In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; (...) its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers. (shrink)

A moderately risk averse person may turn down a 50/50 gamble that either results in her winning $200 or losing $100. Such behaviour seems rational if, for instance, the pain of losing $100 is felt more strongly than the joy of winning $200. The aim of this paper is to examine an influential argument that some have interpreted as showing that such moderate risk aversion is irrational. After presenting an axiomatic argument that I take to be the strongest case (...) for the claim that moderate risk aversion is irrational, I show that it essentially depends on an assumption that those who think that risk aversion can be rational should be skeptical of. Hence, I conclude that risk aversion need not be irrational. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes (...) for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

Since independence, at least 28 African countries have legalized some form of gambling. Yet a range of informal gambling activities have also flourished, often provoking widespread public concern about the negative social and economic impact of unregulated gambling on poor communities. This article addresses an illegal South African numbers game called fahfee. Drawing on interviews with players, operators, and regulatory officials, this article explores two aspects of this game. First, it explores the lives of both players and runners, as (...) well as the clandestine world of the Chinese operators who control the game. Second, the article examines the subjective motivations and aspirations of players, and asks why they continue to play, despite the fact that their aggregate losses easily outstrip their aggregate gains. In contrast with those who reduce its appeal simply to the pursuit of wealth, I conclude that, for the (mostly) black, elderly, working class women who play fahfee several times a week, the associated trade-off—regular, small losses, versus the social enjoyment of playing and the prospect of occasional but realistic windfalls—takes on a whole new meaning, and preferences for relatively lumpy rather than steady consumption streams help explain the continued attraction of fahfee. This reinforces the need to understand players’ own accounts of gambling utility rather than simply to moralistically condemn gambling or to dismiss gamblers behaviour as irrational. (shrink)

Main results are: (i) number e^{e} is transcendental; (ii) the both numbers e+π and e-π are irrational.

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem (...) and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

In this paper, I introduce an iterative method aimed at exploring numbers within the interval [0, 1]. Beginning with a foundational set, S0, a series of algorithms are employed to expand and refine this set. Each algorithm has its designated role, from incorporating irrational numbers to navigating non-deterministic properties. With each successive iteration, our set grows, and after infinite iterations, its cardinality is proposed to align with that of the real numbers. This work is an initial (...) exploration into this approach, and further investigation is encouraged to understand its full implications and potential. (shrink)

This paper attempts to demonstrate that the conviction about the harmony and order of the world was a fundamental metaphysical principle of the Pythagoreans. This harmony and order were primarily sought in the structures of arithmetics, yet following the discovery of incommensurable magnitudes (irrational numbers, as we now call them), the Pythagoreans began to see geometrical structure as a fundamental part of the world. On the example of the Pythagoreans’ metaphysics and science, the paper shows the mutual relations (...) between metaphysics and science. It demonstrates— on the one hand—the necessity of the first as a guide for the latter, and—on the other—how our scientific research can change its basic metaphysical principles when these are found to be inappropriate. The paper also tries to show the need for a realistic approach in our scientific research by means of the same example of the Pythagoreans, that is, the need to discern something which is below the surface appearance. (shrink)

Pattern recognition is represented as the limit, to which an infinite Turing process converges. A Turing machine, in which the bits are substituted with qubits, is introduced. That quantum Turing machine can recognize two complementary patterns in any data. That ability of universal pattern recognition is interpreted as an intellect featuring any quantum computer. The property is valid only within a quantum computer: To utilize it, the observer should be sited inside it. Being outside it, the observer would obtain quite (...) different result depending on the degree of the entanglement of the quantum computer and observer. All extraordinary properties of a quantum computer are due to involving a converging infinite computational process contenting necessarily both a continuous advancing calculation and a leap to the limit. Three types of quantum computation can be distinguished according to whether the series is a finite one, an infinite rational or irrational number. (shrink)

Natural argument is represented as the limit, to which an infinite Turing process converges. A Turing machine, in which the bits are substituted with qubits, is introduced. That quantum Turing machine can recognize two complementary natural arguments in any data. That ability of natural argument is interpreted as an intellect featuring any quantum computer. The property is valid only within a quantum computer: To utilize it, the observer should be sited inside it. Being outside it, the observer would obtain quite (...) different result depending on the degree of the entanglement of the quantum computer and observer. All extraordinary properties of a quantum computer are due to involving a converging infinite computational process contenting necessarily both a continuous advancing calculation and a leap to the limit. Three types of quantum computation can be distinguished according to whether the series is a finite one, an infinite rational or irrational number. -/- . (shrink)

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

This collection of articles explores a wide range of subject, from Godel’s incompleteness theorem, to possible technocalypse and neutrofuturology. Articles on historical debates on irrational number to electroculture, on vortex particle, or on different Neutrosophic applications are included.

