The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the (...) class='Hi'>continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

There are continuum problems in political economy. There are no objective non-debatable solutions to any of them. All answers to them are arbitrary. Responding to these challenges are, ideally, the responsibility of courts, juries, etc.

The Continuum Companion to Ethics offers a definitive guide to a key area of contemporary philosophy. The book covers all the fundamental questions asked by meta-ethics and normative ethical theory - areas that have continued to attract interest historically as well as topics that have emerged more recently as active areas of research. Fourteen specially commissioned essays from an international team of experts reveal where important work continues to be done in the field and, most valuably, the exciting new (...) directions the field is taking. The Companion explores issues pertaining to moral methodology, moral realism, ethical expressivism, constructivism and the error theory, morality and practical reason, moral psychology, morality and religion, consequentialism, Kantian ethics, virtue ethics, feminist ethics, moral particularism, experimental ethics, and biology, evolution, and ethics. Featuring a series of indispensable research tools, including important technical terms in ethics, a historical chronology, an extensive overview of contemporary meta-ethics and normative ethical theory, a detailed list of internet resources for research in ethics, and a thorough list of recommended works for further study, this is the essential reference tool for anyone working in contemporary philosophical ethics. (shrink)

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 is of course the relational, as opposed to the Newtonian absolutist, theory of space and time. The trouble is, as Clarke indicated several times during the correspondence, and as Russell pointed out in his early study of Leibniz: if continua such as space and time are relations, then it must be shown how a relation can behave as we recognize a continuum to do. How, for example, can a relation be divided or measured as we think space and (...) time can? Russell says. (shrink)

Mesoudi et al. overlook an illuminating parallel between cultural and biological evolution, namely, the existence in each realm of a continuum from intelligent, mindful evolution through to oblivious, mindless evolution. In addition, they underplay the independence of cultural fitness from biological fitness. The assumption that successful cultural traits enhance genetic fitness must be sidelined, as must the assumption that such traits will at least be considered worth having. (Published Online November 9 2006).

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)

Some philosophical accounts imply that masturbation is inferior sexual activity. Against this, Soble argues that masturbation is central. Relying on the physical-anatomical indistinguishability of sexual act-types, he derives a Zeno-style paradox about sexual activity: either all sexual activity (even ordinary coitus) is masturbatory or none of it is (not even solitary masturbation). Soble argues for the first horn of the dilemma, thus ensuring that solitary masturbation is a member of the continuum of sexual activities. Going beyond anatomy, Soble also (...) argues that the central psychological features of paired sexual activity are normally exhibited also by solitary masturbation, which further establishes that solitary masturbation is on the continuum of sexual activity. Altogether, Soble's arguments support a "unary" model of sexuality, compelling us to take seriously the genuine sexual nature of solitary masturbation, a result that is important for the philosophy of sex, the scientific study of sexuality, and the politics of human sexuality. (shrink)

Nurses practising in mental health are faced with challenging decisions concerning confidentiality if a patient is deemed a potential risk to self or others, because releasing pertinent information pertaining to the patient may be necessary to circumvent harm. However, decisions to withhold or disclose confidential information that are inappropriately made may lead to adverse outcomes for stakeholders, including nurses and their patients. Nonetheless, there is a dearth of contemporary research literature to advise nurses in these circumstances. Cognitive Continuum Theory (...) presents a single‐system intuitive‐analytical approach to examining and understanding nurse cognition, analogous to the recommended single‐system approach to decision‐making in mental health known as structured clinical judgement. Both approaches incorporate cognitive poles of wholly intuition and analysis and a dynamic continuum characterised by a ‘common sense’ blending of intuitive and analytical cognition, whereby cues presented to a decision‐maker for judgement tasks are weighed and assessed for relevance. Furthermore, Cognitive Continuum Theory promotes the importance of determining pattern recognition and functional relations strategies, which can be used to understand the operationalisation of nurse cognition. (shrink)

