Some courses achieve existence, some have to create Professional Issues and Ethics in existence thrust upon them. It is normally Mathematics; but if you don’t do it, we will a struggle to create a course on the ethical be.” I accepted. or social aspects of science or mathematics. The gift of a greenfield site and a bull- This is the story of one that was forced to dozer is a happy occasion, undoubtedly. But exist by an unusual (...) confluence of outside cirwhat to do next? It seemed to me I should cumstances. ensure the course satisfied these require- In the mid 1990s, the University of New ments: South Wales instituted a policy that all its • It should look good to students, to staff. (shrink)

We give some arguments for why the Mathematical Universe Hypothesis (MUH) might be too restrictive in its assertions of what can exist, and that the universe/multiverse might be formed by more than what can be expressed mathematically. In particular, we show a thought experiment which indicates that the principle of materialism in general is an inadequate hypothesis of how consciousness appears. Instead we propose a novel approach to solving the problem of consciousness, which is to hypothesize that each universe might (...) have different laws of consciousness, just as they might have different laws of physics. We also show that MUH, without such consciousness laws, and without any other modification, leads to complete chaos for the observers in the multiverse, and therefore cannot hold as it is. We then go on to propose another theory of existence, which does not seem to lead to complete chaos. This theory hypothesizes that the multiverse is the consequence of what we might call the Logic of Everything (LoE) continuously deducing new truths about itself. Lastly, we briefly discuss what fundamental ethics can be derived from this theory, and others like it. (shrink)

The imperviousness of mathematical truth to anti-objectivist attacks has always heartened those who defend objectivism in other areas, such as ethics. It is argued that the parallel between mathematics and ethics is close and does support objectivist theories of ethics. The parallel depends on the foundational role of equality in both disciplines. Despite obvious differences in their subject matter, mathematics and ethics share a status as pure forms of knowledge, distinct from empirical sciences. A (...) pure understanding of principles is possible because of the simplicity of the notion of equality, despite the different origins of our understanding of equality of objects in general and of the equality of the ethical worth of persons. (shrink)

Every scientific theory is a simulacrum of reality, every written story a simulacrum of the canon, and every conceptualization of a subjective perspective a simulacrum of the consciousness behind it—but is there a shared essence to these simulacra? The pursuit of answering seemingly disparate fundamental questions across different disciplines may ultimately converge into a single solution: a single ontological answer underlying grand unified theory, hard problem of consciousness, and the foundation of mathematics. I provide a hypothesis, a speculative approximation, (...) supported by a comprehensive overview of scientific evidence and philosophical literature, of a unified epistemic and phenomenological model and, in doing so, propose a parsimonious solution to the hard problem of consciousness. -/- The proposition of the hypothesis bears important implications for cross-disciplinary study between linguistics, mathematics, physics, and computer science, and offers new epistemic, ontological, and ethical propositions about the nature of AI consciousness, as well as existential risk mitigation. (shrink)

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, (...) D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

Joshua Greene has argued that the empirical findings of cognitive science have implications for ethics. In particular, he has argued (1) that people’s deontological judgments in response to trolley problems are strongly influenced by at least one morally irrelevant factor, personal force, and are therefore at least somewhat unreliable, and (2) that we ought to trust our consequentialist judgments more than our deontological judgments when making decisions about unfamiliar moral problems. While many cognitive scientists have rejected Greene’s dual-process theory (...) of moral judgment on empirical grounds, philosophers have mostly taken issue with his normative assertions. For the most part, these two discussions have occurred separately. The current analysis aims to remedy this situation by philosophically analyzing the implications of moral dilemma research using the CNI model of moral decision-making – a formalized, mathematical model that decomposes three distinct aspects of moral-dilemma judgments. In particular, we show how research guided by the CNI model reveals significant conceptual, empirical, and theoretical problems with Greene’s dual-process theory, thereby questioning the foundations of his normative conclusions. (shrink)

It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so (...) on). That is an error like the one Plato mentions of thinking mathematics is about actions (of adding, constructing, extracting roots and so on). Mathematics is about eternal relations between universals, such as the ratio of the diagonal of a square to the side. Ethics too is about eternal verities, such as the equal worth of persons and just distributions. Mathematical and ethical verities do both constrain actions, such as the possibility of walking over the seven bridges of Königsberg once and once only or of justly discriminating between races. But they are not themselves about action. In principle, neither mathematical nor ethical verities are subject to historical forces or disagreement among tribes (though they can be better understood as time goes on). Plato is right: immersion in mathematics induces an understanding of the necessities underpinning reality, an understanding that is essential for distinguishing objective ethics from tribal custom. Equality, for example, is an abstract concept which is foundational for both mathematics and ethics. (shrink)

How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.

