Given a cardinal κ that is λ-supercompact for some regular cardinal λ⩾κ and assuming GCH, we show that one can force the continuum function to agree with any function F:[κ,λ]∩REG→CARD satisfying ∀α,β∈domα F. Our argument extends Woodinʼs technique of surgically modifying a generic filter to a new case: Woodinʼs key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of (...) j on the ghost coordinates. This work answers a question of Friedman and Honzik [5]. We also discuss several related open questions. (shrink)

We say that a regular cardinal κ,, has the tree property if there are no κ‐Aronszajn trees; we say that κ has the weak tree property if there are no special κ‐Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal,, is consistent with an arbitrary continuum function below which satisfies,. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal,, is consistent (...) with an arbitrary continuum function below which satisfies,. Thus the tree property has no provable effect on the continuum function below except for the trivial requirement that the tree property at implies for every infinite κ. (shrink)

We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V[G]$ that has a symmetric inner model $N$ in which $\mathrm {ZF}+\lnot\mathrm {AC}$ holds, $\kappa$ and $\kappa^{+}$ are both singular, and the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model (...) of $\mathrm {ZF}+\lnot\mathrm {AC}_{\omega}$ in which either $\aleph_{1}$ and $\aleph_{2}$ are both singular and the continuum function at $\aleph_{1}$ can be precisely controlled, or $\aleph_{\omega}$ and $\aleph_{\omega+1}$ are both singular and the continuum function at $\aleph_{\omega}$ can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals $\kappa$ and $\kappa^{+}$ in a model of $\mathrm {ZF}$. Some open questions concerning the continuum function in models of $\mathrm {ZF}$ with consecutive singular cardinals are posed. (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)

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)

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)

Much of the legal literature discusses regulation and regulatory forms with a seemingly implicit assumption that "those to be influenced" are inherently self-interested and thus motivated to comply with legal structures only when there are sufficient external incentives to do so. This view of the person is inconsistent with recent perspectives in the field of psychology. A law and morality perspective, coupled with insights from the field of psychology, asserts that influence, compliance, and motivation are far more complex than this (...) legal literature would suggest. In this Article, we map the varying influence structures, motives, psychological needs, emotional mechanisms, and levels of moral reasoning that various forms of regulation, from hard law to soft law, might appeal to. We provide examples from global banking and one soft law initiative, the Equator Principles, to illustrate reasons psychology would suggest why soft law may be more effective in some circumstances in influencing behavior within the firm than hard law, while recognizing important limits to such influence. (shrink)

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).

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said to contain (...) “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

Our model of time is the classical continuum of real numbers, and our model of other measurable quantities that change over time is that of functions defined on real numbers with real numbers as values. This model is not derived from reality or from our experience of it, but imposed on reality; and the fit is very imperfect. In classical mathematics, the value of a function for any real number as argument is independent of its value for any (...) other argument: the analogue is Hume's doctrine that events are loose and separate. This makes continuity in the change of any quantity a contingent law of physica, rather than a conceptual necessity. The article explores alternatives to this classical model. (shrink)

I applaud Fenton and Wiers' attempt to find a demarcation point between cases of addiction that fall within the range of normal function and those that may count as disease. However, I argue that continua don't offer demarcation points, the mechanisms involved are not demonstrably different, and trying to pin down subjectivity doesn't help.

We give some results concerning various generalized continuum cardinals. The results answer some natural questions which have arisen in preparing a new edition of 5. To make the paper self-contained we define all of the cardinal functions that enter into the theorems here. There are many problems concerning these new functions, and we formulate some of the more important ones.