The Continuum Companion to Philosophy of Language offers the definitive guide to contemporary philosophy of language. The book covers all the fundamental questions asked by the philosophy of language - areas that have continued to attract interest historically as well as topics that have emerged more recently as active areas of research. Ten specially commissioned essays from an international team of experts reveal where important work continues to be done in the area and, most valuably, the exciting new directions (...) the field is taking. The Companion explores issues pertaining to the nature of language, form semantics, theories of meaning, reference, intensional contexts, context-dependence, pragmatics, the normativity of language, analyticity, a priority and modality. Featuring a series of indispensable research tools, including an A to Z of key terms and concepts, a detailed list of resources and a fully annotated bibliography, this is the essential reference tool for anyone working in the philosophy of language. (shrink)

A one volume reference guide to the latest research and future directions in Epistemology, featuring chapters written by leading scholars in the field.

The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based (...) on diagonal construction are caused by superficial understanding of what a name is. (shrink)

A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined.

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)

In a number of works, we have suggested that whereas the ordered field R of real numbers should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), the ordered field No of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo NBG). In the present chapter, as part of a more general exposition of the absolute arithmetic continuum, we will outline some of the properties of the system of surreal numbers (...) that we believe lend credence to this mathematico-philosophical thesis. We will also provide an overview of No’s rich structure as a simplicity-hierarchical (or s-hierarchical) ordered field that recursively emerges from the interplay between its structure as an ordered field and its structure as a lexicographically ordered full binary tree. Finally, we will draw attention to how properties of the system of surreal numbers considered as an s-hierarchical ordered algebraic structure can be appealed to in conjunction with classical relations between ordered algebraic and geometric systems to help resolve, for the surreal case, one of the purported difficulties that lies at the heart of attempts to bridge the gap between the domains of number and of geometrical magnitude. In particular, we will explain how it is possible that despite the fact that every surreal number, considered as a member of an s-hierarchical ordered field, differs from every other in characteristic individual properties, the absolute (Euclidean) geometrical continuum, which is modeled by the Cartesian space over the ordered field No of surreal numbers, appears as an amorphous pulp of points that display little individuality. (shrink)

We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum $2^{\aleph_{0}}$ ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be fewer (...) in models of the Continuum Hypothesis. We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients. (shrink)

This chapter traces the historical and conceptual development of the idea of the continuum and the allied concept of real number. Particular attention is paid to the idea of infinitesimal, which played a key role in the development of the calculus during the seventeenth and eighteenth centuries, and which has undergone a revival in the later twentieth century.

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)

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)

The Continuum Companion to Existentialism offers the definitive guide to a key area of modern European philosophy. The book covers the fundamental questions asked by existentialism, providing valuable guidance for students and researchers to some of the many important and enduring contributions of existentialist thinkers. Eighteen specially commissioned essays from an international team of experts explore existentialism’s relationship to philosophical method; ontology; politics; psychoanalysis; ethics; religion; literature; emotion; feminism and sexuality; cognitive science; authenticity and the self; its significance in (...) Latin American culture; and its contribution to the development of Poststructuralism. In addition, five short chapters summarise the status of canonical figures Kierkegaard, Nietzsche, Heidegger, Sartre and de Beauvoir, delineating the historical approach to their work, while pointing to new directions such research is now taking. Featuring a series of indispensable research tools (A to Z of terms, concepts and thinkers; timeline of existentialism; list of resources and an annotated guide to further reading), this Companion is an essential tool to help the new reader navigate through the heart of Existentialism and modern European philosophy. (shrink)

The Continuum Companion to Continental Philosophy offers the definitive guide to contemporary continental thought.

In this paper we consider a natural generalization of the Principle of Instantial Relevance and give a complete characterization of the probabilistic belief functions satisfying this principle as a family of discrete probability functions parameterized by a single real δ ∊ [0, 1).

We prove some consistency results about and δ, which are natural generalisations of the cardinal invariants of the continuum and . We also define invariants cl and δcl, and prove that almost always = cl and = cl.

