We set out a doctrine about truth for the statements of mathematics—a doctrine which we think is a worthy competitor to realist views in the philosophy of mathematics—and argue that this doctrine, which we shall call ‘mathematical relativism’, withstands objections better than do other non-realist accounts.

This chapter contains sections titled: Abstract Introduction Mathematical Relativism: Does Everything Go In Mathematics? Conceptual, Structural and Logical Relativity in Mathematics Mathematical Relativism and Mathematical Objectivity Mathematical Relativism and the Ontology of Mathematics: Platonism Mathematical Relativism and the Ontology of Mathematics: Nominalism Conclusion: The Significance of Mathematical Relativism References.

I discuss the idea of relativistic causality, i.e., the requirement that causal processes or signals can propagate only within the light-cone. After briefly locating this requirement in the philosophy of causation, my main aim is to draw philosophers' attention to the fact that it is subtle, indeed problematic, in relativistic quantum physics: there are scenarios in which it seems to fail. I set aside two such scenarios, which are familiar to philosophers of physics: the pilot-wave approach, and the Newton-Wigner representation. (...) I instead stress two unfamiliar scenarios: the Drummond-Hathrell and Scharnhorst effects. These effects also illustrate a general moral in the philosophy of geometry: that the mathematical structures, especially the metric tensor, that represent geometry get their geometric significance by dint of detailed physical arguments. (shrink)

The need for explicit theoretical reflection on cross-cultural bioethics continues to grow as the spread of communication technologies and increased human migration has made interactions between medical professionals and patients from different cultural backgrounds much more common. I claim that this need presents us with the following dilemma. On the one hand, we do not want to operate according to an imperialist ethical framework that denies and silences the legitimacy of cultural values other than our own. On the other hand, (...) we do not want to backslide into a form of cultural relativism that is unable to critically appraise cultural practices that are harmful, unjust, or oppressive. I examine two prominent attempts – the principlism of Tom Beauchamp and James Childress and the Contractarianism of Robert Baker – to frame cross-cultural bioethics between these two extremes and argue that both approaches have significant flaws. The principlist approach fails to provide a non-question begging way to identify cross-cultural norms that does not already assume the universal legitimacy of moral principles dominant in North American society. Baker's contractarianism cannot grapple with the realities of political power imbalances that often characterize cross-cultural moral disputes. I suggest that a naturalized feminist framework, though not free of its own theoretical difficulties, provides the best alternative for approaching moral diversity respectfully and critically. (shrink)

ArgumentThis paper investigates the conceptual treatment and mathematical modeling of force in Newton's Principia. It argues that, contrary to currently dominant views, Newton's concept of force is best understood as a physico-mathematical construct with theoretical underpinnings rather than a “mathematical construct” or an ontologically “neutral” concept. It uses various philosophical and historical frameworks to clarify interdisciplinary issues in the history of science and draws upon the distinction between axiomatic systems in mathematics and physics, as well as discovery (...) patterns in science. It also dwells on Newton's “philosophy” of mathematics, described here in terms of mathematical naturalism. This philosophy considers mathematical quantities to be physically significant quantities whose motions are best mapped by geometry. It then shows that to understand the epistemic status of force in the Principia, it is important to scrutinize both Newton's mathematical justificatory strategies and his background assumptions about force – without constructing, however, an overarching metaphysical framework for his science. Finally, the paper studies scientific attempts to redefine or eliminate force from science during the period between Newton and Laplace. From a philosophical standpoint, the paper implicitly suggests that questions about the reality of force be distinguished from questions about the validity of force, and that both sets of questions be distinguished from questions about the utility of the concept of force in science. (shrink)

The main purpose of this paper is to characterize and compare two forms any relativist thesis can take: indexical relativism and genuine relativism. Indexical relativists claim that the implicit indexicality of certain sentences is the only source of relativity. Genuine relativists, by contrast, claim that there is relativity not just at the level of sentences, but also at propositional level. After characterizing each of the two forms and discussing their difficulties, I argue that the difference between the two (...) is significant. (shrink)

I investigate the implication of the truth-relativist’s alleged ‘ faultless disagreements’ for issues in the epistemology of disagreement. A conclusion I draw is that the type of disagreement the truth-relativist claims to preserve fails in principle to be epistemically significant in the way we should expect disagreements to be in social-epistemic practice. In particular, the fact of faultless disagreement fails to ever play the epistemically significant role of making doxastic revision rationally required for either party in a disagreement. That the (...) truth-relativists’ disagreements over centred content fail to play this epistemically significant role that disagreements characteristically play in social epistemology should leave us sceptical that disagreement is what the truth-relativist has actually preserved. (shrink)