In this paper I attempt to study the notion of “folding of a semiotic continuum” in a direction of a possible application to the biological processes. More specifically, the process of obtaining protein structures is compared in this paper to the folding of a semiotic continuum. Consequently, peptide chain is presented as a continuous line potential to be formed in order to create functional units. The functional units are protein structures having certain function in the cell or (...) organism. Moreover, protein folding is analyzed in terms of tension between syntax and semantics. (shrink)

The three-dimensional “AIM model” proposed by Hobson et al. is imaginative. However, many kinds of data suggest that the “dimensions” are not orthogonal, but closely correlated. An alternative view is presented in which mental functioning is considered as a continuum, or a group of closely linked continua, running from focused waking activity at one end, to dreaming at the other. The effect of emotional state is increasingly evident towards the dreaming end of the continuum. [Hobson et al.; Nielsen; (...) Solms]. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 9.3 (2002) 237-241 [Access article in PDF] Practical, Functional, and Natural Kinds Nick Haslam Keywords: Classification, essentialism, natural kinds, practical kinds. I am grateful to the two commentators for giving my paper their serious attention, and for writing such stimulating, clarifying, and challenging responses. In a brief response I can only begin to discuss a select few issues, although both commentaries could generate a great (...) deal of profitable exchange.Zachar's commentary elaborates his practical kinds model, discusses several of the graded series of kinds proposed in my taxonomy of psychiatric categories, and disputes some aspects of my treatment of natural kinds. Zachar and I have no quarrel on his first point. I recognize his practical kinds model as a general theory of psychiatric classification rather than simply as a description of one type of category, the "practical kind." It was perhaps bad form to pilfer a term from Zachar's model and give it a more circumscribed role in my own, although by doing so I intended both to acknowledge the importance of the concept and to situate it among some of its alternatives.All the same, I stand by the account of practical kinds developed in my paper, even if it reflects a narrower sense than Zachar's. In my sense, practical kinds are pragmatically drawn distinctions imposed on continuous dimensions, whereas to Zachar all psychiatric categories are at least to some degree practical. He therefore proposes a broad "continuum" of practical kinds. The difficulty that I have with this proposal is that such a continuum implies a fundamental unity among psychiatric categories: categories are all to be understood first and foremost in pragmatic terms and differences in kind between them in internal structure are to be neglected. In my view, although the pragmatic bases of psychiatric classification are vitally important, and perhaps inadequately formulated in my paper, the neglect of internal structure and of possible breaks in the continuum of kinds are real limitations of the practical kinds model. Although Zachar is partially justified in claiming that his view is pluralistic, allowing as it does for variability in degrees of practicality, by failing to acknowledge distinctly different kinds of categories in the psychiatric domain, his view is less pluralistic than I believe it should be.One example of what I take to be this neglect of structural differences is a conflation of two sorts of practicality in Zachar's work. On the one hand, certain categories such as hypertension or obesity are defined on pragmatic grounds, a binary distinction being drawn in the absence of an internal discontinuity, because individuals identified as category members are adjudged to [End Page 237] be at significantly elevated risk. On the other hand, some categories just happen to have high practical utility. Zachar argues that schizotaxia, for example, is practical because it affords a variety of scientific and clinical inferences.It seems to me that these sorts of practicality—or perhaps these sorts of relations to practicality—are importantly different. Hypertension is defined as a category because of its practical utility—because a certain blood pressure threshold marks an inflection in actuarial risk of other illnesses—and might therefore be said to have "front-end" or constitutive practicality. Schizotaxia, which taxometric research shows to be a discrete kind (Haslam, 2003), is not defined and identified primarily on the basis of its relation to an external criterion, but on the basis of its internal structure. Its features cohere in a way that indicates a category with a determinate prevalence. It has "back-end" or consequential practicality to the degree that this category enables useful predictions. These two proposed forms of practicality are equally important: both mean that categories will be pragmatically valuable. However, they are also different, being characteristic of different kinds of categories and perhaps even negatively related. Kinds with front-end practicality, that is, may tend to have a more restricted range of practical implications, defined as they are in relation to specific external criteria. Kinds with back-end practicality, in contrast, may enable a wider array of predictions to the extent that... (shrink)

The purpose of this research was to explore the concept of collaboration within a specific healthcare context and to include the perspectives of healthcare users, a position largely lacking in previous studies. In applying a critical theoretical approach, the focus was on, as an exemplar, mothers with newborn babies who had spent more than 48 hr in a special care nursery. Semistructured interviews were undertaken with child health nurses, midwives and mothers. The three key theoretical findings on collaboration generated in (...) the study point to layers of meanings around identity, knowledge and institutions of care. Findings from the interview data analysis were further examined through the lens of key policy documents. The research outcomes indicate that the concept of collaboration serves an important function in healthcare in obscuring the complexities and ambiguities that characterise the care continuum. The study concludes the need for a more critical approach to the assumptions that underlie the language of collaboration and the implications for practice in healthcare. (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)

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)

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.