The paper attempts to conciliate the important distinction between what-is, or exists, and what-is-not _thereby supporting Russell’s existential analysis_ with some Meinongian insights. For this purpose, it surveys the varied inhabitants of the realm of ‘non-being’ and tries to clarify their diverse statuses. The position that results makes it possible to rescue them back in surprising but non-threatening form, leaving our ontology safe from contradiction.

Premier ouvrage de l’histoire de la philosophie à être totalement consacré au concept d’entéléchie, il ne s’agira pas de proposer un essai historique allant de la mise en place de cette notion chez Aristote jusqu’à la conception biologique développée par Hans Driesch. Inscrit dans un projet plus global de compréhension du temps aristotélicien, c’est à une véritable refondation métaphysique du concept que ce travail nous convie. Si Kant, à l’ultime fin de sa première critique, avait admis qu’il n’existait que deux (...) voies possibles de fondation de l’a priori - la voie historique (platonicienne) empruntée par Hegel et la voie physiologique (aristotélicienne) qui sera la nôtre - la première partie de ce second volume prendra pour acquis qu’il n’existe pas de théorie qui puisse unifier ces deux pôles du savoir. Cela impliquera une scission entre deux mouvements irrationnels que sont le mouvement entéléchique aristotélicien et le mouvement dialectique platonicien. A partir du corps a priori du surhomme, le premier mouvement sera à la racine de la théorie de la connaissance (De la science) alors que le second sera au fondement de la théorie des valeurs (De l’Idéologie). En circonscrivant le mouvement dialectique dorien autour du noûs perse (iranien), ce travail offrira une première définition totalement négative du mouvement entéléchique appuyé, lui, sur le noûs ionien (« phénicien ») dont la seconde partie sera l’objet. De l’analyse de la substance première en passant par l’étude phénoménologique des schèmes de la camusité et du membre moteur, cette étude reviendra sur l’élasticité du tendon afin d’approcher non-phénoménalement le concept lui-même et tenter ainsi une première symphyse avec les rythmes circadiens de la biologie contemporaine. (shrink)

A number of formal epistemologists have argued that perfect rationality requires probabilistic coherence, a requirement that they often claim applies only to ideal agents. However, in “Rationality as an Absolute Concept,” Roy Sorensen contends that ‘rational’ is an absolute term. Just as Peter Unger argued that being flat requires that a surface be completely free of bumps and blemishes, Sorensen claims that being rational requires being perfectly rational. When we combine these two views, though, they lead to counterintuitive results. If (...) being rational requires being perfectly rational, and only the probabilistically coherent are perfectly rational, then this indicts all ordinary agents as irrational. In this paper, I will attempt to resolve this conflict by arguing that Sorensen is only partly correct. One important sense of ‘rational’, the sanctioning sense of ‘rational’, is an absolute term, but another important sense of ‘rational’, the sense in which someone can have rational capacities, is not. I will, then, show that this distinction has important consequences for theorizing about ideal rationality, developing an account of the relationship between ordinary and ideal rationality. Because the sanctioning sense of ‘rational’ is absolute, it is rationally required to adopt the most rational attitude available, but which attitude is most rational can change depending on whether we are dealing with ideal agents or people more like ourselves. (shrink)

Self-interest voting is irrational when it has even a small cost, but it can be rational for those who care about others; its expected utility (EU) may exceed its cost. For cosmopolitan voters (those who care about outsiders), the EU of voting increases with the number of affected others. The EU of voting for the good of the world now and in the future can thus be large. In some cases, the EU of parochial voting (e.g., considering only one's (...) nation) may not cross the cost threshold, but cosmopolitan voting can cross it. I apply these principles to various kinds of voting. And I compare voting to charity. Voting has the advantage of enlisting the coercive power of the state, but charity can be more focused on effective policies. I discuss campaign contributions, and I conclude with a discussion of the existence of utilitarian voters. (shrink)

What makes beliefs thrive? In this paper, we model the dissemination of bona fide science versus pseudoscience, making use of Dan Sperber's epidemiological model of representations. Drawing on cognitive research on the roots of irrational beliefs and the institutional arrangement of science, we explain the dissemination of beliefs in terms of their salience to human cognition and their ability to adapt to specific cultural ecologies. By contrasting the cultural development of science and pseudoscience along a number of dimensions, we (...) gain a better understanding of their underlying epistemic differences. Pseudoscience can achieve widespread acceptance by tapping into evolved cognitive mechanisms, thus sacrificing intellectual integrity for intuitive appeal. Science, by contrast, defies those deeply held intuitions precisely because it is institutionally arranged to track objective patterns in the world, and the world does not care much about our intuitions. In light of these differences, we discuss the degree of openness or resilience to conceptual change (evidence and reason), and the divergent ways in which science and pseudoscience can achieve cultural “success”. (shrink)