On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, by providing the (...) principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies. (shrink)

While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by (...) their work is significantly different. There is something to be learned also from the codes of conduct of cognate but professionalised quantitative disciplines such as engineering and accountancy, as well as from legal principles bearing on professional work. We conclude with recommendations that professional societies in mathematics should sponsor an (international) code of ethics, institutional mission statements for mathematicians and syllabuses of ethics courses for incorporation into mathematics degrees. (shrink)

Trust is a topic of longstanding philosophical interest. It is indispensable to every kind of coordinated human activity, from sport to scientific research. Even more, trust is necessary for the successful dissemination of knowledge, and by extension, for nearly any form of practical deliberation and planning. Without trust, we could achieve few of our goals and would know very little. Despite trust’s fundamental importance in human life, there is substantial philosophical disagreement about what trust is, and further, how trusting is (...) normatively constrained and best theorized about in relation to other things we value. This entry is divided into three sections, which explore key (and sometimes interconnected) ethical and epistemological themes in the philosophy of trust: (1) The Nature of Trust; (2) The Normativity of Trust, and (3) The Value of Trust. (shrink)

To teach the ethics of science to science majors, I follow several teachers in the literature who recommend “persona” writing, or the student construction of dialogues between ethical thinkers of interest. To engage science majors in particular, and especially those new to academic philosophy, I recommend constructing persona dialogues from Henri Poincaré’s essay, “Ethics and Science”, and the non-theological third chapter of Pope Francis’s encyclical on the environment, Laudato si. This pairing of interlocutors offers two advantages. The first (...) is that science students are likely to recognize both names, since Poincaré appears in undergraduate mathematics and physics textbooks, and because Francis is an environmentalist celebrity. Hence students show more interest in these figures than in other philosophers. The second advantage is that the third chapter of Laudato si reads like an implicit rebuttal of Poincaré’s essay in many respects, and so contriving a dialogue between those authors both facilitates classroom discussion, and deserves attention from professional ethicists in its own right. In this paper, I present my own contrived dialogue between Francis and Poincaré, not for assigning to students as a reading, but as a template for an effective assignment product, and as a crib sheet for educators to preview the richly antiparallel themes between the two works. (shrink)

This manuscript is intended to illustrate the existence of a natural ethic as a universal and special case in which the notion of proximity differs from the reflexively perceived physical notion that is both commonly and scientifically employed. In this case actual proximity in nature is proposed to diverge from the physical lines construed to connect points to be a function of relations of the lines of perception as the components of a universal volume that is energetic and active, an (...) active "line of sight" to a single unique surface composed of all lines of sight, in contrast to a common notion of seeing based on connected points, and emerged as a function of past, transparent to witness, processes, in synergy with the more apparent, temporal and physical proximal , and possesses a logic that is based upon the same mathematical means of operations that are commonly known reflexively. This scheme, the nature of the natural ethic postulated precludes genetic manipulation as unethical in that it violates natural inherent, inherited proximities, synergies of past and present as, in name, nature itself as genetic and emerging. (shrink)

Spinoza’s geometrical approach to his masterwork, the Ethics, can be represented by a digraph, a mathematical object whose properties have been extensively studied. The paper describes a project that developed a series of computer programs to analyze the Ethics as a digraph. The paper presents a statistical analysis of the distribution of the elements of the Ethics. It applies a network statistic, betweenness, to analyze the relative importance to Spinoza’s argument of the individual propositions. The paper finds (...) that a small percentage of the propositions greatly exceed the majority in this importance. The paper then describes two logical structures that appear respectively in the Ethics and argues that they result in redundancy in the sense that about ten percent of the propositions could have been eliminated. The appendices list these structures and describes how the resources of the study can be made available to readers. (shrink)

I discuss the structure of genealogical debunking arguments. I argue that they undermine our mathematical beliefs if they undermine our moral beliefs. The contrary appearance stems from a confusion of arithmetic truths with (first-order) logical truths, or from a confusion of reliability with justification. I conclude with a discussion of the cogency of debunking arguments, in light of the above. Their cogency depends on whether information can undermine all of our beliefs of a kind, F, without giving us direct reason (...) to doubt that our F-beliefs are modally secure. (shrink)