In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s (...) theory of rest and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises. (shrink)

Along with technological progress, vocational education and training (VET) is consistently changing. Workforce disruption has serious consequences for workers and international economies, often requiring adults to transition into different occupations or to upskill to maintain employment. We review recent literature covering VET trends, theoretical considerations for the 21st century, and present an approach to workforce training to help workers not only learn necessary skills but also become adaptable to constant change. We suggest a functional contextualist approach to mastery learning achieves (...) this aim. Specifically, we offer suggestions for pedagogy that not only develop skills but also encourage higher order thinking. Within a novice to expert continuum, we suggest deliberate practice, mental simulation, and reflective meaning making as methods to achieve efficiency and transfer—learning outcomes relevant to a changing workforce. This approach recognizes that learning is context bound and should promote broader human capabilities that support both employability and the continuing development of life literacies. (shrink)

An important insight derived from Kant about the workings of the mind is that conscious activity involves both the selection of relevant information, and the integration of that information, so as to form mental coherency. The conscious mind can then utilize this coherent information to solve problems, invent tools, synthesize concepts, produce works of art, and the like. In this paper, it will be suggested that just as biological processes, in general, exhibit selective and integrative functions, and just as visual (...) integration performs the function of selecting and integrating visual module areas, so too, consciousness emerged as a property of the brain to act as a kind of meta-cognitive process that selects and integrates relevant information from psychological modules. The upshot is to establish a psycho-neuro-biological continuum by suggesting that the conscious psychological properties of selectivity and integration are possible because of similar properties that other neurobiological and biological processes exhibit. (shrink)

This paper examines the roles of indexicals in explicating speakers’ intentions and constructing common ground in the context of a continuum with two extreme endpoints, the intracultural at one end, and the intercultural at the other, within the framework of the socio-cognitive approach proposed and developed by Kecskes and Kecskes and Zhang. Thirteen participants from different linguistic and cultural backgrounds were recruited to represent varying degrees on the intra- and intercultural continuum. They were divided into three groups: American (...) English speakers, speakers from Asian countries, and a group of speakers, each of whom represents linguistically and culturally different countries. Eight extracts were drawn from the data of up to three hours of recordings, including discussions on one topic, and retrospective interviews retrieving the speakers’ intentions for using deixis. The results reveal that the closer the interlocutors were towards the intercultural communicative context endpoint on the continuum, the more they employed four types of indexicals as common ground construction strategies. Those strategies included the explicit manifestation of intentions, clarification, and confirmation of referent identification in actual situational context, elicitation of information, disambiguation and explanation of similar salient specifics in their home culture in an effort to sustain cooperative communication. This study enhances our understanding of different functions of indexicals in interactions on the intra- and intercultural continuum, which resulted from different levels of context interpretation and common ground. (shrink)

The Dirac δ function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac’s discovery by over a century, and illuminating the nature of Cauchy’s infinitesimals and his infinitesimal definition of δ.

Smith asks whether patriotism and cosmopolitanism spring from the same source. If they do, we face two anomalies. First, we should expect a British subject to love France more than Great Britain because France has a larger population than Great Britain. Second, we should expect a British subject to love France more than a far-away country such as China given that the British subject is more familiar with the French than with the Chinese people. Both expectations are factually untrue. This (...) led Smith to reject the patriotism-cosmopolitanism continuum thesis. The love of country must spring from a source that is unrelated to the love of humankind. Nonetheless, neither kind of love can be reduced to substantive utility that informs the economist’s utility function and the social welfare function. Substantive utility appears as self-interest and other-interest (altruism). The altruist preference varies in intensity, depending on familiarity: people are ready to help more familiar people than less familiar ones. What complicates the discussion is that Smith uses the same term “familiarity” to discuss varying degrees of love: people tend to love more familiar people than less familiar ones. This paper sheds light on Smith’s confusing concept “universal benevolence”—which is best understood as the love of humankind. (shrink)