Prevailing opinion—defended by Jason Brennan and others—is that voting to change the outcome is irrational, since although the payoffs of tipping an election can be quite large, the probability of doing so is extraordinarily small. This paper argues that prevailing opinion is incorrect. Voting is shown to be rational so long as two conditions are satisfied: First, the average social benefit of electing the better candidate must be at least twice as great as the individual cost of voting, and (...) second, the chance of casting the decisive vote must be at least 1/N, where N stands for the number of citizens. It is argued that both of these conditions are often true in the real world. (shrink)

The question of the rationality of conspiratorial belief ¬divides philosophers into mainly two camps. The particularists believe that each conspiracy theory ought to be examined on its own merits. The generalist, by contrast, argues that there is something inherently suspect about conspiracy theories that makes belief in them irrational. Recent empirical findings indicate that conspiratorial thinking is commonplace among ordinary people, which has naturally shifted attention to the particularists. Yet, even the particularist must agree that not all conspiracy belief (...) is rational, in which case she must explain what separates rational from non-rational conspiratorial thinking. In this paper, I contrast three strategies to this end: 1) the probabilistic objectivist, who assesses the objective probability of conspiracies; 2) the subjectivist, who rather focuses on the perspective of the believer, and typically views the decision to believe in a conspiracy as a problem of decision making under risk. Approaches 1) and 2) rely on assessments of the probability of conspiracy which, I argue, limits their applicability. Instead, I explore 3) viewing the problem facing the potential believer as a decision problem under uncertainty about probabilities. I argue, furthermore, that focusing solely on epistemic utilities fails to do justice to the particular character of conspiracy beliefs, which are not exclusively epistemically motivated, and I investigate the rationality of such beliefs under a number of standard decision rules. (shrink)

The widely discussed "discursive dilemma" shows that majority voting in a group of individuals on logically connected propositions may produce irrational collective judgments. We generalize majority voting by considering quota rules, which accept each proposition if and only if the number of individuals accepting it exceeds a given threshold, where different thresholds may be used for different propositions. After characterizing quota rules, we prove necessary and sufficient conditions on the required thresholds for various collective rationality requirements. We also consider (...) sequential quota rules, which ensure collective rationality by adjudicating propositions sequentially and letting earlier judgments constrain later ones. Sequential rules may be path-dependent and strategically manipulable. We characterize path-independence and prove its essential equivalence to strategy-proofness. Our results shed light on the rationality of simple-, super-, and sub-majoritarian decision-making. (shrink)

A popular objection to theistic commitment involves the idea that faith is irrational. Specifically, some seem to put forth something like the following argument: (P1) Everyone (or almost everyone) who has faith is epistemically irrational, (P2) All theistic believers have faith, thus (C) All (or most) theistic believers are epistemically irrational. In this paper, I argue that this line of reasoning fails. I do so by considering a number of candidates for what faith might be. I argue (...) that, for each candidate, either (P1) is false or (P2) is false. Then, I make two positive suggestions for how faith can be epistemically rational but nonetheless have a unique relationship to evidence: one, that Jamesian self-justifying attitudes describe a distinctive kind of faith in oneself and others, and two, that faith is not solely based on empirical evidence. (shrink)

Some experts have argued that patients should routinely be told the specific magnitude and absolute probability of potential risks and benefits of screening tests. This position is motivated by the idea that framing risk information in ways that are less precise violates the ethical principle of respect for autonomy and its application in informed consent or shared decisionmaking. In this Perspective, we consider a number of problems with this view that have not been adequately addressed. The most important challenges stem (...) from the danger that patients will misunderstand the information or have irrational responses to it. Any initiative in this area should take such factors into account and should consider carefully how to apply the ethical principles of respect for autonomy and beneficence. (shrink)

The world faces grave global problems. These have been made possible by modern science and technology. We have put knowledge-inquiry into academic practice – a seriously irrational kind of inquiry that seeks knowledge and technological know-how dissociated from a more fundamental concern to seek and promote wisdom. We urgently need to bring about a revolution in academic inquiry, so that knowledge-inquiry becomes wisdom-inquiry – a kind of inquiry rationally designed and devoted to helping humanity make progress towards a wiser (...) world. In this paper I indicate what needs to change if knowledge-inquiry is to become wisdom-inquiry, and I indicate a number of recent developments, mainly in universities in the UK, that can be regarded as first steps towards wisdom-inquiry. (shrink)