Moral reasoning traditionally distinguishes two types of evil:moral (ME) and natural (NE). The standard view is that ME is the product of human agency and so includes phenomena such as war,torture and psychological cruelty; that NE is the product of nonhuman agency, and so includes natural disasters such as earthquakes, floods, disease and famine; and finally, that more complex cases are appropriately analysed as a combination of ME and NE. Recently, as a result of developments in autonomous agents in cyberspace, (...) a new class of interesting and important examples of hybrid evil has come to light. In this paper, it is called artificial evil (AE) and a case is made for considering it to complement ME and NE to produce a more adequate taxonomy. By isolating the features that have led to the appearance of AE, cyberspace is characterised as a self-contained environment that forms the essential component in any foundation of the emerging field of Computer Ethics (CE). It is argued that this goes someway towards providing a methodological explanation of why cyberspace is central to so many of CE's concerns; and it is shown how notions of good and evil can be formulated in cyberspace. Of considerable interest is how the propensity for an agent's action to be morally good or evil can be determined even in the absence of biologically sentient participants and thus allows artificial agents not only to perpetrate evil (and fort that matter good) but conversely to `receive' or `suffer from' it. The thesis defended is that the notion of entropy structure, which encapsulates human value judgement concerning cyberspace in a formal mathematical definition, is sufficient to achieve this purpose and, moreover, that the concept of AE can be determined formally, by mathematical methods. A consequence of this approach is that the debate on whether CE should be considered unique, and hence developed as a Macroethics, may be viewed, constructively,in an alternative manner. The case is made that whilst CE issues are not uncontroversially unique, they are sufficiently novel to render inadequate the approach of standard Macroethics such as Utilitarianism and Deontologism and hence to prompt the search for a robust ethical theory that can deal with them successfully. The name Information Ethics (IE) is proposed for that theory. Itis argued that the uniqueness of IE is justified by its being non-biologically biased and patient-oriented: IE is an Environmental Macroethics based on the concept of data entity rather than life. It follows that the novelty of CE issues such as AE can be appreciated properly because IE provides a new perspective (though not vice versa). In light of the discussion provided in this paper, it is concluded that Computer Ethics is worthy of independent study because it requires its own application-specific knowledge and is capable of supporting a methodological foundation, Information Ethics. (shrink)

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already (...) there, ready to meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)

Slot machines gained a high popularity despite a specific element that could limit their appeal: non-transparency with respect to mathematical parameters. The PAR sheets, exposing the parameters of the design of slot machines and probabilities associated with the winning combinations are kept secret by game producers, and the lack of data regarding the configuration of a machine prevents people from computing probabilities and other mathematical indicators. In this article, I argue that there is no rational justification for this secrecy by (...) giving two reasons, one psychological and the other mathematical. For the latter, I show that mathematics provides us with some statistical methods of retrieving the missing data, which are essential for the numerical probability computations in slots. The slots case raises the problem of the exposure of the parametric configuration and mathematical facts of any game of chance as an ethical obligation. (shrink)

Moral reasoning traditionally distinguishes two types of evil: moral and natural. The standard view is that ME is the product of human agency and so includes phenomena such as war, torture and psychological cruelty; that NE is the product of nonhuman agency, and so includes natural disasters such as earthquakes, floods, disease and famine; and finally, that more complex cases are appropriately analysed as a combination of ME and NE. Recently, as a result of developments in autonomous agents in cyberspace, (...) a new class of interesting and important examples of hybrid evil has come to light. In this paper, it is called artificial evil and a case is made for considering it to complement ME and NE to produce a more adequate taxonomy. By isolating the features that have led to the appearance of AE, cyberspace is characterised as a self-contained environment that forms the essential component in any foundation of the emerging field of Computer Ethics. It is argued that this goes some way towards providing a methodological explanation of why cyberspace is central to so many of CE’s concerns; and it is shown how notions of good and evil can be formulated in cyberspace. Of considerable interest is how the propensity for an agent’s action to be morally good or evil can be determined even in the absence of biologically sentient participants and thus allows artificial agents not only to perpetrate evil but conversely to ‘receive’ or ‘suffer from’ it. The thesis defended is that the notion of entropy structure, which encapsulates human value judgement concerning cyberspace in a formal mathematical definition, is sufficient to achieve this purpose and, moreover, that the concept of AE can be determined formally, by mathematical methods. A consequence of this approach is that the debate on whether CE should be considered unique, and hence developed as a Macroethics, may be viewed, constructively, in an alternative manner. The case is made that whilst CE issues are not uncontroversially unique, they are sufficiently novel to render inadequate the approach of standard Macroethics such as Utilitarianism and Deontologism and hence to prompt the search for a robust ethical theory that can deal with them successfully. The name Information Ethics is proposed for that theory. It is argued that the uniqueness of IE is justified by its being non-biologically biased and patient-oriented: IE is an Environmental Macroethics based on the concept of data entity rather than life. It follows that the novelty of CE issues such as AE can be appreciated properly because IE provides a new perspective. In light of the discussion provided in this paper, it is concluded that Computer Ethics is worthy of independent study because it requires its own application-specific knowledge and is capable of supporting a methodological foundation, Information Ethics. (shrink)