In this paper we provide a mathematical model of Kant’s temporal continuum that yields formal correlates for Kant’s informal treatment of this concept in theCritique of Pure Reasonand in other works of his critical period. We show that the formal model satisfies Kant’s synthetic a priori principles for time and that it even illuminates what “faculties and functions” must be in place, as “conditions for the possibility of experience”, for time to satisfy such principles. We then present a mathematically (...) precise account of Kant’s transcendental theory of time—the most precise account to date.Moreover, we show that the Kantian continuum which we obtain has some affinities with the Brouwerian continuum but that it also has “infinitesimal intervals” consisting of nilpotent infinitesimals; these allow us to capture Kant’s theory of rest and motion in theMetaphysical Foundations of Natural Science.While our focus is on Kant’s theory of time the material in this paper is more generally relevant for the problem of developing a rigorous theory of the phenomenological continuum, in the tradition of Whitehead, Russell, and Weyl among others. (shrink)

Photo by Hannah Busing on Unsplash INTRODUCTION Religion has long provided guidance that has led to standards reflected in some aspects of medical practices and traditions. The recent bioethical literature addresses numerous new problems posed by advancing medical technology and demonstrates an erosion of standards rooted in religion and long widely accepted as almost axiomatic. In the deep soul-searching that pervades the publications on bioethics, several disturbing and dangerous trends neglect some basic lessons of philosophy, logic, and history. The bioethics (...) discourse on medical aid in dying emphasizes similarity over previously recognized important distinguishing features. For example, it overplays a likeness between assistance in dying and the withdrawal of life-saving technology. In many bioethics’ topics, arguments based on a logical continuum are used to question the lines demarcating important moral differences. l. The Line Between Ethical and Not: Logic Based on Continuum Careful case selection, often either end of a continuum, allows the tearing down or ridiculing of many rules and codes across most professions and fields of interest. This situation holds true for traffic laws as well as medical ethics guidelines. It is relatively simple for those who desire to attack a particular viewpoint by selecting a case that makes that position seem untenable. In the ethics realm, good and bad medicine exist at opposite ends of an ethical continuum, with many practices lying in between. For example, much of medical ethics exists between the Nazi criminal physicians and the most sainted nurse or physician. A gradual progression occurred over less than two decades from a utilitarian position that supported limited euthanasia for those with certain mental illnesses to genocide. German society embraced a utilitarian ethic in which the value of human life no longer was intrinsic but instrumental.[1] Many morally significant points on a continuum were then ignored as the misguided utilitarian policy rampantly continued. A point in the continuum to distinguish between ethically justifiable and that which is not can be difficult to identify compared to the two extremes. This continuum is not unique to ethics but can be applied to almost any other aspect of human life and endeavor. Between a severely ill schizophrenic person and a superbly well-adjusted individual, there is a continuum of mental and psychological function. The existence of a continuum should not paralyze thinking and prevent us from drawing lines and identifying moral differences based on objective criteria as well as moral philosophy. Yet, by focusing on a continuum, many bioethicists use logic to disregard dividing lines between an "ethical" and an "unethical" act. Unfortunately, sometimes bioethicists draw revolutionary conclusions that would change the scope of medical practices which is accepted as ethical. There are many examples of similar shifts on the continuum. Many authors argue for the ethical permissibility of abortion by pointing out that the human fetus is no different in various characteristics, one arguing it is as like an ape or chick as it is like a person,[2] and does not achieve unique human and individual characteristics until well into the first year of life.[3] While human fetuses arguably do not have certain distinctive qualities of personhood, most people shy away from the logical next conclusion: permitting infanticide. For example, Joshua Lederberg condemns infanticide, in the face of biological illogic, because of our emotional commitment to infants, to me, a relatively weak explanation. Sir Francis Crick suggests we might consider birth at two days of life in order to decide whether an infant is a "suitable" member of society.[4] Giublini and Minerva suggest that infanticide should be permissible since late pregnancy abortions are permissible, arguing there is no significant difference between a fetus just before birth and an infant just after birth.