It has sometimes been argued that Ludwig Wittgenstein’s philosophy is marked by relativism. According to this view people view general concepts in relative terms due to different considerations and perceptions. More specifically, there is a tendency among some philosophers to assign the Wittgensteinian concept of form of life to the idea of culture. The argument goes like this: as language games and forms of life develop within a particular culture and cultures are different they are no absolute phenomena but (...) rather relative depending on the frame within which they developed. I would like to take up this thought and examine whether we can find any relativistic position in Wittgenstein’s writings. The question is whether the existence of a multitude of forms of life points to any sort of relativism. Many philosophers have speculated over the significant concept of “form of life”. There has been an ongoing debate over the question whether the concept is to be understood in the singular or in the plural. The discussion here revolves around the question of whether there is only one prevailing form of life to which others can be subsumed. Or else, is there a variety of coexisting forms of life of equal value? (shrink)

A total of 378 mathematics undergraduates (selected for being strong at “systemizing”) and 414 students in other (control) disciplines at Cambridge University were surveyed with two questions: (1) Do you have a diagnosed autism spectrum condition? (2) How many relatives in your immediate family have a diagnosed autism spectrum condition? Results showed seven cases of autism in the math group (or 1.85%) vs one case of autism in the control group (or 0.24%), a ninefold difference that is significant. Controlling for (...) sex and general population sampling, this represents a three- to sevenfold increase for autism spectrum conditions among the mathematicians. There were 7 of 1,405 (or 0.5%) cases of autism in the immediate families of the math group vs 2 of 1,669 (or 0.1%) cases in the immediate families of the control group, which again is a significant difference. These results confirm a link between autism and systemizing, and they suggest this link is genetic given the association between autism and first-degree relatives of mathematicians. (shrink)

The thirty-three essays in <I>Relativism: A Contemporary Anthology</I> grapple with one of the most intriguing, enduring, and far-reaching philosophical problems of our age. Relativism comes in many varieties. It is often defined as the belief that truth, goodness, or beauty is relative to some context or reference frame, and that no absolute standards can adjudicate between competing reference frames. Michael Krausz's anthology captures the significance and range of relativistic doctrines, rehearsing their virtues and vices and reflecting on (...) a spectrum of attitudes. Invoking diverse philosophical orientations, these doctrines concern conceptions of relativism in relation to facts and conceptual schemes, realism and objectivity, universalism and foundationalism, solidarity and rationality, pluralism and moral relativism, and feminism and poststructuralism. Featuring nine original essays, the volume also includes many classic articles, making it a standard resource for students, scholars, and researchers. <B>Table of Contents:</B> Foreword by Alan Ryan Preface Introduction Michael Krausz <B>Part I. Orienting Relativism</B> 1. Mapping Relativisms Michael Krausz 2. A Brief History of Relativism Maria Baghramian <B>Part II. Relativism, Truth, and Knowledge</B> 3. Subjective, Objective, and Conceptual Relativisms Maurice Mandelbaum 4. “Just the Facts, Ma’am!” Nelson Goodman 5. Relativism in Philosophy of Science Nancy Cartwright 6. The Truth About Relativism Joseph Margolis 7. Making Sense of Relative Truth John MacFarlane 8. On the Very Idea of a Conceptual Scheme Donald Davidson 9. Truth and Convention: On Davidson’s Refutation of Conceptual Relativism Hilary Putnam 10. Conceptual Schemes Simon Blackburn 11. Relativizing the Facts Paul A. Boghossian 12. Targets of Anti-Relativist Arguments Harvey Siegel 13. Realism and Relativism Akeel Bilgrami <B>Part III. Moral Relativism, Objectivity, and Reasons</B> 14. Moral Relativism Defended Gilbert Harman 15. The Truth in Relativism Bernard Williams 16. Pluralism and Ambivalence David B. Wong 17. The Relativity of Fact and the Objectivity of Value Catherine Z. Elgin 18. Senses of Moral Relativity David Wiggins 19. Ethical Relativism and the Problem of Incoherence David Lyons 20. Understanding Alien Morals Gopal Sreenivasan 21. Value: Realism and Objectivity Thomas Nagel 22. Intuitionism, Realism, Relativism, and Rhubarb Crispin Wright 23. Moral Relativism and Moral Realism Russ Schafer-Landau <B>Part IV. Relativism, Culture, and Understanding</B> 24. Anti Anti-Relativism Clifford Geertz 25. Solidarity or Objectivity? Richard Rorty 26. Relativism, Power, and Philosophy Alasdair MacIntyre 27. Internal Criticism and Indian Rationalist Traditions Martha C. Nussbaum and Amartya Sen 28. Phenomenological Rationality and the Overcoming of Relativism Jitendra N. Mohanty 29. Understanding and Ethnocentricity Charles Taylor 30. Relativism and Cross-Cultural Understanding Kwame Anthony Appiah 31. Relativism, Persons, and Practices Amélie Oksenberg Rorty 32. One What? Relativism and Poststructuralism David Couzens Hoy 33. Must a Feminist Be a Relativist After All? Lorraine Code List of Contributors Index. (shrink)