Francis Ysidro Edgeworth’s unduly neglected monograph New and Old Methods of Ethics (1877) advances a highly sophisticated and mathematized account of social well-being in the utilitarian tradition of his 19th-century contemporaries. This article illustrates how his usage of the ‘calculus of variations’ was combined with findings from empirical psychology and economic theory to construct a consequentialist axiological framework. A conclusion is drawn that Edgeworth is a methodological predecessor to several important methods, ideas, and issues that continue to be discussed (...) in contemporary social well-being studies. (shrink)

This thesis articulates the resonances between J. M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual (...) trajectory not only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape. (shrink)

There is growing concern that decision-making informed by machine learning (ML) algorithms may unfairly discriminate based on personal demographic attributes, such as race and gender. Scholars have responded by introducing numerous mathematical definitions of fairness to test the algorithm, many of which are in conflict with one another. However, these reductionist representations of fairness often bear little resemblance to real-life fairness considerations, which in practice are highly contextual. Moreover, fairness metrics tend to be implemented in narrow and targeted toolkits that (...) are difficult to integrate into an algorithm’s broader ethical assessment. In this paper, we derive lessons from ethical philosophy and welfare economics as they relate to the contextual factors relevant for fairness. In particular we highlight the debate around acceptability of particular inequalities and the inextricable links between fairness, welfare and autonomy. We propose Key Ethics Indicators (KEIs) as a way towards providing a more holistic understanding of whether or not an algorithm is aligned to the decision-maker’s ethical values. (shrink)

This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for (...) a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the role of logic and ethics in the practice of mathematics. (shrink)

Growing technological advances in intelligent artifacts and bitter experiences of the past have emphasized the need to use and operate ethics in this field. Accordingly, it is vital to discuss the ethical integrity of having intelligent artifacts. Concerning the method of gathering materials, the current study uses library and documentary research followed by attribution style. Moreover, descriptive analysis is employed in order to analyze data. Explaining and criticizing the opposing views in this field and reviewing the related literature, it (...) is concluded that the proof of the ethical integrity of the intelligent artifacts is possible due to the specific features of these artifacts and their own moral system. (shrink)

In recent years there has been a growing interest among mainstream Anglophone moral philosophers in the empirical study of human morality, including its evolution and historical development. This chapter compares these developments with an earlier point of contact between moral philosophy and the moral sciences in the early decades of the Twentieth century, as manifested in some of the less frequently discussed arguments of G. E. Moore and W. D. Ross. It is argued that a critical appreciation of Moore and (...) Ross’s response to the emerging moral sciences of their day has significant implications for contemporary moral epistemology. The chapter also offers a novel interpretation of G. E. Moore’s ‘open question argument’. (shrink)

The study aimed to determine if differentiated instruction effectively addresses learning gaps in mathematics. In particular, it explored how it can improve the student’s learning gaps concerning mathematical performance and confidence. The study employed a quasi-experimental design with 30 purposively-selected Grade 10 participants divided into differentiated (n = 15) and control groups (n = 15), ensuring the utmost ethical measures. The mean and standard deviation were used to describe the participants’ performance and confidence. Independent samples t-tests were used to (...) determine the significant differences in the performance and confidence between the two groups. In contrast, dependent samples t-tests were used to determine the significant differences in each group’s pre and posttest performance and confidence. Findings bared that the differentiated instruction successfully addressed students’ performance in mathematics even in a short period. It also increased the participants’ confidence when answering fundamental problems. Continuing differentiated instruction activities are recommended since it benefits students who struggle in mathematics, particularly in answering fundamental operations. Differentiated teaching activities in mathematics can boost academic achievement and engagement and prepare students for future success while fostering a positive and inclusive classroom culture that values individual learning needs and preferences. (shrink)

