known passages highlights a key ontological distinction: a collection is bottom-up, (...) such that the parts are real and the whole is an ens rationis, while a continuum is top-down, such that the whole is real and the parts are entia rationis. Accordingly, five properties are jointly necessary and sufficient for Peirce’s topical continuum: rationality, divisibility, homogeneity, contiguity, and inexhaustibility. (shrink)

This paper explores some philosophical and mathematical relationships between the continuum hypothesis, the axiom of choice, logical independence within mathematics, the existence of a higher being, the size of the universe, and some of the objects that compose it. Different comparisons are made between the nature of various classes of infinities and their consistency with our universe, referencing the viewpoints of Cantor and Peirce.

Research in formal theoretical semantics has recently expanded its scope to include gestural communication, focusing in particular on gestures that contribute to the content of an accompanying utterance, e.g., size gestures (LARGE, WIDE), pointing gestures, and gestures that depict objects (TELESCOPE) or actions (SLAP). At the same time, fruitful inquiries at the intersection of primatology and linguistics have given rise to the hypothesis that human and non-human great apes share a common set of directive (=imperative) gestures. Directive gestures such as (...) STOP or COME-CLOSER, pose non-trivial issues for a semantic analysis: we inherit the challenges that pertain to the analysis of imperative utterances (Come closer!), while adding a further challenge that stems from the underspecified mapping between a directive body movement and its potential counterparts in human language. We begin by outlining the problem and surveying the nascent state-of-the-art with regards to a formal semantics of directive gestures. Particular attention is given to the multifunctionality of directive gestures, which typically have different effects in different contexts; for example, a non-human ape gesture may communicate "Stop that" in some contexts and "Move away" in others, with similar patterns found in humans. We show that this multifunctionality can be derived from a single, rich abstract lexical entry, "Not…!", a candidate for a universal building block of meaning, shared by human and non-human great apes. "Not…!", which incorporates negation/rejection, temporal reference and presuppositionality is more explanatory in comparison to a minimal alternative. This chapter lays out the hypothesis space, addressing the foundational question of how to explain attested gestural overlap between human and non-human great apes. We conclude the chapter by reviewing potential expansions of the analysis to the pragmatic gestures of humans, which are thus placed on the primate gestural meaning continuum and connected to the directive gestures we share with our non-human relatives in the great ape family. (shrink)

According to Grice's “Modified Occam's Razor”, in case of uncertainty between the implicature account and the polysemy account of word uses it is parsimonious to opt for the former. However, it is widely agreed that uses can be partially conventionalised by repetition. This fact, I argue, raises a serious problem for MOR as a methodological principle, but also for the substantial notion of implicature in lexical pragmatics. In order to overcome these problems, I propose to reinterpret implicatures in terms of (...) implicature-like effects delivered by non-inferential processes. (shrink)

Since the 1950`s in Britain, and perhaps in the rest of the world, the term pluralism is almost invariably associated with the name of Isaiah Berlin and his formulation of ‘value pluralism’. The core idea is that values (but also, on some interpretations, ends, duties and obligations) are irreducibly plural and heterogeneous, and nevertheless objective.

In the history of philosophy, the concept of self has been perennially elusive. The philosophical quest to understand the self is rife with phenomenological and metaphysical analyses, often overlooking other kinds of selves present in the biological realm. To systematically explore this question of non-human selves, I categorize the literature on philosophical and biological notions of self into the biogenic, the zoogenic, and the anthropogenic approaches to self. This article attempts to chart the genesis, the continuum, and the lowest (...) bound of the self. Further, I enumerate challenges in developing a biogenic approach to self or taking the concept of self all the way down in the phylogenetic tree. (shrink)

Thaler and Sunstein advocate 'libertarian paternalism'. A libertarian paternalist changes the conditions under which people act so that their cognitive biases lead them to choose what is best for themselves. Although libertarian paternalism manipulates people, Thaler and Sunstein say that it respects their autonomy by preserving the possibility of choice. Conly argues that libertarian paternalism does not go far enough, since there is no compelling reason why we should allow people the opportunity to choose to bring disaster upon themselves if (...) sometimes they will make the wrong decision. She defends 'coercive paternalism'. The present paper argues that errors in reasoning are not due only to cognitive biases. People also make errors because they have an insufficient level of general intelligence. Intelligence is distributed on a continuum. Those who fall on higher levels of the continuum have greater abilities, in certain contexts, to reason about both their own and others' interests. Coercive paternalism may sometimes be appropriate to prevent less intelligent people from engaging in self-destructive behavior due to errors of reasoning. (shrink)

In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding as (...) inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis. (shrink)

This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open (...) versus closed intervals is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)

