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.

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.

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.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate (...) controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

This volume develops a fundamentally different categorical framework for conceptualizing time and reality. The actual taking place of reality is conceived as a “constellatory self-unfolding” characterized by strong self-referentiality and occurring in the primordial form of time, the not yet sequentially structured “time-space of the present.” Concomitantly, both the sequentially ordered aspect of time and the factual aspect of reality appear as emergent phenomena that come into being only after reality has actually taken place. In this new framework, time functions (...) as an ontophainetic [H1] platform, i.e., as the stage on which reality can first occur. Events are merely the “tracks” that the actual taking place of reality leaves behind on the co-emergent “canvas’’ of local spacetime. -/- The view of time proposed here is particularly relevant to the recent debate over the “ER=EPR” conjecture targeting the relation between quantum physics and general relativity theory. The novelty of this radically different framework is that it allows quantum reduction and singularities to be addressed as inverse transitions into and out of the factual layer of reality: In quantum physical state reduction, reality “gains” the chrono-ontological format of facticity, and the sequential aspect of time becomes applicable. In singularities, by contrast, the opposite happens: Reality loses its local spacetime formation and reverts back to its primordial, pre-local shape – making the use of causality relations, Boolean logic and the dichotomization of subject and object obsolete in the process. -/- For our understanding of the relation between quantum and relativistic physics, this new view opens up fundamentally new perspectives: Both are legitimate views of time and reality; they simply address very different chrono-ontological portraits, and thus should not lead us to erroneously prefer one view over the other. -/- The task of the book is to provide a formal framework in which this radically different view of time and reality can be suitably addressed. The mathematical approach is based on the logical and topological features of the Borromean Rings, and draws upon concepts and methods from algebraic and geometric topology – especially the theory of sheaves and links, group theory, logic and information theory in relation to the standard constructions employed in quantum mechanics and general relativity, shedding new light on the problems of their compatibility. The intended audience includes physicists, mathematicians and philosophers with an interest in the conceptual and mathematical foundations of modern physics. (shrink)

Timm Triplett argues (Inquiry 29 [1986], no. 4) that David Bloor does not succeed in justifying a relativistic interpretation of mathematics. It is objected that Triplett has focused his attention on the wrong chapter of Bloor's Knowledge and Social Imagery, and that the examples which Triplett demands Bloor provide to make the case do appear in the subsequent chapter. Moreover, Bloor has anticipated and refuted Triplett's brief criticism of the examples that make Bloor's case for the relativism of mathematics. (...) Finally, Triplett's own example of a basic mathematical truth can be shown to be socially relative. (shrink)

What is the real issue at stake in discussions of "moral explanation"? There isn't one; there are many. The standing of purported moral properties and problems about our epistemic or semantic access to them are of concern both from within and without moral practice. An account of their potential contribution to explaining our values, beliefs, conduct, practices, etc. can help in these respects. By examining some claims made about moral explanation in Judith Thompson's and Gilbert Harman's Moral Relativism and (...) Moral Objectivity, I try to suggest a worthwhile account of this potential explanatory contribution. (shrink)

Do moral questions have objective answers? In this great debate, Gilbert Harman explains and argues for relativism, emotivism, and moral scepticism. In his view, moral disagreements are like disagreements about what to pay for a house; there are no correct answers ahead of time, except in relation to one or another moral framework. Independently, Judith Jarvis Thomson examines what she takes to be the case against moral objectivity, and rejects it; she argues that it is possible to find (...) out the correct answers to some moral questions. In her view, some moral disagreements are like disagreements about whether the house has a ghost. Harman and Thomson then reply to each other. This important, lively accessible exchange will be invaluable to all students of moral theory and meta-ethics. (shrink)