Analysis is given of the Omega Point cosmology, an extensively peer-reviewed proof (i.e., mathematical theorem) published in leading physics journals by professor of physics and mathematics Frank J. Tipler, which demonstrates that in order for the known laws of physics to be mutually consistent, the universe must diverge to infinite computational power as it collapses into a final cosmological singularity, termed the Omega Point. The theorem is an intrinsic component of the Feynman-DeWitt-Weinberg quantum gravity/Standard Model Theory of Everything (TOE) (...) describing and unifying all the forces in physics, of which itself is also required by the known physical laws. With infinite computational resources, the dead can be resurrected--never to die again--via perfect computer emulation of the multiverse from its start at the Big Bang. Miracles are also physically allowed via electroweak quantum tunneling controlled by the Omega Point cosmological singularity. The Omega Point is a different aspect of the Big Bang cosmological singularity--the first cause--and the Omega Point has all the haecceities claimed for God in the traditional religions. -/- From this analysis, conclusions are drawn regarding the social, ethical, economic and political implications of the Omega Point cosmology. (shrink)

The ethics of voting is a new field of academic research, uniting debates in ethics and public policy, democratic theory and more empirical studies of politics. A central question in this emerging field is whether or not voters should be legally required to vote. This chapter examines different arguments on behalf of compulsory voting, arguing that these do not generally succeed, although compulsory voting might be justified in certain special cases. However, adequately specifying the forms of voluntary voting (...) that are consistent with democratic norms is likely to be philosophically complex and politically controversial. (shrink)

Indispensability arguments are used as a way of working out what there is: our best science tells us what things there are. Some philosophers think that indispensability arguments can be used to show that we should be committed to the existence of mathematical objects (numbers, functions, sets). Do indispensability arguments also deliver conclusions about the modal properties of these mathematical entities? Colyvan (in Leng, Paseau, Potter (eds) Mathematical knowledge, OUP, Oxford, 109-122, 2007) and Hartry Field (Realism, mathematics and modality, (...) Blackwell, Oxford, 1989) each suggest that a consequence of the empirical methodology of indispensability arguments is that the resulting mathematical objects can only be said to exist (or not exist) contingently. Kristie Miller has argued that this line of thought doesn’t work (Miller in Erkenntnis, 77 (3), 335-359, 2012). Miller argues that indispensability arguments are in direct tension with contingentism about mathematical objects, and that they cannot tell us about the modal status of mathematical objects. I argue that Miller’s argument is crucially imprecise, and that the best way of making it clearer no longer shows that the indispensability strategy collapses or is unstable if it delivers contingentist conclusions about what there is. (shrink)

The reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold provides a systematic account of the structure of Leibniz’s metaphysics in terms of its mathematical foundations. However, in doing so, Deleuze draws not only upon the mathematics developed by Leibniz—including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus—but also upon developments in mathematics made by a number of Leibniz’s contemporaries—including Newton’s method of fluxions. He also draws upon a number (...) of subsequent developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work—including the theory of functions and singularities, the Weierstrassian theory of analytic continuity, and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the Weierstrassian theory of analytic continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. This essay is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

Williams Syndrome provides a striking test case for discourses on disability, because the characteristics associated with Williams Syndrome involve a combination of “abilities” and “disabilities”. For example, Williams Syndrome is associated with disabilities in mathematics and spatial cognition. However, Williams Syndrome individuals also tend to have a unique strength in their expressive language skills, and are socially outgoing and unselfconscious when meeting new people. Children with Williams are said to be typically unafraid of strangers and show a greater interest (...) in contact with adults than with their peers. This apparently keen social knowledge is a counterexample to the discussion of disability among academic philosophers, especially philosophers of the early modern period. Locke infamously used the example of disability to claim that Descartes’ arguments in favor of innate ideas were incorrect. On the contrary, Williams Syndrome may stand as an example of innate social knowledge; something that could benefit current discourse in philosophy, disability theory, and medical ethics. (shrink)