[5] Clearly the continuum approach would allow for subjective arguments in favor of later infanticide at other points many days post-birth. Years ago, with a cynical tone, I mentioned infanticide as a further step on the continuum beyond abortion, and I was rightly shouted down as being deliberately provocative to assert the logic would ever stretch so far. While it is not an accepted mainstream position, the movement in academic settings from widespread condemnation to limited possible acceptance of infanticide has taken place in an incredibly short time. Public opinion and medical opinion in these areas have shifted dramatically in a short time. In another area, from a biological and chemical point of view, there is a continuum from man down to a single carbon atom. Yet, it would not seem logical to ignore the emotional differences, the meaning of personhood, or the moral distinction between killing an insect and killing a person. ll. A False Continuum: Medical Aid in Dying I assert that there has been an erosion of ethical guidelines in recent years attributable to using continuums to camouflage important distinctions. James Rachels’ work on active and passive euthanasia, which contends that the two are ethically identical, exemplifies that logic.[6] He illustrates this thesis, using a continuum to compare different scenarios with like consequences as morally equivalent, by comparing the deliberate drowning of a child with a deliberate failure to rescue a drowning child when easily able to do so. The author's comparison proposes that since much of the medical profession has already made peace with withholding treatment in order to hasten death, consistency inexorably demands that we permit active euthanasia as well.[7] When permission for active euthanasia was first introduced, it was limited exclusively to patients suffering severely from an intractable, incurable, and irreversible disease. These guidelines have been continuously eroded. There is now a substantial serious consideration for permitting active euthanasia of healthy elderly individuals who feel that they have completed their lives and are "tired of living."[8] There are many moral and factual differences along the ethical continuum. In human life, there is a difference between a live baby and a fetus, between a viable fetus and one that is not, between a fetus and a zygote, and between a zygote and a sperm cell. Similarly, there is a difference between pulling a trigger to kill someone and not interfering in preventing his death, which is reprehensible though both may be. There is a difference between not resuscitating an 80-year-old man with cancer when his heart stops and injecting him with a fatal dose of potassium chloride. I argue that an overt act of taking life repels civilized human beings is to be commended and encouraged as the reverence for human life or even for just a moment of human life is one of the great contributions of our civilization. CONCLUSION As an orthodox Jew, I feel that divinely inspired guidelines that have stood the test of centuries shape my beliefs, and such guidelines contradict medical aid in dying. I cannot speak to the viewpoint of those who do not access religion in defining their moral stance, nor do I implicate them in the current bioethics' trends, as I am not aware of the personal role of religion in the lives of most such authors. While many nonreligious people have a firm philosophical grounding and oppose medical aid in dying, I suggest that in the absence of any religious or other absolute standards, developing logically defensible ethical guidelines may be challenging. At the least, religion may play a role in defining the points on the continuums that are ethically meaningful and refuting the trending beliefs that if the endpoint is the same, allowing different methods of arriving at that end are somehow ethically equal. The continuum of ways death may result does not negate analysis of whether death is brought about in ways that recognize the importance of life. The German philosopher Hans Jonas said, "It is a question whether without restoring the category of the sacred, the category most thoroughly destroyed by the scientific enlightenment, we can have an ethics able to cope with the extreme powers that we possess today and constantly increase and are compelled to use."[9] While countries vary on the role of religion in policy, with many emphasizing freedoms of religion, a recent position paper released by a group of Jewish, Christian, and Moslem leaders suggested the need for agreement on the unique sanctity of human life.[10] I would recommend that such a document serve as an example of consensus on critical foundational bioethical guidelines for democratic secular societies. - [1] Alexander L Medical science under dictatorship. New England Journal of Medicine, 241, p39-47 DOI10.1056/NEJM194907142410201 [2] Lederberg J. A geneticist looks at contraception and abortion, Annals of Internal Medicine 67, sup 2, 25-27. https:/doi.org/10.7326/0003-4819-67-3-25 [3] Ibid. [4] Editorial, Sociology: Logic of biology. Nature 220, 429 https://www.nature.com/articles/220429b0 [5] Giublini A Minerva F After-birth abortion: why should the baby live. J Med Ethics 39, 261- [6] Rachels J Active and passive euthanasia. New England Journal of Medicine 292, 78-80 [7] Ibid. [8] Cohen-Almagor R Euthanizing people who are "tired of life". in Euthanasia and Assisted Suicide-Lessons from Belgium. Ch 11 of Euthanasia and Assisted Suicide, Cambridge University Press pp173-187. 2017 and DOI; https://doi.org/10.1017/9781108182799.012 [9] Hans Jonas, Technology and Responsibility: Reflections on the New Tasks of Ethics, 1972, found as Chapter IX, Philosophical Essays, 1980. https://inters.org/jonas-technology-responsability [10] A position paper of the Abrahamic Monotheistic religions on matters concerning the end-of-life. Vatican Press 28 October 2019 https://press.vatican.va/content/salastampa/en/bollettino/pubblico/2019/10/28/191028f.html. (shrink)