In this paper, I propose a logical-cognitive approach to argumentation and advocate an idea that argumentation presupposes that intelligent agents engaged in it are cognitively diverse. My approach to argumentation allows drawing distinctions between justification, conviction and persuasion as its different kinds. In justification agents seek to verify weak or strong coherency of an agent’s position in a dialogue. In conviction they argue to modify their partner’s position by means of demonstrating weak or strong cogency of their positions before a (...) ‘rational judge’. My approach to argumentation employs a ‘light’ version of Dung’s abstract argumentative frameworks. It is based on Stich’s idea of agents’ cognitive diversity the epistemic aspect of which is argued to be close to Pavilionis’s conception of meaning continuum. To illustrate my contributions I use an example based on the Kitchen Debate (1959) between Khrushchev and Nixon. (shrink)

In this paper I reconstruct and discuss Antonio Rubio (1546-1615)’s theory of the composition of the continuum, as set out in his Tractatus de compositione continui, a part of his influential commentary on Aristotle’s Physics, published in 1605 but rewritten in 1606. Here I attempt especially to show that Rubio’s is a significant case of Scholastic overlapping between Aristotle’s theory of infinitely divisible parts and indivisibilism or ‘Zenonism’, i.e. the theory that allows for indivisibles, extensionless points, lines, and surfaces, (...) which are supposed to take part in the composition of the continuum. Even if such a syncretic tendency was, in many different ways, already developing in the medieval period and then at the end of the sixteenth century, Rubio’s position is indeed peculiar. He maintains that indivisibles are real and actual, infinite in act, really distinct from each other, and that, although they indwell in substance, indivisibles do not contribute directly to the constitution of the continuum. In this reconstruction I emphasize notably Rubio’s usage of mereological notions like those of part, whole, completeness and incompleteness. (shrink)

We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the language with just a single predicate.

Governments standardly deploy a distinction between goods and services in assessing economic health and tracking national income statistics, of which medical goods and services carry significant importance. In what follows we draw on Basic Formal Ontology (BFO) to introduce a third kind of entity called patterns, which help capture the various ways in which goods and services are intertwined and help also to show how many services generate a new kind of non-goods-related products. Patterns are an overlooked yet essential features (...) of many economic sectors including medicine. Studying patterns offers new insights into various components of economic analysis, including outcomes-oriented evaluations of medical services and the value of human capital in the medical sphere. (shrink)

La conciencia falogocéntrica se funda en el a priori incondicional de carácter ético-metafísico que sostiene la criminalización universal del ser como efecto de una culpa o caída original. Nacer supone para esta conciencia un pecado y la realidad material representa para ella un lugar de exilio, extrañeza y alienación. Sin embargo, cuando uno retrocede sobre la protohistoria de la conciencia humana, sorprende la determinación de integridad y sacralidad que el pensamiento primitivo le atribuye a lo real. Para la conciencia primitiva, (...) la realidad nace en el parto de un seno infinito que la alimenta, cobija y bendice sus propios frutos. En lo que sigue, intentaremos remontarnos a la protohistoria de esta conciencia a fin de reconsiderar las condiciones de posibilidad de la inocencia metafísica y ética, o bien, el modo de ser de la inocencia original. (shrink)

In this paper a new model of mind is proposed, to do so, at first it was assumed that our physical world a new structure and the mind defined in this context. In this model, the planets are massive curvature of time-space continuum that has made a trapping physical reality that we are located within. Then the mind is defined as an hourglass structure with half bulb within the physical reality and half out of it. This model with attention (...) to figures perhaps might be able to explain some mental states and disorders. (shrink)

When so much is being written on conscious experience, it is past time to face the question whether experience happens that is not conscious of itself. The recognition that we and most other living things experience non-consciously has recently been firmly supported by experimental science, clinical studies, and theoretic investigations; the related if not identical philosophic notion of experience without a subject has a rich pedigree. Leaving aside the question of how experience could become conscious of itself, I aim here (...) to demonstrate that the terms experience and consciousness are not interchangeable. Experience is a notoriously difficult concept to pin down, but I see non-conscious experience as based mainly in momentary sensations, relational between bodies or systems, and probably common throughout the natural world. If this continuum of experience — from non-conscious, to conscious, to self-transcending awareness — can be understood and accepted, radical constructivism (the “outside” world as a construct of experience) will gain a firmer foundation, panexperientialism (a living universe) may gain credibility, and psi will find its medium. (shrink)

Feeding is a fundamental physiological need that falls within the realm of what we consider natural. However, throughout history, human beings have developed and employed several techniques to produce and manipulate food. Therefore, if the classical metaphysical distinction between natural and artefactual entities must be accepted, our food seems to predominantly fall into the latter category. Yet, discussions of artefacts do not typically involve organisms but rather material objects that cannot be found “in nature”. Thus, food may appear as a (...) sui generis artefact. This paper, after an analysis of the relationships holding between food and human beings, suggests that the former, rather than being an unusual artifact is the quintessential artifact, one that seems to reflect better than any other the complex connections between Homo sapiens and its environment. (shrink)