A strong motivation for the human genome project was to relate biological features to the structure and function of small sets of genes, and ideally to individual genes. However, it is now increasingly realized that many problems require a "systems" approach emphasizing the interplay of large numbers of genes, and the involvement of complex networks of gene regulation. This implies a new emphasis on integrative, systems theoretical approaches. It may be called 'holistic' if the term is used without (...) class='Hi'>irrational overtones, in the general sense of directing attention to integrated features of organs and organisms. In the history of biology, seemingly conflicting reductionist and holistic notions have alternated, with bottom-up as well as top-down approaches eventually contributing to the solutions of basic problems. By now, there is no doubt that biological features and phenomena are rooted in physico-chemical processes of the molecules involved; and yet, integrated systems aspects are becoming more and more relevant in developmental biology, brain and behavioural science, and socio-biology. -/- . (shrink)

How do people reason about their opponent in turn-taking games? Often, people do not make the decisions that game theory would prescribe. We present a logic that can play a key role in understanding how people make their decisions, by delineating all plausible reasoning strategies in a systematic manner. This in turn makes it possible to construct a corresponding set of computational models in a cognitive architecture. These models can be run and fitted to the participants’ data in terms of (...) decisions, response times, and answers to questions. We validate these claims on the basis of an earlier game-theoretic experiment about the turn-taking game “Marble Drop with Surprising Opponent”, in which the opponent often starts with a seemingly irrational move. We explore two ways of segregating the participants into reasonable “player types”. The first way is based on latent class analysis, which divides the players into three classes according to their first decisions in the game: Random players, Learners, and Expected players, who make decisions consistent with forward induction. The second way is based on participants’ answers to a question about their opponent, classified according to levels of theory of mind: zero-order, first-order and second-order. It turns out that increasing levels of decisions and theory of mind both correspond to increasing success as measured by monetary awards and increasing decision times. Next, we use the logical language to express different kinds of strategies that people apply when reasoning about their opponent and making decisions in turn-taking games, as well as the ‘reasoning types’ reflected in their behavior. Then, we translate the logical formulas into computational cognitive models in the PRIMs architecture. Finally, we run two of the resulting models, corresponding to the strategy of only being interested in one’s own payoff and to the myopic strategy, in which one can only look ahead to a limited number of nodes. It turns out that the participant data fit to the own-payoff strategy, not the myopic one. The article closes the circle from experiments via logic and cognitive modelling back to predictions about new experiments. (shrink)

Most prominent arguments favoring the widespread discretionary business practice of sending jobs overseas, known as ‘offshoring,’ attempt to justify the trend by appeal to utilitarian principles. It is argued that when business can be performed more cost-effectively offshore, doing so tends, over the longterm, to achieve the greatest good for the greatest number. This claim is supported by evidence that exporting jobs actively promotes economic development overseas while simultaneously increasing the revenue of the exporting country. After showing that offshoring might (...) indeed be justified on utilitarian grounds, I argue that according to Rawlsian social-contract theory, the practice is nevertheless irrational and unjust. For it unfairly expects the people of a given society to accept job-gain benefits to peoples of other societies as outweighing job-loss hardships it imposes on itself. Finally, I conclude that contrary to socialism, which relies much more on government control, capitalism constitutes a social contract that places a particularly strong moral obligation on corporations themselves to refrain from offshoring. (shrink)