The purpose of this study is to investigate a relationship between egalitarian equivalence and envy-freeness in a continuum-agent economy, where tastes vary continuously across individuals. Under efficiency, the two criteria of equity are not compatible, except in the knife-edge case. In particular, when individual utility functions are restricted to the class of Cobb–Douglas-type functions, there exists an efficient, egalitarian-equivalent, and envy-free allocation if and only if all individuals have the same taste over commodities.

We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing S and we obtain a cardinal invariant yω such that S collapses the continuum to yω and y ≤ yω ≤ b. Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of y = yω < b. We define two relations $\leq _{0}^{\ast}$ and $\leq _{1}^{\ast}$ on the set $(^{\omega}\omega)_{{\rm (...) Fin}}$ of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if $\germ{F}\subseteq (^{\omega}\omega)_{Fin}$ is $\leq _{1}^{\ast}$ -unbounded, well-ordered by $\leq _{1}^{\ast}$ , and not $\leq _{0}^{\ast}$ -dominating, then there is a nonmeager p-ideal. The existence of such a system F follows from Martin's axiom. This is an analogue of the results of [3], [9, 10] for increasing functions. (shrink)

The canonical function game is a game of length ω1 introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ22 absoluteness, cardinality spectra and Π2 maximality for H(ω2) relative to the Continuum Hypothesis.

This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical modelling, (...) the formation of mathematics on the basis of internal mathematical principles and the mathematical theory of experiment. (shrink)

A function f from ω1 to the ordinals is called a canonical function for an ordinal α if f represents α in any generic ultrapower induced by forcing with math formula. We introduce here a method for coding sets of ordinals using canonical functions from ω1 to ω1. Combining this approach with arguments from, we show, assuming the Continuum Hypothesis, that for each cardinal κ there is a forcing construction preserving cardinalities and cofinalities forcing that every subset (...) of κ is an element of the inner model math formula. (shrink)

We suggest that morbid jealousy falls on the extreme end of a jealousy continuum. Thus, many features associated with normal jealousy will be present in individuals diagnosed with morbid jealousy. We apply Boyer & Lienard's (B&L's) prediction one (P1; target article, sect. 7.1) to morbid jealousy, suggesting that fitness-related life-cycle dimensions predict sensitivity to cues, and frequency, intensity, and content of intrusive thoughts of partner infidelity. (Published Online February 8 2007).

We show that the members of a certain class of semi-proper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω1 to ω1 is bounded on a club by a canonical function for an ordinal less than ω2.

In this paper we show that if the real line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\Bbb R}$\end{document} is not a union of less than continuum many of its meager subsets then there exists an almost continuous Sierpiński–Zygmund function having a perfect road at each point. We also prove that it is consistent with ZFC that every Darboux function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $f\colon{\Bbb R}\to{\Bbb R}$\end{document} is continuous (...) on some set of cardinality continuum. In particular, both these results imply that the existence of a Sierpiński–Zygmund function which is either Darboux or almost continuous is independent of ZFC axioms. This gives a complete solution of a problem of Darji [4]. The paper contains also a construction (in ZFC) of an additive Sierpiński–Zygmund function with a perfect road at each point. (shrink)