What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock (...) of higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)

This thesis is about experiential content: what it is; what kind of account can be given of it. I am concerned with identifying and attacking one main view - I call it the inferentialist proposal. This account is central to the philosophy of mind, epistemology and philosophy of science and perception. I claim, however, that it needs to be recast into something far more subtle and enriched, and I attempt to provide a better alternative in these pages. The inferentialist proposal (...) holds that experiential content is necessarily under¬pinned by sophisticated cognitive influences. My alternative, the continuum theory, holds that these influences are relevant to experience only at certain levels of organisation and that at other levels there are contents which such features do not capture at all. Central to my account is that there are degrees to which cognitive influences affect experiential content; indeed, for the most part, experience is an amalgam of both inferential and non-inferential features. I claim that the inferentialist proposal is fundamentally flawed and deserves replacement, and I argue that my alternative fills the hollow that remains. The thesis is divided into four sections. In Part I, Chapter 1, I introduce two traditionally rival views of experiential content. In Chapter 2, I develop my continuum alternative. Chapter 3 assesses the relationship between experience and language, while Chapter 4 explores the relationship between beliefs and experience. The overall argument is that it has been a mistake to understand experience simply in inferential or non-inferential terms. In Part II, I examine the structure of mental content. Chapter 5 is concerned with the kinds of experiences which escape the inferentialist analysis. Chapter 6 considers Kant’s metaphysic of experience counterpointed to Lorenz’s reading of his work in the light of evolutionary biology. Chapter 7 treats animal experience in relation to the continuum view I am developing, while Chapter 8 reviews Fodor’s contribution to perceptual psychology. It is argued that the view of experiential content being developed is both consistent with empirical data on informationally local perceptual sub-systems, but also accords well with evolutionary theory and a naturalist interpretation of Kant’s taxonomy. Part III deals with inferentialism in the philosophy of science. In Chapter 9, I assess the theory dependence of observation thesis as it is advanced by Paul Feyerabend. I bring out of his account a subtle confusion concerning the importance of inference in the context of scientific inquiry. Part IV deals with the issue of experience in the philosophy of mind. In Chapter 10, I look at Wilfred Sellars’s attack on sense data theories. Chapter 11 confronts Paul Churchland’s treatment of ‘folk psychology’ while Chapter 12 isolates the issue of experiential qualia and the position of property dualism. I offer a critical review of Thomas Nagel’s work in this chapter and claim that his position can be read in a way which is consistent with the continuum account I am developing. I conclude the thesis in the usual fashion with a summary of the central claims. (shrink)

This dynamical interpretation of the continuum is based on a threefold perspective. First, detailed differentiation of all standard realms of Leibnizian Weltanschauung – (R real), (P phenomenal), (I ideal). Second, analysis of the scope of the Law of Continuity famously formulated by Leibniz and mapping it onto this (RPI) structure. Third, finding the precise place of dynamics and force in this (RPI) continuum.

We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...) sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (shrink)

A non-relativistic quantum mechanical theory is proposed that describes the universe as a continuum of worlds whose mutual interference gives rise to quantum phenomena. A logical framework is introduced to properly deal with propositions about objects in a multiplicity of worlds. In this logical framework, the continuum of worlds is treated in analogy to the continuum of time points; both “time” and “world” are considered as mutually independent modes of existence. The theory combines elements of Bohmian mechanics (...) and of Everett’s many-worlds interpretation; it has a clear ontology and a set of precisely defined postulates from where the predictions of standard quantum mechanics can be derived. Probability as given by the Born rule emerges as a consequence of insufficient knowledge of observers about which world it is that they live in. The theory describes a continuum of worlds rather than a single world or a discrete set of worlds, so it is similar in spirit to many-worlds interpretations based on Everett’s approach, without being actually reducible to these. In particular, there is no splitting of worlds, which is a typical feature of Everett-type theories. Altogether, the theory explains (1) the subjective occurrence of probabilities, (2) their quantitative value as given by the Born rule, and (3) the apparently random “collapse of the wavefunction” caused by the measurement, while still being an objectively deterministic theory. (shrink)

ABSTRACT In previous work I explored how Integral Theory can be applied as a metatheoretical and transdisciplinary framework, in an attempt to arrive at an integrally informed metatheory of addiction. There was an overemphasis on Integral Methodological Pluralism in that thread of research, without clarifying the ontological pluralism of addiction as a multiple object enacted by various methodologies. To arrive at a comprehensive integral metatheory and integral ontology of addiction, I believe it is necessary to include the conception of Integral (...) Pluralism and Integral Enactment Theory as posited by Sean Esbjörn-Hargens. Integral Enactment Theory highlights the phenomenon of addiction as a multiple and dynamic object arising along a continuum of ontological complexity; it adeptly points out how etiological models “co-arise” in relation to methodology (methodological pluralism) to enact a particular reality of addiction (ontological pluralism) while being mediated by the worldview of the subject (epistemological pluralism) applying the method. This article briefly explores the significance of including Integral Pluralism and Integral Enactment Theory in the quest of an integral metatheory and integral ontology of addiction. (shrink)