Empirical findings may be relevant for aesthetic evaluation in at least two ways. First — within criticism — they may help us to identify the aesthetic value of objects. Second— whithin philosophy — they may help us to decide which theory of aesthetic value and evaluation to prefer. In this paper, I address both kinds of relevance. My focus is thereby on empirical evidence gathered, not by means of first-personal experiences, but by means of third-personal scientific investigations of individual artworks (...) or, more generally, our interaction with art. The main thesis to be defended is that third-personal empirical findings are of limited significance for both critical and philosophical aesthetics. Indeed, they matter only to the extent to which they draw our attention to features or facts that we then iden- tify, from our first-personal perspective, as aesthetically relevant — for instance, as reasons counting for or against certain ascriptions of aesthetic value, or as factors that causally influence our actual as- sessments and thus render them partly inadequate or irrational. This limited significance of empirical findings is in line with the rationalist approach to the formation and justification of aesthetic judgements, that I have already started to defend elsewhere. -/- With respect to critical aesthetics, one problem is that empirical investigations cannot capture the normativity of the aesthetic relevance of lower-level features or facts: the respective studies can tell us what we take to be reason-giving or valuable, but not what is reason-giving or valuable. Furthermore, we cannot use empirically knowable principles to infer the presence of aesthetic values or reasons on the basis of recognising measurable lower-level features because the only available principles are too specific to allow for their actual application to more than one existing object, or even for their actual formulation. For their antecedents make reference to a large number of very determinate properties, as well as sometimes to particular events of creation. The only exception are conceptual principles (such as ‘something symmetrical is balanced’) or default principles (such as ‘something elegant is, by default, beautiful’). But knowledge of these principles is of little inferential use due to the holistic character of the justificatory power of aesthetic reasons and, moreover, does not allow for empirical acquisition. -/- With respect to philosophical aesthetics, I start with considering the limits of evolutionary accounts of aesthetic value. Even if it is true that humans originally came to value artworks because they recognised that artworks reveal certain skills or features of artists (such as imaginativeness or resourcefulness) that are desirable in sexual partners, it does not value that our reasons for valuing art have not changed, or at least not become much more complex. In addition, the answer to the question of why we value art from an evolutionary perspective has no obvious bearing on the issue of how we should appreciate art from an aesthetic perspective, or what such aesthetic appreciation would consist in. On the other hand, experimental studies showing that our actual aesthetic evaluations are influenced by factors (such as exposure e ects, or knowledge of the prices of objects), that undermine the good standing of our responses, just reveal that it is more difficult than perhaps expected to form sound aesthetic judgements; and that we need to improve our future aesthetic judgements by diminishing as much as possible the impact of those factors. (shrink)

‘Intentionalist’ approaches portray self-deceivers as “akratic believers”, subjects who deliberately choose to believe p despite knowing that p is false. In this paper I argue that the intentionalist model leads to a number of paradoxes that seem to undermine it. I claim that these paradoxes can nevertheless be overcome in light of the rival hypothesis that self-deception is a non-intentional process that stems from the influence of emotions upon cognitive processes. Furthermore, I propose a motivational interpretation of the phenomenon of (...) ‘hyperbolic discounting bias’ which highlights the role of emotional biases in akratic behavior. Finally, I argue that we are not the helpless victims of our irrational attitudes, insofar as we have the ability – and arguably the epistemic obligation – to counteract motivational biases. (shrink)

In 2012, a German court ruled that religious circumcision of male minors constitutes criminal bodily assault. Muslim and Jewish groups responded with outrage, with some commentators pegging the ruling to Islamophobic and anti-Semitic motivations. In doing so, these commentators failed to engage with any of the legal and ethical arguments actually given by the court in its landmark decision. In this brief commentary, I argue that a firm distinction must be drawn between criticisms of religious practices that stem from (...) class='Hi'>irrational prejudice and bigoted attitudes and those that are grounded in sound moral reasoning. Given that ritual circumcision is a pre-Enlightenment custom that elevates the inclinations of the community over the rights of the individual, it is hardly surprising that a growing number of post-Enlightenment philosophers and legal scholars are taking an ethical stand against it. As the “circumcision debate” continues, parties on all sides of the issue must remember to reason through the relevant considerations with care and respect. (shrink)

Cognitive science of religion (CSR) raises a number of issues that are of interest to theologians and philosophers of religion. The latter have focused primarily on the epistemological implications of CSR, that is, whether science shows religious belief to be irrational or unjustified. Another broad question is whether CSR is compatible with theism and Christian theology. Theological doctrines, such as Calvin’s views about sensus divinitatis and the noetic effects of sin, play an important part in these conversations. Less attention (...) has been directed to how CSR can function as a source for theological and philosophical inquiry. So far, CSR has been invoked in discussions on the natural knowledge of God, classical theism and divine hiddenness. This article provides an overview of the recent philosophical and theological engagements with CSR. (shrink)

