According to a popular ‘explanationist’ argument for moral or mathematical realism the best explanation of some phenomena are moral or mathematical, and this implies the relevant form of realism. One popular way to resist the premiss of such arguments is to hold that any supposed explanation provided by moral or mathematical properties is in fact provided only by the non-moral or non-mathematical grounds of those properties. Many realists have responded to this objection by urging that the explanations provided by the (...) higher-level, moral and mathematical, properties are informative in a way in which their supposed replacements are not, and that this is because moral and mathematical properties are multiply realizable. This chapter responds to this manoeuvre by proposing a way in which moral and mathematical explanations might preserve the distinctive import that multiple realizability brings, without committing those who accept them to the existence of moral and mathematical properties. (shrink)

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)

I offer several reasons for rejecting naturalism as a philosophical viewpoint or program envisaged for two paradigm cases: the case of mathematics and the case of ethics. Semantical, epistemological and metaphysical similarities between the two are investigated and assessed. I then offer a sketch of a different way of understanding the nature of mathematical difficulties and that of ethical puzzles.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.3 (2003) 426-427 [Access article in PDF] Timothy Smiley, editor. Mathematics and Necessity: Essays in the History of Philosophy. New York: Oxford University Press, 2000. Pp. ix + 166. Cloth, $35.00.Mathematics and Necessity contains essays by M. F. Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given to the British Academy in 1998. All concern the history of the philosophical (...) treatment of mathematics, necessity, or both, although there is no single thread running through all three essays.Burnyeat focuses on Plato's understanding of mathematics in the Republic, making use of the Timeaus in particular to defend his interpretation. Plato claimed that in the ideal society mathematics should be studied for a full ten years, which many have thought excessive. To explain it, Burnyeat argues that the role of mathematics is pervasive: ethics and politics are mathematical, because in Plato's view mathematics is a constitutive part of ethical understanding. On Burnyeat's interpretation, goodness is part of the objective world, and mathematics is a science of values. The link between mathematics and goodness in the objective world is forged by the concord, harmony, and unity of proportions, which express the goodness of the Divine Craftsman's beneficent design. Proportions are both intrinsically good and the proper study of mathematics.Burnyeat explains why Plato thinks abstract kinematics (the pure non-sensible counterpart to astronomy) and abstract harmonics lead to knowledge of the good. Abstract kinematics concerns the "harmony of the spheres" described in the Republic. Drawing heavily on the Timeaus, Burnyeat argues that the circular motions of the spheres are motions in the intelligence of the world soul and that there are similar motions in human souls. The world soul and human souls are spatial and have circular motions (invisibility and intangibility make them incorporeal). Souls are the most important subject matter of abstract harmonics—the study of good proportions. A soul that understands the various mathematical disciplines and their relations takes on the harmony and hence goodness of the world soul. Such a soul serves as a model for organizing the social world, a model which properly trained rulers can use to shape a community. Burnyeat's well-argued essay is compelling and exciting. [End Page 427]Ian Hacking's views provide a great contrast to Burnyeat's Plato. Hacking conducts a phenomenological inquiry into why some philosophers have become obsessed by mathematics as a source of philosophical inspiration. He holds that mathematics concerns a relatively minor part of human culture and that the influence of mathematics on philosophy is pernicious; he even describes it as an infection.Hacking examines the views of six philosophers: Plato, Leibniz, Mill, Descartes, Lakatos, and Wittgenstein. Hacking's assessment of the six philosophers is at times disputable, but it is also interesting and stimulating, and serves his phenomenological inquiry.Hacking traces philosophy's infection by mathematics to two factors. A mathematical proof can be perspicuous; that is, a proof can be "grasped," which requires seeing directly why something must be true and being able to recapitulate it. A mathematical proof can also anticipate facts about the world. He thinks that these two factors affect us in two ways: they give us the idea of a priori knowledge, and they dazzle us into unreasoned conviction in proofs. One might respond that, setting aside the feeling proofs give us, we are still justified in being impressed by them. Hacking argues that much of what impresses is based on a misunderstanding of their nature, which he attempts to correct.According to Hacking, a mathematical demonstration involves shifts of meaning that analytically bind the proof and the theorem it establishes. They thereby mutually support each other and provide what he calls self-authentication. The correctness of the proof is supported by the "analytified" truth of the theorem and the theorem is considered true because proven.This bootstrapping feature of mathematical proofs supports Hacking's deflationary account of necessity. Hacking suggests that the "must" of logical necessity rests upon our unwillingness to accept empirical counter-examples to the... (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)