Moral relativism is an attractive position, but also one that it is difficult to formulate. In this paper, we propose an alternative way of formulating moral relativism that locates the relativity of morality in the property that makes moral claims true. Such an approach, we believe, has significant advantages over other possible ways of formulating moral relativism. We conclude by considering a few problems such a position might face.

Consider the following facts about the average, philosophically untrained moral relativist: (1.1) The average moral relativist denies the existence of “absolute moral truths.” (1.2) The average moral relativist often expresses her commitment to moral relativism with slogans like ‘What’s true (or right) for you may not be what’s true (or right) for me’ or ‘What’s true (or right) for your culture may not be what’s true (or right) for my culture.’ (1.3) The average moral relativist endorses relativistic views of (...) morality without endorsing relativistic views about science or mathematics. (1.4) The average moral relativist takes moral relativism to be non-relatively true and does not think there is anything contradictory about doing so. (1.5) The average moral relativist adopts an egalitarian attitude toward a wide range of moral values, practices and beliefs, claiming they are all equally legitimate or correct. (1.6) The average moral relativist often admonishes others to be more tolerant of those who engage in alternative ethical practices and to refrain from making negative moral judgments about them. (1.7) The average moral relativist sometimes makes negative moral judgments about the behavior of others—e.g., by harshly judging moral absolutists to be intolerant—but is less inclined to do so when the relativist’s metaethical views are salient in a context of judgment. (1.8) The average moral relativist takes anthropological evidence concerning the worldwide diversity of ethical views and practices to support moral relativism. (shrink)

It was shown in Dirac A117, 610; A118, 351, 1928) that the ultra-violet divergence in quantum electrodynamics is caused by a violation of the time-energy uncertainly relationship, due to the implicit assumption of infinitesimal time information. In Wheeler et al. it was shown that Einstein’s special theory of relativity and Maxwell’s field theory have mathematically equivalent dual versions. The dual versions arise from an identity relating observer time to proper time as a contact transformation on configuration space, which leaves phase (...) space invariant. The special theory has a dual version in the sense that, for any set of n particles, every observer has two unique sets of global variables \\) and \\) to describe the dynamics, where \ is the canonical center of mass. In the \\) variables, time is relative and the speed of light is unique, while in the \\) variables, time is unique and the speed of light is relative with no upper bound. In the Maxwell case, the two sets of particle wave equations are not equivalent. The dual version contains an additional longitudinal radiation term that appears instantaneously with acceleration, leading to the prediction that radiation from a cyclotron will not produce photoelectrons. A major outcome is the dual unification of Newtonian mechanics and classical electrodynamics with Einstein’s special theory of relativity, without the need for point particles, without a self-energy divergency and, without need for the problematic Lorentz–Dirac equation. The purpose of this paper is to introduce the dual theory of relativistic quantum mechanics. In our approach, we obtain three distinct dual relativistic wave equations that reduce to the Schrödinger equation when minimal coupling is turned off. We show that the dual Dirac equation provides a new formula for the anomalous magnetic moment of a charged particle. We can obtain the exact value for the electron g-factor and phenomenological values for the muon and proton g-factors. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are (...) vulnerable to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