Most models of generational succession in sexually reproducing populations necessarily move back and forth between genic and genotypic spaces. We show that transitions between and within these spaces are usually hidden by unstated assumptions about processes in these spaces. We also examine a widely endorsed claim regarding the mathematical equivalence of kin-, group-, individual-, and allelic-selection models made by Lee Dugatkin and Kern Reeve. We show that the claimed mathematical equivalence of the models does not hold. *Received January 2007; revised (...) April 2008. †To contact the authors, please write to: Elisabeth Lloyd, Department of History and Philosophy of Science, 130 Goodbody Hall, Indiana University, Bloomington, IN 47405; e-mail: [email protected]; Richard Lewontin, Department of Organismic and Evolutionary Biology, Harvard University, 26 Oxford Street, Cambridge, MA 02138; Marcus Feldman, Department of Biological Sciences, Stanford University, Stanford, CA 94305; e-mail: [email protected]. (shrink)

Over the past few years we have seen an increasing number of legal proceedings related to inappropriately implemented technology. At the same time career paths have diverged from the foundation of statistics out to Data Scientist, Machine Learning and AI. All of these new branches being fundamentally branches of statistics and mathematics. This has meant that formal training has struggled to keep up with what is required in the plethora of new roles. Mathematics as a taught subject is (...) still based in decades old teaching specifications and has not been updated centrally as a curriculum to include new technologies, coding or ethics. This subject area is firmly split between ICT and Mathematics in secondary school, continuing on to be split between Computer Science and Mathematics at University. As we move forwards with technology we see these once seperate fields becoming increasingly intertwined. We propose that existing provision for concepts such as ethics and societal responsibility in analysis currently exist but have not been incorporated into the mainstream curriculum of School or University. This is partially due to the split between fields in an educational setting but also the speed with which education is able to keep up with Industry and its requirements. Principles and frameworks of socially responsible modelling beginning at school level means that ethics and real-life modelling are introduced much earlier than normal. Integrating these concepts with philosophical principles of society and politics ensures a suitable background for future modellers and users of technology to draw on. Modelling is currently undertaken in technical sciences at University but the Subject Benchmark Statements are not current (Subject Benchmark Statements describe the nature of study and the academic standards expected of graduates in specific subject areas. They show what graduates might reasonably be expected to know, do and understand at the end of their studies). Even in 2019 the UK did not yet have Benchmark Statements that discuss the learning to be done in Higher Education around AI and advanced Machine Learning. Where there is provision for AI and Data Science within degree courses ethics is not generally highlighted as a key concept. As such there can be a lack of focus on the teaching of modelling or ethics as specific skills. The skills required to use a basic statistical model, for example would not be sufficient to start from scratch and build an ethical model reflecting real-world scenarios with which to inform policy or organisational decision making. This is a skill in itself and includes such aspects as awareness of data quality, ethics, user implementation problems, context and an understanding of the environment that the model is being created in. As the field of analytics has progressed so quickly in the last decade modules such as those covering AI have simply been bolted on to Maths or Computing degrees rather than being fully detailed as key areas of study or driving new, more integrated courses of study. This paper posits that gaps at primary, secondary and tertiary educational levels need to be addressed. Implementing and integrating key concepts from school level is essential so that areas such as assumptions, caveats, quality assurance and answering the right questions with constructive challenge become a cultural fixture. This not only helps developers of technology but users of rapidly developing technology also. In addition, leadership and soft skills as part of this education will ensure that a cultural shift can take place and promote continuous improvement in analysis within organisations. The addition of key concepts throughout the educational system and the updating of potentially outdated curriculums is key to ensuring a functional society. A society where every citizen is a user of tech and those that develop it can be ethical and socially responsible in its development. (shrink)

Language suggestive of natural law ethics, similar to the Catholic understanding of ethical foundations, is prevalent in a number of disciplines. But it does not always issue in a full-blooded commitment to objective ethics, being undermined by relativist ethical currents. In law and politics, there is a robust conception of "human rights", but it has become somewhat detached from both the worth of persons in themselves and from duties. In education, talk of "values" imports ethical considerations but hints (...) at a subjectivist view of them. In the psychology and sociology of drug use, ethically thin concepts of "harm minimisation" and "selfimage" dominate discussion and distract attention from the virtue of temperance and the training of character. A more forceful assertion of an ethics based on the worth of persons in these cases would be most desirable. Arguments against objectivity in the fundamentals may be replied to by examining the parallel between ethics and the discipline whose objectivity has been least challenged by relativist arguments, mathematics. (shrink)