The idea of absolute goodness and the idea of an absolute requitement tend nowadays to be viewed with suspicion in the world of English-speaking philosophy. The tendency is well rooted and has not just arisen by osmosis from the temper of the times. There are various lines of thought, all of them attractive, by which a recent or contemporary academic practitioner of the subject could have been induced into scepticism about an ethics of absolute conceptions.

The idea of absolute goodness and the idea of an absolute requitement tend nowadays to be viewed with suspicion in the world of English-speaking philosophy. The tendency is well rooted and has not just arisen by osmosis from the temper of the times. There are various lines of thought, all of them attractive, by which a recent or contemporary academic practitioner of the subject could have been induced into scepticism about an ethics of absolute conceptions.

How far should our realism extend? For many years philosophers of mathematics and philosophers of ethics have worked independently to address the question of how best to understand the entities apparently referred to by mathematical and ethical talk. But the similarities between their endeavours are not often emphasised. This book provides that emphasis. In particular, it focuses on two types of argumentative strategies that have been deployed in both areas. The first—debunking arguments—aims to put pressure on realism by (...) emphasising the seeming redundancy of mathematical or moral entities when it comes to explaining our judgements. In the moral realm this challenge has been made by Gilbert Harman and Sharon Street; in the mathematical realm it is known as the 'Benacerraf-Field' problem. The second strategy—indispensability arguments—aims to provide support for realism by emphasising the seeming intellectual indispensability of mathematical or moral entities, for example when constructing good explanatory theories. This strategy is associated with Quine and Putnam in mathematics and with Nicholas Sturgeon and David Enoch in ethics. Explanation in Ethics and Mathematics addresses these issues through an explicitly comparative methodology which we call the 'companions in illumination' approach. By considering how argumentative strategies in the philosophy of mathematics might apply to the philosophy of ethics, and vice versa, the papers collected here break new ground in both areas. For good measure, two further companions for illumination are also broached: the philosophy of chance and the philosophy of religion. Collectively, these comparisons light up new questions, arguments, and problems of interest to scholars interested in realism in any area. (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)

I examine the role that mathematics plays in understanding and modelling finance, especially stock markets, and how philosophy affects it. To this end, I explore how mathematics penetrates finance via physics, constructing a ‘financial physics’, and I outline the philosophical backgrounds of this process, in particular the ‘philosophy of equilibrium’ and that of critical points or ‘out-of-equilibrium’. I discuss the main characteristics and a few weaknesses of these mathematizations of financial systems, notably econometrics and econophysics, and I compare (...) the two ways, top-down and bottom-up, of building mathematical approaches to finance. The top-down approach is the most used and the most conservative. I argue that it guarantees a mathematical account, but it is less effective than the more difficult bottom-up approach, which is more appropriate and which may end up without a mathematical account of financial phenomena. I then consider two important issues raised by a mathematical approach to finance, that is, the performative and the reversing side of mathematics, and an ethics of mathematics. In the first case I argue that mathematics not only can be a performative device, but it also enables ‘reversing’ dynamics, that is, a mathematical model may become not a representational tool, but a device of social engineering—a mathematical way of investigating what are the initial conditions and processes needed to obtain an intended result. In the second case I argue that this specific relation between mathematical modelling and prediction raises an ethical question in finance, namely, a responsible construction and use of the mathematical models provided by a financial physics. (shrink)

A twenty‐person working group convened to discuss the ethical and policy considerations of the controversial intervention called “growth attenuation,” and if possible to develop practical guidance for health professionals. A consensus proved elusive, but most of the members did reach a compromise.