This paper presents an alternative conceptual foundation for biological evolution. First the causal and statistical perspectives on evolutionary fitness are analyzed, finding them to implicitly depend on each other, and hence cannot be individually fundamental. It is argued that this is an instance of a relativistic perspective over evolutionary phenomena. New accounts of fitness, the struggle for life, and Natural Selection are developed under this interpretation. This biological relativism is unique in that it draws from General Relativity in physics, (...) unlike previous theories that drew upon statistical mechanics or Newtonian dynamics. A mathematical law of evolutionary change, as well as new theoretical biological concepts to underpin it, are likewise developed. The law and theory are then applied to give examples of how both cornerstones and edge cases can be understood using these new methods. Using General Relativistic Biology provides fresh insight into evolution, all while preserving the core, canonical scientific research program. (shrink)

The physical motivations for the dynamical group are presented and it is shown how Barut's mathematical speculations were combined with the idea of an elementary length to provide group theoretical models of relativistic extended objects. Then the simplest nonrelativistic and relativistic models are described.

The relativistic three-particle systems are studied within the framework of Relativistic Schrödinger Theory, with emphasis on the determination of the energy functional for the stationary bound states. The phenomenon of entanglement shows up here in form of the exchange energy which is a significant part of the relativistic field energy. The electromagnetic interactions become unified with the exchange interactions into a relativistic U gauge theory, which has the Hartree–Fock equations as its non-relativistic limit. This yields a general framework for treating (...) entangled states of relativistic many-particle systems, e.g., the N-electron atoms. (shrink)

Both decentralization of state law and cultural relativism have been fundamentally embedded in legal pluralism. As a scholarly trend in law and society, it has insightfully challenged the underpinnings of analytical positivist jurisprudence. Nevertheless, a theoretical concept of political power has significantly been missing in research on the plurality of legal practices in various jurisdictions. This Article aims to critically offer a theoretical concept of political power that takes legal decentralization and cultural relativism seriously and yet points to (...) how and where we should look into political power, assuming that legal pluralism itself may be a strategy of elites and nation-states amid globalization. First, the Article explores the contributions of legal pluralism, and its limits, in intellectually revolting against analytical positivist jurisprudence. Second, it explicates why a concept of political power has been missing, and why such a concept is required for better comprehension of legal pluralism. Third, it calls for a look into three sites of political power in the praxis of legal pluralism: politics of identities, non-ruling communities, and neo-liberal globalization. Last, the Article constructs a concept of political-legal transformations that enables us to unveil political power in the context of de-centralized legal pluralities. Power is produced in, resides in and is generated in the dynamic interactions between nation-states, localities and global agents. Transformative relations along these dimensions allow the nation-state to forfeit some elements of power, both in economics and in law, but they also enable it to maintain some essential ingredients of political power that are often veiled in the rhetoric of globalized pluralism. (shrink)

This essay discusses ways in which contemporary mathematical logic may be reconciled with John Dewey’s logical theory. Standard formal techniques drawn from dynamic modal logic, situation theory, generative grammar, generalized quantifier theory, category theory, lambda calculi, game theoretic semantics, network exchange theory, etc., are accommodated within a framework consistent with Dewey’s Logic: The Theory of Inquiry (1938). This essay outlines some basic features of Dewey’s logical theory, working in a top-down fashion through various technical notions pertaining to existential and (...) ideational aspects of inquiry, from contextual notions of situations and ideologies, to primitive operational and empirical notions of modes of action and particular qualities. The aim here is to display the scope of Dewey’s logical theory and thus establish a framework relative to which details may be explored without losing sight of their larger significance. (shrink)

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which (...) are essential to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)

Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Godel's Incompleteness theorems; and semantic content and designation. Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians.

Recent no-go theorems for absolute facts in single-world interpretations are widely considered the strongest arguments in favour of ‘quantum relativism’: interpretations according to which measurement results are observer-relative. In this paper, however, I show that relativist interpretations are themselves vulnerable to mathematically identical ‘revenge’ theorems, unless they assume a particularly radical form. To this end, a novel distinction between tame and feral varieties of quantum relativism is introduced, where a relativist interpretation counts as tame if and only if (...) it postulates i) absolute facts about relative facts (absolute facts of second order) and ii) the possibility of communication across perspectives. While at first sight preferable to feral views, tame relativism turns out to be subject to revenge problems: arguments against absolute measurement outcomes can be modified and turned against absolute facts about which outcome a measurement has relative to a particular observer. This observation is developed in detail for the case of the ‘Local Friendliness’ EWF theorem. Moreover, it is shown to generalise to absolute facts about the facts that obtain relative to sequences of specified reference systems of arbitrary length. This suggests that only the most radical types of relativist interpretations have a hope of being motivated with recourse to EWF scenarios. (shrink)

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein's famous E = mc² . The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.