Most studies on Nietzsche seldom associate him with the dialectic method. We readily think of Socrates, Hegel, and Marx when we hear of dialectic, but very rarely, if at all, of Nietzsche. To date, very few studies on Nietzsche have claimed that one of the German philosopher's underpinning philosophical methodologies in his literary oeuvre is the dialectic. This paper thus intends to show that Nietzsche has been employing the dialectic throughout his writings, especially in his treatment of the "free spirits"-a (...) recurring subject in his different compositions. To do this, I will first revisit the meaning of dialectic in Socrates, Hegel, and Marx and subsequently argue that in Nietzsche, a type of dialectic akin to Hegel's permeates his writings. In the next parts of this paper, I will discuss in-depth how for Nietzsche, the incarnation of the free spirits involves not just one, but a continuum of triple dialectic. I will argue that to become a free spirit, one must constantly apply the dialectics of (1) criticality and openness, (2) unlearning and relearning, and (3) overcoming and becoming. These dialectics, in my view, are the necessary conditions for any individual to be truly free. In this paper, I will importantly highlight that these dialectics run through Nietzsche's works starting from the early up to the final period of his literary productivity. (shrink)

Noise and silence as social phenomena with certain depths in terms of their cultural value, when viewed through the lens of the mindsponge theory, become very interesting and often contain many underlying educational implications.

Albert Lautman. Mathematics, Ideas and the Physical Real. Simon B. Duffy, trans. London and New York: Continuum, 2011. 978-1-4411-2344-2 (pbk); 978-1-44114656-4 (hbk); 978-1-44114433-1 (pdf e-bk); 978-1-44114654-0 (epub e-bk). Pp. xlii + 310.

This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.

It is often supposed that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a (...) mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH. (shrink)

In this book, for the first time, authors try to introduce the concept of linguistic variables as a continuum of linguistic terms/elements/words in par or similar to a real continuum. For instance, we have the linguistic variable, say the heights of people, then we place the heights in the linguistic continuum [shortest, tallest] unlike the real continuum (–∞, ∞) where both –∞ or +∞ is only a non-included symbols of the real continuum, but in case (...) of the linguistic continuum we generally include the ends or to be more mathematical say it is a closed interval, where shortest denotes the shortest height of a person, maybe the born infant who is very short from usual and the tallest will denote the tallest one usually very tall; however this linguistic continuum [shortest, tallest] in the real continuum will be the closed interval say [1 foot, 8 feet] or [1, 8] the measurement in terms of feet. So, the real interval is a subinterval with which we have associated the real continuum in terms of qualifying unit feet and inches in this case. (shrink)

Although he is best known as the founder of pragmatism, the name that Charles Sanders Peirce prefers to use for his comprehensive system of thought is "synechism" because the principle of continuity is its central thesis. This paper arranges and summarizes numerous quotations and citations from his voluminous writings to formalize and explicate his distinctive mathematical conceptions of hyperbolic and topical continuity, both of which are derived from the direct observation of time as their paradigmatic manifestation, and then apply them (...) in normative science and metaphysics, especially logic as semeiotic and cosmology. The resulting conclusion is that the intelligibility of the universe is plausibly explained by conceiving it as one immense sign, a vast inferential process: a semiosic continuum whose connected constituent signs are indefinite until deliberately marked off, with God the Creator as its overall dynamical object in the infinite past and God completely revealed as its overall final interpretant in the infinite future. (shrink)

How we ought to diagnose, categorise and respond to spectrum disabilities such as autism and Attention Deficit/Hyperactivity Disorder (ADHD) is a topic of lively debate. The heterogeneity associated with ADHD and autism is described as falling on various continua of behavioural, neural, and genetic difference. These continua are varyingly described either as extending into the general population, or as being continua within a given disorder demarcation. Moreover, the interrelationships of these continua are likewise often vague and subject to diverse interpretations. (...) -/- In this paper, I explore geneticists' and self-advocates’ perspectives concerning autism and ADHD as continua. These diagnoses are overwhelmingly analysed as falling on a continuum or continua of underlying traits, which supports the notion of “the neurodiversity spectrum”, i.e., a broader swath of human neural and behavioural diversity on which some concentrations of different functioning are diagnosed. I offer a taxonomy of conceptions of the genetic, phenotypic, and endophenotypic dimensionality within and beyond these diagnostic categories, and suggest that the spectrum of neurodiversity is characteristically endophenotypic. (shrink)

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which (...) includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. (shrink)