The field of quantitative analysis is often mistaken to be a discipline free from ethical burdens. The quantitative financial analyst or “quant” profession holds a position of significant responsibility as the keeper of mathematical models used in complex derivative security pricing and risk management. Despite this responsibility very few postgraduate programs address the teaching of ethics and professional standards in their curriculum, and the credibility of the profession has suffered as a result of several high-profile financial losses. Some of (...) these failures could have been avoided and their impacts diminished if ethical considerations were integrated with quantitative method. Appropriate development in ethics education for quants is needed to identify points in the decision-making process where ethical questions can arise, and to explain how quants can protect stakeholders from the costs of unethical behaviour. An approach to ethics education needs to be flexible and allow for different methods to infuse ethical coverage into the course. Such an approach will go some way towards aligning the profession with other specialisations in banking and avoid the need for complex and unnecessary regulation. (shrink)

This book offers the most complete and up-to-date overview of the philosophical work of Evandro Agazzi, presently the most important Italian philosopher of science, and one of the most influential in the world. Scholars from seven countries explore his contributions in areas ranging from philosophy of physics and general philosophy of science to bioethics, philosophy of mathematics and logic, epistemology of the social sciences and history of science, philosophy of language and artificial intelligence, education and anthropology, metaphysics, and philosophy (...) of religion. Agazzi developed a complete and coherent philosophical system, anticipating some of the turns in the philosophy of science after the crisis of logical empiricism and exerting an equal influence on continental hermeneutic philosophy. His work is characterized by an original synthesis of contemporary analytic philosophy, phenomenology, and classical philosophy, including the scholastic tradition, and these threads are reflected in the different backgrounds of the contributors to this book. While upholding the epistemological value of science against scepticism and relativism, Agazzi eschews scientism by stressing the equal importance of non-scientific forms of thought, such as metaphysics and religion. While defending the freedom of research as a cognitive enterprise, he argues that as a human and social practice it must nonetheless respect ethical constraints. (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)

Philosophers have often wanted to state a principled way of demarcating empirical from non-empirical thought. This was a major concern of the Vienna Circle. In my view, this is an important intellectual project. Although it is not so common now to address the issue directly, it hovers in the background of many discussions. Non-empirical thought comes in different kinds. Perhaps some is a priori. Common candidates are mathematical, logical, modal and moral thought. Some non-empirical thought might be non-cognitive. Common candidates (...) are moral, aesthetic and religious thought. Or some non-empirical thought might be pseudo-empirical. Common candidates are astrological, theistic, Marxist or psychoanalytic thought; such thought masquerades as empirical, but in fact is not. These are all ways that thought can fail to be empirical. Here I shall focus for the most part on the case of morality, where there has been a lively debate about the empirical status of moral thought. I shall probe the prospects for a demarcation principle that can be used to decide whether or not moral thought is empirical. Some philosophers have recently presented what seems like a compelling case for thinking that moral thought is empirical, and so engaging with their arguments will raise the issue of what the empirical/nonempirical distinction amounts to, and also the issue of how to tell which forms of thought are empirical and which are not. (shrink)

A traditional problem of ethics in mathematics is the denial of social responsibility. Pure mathematics is viewed as neutral and value free, and therefore free of ethical responsibility. Applications of mathematics are seen as employing a neutral set of tools which, of themselves, are free from social responsibility. However, mathematicians are convinced they know what constitutes good mathematics. Furthermore many pure mathematicians are committed to purism, the ideology that values purity above applications in mathematics, (...) and some historical reasons for this are discussed. MacIntyre’s virtue ethics accommodates both the good mathematician and the ethics of the social practice of mathematics. It demonstrates that purism is compatible with acknowledging the social responsibility of mathematics. Four aspects of this responsibility are mentioned, two concerning the impact of mathematics via education, and two concerning explicit and implicit applications of mathematics. The last of these opens up the performativity of mathematical and measurement applications in society, which change the very processes they are supposed to measure. Although these applications are not explored in detail, they illustrate the importance of considering the ethics and social responsibility of mathematics in society. MacIntyre’s virtue theory opens a broad approach to the controversial topic of the ethics of mathematics encompassing purism, and absolutist and social constructivist philosophies of mathematics, but still enabling ethical critiques of the impact of mathematics on society. (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.