This paper deals with a number of technical achievements that are instrumental for a dis-solution of the so-called "Hole Argument" in general relativity. Such achievements include: 1) the analysis of the "Hole" phenomenology in strict connection with the Hamiltonian treatment of the initial value problem. The work is carried through in metric gravity for the class of Christoudoulou-Klainermann space-times, in which the temporal evolution is ruled by the "weak" ADM energy; 2) a re-interpretation of "active" diffeomorphisms as "passive and metric-dependent" (...) dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (up to now unknown) connection to gauge transformations on-shell; understanding such connection also enlightens the real content of the Hole Argument or, better, dis-solves it together with its alleged "indeterminism"; 3) the utilization of the Bergmann-Komar "intrinsic pseudo-coordinates", defined as suitable functionals of the Weyl curvature scalars, as tools for a peculiar gauge-fixing to the super-hamiltonian and super-momentum constraints; 4) the consequent construction of a "physical atlas" of 4-coordinate systems for the 4-dimensional "mathematical" manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent "Dirac observables"). Such construction embodies the "physical individuation" of the points of space-time as "point-events", independently of the presence of matter, and associates a "non-commutative structure" to each gauge fixing or four-dimensional coordinate system; 5) a clarification of the multiple definition given by Peter Bergmann of the concept of "(Bergmann) observable" in general relativity. This clarification leads to the proposal of a "main conjecture" asserting the existence of i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). A by-product of this achievements is the falsification of a recently advanced argument asserting the absence of (any kind of) "change" in the observable quantities of general relativity. 6) a clarification of the physical role of Dirac and gauge variables as their being related to "tidal-like" and "inertial-like" effects, respectively. This clarification is mainly due to the fact that, unlike the standard formulations of the equivalence principle, the Hamiltonian formalism allows to define notion of "force" in general relativity in a natural way; 7) a proposal showing how the physical individuation of point-events could in principle be implemented as an experimental setup and protocol leading to a "standard of space-time" more or less like atomic clocks define standards of time. We conclude that, besides being operationally essential for building measuring apparatuses for the gravitational field, the role of matter in the non-vacuum gravitational case is also that of "participating directly" in the individuation process, being involved in the determination of the Dirac observables. This circumstance leads naturally to a peculiar new kind of "structuralist" view of the general-relativistic concept of space-time, a view that embodies some elements of both the traditional "absolutist" and "relational" conceptions. In the end, space-time point-events maintain a "peculiar sort of objectivity". Some hints following from our approach for the quantum gravity programme are also given. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the (...) concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

I assume there is no point in speaking of objective interpretation without explaining what it is about the nature of things that fits them for interpretation and how we may be said to know that particular interpretations do indeed fit interpretable things objectively. On that assumption I provide an argument, largely in accord with post‐Kantian themes, to show that a relativistic account of interpretation is not likely to be ruled out by any objectivism or by any exceptionless rule of bivalence. (...) I offer arguments to distinguish between, and account for, what I call interpretation in the “constituting” sense, which bears on the constructed nature of perception and the distinction between description and interpretation, and interpretation in the “ampliative” sense, which bears primarily on the distinctive properties of intrinsically interpretable things. By this strategy I show the sense in which the interpretation of physical phenomena under explanatory conditions invites epistemological forms of incommensurabilism and the Intentional properties of cultural phenomena provide metaphysical grounds for viable forms of relativistic interpretation. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

Needless to say, Charles Parsons’s long awaited book1 is a must-read for anyone with an interest in the philosophy of mathematics. But as Parsons himself says, this has been a very long time in the writing. Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years. What we are reading is thus a multi-layered text with different passages added at different times. And this makes for (...) a rather bumpy read. There is another route Parsons could have taken: he could have reprinted the relevant papers with postscripts, and then top-and-tailed the collection with a preface and added concluding reflections. It must sound very ungrateful, but I rather suspect that that might have worked better. Much of the book is about arithmetic. But Parsons has woven into the discussion claims about mathematics more generally and about set theory in particular. We might well have a basic worry about this structure: for do defensible claims about the ontology and epistemology of arithmetic have to be generalizable to apply to more infinitary mathematics? For example, suppose you are attracted to a Hellman-like modal structuralist account of arithmetic: then should you think it a problem if you suspect that such an account can’t readily be extended to cope e.g. with set theory (since you boggle at the idea of possible worlds free of abstracta but with enough structure to somehow model ZFC)? Parsons himself seems to waver over such questions. So for present purposes, I’ll focus just on arithmetic, and in what follows I’ll revisit two of the most familiar Parsonian themes, his views on structuralism as an account of the ontology of arithmetic, and his exploration of the role of intuition in grounding arithmetical knowledge. (shrink)

This essay explores theories of place, or lived-space, as regards the role of objectivity and the problem of relativism. As will be argued, the neglect of mathematics and geometry by the lived-space theorists, which can be traced to the influence of the early phenomenologists, principally the later Husserl and Heidegger, has been a major contributing factor in the relativist dilemma that afflicts the lived-space movement. By incorporating various geometrical concepts within the analysis of place, it is demonstrated that (...) the lived-space theorists can gain a better insight into the objective spatial relationships among individuals and within groups—and, more importantly, this appeal to mathematical content need not be construed as undermining the basic tenants of the lived-space approach. (shrink)