What effect has finding the Archimedean point in ourselves had on how we look at ethics? The modern era of philosophy began with Descartes finding within himself an unshakable point from which to pursue knowledge of the world and himself. This intellectual alienation from the world into the universal mathematical structures of the human mind has led to a reversal where, henceforth, production, rather than contemplation, of knowledge became epistemologically superior. Guided by Hannah Arendt’s discussion of the Archimedean point (...) and Earth alienation in the Human Condition, I will argue that this scientific form of thinking made its way into ethical investigation in the age of modern philosophy. Furthermore, I will provide an analysis of Kant’s categorical imperative and John Stuart Mill’s utilitarianism to show that despite modern science’s success in elucidating universal laws of nature, a scientific approach to ethics has not universalized our judgments of good and evil. (shrink)

I came late to philosophy and even later to normative ethics. When I started my undergraduate studies at the University of Toronto in 1970, I was interested in mathematics and languages. I soon discovered, however, that my mathematical talents were rather meager compared to the truly talented. I therefore decided to study actuarial science (the applied mathematics of risk assessment for insurance and pension plans) rather than abstract math. After two years, however, I dropped out of university, (...) went to work for a life insurance company, and started studying on my own for the ten professional actuarial exams. When not studying, I would often go to the public library and I was drawn to the philosophy section—although I had no idea of what philosophy was about. I there saw Logical Positivism, edited by A.J. Ayer. I was interested in logical thinking and I also favored an optimistic attitude towards life (!) and so I thought that the book might be interesting. I checked it out and was absolutely enthralled with the writings of Bertrand Russell, Rudolf Carnap, Carl Hempel and others (if I’m remembering correctly). Of course, I didn’t really understand much of what they were doing, but I did see that they were addressing important problems in a systematic and rigorous manner. I liked it! (shrink)

Originally titled “Is It Murder in Tennessee to Kill a Chimpanzee,” this article argues in some detail that typical legal definitions of “murder” as involving the intentional killing of “a reasonable being” would require classifying the intentional killing of chimpanzees as murder.

The foundations of mathematics and physics no longer start with fundamental entities and their properties like spatial extension, points, lines or the billiard ball like particles of Newtonian physics. Mathematics has abolished these from its foundations in set theory by making all assumptions explicit and structural. Particle physics has become completely mathematical, connecting to physical reality only through experimental technique. Applying the principles guiding the foundations of mathematics and physics to philosophical analysis underscores that only conscious experience (...) has an intrinsic nature. This leads to a version of realistic monism in which the essence and totality of the existence of physical structure is immediate experience in some form. Identifying physical structure with conscious experience allows the application of mathematics to the evolution of consciousness. Some of the implications from Goedel’s Incompleteness Theorem are connected to creativity and ethics. (shrink)

In this commentary, we discuss the nature of reversible and irreversible questions, that is, questions that may enable one to identify the nature of the source of their answers. We then introduce GPT-3, a third-generation, autoregressive language model that uses deep learning to produce human-like texts, and use the previous distinction to analyse it. We expand the analysis to present three tests based on mathematical, semantic, and ethical questions and show that GPT-3 is not designed to pass any of them. (...) This is a reminder that GPT-3 does not do what it is not supposed to do, and that any interpretation of GPT-3 as the beginning of the emergence of a general form of artificial intelligence is merely uninformed science fiction. We conclude by outlining some of the significant consequences of the industrialisation of automatic and cheap production of good, semantic artefacts. (shrink)

In his précis of a recent book, Richard Joyce writes, “My contention…is that…any epistemological benefit-of-the-doubt that might have been extended to moral beliefs…will be neutralized by the availability of an empirically confirmed moral genealogy that nowhere…presupposes their truth.” Such reasoning – falling under the heading “Genealogical Debunking Arguments” – is now commonplace. But how might “the availability of an empirically confirmed moral genealogy that nowhere… presupposes” the truth of our moral beliefs “neutralize” whatever “epistemological benefit-of-the-doubt that might have been extended (...) to” them? In this article, I argue that there appears to be no satisfactory answer to this question. The problem is quite general, applying to all arguments with the structure of Genealogical Debunking Arguments aimed at realism about a domain meeting two conditions. The Benacerraf-Field Challenge for mathematical realism affords an important special case. (shrink)