This paper addresses the notorious logic and semantic difficulties encountered by Lévinas in articulating his ethics of alterity. Tracing the philosophical genesis of this question in Descartes and Heidegger, it recognises Lévinas's claim that there can be no ontological foundation for ethics because ontology would reduce ethics to a form of mathematical ratio. Lévinas is unwilling to deny his phenomenological experience of a desire for goodness and unable to deny his despair at his ontological alienation from the (...) good and so he is driven to seek an irrational link between the human being and a metaphysical ‘good beyond being’. Retrieving an ancient gnostic neo-Platonist working of the same problem from the work of Hans Jonas, the paper reconsiders Lévinas's reading of Plato, specifically his understanding of Socrates’ exposition of erosin the Phaedo, in order to illustrate how his ethics of alterity may allow the human being to refer irrationally to the good beyond being. (shrink)

Edgeworth's original mathematical formalization of utilitarianism as presented in his works of 1877–81 illustrates an intriguing phase in the mutually intertwined history of economics and utilitarianism. In this article I analyse Edgeworth's motivations and point to its interesting implications. In particular, it is pointed out that the starting point of Edgeworth's project had little to do with the field of economics, but formed part of an attempt to present utilitarianism in the most scientific way possible; an attempt made in the (...) context of intensive dispute between the three opposing camps in the field of ethics at the time. Nevertheless, the project concluded with the monograph Mathematical Psychics (1881) that embodied an original relationship between economics and utilitarianism. (shrink)

A translation of the 1913 volume Dernières Pensées, this collection of papers contains much material of interest to the logician and the philosopher of science. In "The Logic of Infinity" Poincaré clarifies the notion of "predicative set" and discusses Zermelo's and Russell's approaches to set theory. "The Evolution of Laws" attempts to formulate the question "do laws of nature evolve?" Two papers concern space and time, two others, the electrostatic and quantum theories of matter. The collection concludes with a pair (...) of papers on the ethical import of science.—P. J. M. (shrink)

This paper addresses the notorious logic and semantic difficulties encountered by Lévinas in articulating his ethics of alterity. Tracing the philosophical genesis of this question in Descartes and Heidegger, it recognises Lévinas's claim that there can be no ontological foundation for ethics because ontology would reduce ethics to a form of mathematical ratio. Lévinas is unwilling to deny his phenomenological experience of a desire for goodness and unable to deny his despair at his ontological alienation from the (...) good and so he is driven to seek an irrational link between the human being and a metaphysical ‘good beyond being’. Retrieving an ancient gnostic neo-Platonist working of the same problem from the work of Hans Jonas, the paper reconsiders Lévinas's reading of Plato, specifically his understanding of Socrates’ exposition of erosin the Phaedo, in order to illustrate how his ethics of alterity may allow the human being to refer irrationally to the good beyond being. (shrink)

This book presents an exploration of the idea of the common or social good, extended so that alternatives with different populations can be ranked. The approach is, in the main, welfarist, basing rankings on the well-being, broadly conceived, of those who are alive. The axiomatic method is employed, and topics investigated include: the measurement of individual well-being, social attitudes toward inequality of well-being, the main classes of population principles, principles that provide incomplete rankings, principles that rank uncertain alternatives, best choices (...) from feasible sets, and applications. The chapters are divided, with mathematical arguments confined to the second part. The first part is intended to make the arguments accessible to a more general readership. Although the book can be read as a defense of the critical-level generalized-utilitarian class of principles, comprehensive examinations of other classes are included. (shrink)