There are a bewildering variety of ways the terms "realism" and "anti-realism" have been used in philosophy and furthermore the different uses of these terms are only loosely connected with one another. Rather than give a piecemeal map of this very diverse landscape, the authors focus on what they see as the core concept: realism about a particular domain is the view that there are facts or entities distinctive of that domain, and their existence and nature is in some important (...) sense objective and mind-independent. The authors carefully set out and explain the different realist and anti-realist positions and arguments that occur in five key domains: science, ethics, mathematics, modality and fictional objects. For each area the authors examine the various styles of argument in support of and against realism and anti-realism, show how these different positions and arguments arise in very different domains, evaluate their success within these fields, and draw general conclusions about these assorted strategies. Error theory, fictionalism, non-cognitivism, relativism and response-dependence are taken as the most important positions in opposition to the realist and these are explored in depth. Suitable for advanced level undergraduates, the book offers readers a clear introduction to a subject central to much contemporary work in metaphysics, epistemology and philosophy of language. (shrink)

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might (...) take to meet this challenge, and we argue that all these strategies are untenable. (shrink)

Antony Flew's ?A Strong Programme for the Sociology of Belief (Inquiry 25 {1982], 365?78) critically assesses the strong programme in the sociology of knowledge defended in David Bloor's Knowledge and Social Imagery. I argue that Flew's rejection of the epistemological relativism evident in Bloor's work begs the question against the relativist and ignores Bloor's focus on the social relativity of mathematical knowledge. Bloor attempts to establish such relativity via a sociological analysis of Frege's theory of number. But this (...) analysis only succeeds if the rejection of an explanatory theory entails that there are reasonable grounds for the rejection of the set of propositions which that theory was intended to explain. I argue against such an entailment, and thus against Bloor's attempt to relativize mathematical knowledge. (shrink)

This chapter considers the status and coherence of modern humanism as a secular and ethical philosophy. As secular, humanism prioritizes the naturalistic worldview, and privileges information from the social and cognitive sciences about human sociality and morality. As ethical, humanism does more than recommend specific moral virtues and rules, by proposing methods to evaluate moralities and recommend ideals of moral progress for all peoples around the world. Moral relativism is one of most talked‐about yet least understood notions around today. (...) Despite what one might hear, morality is neither a matter of taste nor preference, it is not arbitrary or immune from criticism, and we don't have to tolerate just anybody's moral opinions. Morality can be objectively studied, morality can be rationally debated and evaluated, there are justifiable ways to determine better and worse moral norms, and important moral norms can be reasonably recommended for all people. (shrink)

This paper, written in Romanian, compares fictionalism, nominalism, and neo-Meinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance.

Formulating quantum gravity is the greatest challenge that 21st century physics must address. If quantum physics refuses to blend with general relativity, it may be that relativity does not represent a good description of the universe in line with gravity and all its effects. This opens a path for us: first understanding quantum physics and what it reveals about nature and then analyzing the boundaries of relativistic cosmology and reconsidering the whole matter. Physicists consider entanglement as a fundamental property derived (...) from quantum mechanics, which is a physics of communication. The involvement of quantum computing shows that the entanglement/geometry duality works by employing a "quantum algorithm" that corrects the information. Most physical theories make use of gauges. Gauges give us evidence about two things: how the cosmos is organized; and why and how nature is "captured" by mathematical theories, opening a line of inquiry about the relationship between man and the universe. (shrink)

Richard Rorty's collected papers, written during the 1980s and now published in two volumes, take up some of the issues which divide Anglo-Saxon analytic philosophers and contemporary French and German philosophers and offer something of a compromise - agreeing with the latter in their criticisms of traditional notions of truth and objectivity, but disagreeing with them over the political implications they draw from dropping traditional philosophical doctrines. In this volume Rorty offers a Deweyan account of objectivity as intersubjectivity, (...) one that drops claims about universal validity and instead focuses on utility for the purposes of a community. The sense in which the natural sciences are exemplary for inquiry is explicated in terms of the moral virtues of scientific communities rather than in terms of a special scientific method. The volume concludes with reflections on the relation of social democratic politics to philosophy. (shrink)

This paper comments on the conflict between ethical relativism and anthropologists’ concerns with rights, and tries to show that neither scientific objectivity nor respect for cultural diversity require denying an extracultural stance for ethical judgments.