Many wrongly believe that emotion plays little or no role in legal reasoning. Unfortunately, Langdell and his “scientific” case method encourage this error. A careful review of analysis in the real world, however, belies this common belief. Emotion can be cognitive, and cognition can be emotional. Additionally, modern neuroscience underscores the “co-dependence” of reason and emotion. Thus, even if law were a certain science of appellate cases (which it is not), emotion could not be torn from such “science.” -/- As (...) we reform legal education, we must recognize the role of cognitive emotion in law and legal analysis. If we fail to do this, we shortchange law schools, students, and the bar in grievous ways. We shortchange the very basics of true and best legal analysis. We shortchange at least half the universe of expression (the affective half). We shortchange the importance of watching and guarding the true interests of our clients, which interests are inextricably intertwined with affective experience. We shortchange the importance of motivation in law, life, and legal education. How can lawyers understand the motives of clients and other relevant parties without understanding the emotions that motivate them? How can lawyers hope to persuade judges, other advocates, or parties across the table in a transaction without grasping affective experience that motivates them? How can law professors fully engage students while ignoring affective experience that motivates students? Finally, we shortchange matters of life and death: emotions affect health and thus the very vigor of the bar. -/- Using insights from practice, modern neuroscience, and philosophy, I therefore explore emotion and other affective experience through a lawyer’s lens. In doing this, I reject claims that emotion and other affective experience are mere feeling (though I do not discount the importance of feeling). I also reject claims that emotion and other affective experience are necessarily irrational or beyond our control. Instead, such experience is often intentional and quite rational and controllable. After exploring law and affective experience at more “macro” levels, I consider three more specific examples of the interaction of law and emotion: (i) emotion, expression, and the first amendment, (ii) emotion in legal elements and exceptions, and (iii) emotion and lawyer mental health. To provide lawyers and legal scholars with a “one-source” overview of emotion and the law, I have also included an Appendix addressing a number of particular emotions. -/- Keywords: Emotion, Cognitive Emotion, Reason, Logos, Pathos, Legal Analysis, Legal Reasoning, Neuroscience, Philosophy, Persuasion, Rhetoric, Expression, First Amendment, Free Speech, Common Law Murder, Mental Health, NAACP v. McCrory, Anger, Contempt, Disgust, Hate, Malice. (shrink)

The strings of physics’ string theory are the binary digits of 1 and 0 used in computers and electronics. The digits are constantly switching between their representations of the “on” and “off” states. This switching is usually referred to as a flow or current. Currents in the two 2-dimensional programs called Mobius loops are connected into a four-dimensional figure-8 Klein bottle by the infinitely-long irrational and transcendental numbers. Such an infinite connection translates - via bosons being ultimately composed (...) of 1’s and 0’s depicting pi, e, √2 etc.; and fermions being given mass by bosons interacting in matter particles’ “wave packets” – into an infinite number of 8-Kleins. Each Klein 1) is one of the universe’s subuniverses (our own is 13.7 billion years old), 2) is made flexible through its binary digits which seamlessly, or almost seamlessly, join it to surrounding subuniverses and eliminate its central hole, and 3) possesses warped time and space because its foundation is the programmed curves in its mathematical Mobius loops (along with the twists they generate [p.7]). The universe functions according to the rules of fractal geometry. So the Mobius does not exist only at the cosmic level. It also manifests at the quantum scale, giving us photons and protons etc. Space and time are no longer separate, but are an indivisible space-time. So if space and the universe are infinite, how can time not be eternal? The past and the future must both extend forever (the idea of time being finite arises from confusion of our subuniverse with the one infinite universe). -/- BITS (Binary digiTS) only suggest existence of the divine if time is linear. Although a non-supernatural God is proposed via the inverse-square law coupled with eternal quantum entanglement, Einstein taught us that time is warped. Warped time is nonlinear, making it at least possible that the BITS composing space-time and all particles originate from the computer science of humans. -/- I suspect many readers will be content with reading this abstract. While there are more details, and mathematics, in the content; my natural style of writing is to avoid jargon and maths. I also tend to get philosophical. While I personally feel that there’s a lot of precious information in the content, I realize it won’t all be to everyone’s liking. Other subjects dealt with in this article are - the “Pioneer anomaly”, refinement of gravitational physics, dark energy and dark matter, quantum phenomena like mass and electric charge and quantum spin, Kepler’s laws of planetary motion, deflection of starlight by the sun, tides, falling bodies, Earth’s orbit, ancient Greek philosophers, Newton, Kepler, Galileo, Aristotle, Parmenides, Zeno of Elea, time travel into the past as well as the future, the elimination of distances in space, humanity’s construction of this universe we live in, The Law of Conservation of Matter-Energy, and support for the science-fiction-like idea of the electronic binary digits of 1 and 0 being the building blocks of our universe. (shrink)