A mechanism describing state reduction dynamics in relativistic quantum field theory is outlined. The mechanism involves nonlinear stochastic modifications to the standard description of unitary state evolution and the introduction of a relativistic field in which a quantized degree of freedom is associated to each point in spacetime. The purpose of this field is to mediate in the interaction between classical stochastic influences and conventional quantum fields. The equations of motion are Lorentz covariant, frame independent, and do not result in (...) divergent behavior. It is shown that the mathematical framework permits the specification of unambiguous local properties providing a connection between the model and evidence of real world phenomena. The collapse process is demonstrated for an idealized example. (shrink)

Through the explicit introduction of hyperplane dependence as a form of relativistic dynamical evolution, we construct a manifestly covariant description of a single positive energy particle interacting with any one of a large class of “moving” external potentials. In1+1 dimensions, the simplified mathematics allows us to display a number of general properties of solutions to the equations of motion for evolution on hyperplanes.

Relativism is one of the most tenacious theories about truth, with a pedigree as old as philosophy itself. Nearly as ancient is the chief criticism of relativism, namely the charge that the theory is self-refuting. This paper develops a logic of relativism that (1) illuminates the classic self-refutation charge and shows how to escape it; (2) makes rigorous the ideas of truth as relative and truth as absolute, and shows the relations between them; (3) develops an intensional (...) logic for relativism; (4) provides a framework in which relativists can consistently promote ethical, mathematical, scientific, religious, and political truths (among others) as being relative; (5) argues that the notion of incommensurability is far less troubling than is commonly thought; and (6) argues that the concept of a perspective as needed by the theory is not prey to Davidson's well-known critique of conceptual schemes. The paper will not defend relativism as the correct theory of truth, nor will it provide a fully satisfying theory about the nature of a perspective. The logic of relativism is primarily meant to provide a formal framework in which relativists can consistently develop their theories. This alone is a considerable step forward, since the debate about relativism often founders upon the rock of self-refutation. It is argued that while 'everything is relative' is inconsistent, 'everything true is relatively true' is not. The latter is all a relativist really needs. (shrink)

In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of argumentative products, but little attention has been paid to the creative process of arguing. Mathematics can be used as a clear example to illustrate some significant theoretical differences between mathematical practice and the products of it, to differentiate the distinct components of the arguments, and to emphasize the need to address the different types of argumentative discourse and argumentative situation in the practice. (...) I consider some issues of recent papers associated with mathematical argumentation in an attempt to contribute to the discussion about the role of arguing in mathematical practice and in the evaluation of the products of this practice. I apply this discussion to learning environments to defend the thesis that argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity. (shrink)

A rank function for a directed graph G assigns elements of a well ordering to the vertices of G in a fashion that preserves the order induced by the edges. While topological sortings require a one-to-one matching of vertices and elements of the ordering, rank functions frequently must assign several vertices the same value. Theorems stating basic properties of rank functions vary significantly in logical strength. Using the techniques of reverse mathematics, we present results that require the subsystems ${\ensuremath{\vec{RCA}_0}}$ , (...) ${{\vec{ACA}_0}}$ , ${{\vec{ATR}_0}}$ , and ${{\vec{\Pi^1_1 - CA}_0}}$. (shrink)

Retention of structure across theory change has been invoked in support of a ‘structural’ alternative to more traditional entity-based scientific realism. In that context the transition from Newtonian mechanics to the Special Theory of Relativity is often regarded as a very significant instance of structural preservation, or retention, associated with correspondence-based recovery. The joint derivation, from a small set of elementary and ontologically neutral assumptions, of both the Galilei and the Lorentz transformation exemplifies the virtues of structural approaches to the (...) foundations of physical theories. The common origination of the resulting two relativistic frameworks sheds light on both the successes and the limitations of correspondence claims. However, the cognitive-operational character of the basic assumptions lends no support to the structural realist’s ‘inference to the best explanation’. (shrink)