This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on (...) Riemann’s work on complex functions, I propose that mathematicians deal with systems of representations and that truth—or what we can prove—depends on available representations in some context where the problem can be solved. (shrink)

This paper is a discussion of Gödel's arguments for a Platonistic conception of mathematical objects. I review the arguments that Gödel offers in different papers, and compare them to unpublished material (from Gödel's Nachlass). My claim is that Gödel's later arguments simply intend to establish that mathematical knowledge cannot be accounted for by a reflexive analysis of our mental acts. In other words, there is at the basis of mathematics some data whose constitution cannot be explained by introspective (...) analysis. This does not mean that mathematics is independent of the human mind, but only that it is independent of our ?conscious acts and decisions?, to use Gödel's own words. Mathematical objects may then have been created by the human mind, but if so, the process of creation cannot be completely analysed and re-enacted. Such a thesis is weaker than some of the statements that Gödel made about his conceptual realism. However, there is evidence that Gödel seriously considered this weak thesis, or a position depending only on this weak thesis. He also criticized Husserl's Phenomenology from this point of view. (shrink)

Context: the position of pure and applied mathematics in the epistemic conflict between realism and relativism. Problem: To investigate the change in the status of mathematical knowledge over historical time: specifically, the shift from a realist epistemology to a relativist epistemology. Method: Two examples are discussed: geometry and number theory. It is demonstrated how the initially realist epistemic framework – with mathematics situated in a platonic ideal reality from where it governs our physical world – became untenable, with (...) the advent of non-Euclidean geometry and the increasing abstraction of the number concept. Results: Radical constructivism offers an alternative relativist epistemology, where mathematical knowledge is constructed by the individual knower in a context of an axiomatic base and subject items chosen at her discretion, for the purpose of modelling some part of her personal experiential world. Thus it can be expedient to view the practice of mathematics as a game, played by mathematicians according to agreed-upon rules. Constructivist content: The role played by constructivism in the formulation of mathematics is discussed. This is illustrated by the historical transition from a classical (platonic) view of mathematics, as having an objective existence of its own in the “realm of ideal forms,” to the now widely accepted modern view where one has a wide freedom to construct mathematical theories to model various parts of one’s experiential world. (shrink)

Baker claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating (...) mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework. (shrink)

It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is in fact not even disagreement in any interesting sense, let alone a faultless one, in spite of appearances and claims to the contrary. A position I call classical pluralism is (...) sketched, intended to provide a coherent methodological stance towards the issue. Some reasons to recommend this stance are given, as well as some speculations as to why not everyone might want to follow the recommendation. (shrink)

Wright's _Truth and Objectivity_ seeks to systematise a variety of anti-realist positions. I argue that many objections to the system are avoided by transposing its talk of truth into talk of warrant. However, a problem remains about debates involving 'direction-of-fit'. -/- Dummett introduced 'anti-realism' as a philosophical view informed by mathematical intuitionism. Subsequently, the term has been associated with many debates, ancient and modern. _Truth and Objectivity_ proposes that truth admits of different characteristics; these various debates then concern which (...) characteristics truth has, in a given area. This pluralism of truth is at odds with deflationism. I find fault with Wright's argument against deflationism. However, transmission of warrant across the Disquotational Schema suffices to ground Wright's proposal, which survives as a pluralism of classes of warrant. -/- The two main debates concern whether truths are always knowable (Epistemic Constraint) and whether disagreements in an area must be down to some fault of one of those involved (Cognitive Command). I introduce Assertoric Constraint, relating to Epistemic Constraint, where truths cannot outstrip the availability of warrant for their assertion. I solve a structural problem by a comparison with a constitutive analysis of Moore's Paradox. The relativism of blameless disagreement is problematic. Wright's response invokes a sort of ignorance which he calls `Quandary'. I criticise this before proposing an alternative. -/- I agree with Wright that Dummett's original anti-realism does not belong among the positions which Wright seeks to systematise. However, two candidates show that the proposal suffers a weakness. Wright thinks Expressivism misguided, and implicitly rules out his earlier non-cognitivism about necessity. I argue that Expressivism has promise, and I endorse Wright's Cautious Man argument for non-cognitivism about necessity; both involve play with `direction-of-fit'. I conclude that this sort of anti-realist debate needs to be accommodated by the proposal. (shrink)