In this paper, I am generally concerned with certain mental disorders and the doxastic attitudes that sometimes characterize them. According to recent Anglo-American philosophical studies on this topic, the latter involve beliefs that have somehow “gone wrong”: strange or irrational beliefs and cases of “motivated irrationality”. I aim to focus on pathological and deceptive phenomena such as delusion and self-deception. From a phenomenological perspective, these can also be investigated with regard to their experiential content. Adopting this approach, and starting (...) in particular with the eidetic connection between the living body and its own environment, I will examine disorders of the ecological self. Psychiatric delusion, which is at the core of this examination, is a symptom of a great number of mental disorders, such as Capgras disorder, Cotard disorder and schizophrenia. Integrating dominant (propositional) models of delusion which reduce this pathological phenomenon to a certain kind of belief, I propose an eco-phenomenological understanding of its contours. From this perspective, delusion reveals itself primarily as an illusory experience: a delusional value-perception. In particular, I offer a Schelerian-based description of a peculiar case of delusion: Capgras delusion. People who suffer from Capgras delusion claim that an otherwise familiar individual has been replaced by an impostor. I consider, although in its negative variations, the contextual (rather than internalist) character of this type of delusion. (shrink)

In 1980 F. Wattenberg constructed the Dedekind completion

d of the Robinson non-archimedean field

and established basic algebraic properties of

d [6]. In 1985 H. Gonshor established further fundamental properties of

d [7].In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion

d in transcendental number theory were considered. We dealing using set theory ZFC
(
-model of (...) ZFC).Given an class of analytic functions of one complex variable f  

z, we investigate the arithmetic nature of the values of fz at transcendental points en, n  .Main results are: (i) the both numbers e
 and e   are irrational, (ii) number ee is transcendental. Nontrivial generalization of the Lindemann- Weierstrass theorem is obtained. (shrink)

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, (...) along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (shrink)

One of the most important abilities we have as humans is the ability to think about number. In this chapter, we examine the question of whether there is an essential connection between language and number. We provide a careful examination of two prominent theories according to which concepts of the positive integers are dependent on language. The first of these claims that language creates the positive integers on the basis of an innate capacity to represent real numbers. The second (...) claims that language’s function is to integrate contents from modules that humans share with other animals. We argue that neither model is successful. (shrink)

The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring to (...) class='Hi'>numbers as abstract objects. (shrink)

This paper offers a practical argument for metaphysical emergence. The main message is that the growing reliance on so-called irrational scientific methods provides evidence that objects of science are indecomposable and as such, are better described by metaphysical emergence as opposed to the prevalent reductionistic metaphysics. I show that a potential counterargument that science will eventually reduce everything to physics has little weight given where science is heading with its current methodological trend. I substantiate my arguments by detailed examples (...) from biological engineering, but the conclusions are extendable beyond that discipline. (shrink)

Many believe that the rational evaluation of actions depends on the rational evaluation of even basic desires. Hume, though, viewed desires as "original existences" which cannot be contrary to either truth or reason. Contemporary critics of Hume, including Norman, Brandt and Parfit, have sought a basis for the rational evaluation of desires that would deny some basic desires reason-giving force. I side with Hume against these modern critics. Hume's concept of rational evaluation is admittedly too narrow; even basic desires are, (...) despite their nonrepresentational nature, subject to rational evaluation. But this evaluation is not relevant to their tendency to generate reasons for action, and does not undermine the spirit of the Humean view of rational action. (shrink)

The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in (...) a more Kantian sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori. (shrink)

There at least three ways of thinking about rationality: instrumental, substantive, and intentional. By far, the instrumental account is most influential. This essay proposes that intentional rationality can provide substantive accounts with room to breathe, and in a way that is facially distinct from instrumental accounts. I suggest that the intentionality of a judgment is made up of what it is about and the orientation through which it is judged, while irrationality is the subversion of a strict supporting connection between (...) the judgment and its corresponding set of coordinated attitudes (reasons). It follows that irrational intentionality is made up of episodic states where a judgment is subverted because of a misalignment between what it is about and the way the judge is oriented towards its contents. Four examples of irrational intentionality are considered: passivity towards ends, constitutive ignorance towards facts, delirious paranoia towards objects, and a disjunctive orientation towards categorizations. (shrink)

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide evidence (...) that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

It is often assumed that adaptation — a temporary change in sensitivity to a perceptual dimension following exposure to that dimension — is a litmus test for what is and is not a “primary visual attribute”. Thus, papers purporting to find evidence of number adaptation motivate a claim of great philosophical significance: That number is something that can be seen in much the way that canonical visual features, like color, contrast, size, and speed, can. Fifteen years after its reported discovery, (...) number adaptation’s existence seems to be nearly undisputed, with dozens of papers documenting support for the phenomenon. The aim of this paper is to offer a counterweight — to critically assess the evidence for and against number adaptation. After surveying the many reasons for thinking that number adaptation exists, we introduce several lesser-known reasons to be skeptical. We then advance an alternative account — the old news hypothesis — which can accommodate previously published findings while explaining various (otherwise unexplained) anomalies in the existing literature. Next, we describe the results of eight pre-registered experiments which pit our novel old news hypothesis against the received number adaptation hypothesis. Collectively, the results of these experiments undermine the number adaptation hypothesis on several fronts, whilst consistently supporting the old news hypothesis. More broadly our work raises questions about the status of adaptation itself as a means of discerning what is and is not a visual attribute. (shrink)