We advance an approach to logical contexts that grounds the claim that logic is a local matter: distinct contexts require distinct logics. The approach results from a concern about context individuation, and holds that a logic may be constitutive of a context or domain of application. We add a naturalistic component: distinct domains are more than mere technical curiosities; as intuitionistic mathematics testifies, some of the distinct forms of inference in different domains are actively pursued as legitimate fields of research (...) in current mathematics, so, unless one is willing to revise the current scientific practice, generalism must go. The approach is advanced by discussing some tenets of a similar argument advanced by Shapiro, in the context of logic as models approach. In order to make our view more appealing, we reformulate a version of logic as models approach following naturalistic lines, and bring logic closer to the use of models in science. (shrink)

In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less prevalent (...) in mathematics papers than in physics papers or in general English. Furthermore, we find that the proportion with which mathematicians use expressions related to ‘explaining why’ and ‘explaining how’ is significantly different to the equivalent proportion in physics and in general English. We discuss possible accounts for these differences. (shrink)

The mathematical framework of Relativistic Schrödinger Theory (RST) is generalized in order to include the self-interactions of the particles as an integral part of the theory (i.e. in a non-perturbative way). The extended theory admits a Lagrangean formulation where the Noether theorems confirm the existence of the conservation laws for charge and energy–momentum which were originally deduced directly from the dynamical equations. The generalized RST dynamics is applied to the case of some heavy helium-like ions, ranging from germanium (Z=32) (...) to bismuth (Z=83), in order to compute the interaction energy of the two electrons in their ground-state. The present inclusion of the electron self-energies into RST yields a better agreement of the theoretical predictions with the experimental data. (shrink)

What is truth, goodness, or beauty? Can we really define these concepts without the idea of a frame of reference? In the newest addition to the New Dialogues in Philosophy series, Michael Krausz presents fictional dialogues between four former classmates who hold significantly different views about these questions. As they travel in India, a place with unfamiliar concepts and customs, these four friends debate the rightness of relativism and absolutism. Are these concepts irreconcilable? Might there be a better view (...) that goes beyond both of them? These lively discussions provide students with an accessible introduction to one of the most enduring and far-reaching philosophical problems of our age. (shrink)

The significance of the de Broglie/Bohm hidden-particle position in the relativistic regime is addressed, seeking connection to the single-particle Newton–Wigner position. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here for free spin-0 particles in 1 + 1 dimensions and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohm-theoretic description of the acausal explosion (...) of a specific Newton–Wigner-localized state is presented in detail. The presence of virtual pairs found is interpreted as the Bohm picture of the spatial extension beyond single point particles proposed in the 1960s as to why space-like hyperplane dependence of the Newton–Wigner wavefunctions may be needed to achieve Lorentz covariance. For spin-1/2 particles the convective current is speculatively utilized for achieving parity with the spin-0 theory. The spin-0 improper quantum potential is generalized to an improper stress tensor for spin-1/2 particles. (shrink)

Philosophers have developed several detailed accounts of what makes some mathematical proofs explanatory. Significantly less attention has been paid, however, to what makes some proofs more explanatory than other proofs. That is problematic, since the reasons for thinking that some proofs explain are also reasons for thinking that some proofs are more explanatory than others. So in this paper, I develop an account of comparative explanation in mathematics. I propose a theory of the `at least as explanatory as' relation (...) among mathematical proofs. (shrink)

This article explains that and how Robert S. Hartman and Brand Blanshard, two of the most insightful philosophers of the 20th Century, were complete rationalists in their approach to philosophical problems, especially those in value theory. They both rejected emotive, subjectivist, and relativistic approaches to ethical values. Both were convinced that “intrinsic goodness” is the most important, meaningful, and basic of all ethical or moral concepts. Just how they understood reasonableness and the task of philosophers is explored. Significant differences between (...) the two on the meaning of “good” are examined. Blanshard’s equating of “intrinsically good” with “fulfillment and pleasure” is critiqued, and Hartman’s alternative to this is presented and defended. (shrink)

This volume gathers papers by many of the best-known philosophers now at work on issues of realism and relativism across the field of philosophy. The result is representative of the best cutting-edge work in the area. Includes work both on the ontology and epistemology of the normative as well as on more general metaphysical and epistemological issues of relativism and realism Essays include discussions of relativism and the first-person perspective, underdetermination and realism, and mathematical realism and (...) epistemic relativism, plus many other points of reflection. (shrink)