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non professionals.

The Continuum Companion to Hume is a comprehensive and accessible guide to Hume's life and work includes 21 specially commissioned essays, written by a team of leading experts, covering every aspect of Hume's thought. The Companion presents details of Hume's life, historical and philosophical context, a comprehensive overview of all the key themes and topics apparent in his work, including his accounts of causal reasoning, scepticism, the soul and the self, action, reason, free will, miracles, natural religion, politics, human (...) nature, women, economics and history, and an account of his reception and enduring influence. This is an essential reference tool for anyone working in the fields of Hume Studies and Eighteenth-Century Philosophy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:240 Feminist Studies 45, no. 1. © 2019 by Mary Anna Evans Mary Anna Evans Personal Continuum The scent of gasoline is neither attractive nor repulsive. It falls somewhere on the continuum between. It is medicinal, but without the acrid bitterness of medicine. It draws children like a drug, but only when their parents aren’t hovering to warn of danger. Adults know about fire and poisoning and (...) long-range carcinogenic doom. Children only know that the odor around gas pumps smells naughty and fun, until their lips open and the flavor of petroleum turns their stomachs. Why do some things smell so much better than they taste? Gasoline, unlike honey, draws humans but not flies. Sarah was not thinking of fire, poisoning, or doom when she fitted the long metal nozzle into the mouth of her car’s fuel tank. She was trying not to think at all, though the customary thoughts slipped in. None of them were about gasoline. The pump’s air pollution control system was working well but not perfectly, so it took a moment for the odor to reach her nose. Her body roused and snarled at her. The baby. What are you thinking? If you breathe those fumes, what will they do to the baby? A moment passed while her body checked itself, remembering what it already knew. There was no baby. Where is the baby? Mary Anna Evans 241 The frantic voice didn’t echo between her ears. It had nothing to do with her brain at all. It was more than thought. It came from a place deeper than thought, and it wasn’t hers. Her body said things she didn’t want to say, and it needed things she didn’t need. Her body wanted things that were not good for her. Again, Where is the baby? And then, the darker question. What did you do? “Nothing,” she said aloud, since there was no one to hear. I didn’t do anything but bleed. I’m still bleeding. She resisted the instinct to clap her hands over her ears to shut out the voice that couldn’t be shut out. Her defiant hold on the nozzle kept her upright. She listened to the rhythmic throb of the pump. Refusing to turn her face from the gasoline’s stench, she opened her mouth and let the fumes seep in. * * * “Hello, sweetheart,” Frederick said as he came in the door. His peck on the cheek didn’t linger, so Sarah knew there would be no more expressions of affection that day. He had read in a self-help book that it was important for couples to use terms of endearment as a form of bonding. Physical displays of affection were important, too, or so said the self-affirmed relationship expert who wrote the book. In public, displays of affection affirmed the pair bond. In private, even a quick smooch forced men and women to secrete an attachment chemical called oxy-whatever. Oh, hell. She was a chemist. She knew what oxytocin was. She was just mystified by the notion of reading a book to find out how to get some. He turned away and went into the kitchen to fetch the marinating chicken. He always grilled when it was his night to cook. She had come to wonder whether modern men were drawn to cook with fire by an ageold hunter instinct, or whether they were attracted to an activity that put a well-insulated exterior wall between them and the women who remained inside to toss the salads. All across America, every warm summer night, lonely women toss salads and hope for romance. His head poked in the back door. “I bought slivered almonds and croutons.” 242 Mary Anna Evans Her “Thanks for remembering, sweetheart,” bounced off the closing door. Shit. She could have spared herself that soulless term of endearment. Leaving the lettuce in the refrigerator, she thrust her soapy hands under warm running water, because even thinking the word “shit” made the idea of handling raw food appalling. Her body, which had forgotten again that she’d lost the baby, approved of her caution. She guarded her thoughts against disturbing its... (shrink)

Business and Society Review, Volume 127, Issue 1, Page 3-22, Spring 2022.