This is a chapter written for a popular audience, in which I use poker as a convenient illustration of probability, determinism and counterfactuals. More originally, I also discuss the roles of rationality versus psychological hunches, and explain why even in principle game theory cannot provide us the panacea of a perfect winning srategy. (N.B. The document I have uploaded here is slightly longer than the abbreviated version that appears in the book, and also differs in a few other minor details.) (...) . (shrink)

Proponents of epistocracy worry that high levels of voter ignorance can harm democracies. To combat such ignorance, they recommend allocating comparatively more political power to more politically knowledgeable citizens. In response, some recent critics of epistocracy contend that epistocratic institutions risk causing even more harm, since much evidence from political psychology indicates that more politically knowledgeable citizens are typically more biased, less open-minded, and more prone to motivated reasoning about political matters than their less knowledgeable counterparts. If so, perhaps epistocratic (...) institutions will perform worse epistemically than corresponding democratic institutions. Call this 'the problem of epistocratic irrationality'. This paper argues that the problem of epistocratic irrationality can be overcome. First, I argue that critics of epistocracy have overlooked several complications regarding the psychological data they claim shows that more knowledgeable citizens are less politically rational. Second, I argue that appropriately designed epistocratic institutions could overcome the problem of epistocratic irrationality even if such critics have interpreted the data correctly. I first explore whether refined selection mechanisms could allow epistocrats to avoid empowering less rational citizens, before assessing the prospects of implementing only those epistocratic institutions with a solid track record of reliable performance. (shrink)

Suppose we can save either a larger group of persons or a distinct, smaller group from some harm. Many people think that, all else equal, we ought to save the greater number. This article defends this view (with qualifications). But unlike earlier theories, it does not rely on the idea that several people's interests or claims receive greater aggregate weight. The argument starts from the idea that due to their stakes, the affected people have claims to have a say in (...) the rescue decision. As rescuers, our primary duty is to respect these procedural claims, which we must do by doing what these people would decide, in a process where each is given an equal vote on the matter. So in cases where each votes in their own self-interest, respect for their equal right to decide, or their autonomy, will lead us to save the greater number. The argument is explained in detail, with special attention to the questions of how, exactly, it avoids aggregation, and of why majority rule is superior to lottery procedures. The view has further advantages. Especially, it explains the “partial” relevance of numbers in cases involving unequal harms, and it does so in a way that dissolves the appearance of paradox that besets theories of “partial aggregation.”. (shrink)

One of the explanations for the Great Crisis of 2007-2008 was that financial authorities should have issued stricter regulations to prevent the housing bubble. However, according to Alan Greenspan, President of the Federal Reserve System (FED) from 1987 to 2006, this is to judge with hindsight. No one can guess when a “bubble” begins, nor when it ends; they happen because of the “irrational exuberance” in investors’ behavior, which causes boom and bust cycles. Regulators are not in a better (...) situation for assessing risks, though: since market participants supposedly know their own risks better than the regulator (a kind of informational asymmetry), an intervention (except to ensure law-enforcement) would imply unjustified paternalism. However, a regulator does not have to be conceived as a paternalistic authority. We sketch an objection to Greenspan's argument, arguing that crises don’t require a defective reasoning such as the “irrational exuberance” – our usual bounded rationality might be enough to provide the kind of “self-fulfilling prophecy” observed in the rise and fall of bubble assets value. Given the possibility of grave externalities, authorities are justified in adopting measures to ensure investors behave in a prudent way, even if they supposedly know better their own risks. (shrink)

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

This paper criticizes the view that number words in argument position retain the meaning they have on an adjectival or determiner use, as argued by Hofweber :179–225, 2005) and Moltmann :499–534, 2013a, 2013b). In particular the paper re-evaluates syntactic evidence from German given in Moltmann to that effect.

According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the (...) concrete world is just as it in fact is, then T’ bear on this claim. It concludes that, while analogies with theories that attempt to eliminate unobservable concrete entities provide good reason to regard theories of the former sort as explanatorily bad, this reason does not apply to theories of the latter sort. (shrink)