In his paper, “Higher‐order truths about chmess,” Daniel Dennett argues that “[m]any projects in contemporary philosophy are artifactual puzzles of no abiding significance.” In other words, much contemporary academic philosophy is a waste of time. In this chapter, we use mathematics, models, and metaphysics, to expand and clarify Dennett's chmess analogy. We further the argument that some contemporary academic philosophy loses its way and chases chmess‐like endeavors – arguing that philosophy is bloated by extraneous, esoteric, and bizarre philosophical projects (...) that aren't detached from reality but only related to it tangentially. Chmess‐like games, like some contemporary academic philosophy, nick reality and then shoot off on their own trajectory, often down intellectual rabbit holes. (shrink)

The International research Library of Philosophy collects in book form a wide range of important and influential essays in philosophy, drawn predominantly from English-language journals. Each volume in the library deals with a field of enquiry which has received significant attention in philosophy in the last 25 years and is edited by a philosopher noted in that field.

Mark Wilson presents a highly original and broad-ranging investigation of the way we get to grips with the world conceptually, and the way that philosophical problems commonly arise from this. He combines traditional philosophical concerns about human conceptual thinking with illuminating data derived from a large variety of fields including physics and applied mathematics, cognitive psychology, and linguistics. Wandering Significance offers abundant new insights and perspectives for philosophers of language, mind, and science, and will also reward the interest of (...) psychologists, linguists, and anyone curious about the mysterious ways in which useful language obtains its practical applicability. (shrink)

These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is _a priori_ or _a posteriori_ in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification.

In 1935, a group of mathematical statisticians in the United States announced the founding of a new professional organization, the Institute of Mathematical Statistics. In the coming years, this institution became a significant source of support for the growing community of mathematical statisticians. Thus it would seem that the formation of the Institute marked the beginning of the process of the creation of that community. This paper, however, argues that those who initially participated in the Institute constituted (...) a small, loosely organized, but wellconnected group within the broader scientific world even before they joined together formally to promote their discipline. We will examine some aspects of their education, employment, and interests in order to determine how American mathematical statisticians set themselves apart from other scientists and helped to define their discipline before the community acquired its official status. (shrink)

It has been almost eighty years since Paul Dirac delivered a lecture on the relationship between mathematics and physics and since 1960 that Eugene Wigner wrote about the unreasonable effectiveness of mathematics in the natural sciences. The field of cosmology and efforts to define a more comprehensive theory have changed significantly since the 1960s; thus, it is time to refocus on the issue. This paper expands on ideas addressed by these two great physicists, specifically, the ultimate effectiveness of mathematics to (...) describe nature. After illustrating how theoretical physic predicted discoveries, the concepts of established theories are summarized. Then, the “world equation” which combines all key physics theories is conceptually described. As the possible last piece of the puzzle, String Theory is briefly defined. And last, a cursory overview of an extreme proposal is presented — the world as a mathematical object. However, there is no fundamental reason to believe that math will allow mankind to completely comprehend nature. If math does not provide a comprehensive theory, nature will retain her secrets. (shrink)