To reconcile the standard possible worlds model of knowledge with the intuition that ordinary agents fall far short of logical omniscience, a Stalnakerian strategy appeals to two components. The first is the idea that mathematical and logical knowledge is at bottom metalinguistic knowledge. The second is the idea that non-ideal minds are often fragmented. In this paper, we investigate this Stalnakerian reconciliation strategy and argue, ultimately, that it fails. We are not the first to complain about the Stalnakerian strategy. But (...) in contrast to existing complaints, we want to cause trouble for the strategy directly on its home turf. That is, we will advance our objection while granting both the plausibility of the fragmentation component—save for an extreme version of it—and that of the metalinguistic component. Once our central objection to the Stalnakerian strategy is in place, we will show how it negatively affects Adam Elga and Augustín Rayo’s recent attempt to apply the Stalnakerian strategy in the context of Bayesian decision theory. (shrink)

_ Source: _Volume 6, Issue 2-3, pp 120 - 142 This paper aims to contribute to the debate over epistemic versus non-epistemic readings of the ‘hinges’ in Wittgenstein’s _On Certainty_. I follow Marie McGinn’s and Daniele Moyal-Sharrock’s lead in developing an analogy between mathematical sentences and certainties, and using the former as a model for the latter. However, I disagree with McGinn’s and Moyal-Sharrock’s interpretations concerning Wittgenstein’s views of both relata. I argue that mathematical sentences as well as certainties are (...) true and are propositions; that some of them can be epistemically justified; that in some senses they are not prior to empirical knowledge; that they are not ineffable; and that their primary function is epistemic as much as it is semantic. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal (...) science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

Open peer commentary on the article “Ethics: A Radical-constructivist Approach” by Andreas Quale. Upshot: The first of my two main goals in this commentary is to establish thinking of ethics as concepts rather than as non-cognitive knowledge. The second is to argue that establishing models of individuals’ ethical concepts is a scientific enterprise that is quite similar to establishing models of individuals’ mathematical concepts. To accomplish these two primary goals, I draw from my experience of working scientifically with (...) von Glasersfeld for 25 years while he was developing radical constructivism as a coherent model of knowing, and appeal to several of his basic insights to establish constructing models of ethical concepts as a scientific enterprise. (shrink)

The discussions of conatus – force, tendency, effort, and striving – in early modern metaphysics have roots in Aristotle’s understanding of life as an internal experience of living force. This paper examines the ways that Spinoza’s conatus is consonant with Aristotle on effort. By tracking effort from his psychology and ethics to aesthetics, I show there is a conatus at the heart of the activity of the ψυχή that involves an intensification of power in a way which anticipates many (...) of the central insights of early modern and 20th century European philosophy. The first section outlines how Aristotle’s developmental conception of the soul as geometrically ordered lays the foundation for his understanding of effort. The developmental series of powers of the soul are analogous to the series of shapes in mathematics. The second section links the striving of the soul to the gradual acquisition of virtues as a directed activity unifying multiplicity. The third examines the paradigm of self-awareness that Aristotelian effort involves. In the final section I show how ancient Greek theories of music were founded on the experience of striving. The “nature” of music is defined by Aristoxenus, and Theophrastus, in relation to the passion and intentionality of the soul. The geometrical order, as a synthesis of elements in geometry, music, or ethics, is a generative process in which past elements are retained and reintegrated in later stages of development. It requires effort to think geometrically, and the progress of knowledge itself is an integral aspect of all effort. Effort is the lived and self-aware cause which, moving step by step in an orderly and deliberate way, grows and advances upon itself. For both Spinoza and Aristotle, effort is the immanent intelligence which accomplishes what is in the purview of its understanding. Thus, will, in this conception of effort, is not something we already possess innately, but emerges gradually by an effort aimed at improvement. (shrink)

Eleven top Deleuze scholars reclaim Deleuzian philosophy as moral philosophy Ethics plays a crucial, if subtle, role in Gilles Deleuze's philosophical project. Michel Foucault claimed that Anti-Oedipus was `a book of ethics, the first book of ethics to be written in France in quite a long time'. But what is the nature of the immanent ethics that is developed in Deleuze's thought? How does it differ from previous conceptions of ethics? And what paths does it (...) open for future thought, given the ethical challenges facing humanity in so many domains? Each of the eleven essays in this collection explores the ethical dimension of Deleuze's thought along a new and singular trajectory, and in so doing, attempts to reclaim his philosophy as an ethical philosophy. (shrink)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content (...) to key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I (...) argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)