Philosophers since antiquity have argued the merits of mathematics as a normative aid in ethical decision-making and of the mathematization of ethics a theoretical discipline. Recently, Anagnostopoulos, Annas, Broadie and Hutchinson have probed such issues said to be of interest to Aristotle. Despite their studies, the sense in which Aristotle either opposed or proposed a mathematical ethics in subject-matter and method remains unclear. This paper attempts to clarify the matter. It shows Aristotle’s matrix of exactness and inexactness (...) for ethical subject-matter and ethical method in the Nicomachean Ethics. Then it probes a resultant puzzle from the matrix, namely, the HL model of the happy life without consideration of mathematical justice and the HJL model of the happy life with such consideration. Finally, it examines Aristotle’s twofold rationale for differentiating these two models in his overall moral feedback loop system: differences in the intellectual virtue of good deliberation; the priority of friendship over justice for the happy life. This suggests Aristotle saw no objection either to using mathematics as an aid to ethical decision-making for a happy life, or to mathematizing at least some parts of an ethical theory of eudaimonism. (shrink)

In addition to the obvious social and ethical risks, there are philosophical hazards behind artificial intelligence and machine learning. I try to raise here some critical points that might counteract some naive optimism, and warn against the possibility that synthetic intelligence may surreptitiously influence the agenda of science before we can realize it.

This project explored what constitutes “ethical practice of mathematics”. Thematic analysis of ethical practice standards from mathematics-adjacent disciplines (statistics and computing), were combined with two organizational codes of conduct and community input resulting in over 100 items. These analyses identified 29 of the 52 items in the 2018 American Statistical Association Ethical Guidelines for Statistical Practice, and 15 of the 24 additional (unique) items from the 2018 Association of Computing Machinery Code of Ethics for inclusion. Three of (...) the 29 items synthesized from the 2019 American Mathematical Society Code of Ethics, and zero of the Mathematical Association of America Code of Ethics, were identified as reflective of “ethical mathematical practice” beyond items already identified from the other two codes. The community contributed six unique items. Item stems were standardized to, “The ethical mathematics practitioner…”. Invitations to complete the 30-min online survey were shared nationally (US) via Mathematics organization listservs and other widespread emails and announcements. We received 142 individual responses to the national survey, 75% of whom endorsed 41/52 items, with 90–100% endorsing 20/52 items on the survey. Items from different sources were endorsed at both high and low rates. A final thematic analysis yielded 44 items, grouped into “General” (12 items), “Profession” (10 items) and “Scholarship” (11 items). Moreover, for the practitioner in a leader/mentor/supervisor/instructor role, there are an additional 11 items (4 General/7 Professional). These results suggest that the community perceives a much wider range of behaviors by mathematicians to be subject to ethical practice standards than had been previously included in professional organization codes. The results provide evidence against the argument that mathematics practitioners engaged in “pure” or “theoretical” work have minimal, small, or no ethical obligations. (shrink)

Ethics and mathematics have long invited comparisons. On the one hand, both ethical and mathematical propositions can appear to be knowable a priori, if knowable at all. On the other hand, mathematical propositions seem to admit of proof, and to enter into empirical scientific theories, in a way that ethical propositions do not. In this article, I discuss apparent similarities and differences between ethical (i.e., moral) and mathematical knowledge, realistically construed -- i.e., construed as independent of human mind (...) and languages. I argue that some are are merely apparent, while others are of little consequence. There is a difference between the cases. But it is not an epistemological difference per se. The difference, surprisingly, is that ethical knowledge, if it is practical, cannot fail to be objective in a way that mathematical knowledge can. One upshot of the discussion is radicalization of Moore’s Open Question Argument. Another is that the concepts of realism and objectivity, which are widely identified, are actually in tension. (shrink)