Metaethical relativism is the view that whether a moral claim is true depends on the standards endorsed by an individual or society. This view is attractive because it allows one to hold that moral claims can be true or false in an ordinary correspondence sense, without being committed to the view that moral claims state objective facts. But what could it mean to say that a whether a moral claim is true depends on an individual or society’s standards? How (...) could a single claim be true or false in an ordinary sense, and yet be true for some people while false for others? This thesis defends a certain answer to this question: metaethical truth relativism. Metaethical relativists have historically endorsed a different view: content relativism, which is the view that the semantic contents of moral claims vary across contexts of use. However, semanticists working outside metaethics have recently shown that an alternative exists, namely truth relativism: the view that the truth values of moral contents vary across contexts. Metaethicists have been slow to investigate the significance of this discovery. This is surprising, because truth relativism appears to avoid some of the most influential objections to content relativist views. Thus, this development in semantics appears to offer an exciting new direction for research on relativism in metaethics. The scope of this thesis is not narrowly semantic, however. Questions in moral semantics often interconnect with questions in substantive metaethics, and the question of whether ethical thought and talk is semantically relativistic is no exception. Therefore, while the core of the this thesis focuses on moral semantics, it also contains a fair amount of substantive metaethical material, located primarily in the first and final chapters. The plan of the thesis is as follows. Chapters 1 & 2 begin with a discussion of the most historically influential argument for metaethical relativism: the argument from disagreement. Chapter 1 discusses an epistemic version of this argument, but concludes that this version provides only limited evidence for relativism. However, chapter 2 introduces a different version of the argument, with a more semantic focus, and begins to make the case that this argument provides evidence not only for metaethical relativism broadly construed, but for truth relativism in particular. The broad strokes of this argument are that there can be moral disagreements which are faultless and furthermore that only truth relativism can vindicate the sense that these are genuine disagreements, rather than mere differences of opinion. Chapter 2 presents empirical evidence for the claim that putative moral disagreements can be faultless. Chapters 3 and 4 then take a deep dive into the topic of disagreement. Chapter 3 develops a positive account of disagreement which is available only to truth relativists. And chapter 4 advances detailed arguments against contextualist accounts of disagreement. Chapters 3 and 4 together thus defend in detail the claim that truth relativism can, while content relativism cannot, account for the sense that there is genuine disagreement in the relevant cases. Chapters 2 through 4, therefore, form a single long argument for truth relativism. They show that there is a compelling argument for relativism broadly construed, and that deep analysis of the issues surrounding this argument favour truth relativism over content relativism. Chapter 5 zooms back out to the broader metaethical landscape, discussing two remaining objections to relativism. The first objection is that relativism has objectionably subjectivistic implications, and the second is that moral standards cannot helpfully be understood as determined by societies. In the course of responding to these objections, natural opportunities to flesh out the substantive metaethical commitments of the theory will arise. And this will set us up to conclude with a brief comparison of truth relativism with some of the other main options in the space of substantive metaethical theory. (shrink)

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of (...) his latest thoughts on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained). (shrink)

In 1908, Henri Poincar? claimed that: ...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Towards the end of the twentieth century, with many (...) more mathematical facts since discovered, several mathematicians proposed overarching schemes to organise the facts they considered most significant. In this paper, I shall briefly discuss three of these schemes (those of Arnold, Atiyah, and Baez and Dolan), before drawing some philosophical consequences from their attempts. Rather than the kind of claim made by Frege that with the entry of imaginary numbers we reach the ?natural end of the domain of numbers?, we are dealing here with a more open-ended sense of conceptual growth. I shall illustrate this theme by discussing the elaboration of algebraic structures designed to measure symmetry. What emerges is that at any one time mathematicians are operating with a notion very similar to that discussed in the philosophy of science under the heading of ?natural kinds?. We find ?quasi-causal? talk of properties being ?responsible? for a phenomenon, projectability, the transfer of robust mechanisms between domains, and reference to entities not yet fully determined. Debates in philosophy of science prompt further questioning as to the ?naturalness? of these mathematical kinds, whether one should expect them to be dependent on varying human interests, whether there is a distinction between artefactual and real kinds, and whether there is convergence of kinds. I believe that these questions present a wonderful opportunity for a philosophy of mathematics to treat ?real? mathematics, while making powerful points of contact with philosophy of science, and philosophy in general. (shrink)

In her book Extended Rationality: A Hinge Epistemology (2015), Annalisa Coliva puts forward an anti-sceptical proposal based on the idea that the notion of rationality extends to the unwarrantable presuppositions “that make the acquisition of perceptual warrants possible” (2015: 150). These presuppositions are commonly the target of sceptical arguments, and by showing that they are on the one hand unwarrantable, but on the other are constitutive components of rationality itself, she reveals that they are beyond rational doubt and thus avoids (...) scepticism. Part of Coliva’s defence of using extended rationality in this way involves distancing it from relativist interpretations, which she describes as “devastating” (2015: 120). She proposes two ways that someone might attempt to formulate epistemic relativism from her extended rationality view, and argues that both would be unsuccessful because they require describing alternative conceptions of rationality which are “unintelligible” (in the first case), or “inconceivable” (in the second). The superficial claim of this chapter is that Coliva’s arguments, as they stand, are insufficient to dismiss relativism. She fails to show that the conception of rationality needed for the first formulation of relativism is unintelligible, and overestimates the significance of the second formulation’s conception turning out to be inconceivable. But I also highlight a deeper problem: neither formulation follows a realistic blueprint for relativism, and so these were the wrong possibilities to consider in the first place. To address this issue, I propose a third, more plausible, way in which someone might attempt to formulate relativism on the basis of Coliva’s account. I leave open whether this strategy is successful, or whether a version of Coliva’s criticisms apply to it too. (shrink)