Business and Society Review, Volume 127, Issue 1, Page 3-22, Spring 2022.

There inevitably comes a time when even the best informed minds are tempted to yield to the lure of binarism: when we are no longer concerned with the details of a system, but rather with our vision of the relationship between man and the world. Without going any further it can already be said that the problem of existence is presented to us in terms of a man/world duality, as though we were unable to exceed our subjective vision of things (...) so as to assess their intrinsic reality.When linguists discovered that they could isolate discrete units called phonemes, they were tempted to oppose the discontinuity of these phonemes – reflecting the underlying discontinuity of the signifiers – to a continuum of prelinguistic experience, a continuum that is not arranged as discrete elements except with reference to any significant units of the language selected for the purpose of communicating this experience to others. At the outset we have nothing but a nebulous mass where only the application of a linguistic grid allows us to discern identifiable units. Even if, in the final analysis, it will prove necessary to doubt the universal validity of this vision of things, we must recognize its usefulness in concretizing a concept of linguistic facts – a concept that has for some time played a beneficent role in research. At least for some of us, this vision should reflect the manner in which the world is gradually organized in our minds each time a perceptible reality is isolated from its context at the instant it receives a name. Allow me to introduce an illustration from personal experience. It was the last day of July in 1914. In our little village in the Savoy the international tension – which would eventually end in general mobilization – was very perceptible; our classes were idle, and teachers and pupils had settled down in a field near the school. I was six. Sitting on the grass, my attention was attracted by a plant with large flat leaves. At that moment someone said the word plantain “plantain.” The plant had not been heretofore exactly unknown to me; it didn't strike me, at that moment, as a discovery; but it acquired a distinct existence only at that instant in the green carpet of the field when a signifier had definitively isolated it. It was most assuredly the gravity of the hour that engraved this incident in my memory. (shrink)

Life -- Influences -- Early critics -- Glossary -- Short synopses -- Spinoza scholarship.

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)

Unitary field theories and “SUPER-GUT” theories work with an universal continuum, the structured spacetime of R. Descartes, B. Spinoza, B. Riemann, and A. Einstein, or a (Machian (1–3) ) structured vacuum according the quantum theory of unitary fields (Dirac, (4,5) and Heisenberg (6–8) ). The atomistic aspect of the substantial world is represented by the fundamental constants which are invariant against “all transformations” and which “depend on nothings” (Planck (9–11) ). A satisfactory unitary theory has to involve these constants (...) like the mathematical numbers. Today, Planck's conception of the three elementary constants ħ, c, and G may be the key to general relativistic quantum field theory like unitary theory. However, the elementary constants are a question of measurement-theory, also.According to Popper's theory (12–16) of induction, such unitary theories are “universal explaining theories.” The fundamental constants involve the complementarity between the universal statements in unitary theory and the “basic statements” in the language of classical observables. (shrink)

What does world peace mean? Peace is more than the absence and prevention of war, whether international or civil, yet most of our ways of conceptualizing and measuring peace amount to just that definition. In this essay, as part of the roundtable “World Peace,” I argue that any vision of world peace must grapple not only with war but with the continuums of violence and peace emphasized by feminists: running from the home and community to the public spaces of international (...) relations. Breaking free of the constraints of the last century's intellectual boundaries, I suggest that war and peace are not a dichotomy but rather are intimately related. Yet the dearth of feminist perspectives in global debates prevents us from seeing how violence and harm are exacerbated in households and through the global economy under conditions of both “war” and “peace.” To understand the possibilities for world peace, we must understand these varieties of violence and harm that threaten peace. And tosustainpeace we must address the harmful gendered identities, ideologies, and social dynamics that support violence in every society. A narrow understanding of peace as merely the absence of organized violence does not engender the kind of nuanced and rich understanding of human history and human relations needed to bring an end to the structural and physical violence that remains pervasive worldwide. (shrink)