That mathematics makes for poor literature is a conclusion as uninteresting as it is inevitable—inevitable because were mathematical prose to score high on a scale of literary value, this result would do more to discredit the scale than glorify the prose. It may, however, help us better understand our cultural landscape if, without attempting a literary appraisal of mathematics or a mathematical appraisal of literature, we search for some community of interest between the formal sciences and the literary (...) arts. This search for common ground will also be an opportunity to learn from some voices seldom heard in conversations about mathematics. Even at the risk of giving our discussion an antiquated flavor, we will... (shrink)

This book is for anyone interested in how to build a teacher education program utilizing the arts as one central modality for teaching and learning or for those interested in building some of their program along these lines. Throughout the book you will find reference to the intersection of ethics, aesthetics, and teaching. We provide an integrated program devoted to good learning and the good society. In the book we discuss how the program came to be and the underlying (...) educational thinking that informs the whole program. This section of the book is invaluable for understanding how the reader can build her/his own arts approach to teacher education. The central section of the book is devoted to the specific coursework of the program. Each author describes in detail how she/he leverages aesthetics and art to expand the possibilities of learning and teaching (including a chapter focused on the core competency course, Teaching, Imagination, Creativity) in language and literacy, psychology of education, science education, mathematics education, social studies education, and classroom management including many examples from our teaching. The book ends with a focus group discussion about the program by former students. (shrink)

This paper considers Peirce's striking remarks about mathematics in a little-known review of Spinoza's Ethics within the larger context of his philosophy of mathematics. It argues that, for Peirce, true mathematical reasoning is always at the vanguard of thought, and resists logical demonstration. Through diagrammatic thought and her pre-theoretical innate faculty of logica utens, the great mathematician is able to see a theorem as true long before the logical apparatus necessary to demonstrate its truth exists. For Peirce, (...) true mathematical thought is in some sense pre-logical, and thus, the logical demonstration of this thought in the form of mathematical proofs is in fact "merely a veil over the living thought.". (shrink)

Ovaj rad pokazuje kako pojam mjere, koji je jedan od najmnogoznačnijih filozofijskih pojmova, počinje dobivati etički smisao tijekom povijesnog razvoja grčke filozofije. U prvom dijelu pokazuje se kako se u Homerovim epovima riječ ‘mjera’ i srodne riječi odnose isključivo na konkretno mjerljive količine. Drugi se dio bavi mijenom značenja, koja se zbiva u pitagorovskom mišljenju. Tu se najviše pažnje poklanja povijesnim okolnostima nastanka grčkog polisa, za što je vrlina umjerenosti bila neophodno potrebna. U tom kontekstu pobliže se razmatraju Filolajevi i (...) Arhitini fragmenti, budući da oni ukazuju na političku važnost mjere i mjerenja. Treći dio posvećen je analizi relevantnog odjeljka iz Platonova Državnika u kojem Platon razlikuje dvije vrste mjerenja: brojčano i drugo, koje se može nazvati ‘deontičkim’, koje mjeri ono više i manje u odnosu na ‘pravu mjeru’ . Druga vrsta mjerenja je također i etičko mjerenje utoliko što uključuje različite prosudbe svrhe, ciljeva i prirode čovjeka.The aim of this article is to show how the concept of »measure«, which is one of the most intricate and manifold philosophical concepts, started acquiring ethical meanings in the course of Greek philosophical thought. In the first part it is discussed how the word »measure« and its cognates refer exclusively to concretely measurable quantities in Homer’s epics. The second part deals with the switch of meaning that started happening in the Pythagorean thought. In this part more attention is paid to historical references of emergence of the Greek polis for which the virtue of temperance was indispensable. In this context fragments of Philolaus and Archytas are more closely observed because they point to political importance of measure and measurement. The third part is dedicated to analysis of the relevant passage in Plato’s Statesman in which Plato distinguishes between two concepts of measurement: one numerical and the other, which can be called »deontic«, which measures more and less with respect to the »right measure« . The second kind of measurement is also ethical because it includes various judgments of purpose, goals, and human nature. (shrink)

Name der Zeitschrift: Jahrbuch für Wissenschaft und Ethik Jahrgang: 20 Heft: 1 Seiten: 293-302.