��This paper investigates how an evolutionary al- gorithm with an indirect encoding exploits the property of phenotypic regularity, an important design principle found in natural organisms and engineered designs. We present the first comprehensive study showing that such phenotypic regularity enables an indirect encoding to outperform direct encoding con- trols as problem regularity increases. Such an ability to produce regular solutions that can exploit the regularity of problems is an important prerequisite if evolutionary algorithms are to scale to high-dimensional real-world (...) problems, which typically contain many regularities, both known and unrecognized. The indirect encoding in this case study is HyperNEAT, which evolves artificial neural networks (ANNs) in a manner inspired by concepts from biological development. We demonstrate that, in contrast to two direct encoding controls, HyperNEAT produces both regular behaviors and regular ANNs, which enables HyperNEAT to significantly outperform the direct encodings as regularity increases in three problem domains. We also show that the types of regularities HyperNEAT produces can be biased, allowing domain knowledge and preferences to be injected into the search. Finally, we examine the downside of a bias toward regularity. Even when a solution is mainly regular, some irregularity may be needed to perfect its functionality. This insight is illustrated by a new algorithm called HybrID that hybridizes indirect and direct encodings, which matched HyperNEAT’s performance on regular problems yet outperformed it on problems with some irregularity. HybrID’s ability to improve upon the performance of HyperNEAT raises the question of whether indirect encodings may ultimately excel not as stand-alone algorithms, but by being hybridized with a further process of refinement, wherein the indirect encoding produces patterns that exploit problem regu-. (shrink)

Consciousness is seen as a difficult “binding” problem. Binding, a process where different sensations evoked by an item are associated in the nervous system, can be viewed as a process similar to associative learning. Several reports that consciousness is associated with some form of memory imply that different forms of memories have a common feature contributing to consciousness. Based on a proposed synaptic mechanism capable of explaining different forms of memory, we developed a framework for consciousness. It is based on (...) the formation of semblance of sensory stimulus from (1) synaptic semblances when excitatory postsynaptic potentials arrive at functionally LINKed postsynaptic membranes, and (2) network semblances when these potentials summate to elicit action potential initiating activity in a network of neurons. It is then possible to derive a framework for consciousness as a multi-dimensional semblance. According to this framework, a continuum of semblances formed from background sensory stimuli and oscillating neuronal activities serve to maintain consciousness. Feasibility of this framework to explain various physiological and pathological states of consciousness, its subjective nature and qualia is examined. (shrink)

In his writings about hypergeometric functions Gauss succeeded in moving beyond the restricted domain of eighteenth-century functions by changing several basic notions of analysis. He rejected formal methodology and the traditional notions of functions, complex numbers, infinite numbers, integration, and the sum of a series. Indeed, he thought that analysis derived from a few, intuitively given notions by means of other well-defined concepts which were reducible to intuitive ones. Gauss considered functions to be relations between continuous variable quantities while he (...) regarded integration and summation as appropriate operations with limits. He also regarded infinite and infinitesimal numbers as a façon de parler and used inequalities in order to prove the existence of certain limits. He took complex numbers to have the same legitimacy as real quantities. However, Gauss’s continuum was linked to a revised form of the eighteenth-century notion of continuous quantity: it was not reducible to a set of numbers but was immediately given. (shrink)