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)

This paper considers Peirce's striking remarks about mathematics in a little-known review of Spinoza's Ethics within the larger context of his philosophy of mathematics. It argues that, for Peirce, true mathematical reasoning is always at the vanguard of thought, and resists logical demonstration. Through diagrammatic thought and her pre-theoretical innate faculty of logica utens, the great mathematician is able to see a theorem as true long before the logical apparatus necessary to demonstrate its truth exists. For Peirce, (...) true (theoremic) mathematical thought is in some sense pre-logical, and thus, the logical demonstration of this thought in the form of mathematical proofs is in fact "merely a veil over the living thought.". (shrink)

Dominic O'Meara has produced a scholarly and sympathetic account of a most enigmatic subject, namely, the role of mathematics in late Greek Platonic thought. O'Meara traces the path of mathematical philosophy from the Neopythagoreanism of the second and third centuries A.D. through that master of Athenian Neoplatonism, Proclus. Without this study few would recognize the paradigmatic role that mathematics played in Platonic thinkers throughout this period, for mathematics became the model for many forms of philosophical inquiry--not only (...) theology and physics, but even ethics. (shrink)

This chapter examines the role and need for ethics in mathematical practice. Mathematics is one of the few areas of study in which ethics is widely perceived as irrelevant. Many mathematicians and others resist the idea that we need to consider the ethics of both pure and applied mathematics. The foundations of this resistance are analyzed and located in background philosophies and ideologies of purism and neutrality. The range of social practices is investigated, and different (...) ethical problems and issues are brought to light. It is argued that virtuous mathematicians can legitimately pursue mathematics for its own sake, but as citizens they also have a responsibility to care about the social impacts of mathematics. A review of the literature on the social responsibility of science and mathematics reveals that, although long neglected, concerns about the ethics of mathematics are starting to emerge in publications and training practices. Some of the more ethically sensitive areas are explored and three problematic categories are distinguished and exemplified. These are (1) mathematics in public communications, (2) overt applications of mathematics with powerful social impacts, and (3) the hidden performativity of mathematics in restructuring society, institutions, and social practices. (shrink)

Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality (...) condition (roughly: absence of independent explanation). The paper concludes with a moral concerning the judgment-dependence of a posteriori areas of discourse that emerges from consideration of these two a priori cases. (shrink)

Are moral properties intellectually indispensable, and, if so, what consequences does this have for our understanding of their nature, and of our talk and knowledge of them? Are mathematical objects intellectually indispensable, and, if so, what consequences does this have for our understanding of their nature, and of our talk and knowledge of them? What similarities are there, if any, in the answers to the first two questions? Can comparison of the two cases shed light on which answers are most (...) plausible in either case? This chapter – the introduction to the volume Explanation in Ethics and Mathematics – elucidates these questions and sketches their history. (shrink)

It is commonly suggested that evolutionary considerations generate an epistemological challenge for moral realism. At first approximation, the challenge for the moral realist is to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. An important question surrounding this challenge is the extent to which it generalizes. In particular, it is of interest whether the Evolutionary Challenge for moral realism is equally a challenge for (...) mathematical realism. It is widely thought not to be. In this paper, I argue that the Evolutionary Challenge for moral realism is equally a challenge for mathematical realism. Along the way, I substantially clarify the Evolutionary Challenge, discuss its relation to more familiar epistemological challenges, and broach the problem of moral disagreement. The paper should be of interest to ethicists because it places pressure on anyone who rejects moral realism on the basis of the Evolutionary Challenge to reject mathematical realism as well. And the paper should be of interest to philosophers of mathematics because it presents a new epistemological challenge for mathematical realism that bears, I argue, no simple relation to Paul Benacerraf's familiar challenge. (shrink)