Consciousness is seen as a difficult “binding” problem. Binding, a process where different sensations evoked by an item are associated in the nervous system, can be viewed as a process similar to associative learning. Several reports that consciousness is associated with some form of memory imply that different forms of memories have a common feature contributing to consciousness. Based on a proposed synaptic mechanism capable of explaining different forms of memory, we developed a framework for consciousness. It is based on (...) the formation of semblance of sensory stimulus from (1) synaptic semblances when excitatory postsynaptic potentials arrive at functionally LINKed postsynaptic membranes, and (2) network semblances when these potentials summate to elicit action potential initiating activity in a network of neurons. It is then possible to derive a framework for consciousness as a multi-dimensional semblance. According to this framework, a continuum of semblances formed from background sensory stimuli and oscillating neuronal activities serve to maintain consciousness. Feasibility of this framework to explain various physiological and pathological states of consciousness, its subjective nature and qualia is examined. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 12.4 (2005) 343-348 [Access article in PDF] Doomed by Nature: The Inevitable Failure of our Naturally Selected Functions Andreas De Block Keywords psychoanalysis, Darwinism, evolutionary psychiatry, pathogenic metaphysics In their very thoughtful and stimulating replies, the three commentators foreground several topics crucial for both psychoanalysis and philosophical psychiatry. In my short response, I focus primarily on what the commentators believe to be the paper's main (...) shortcomings. The first critique reads as follows: Freud never claimed that the Homo sapiens is a fundamentally sick animal, but in fact thought that normalcy is possible and that normal behavior need not be interpreted as neurotic, perverse or psychotic. The second is that current Darwinian psychiatry does not substantiate the view that Homo sapiens is essentially an ill animal, whatever Freud's position might be. Third and last, the proposal that all human beings are ill to the core seems to be more of a religious slogan than a scientifically useful description. What Freud Really Said A transhistorical approach to Freud's work is far from simple, considering that it spans a period of over forty years and that it underwent considerable changes during that time. Not only did Freud introduce some radically new concepts during that time, such as narcissism and the death drive. Also, he regularly changed his position on the aim of psychoanalytic therapy, the ontogenesis of anxiety, and the nature of human sexuality. That Freud sometimes saw homosexuality as an inborn drive did not keep him on other occasions from thinking that homosexual desire was a result of the repression or projection of an oedipal (heterosexual) drive. Examples like these show that any exegesis of Freud's writings inevitably has to deal with some tensions. Freud's work contains numerous apparent, as well as real, contradictions, so that it is no surprise at all that Katherine Morris can produce textual evidence "proving" that Freud never held the view I attribute to him. According to her, Freud thought that we are not all neurotics and that normalcy is possible.I do not deny that several passages in Freud's work underscore the possibility of non-neurotic normalcy. Indeed, Freudian theory even describes processes—sublimation, the transformation from Id to Ego, the resolution of the oedipal complex, among others—that explain how human beings can escape neurosis and other mental disorders. But Freud's illustrations of these mechanisms or processes make clear that he could not conceive [End Page 343] of a nonpathogenic psychodynamics. Consider, for instance, how Freud emphasizes throughout his study of Leonardo da Vinci that Leonardo's sublimations make him extremely vulnerable to psychic exhaustion or "neurasthenia" (1910). Freud never tires of repeating that there is only a gradual difference between a neurotic patient and Leonardo da Vinci, the sublimating ("healthy") artist and scientist.Morris (2005) is right, of course, in arguing that a gradual difference between normality and mental illness does not imply that there is no difference at all. Quantity should not be thought the opposite of quality, because small quantitative differences can transform an "articulation" into a "breach." To use Hegel's famous example, there is a continuum between 110° C and -10° C, but everybody knows that that water evaporates above 100° C and freezes below 0° C. Applied to mental disorders, quantitative differences in levels of testosterone and serotonin may lead to qualitative differences in behavior; in other words, categorical and dimensional approaches to mental illnesses are not mutually exclusive. Having established that, we now need to take a closer look at how Freud articulated the differences between seemingly normal individuals and the mentally ill.Freud held that neurosis, psychosis, and perversion are caused by the interplay between instincts and defense mechanisms. Further, he thought that the pathogenic instincts were almost exclusively sexual. Now, although it is true that he sometimes recognized that the fear of death and the death drive are important factors for psychic life (Deeley 2005) and for the etiology of psychosis, more often than not he interpreted the fear of death as a symbolization of... (shrink)

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega ^\omega \to \omega ^\omega $ introduced by the second author in [8]. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg \mathsf {CH}$ such as the Cohen, random and Sacks models and, as a byproduct show (...) that, with one exception, for the bounding numbers there are no $\mathsf {ZFC}$ relations between them beyond those in the higher dimensional Cichoń diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak {d}^\lambda _\kappa $. (shrink)

The social sciences in particular operate on this continuum in flexible manner, sometimes close to the natural-scientific pole as in the case of experimental psychology or econometrics, sometimes close to the natural-scientific pole as in the case of experimental psychology or econometrics, sometimes close to the cultural-scientific approach, as in the case of cultural sociology or cultural history. Thus there is in Rickert's logic of science no room for any methodological quarrel."--BOOK JACKET.