Citations of:
Add citations
You must login to add citations.


This paper proposes a reading of the history of equivalence in mathematics. The paper has two main parts. The first part focuses on a relatively short historical period when the notion of equivalence is about to be decontextualized, but yet, has no commonly agreedupon name. The method for this part is rather straightforward: following the clues left by the others for the ‘first’ modern use of equivalence. The second part focuses on a relatively long historical period when equivalence is experienced (...) 

We have two aims in this paper. The first is to provide the reader with a critical guide to recent work on relativity and persistence by Balashov, Gilmore and others. Much of this work investigates whether endurantism can be sustained in the context of relativity. Several arguments have been advanced that aim to show that it cannot. We find these unpersuasive, and will add our own criticisms to those we review. Our second aim, which complements the first, is to demarcate (...) 

In this critical notice we argue against William Craig's recent attempt to reconcile presentism (roughly, the view that only the present is real) with relativity theory. Craig's defense of his position boils down to endorsing a ‘neoLorentzian interpretation’ of special relativity. We contend that his reconstruction of Lorentz's theory and its historical development is fatally flawed and that his arguments for reviving this theory fail on many counts. 1 Rival theories of time 2 Relativity and the present 3 Special relativity: (...) 

This paper argues that Carnap's epistemological project in the Aufbau is best considered as an attempt to determine the conditions for both objectivity and understanding, thus aiming at refuting the skeptical contention that objectivity and understanding are incompossible ideals of science. 

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







According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) 

Pour le philosophe intéressé aux structures et aux fondements du savoir théorétique, à la constitution d'une « métathéorétique «, θεωρíα., qui, mieux que les « Wissenschaftslehre » fichtéenne ou husserlienne et pardelà les débris de la métaphysique, veut dans une intention nouvelle faire la synthèse du « théorétique », la logique mathématique se révèle un objet privilégié. 

This paper offers a new interpretation for Wittgenstein`s treatment of mathematical identities. As it is widely known, Wittgenstein`s mature philosophy of mathematics includes a general rejection of abstract objects. On the other hand, the traditional interpretation of mathematical identities involves precisely the idea of a single abstract object – usually a number –named by both sides of an equation. 



In a recent PSA paper (2001a) as well as some other papers ((1995), (2000), (2001b)) and a book chapter (1999, ch. 7), Stathis Psillos raised a number of objections against structural realism. The aim of this paper is threefold: 1) to evaluate part of Psillos’ offence on the Russellian version of epistemic structural realism (ESR for short), 2) to elaborate more fully what Russellian ESR involves, and 3) to suggest improvements where it is indeed failing. 

I develop an original view of the structure of spacecalled "infinitesimal atomism"as a reply to Zeno's paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson's (1966) nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a finite region is the (...) 

In a recent article, van Fraassen has taken issue with the use to which Perrin’s experiments on Brownian motion have been put by philosophers, especially those defending scientific realism. He defends an alternative position by analysing the details of Perrin’s case in its historical context. In this reply, I argue that van Fraassen has not done the job well enough and I extend and in some respects attempt to correct his claims by close attention to the historical details. 

It has been contended that it is unjustified to believe, as Weyl did, that formalism's victory against intuitionism entails a defeat of the phenomenological approach to mathematics. The reason for this contention is that, unlike intuitionistic Anschauung, phenomenological intuition could ground classical mathematics. Against this, I argue that Weyl did not take formalism to prevail over intuitionism with respect to grounding classical mathematics, and that if intuitionism fails in the way he thought it did, i.e., with respect to supporting scientific (...) 

Weyl's tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called "the mixed account," according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To motivate this account as a genuine solution to (...) 





I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. / It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) 

When we begin to investigate the persistence of objects through time, we find immediately that the sort of concerns embodied in Leibniz's Law cause philosophers to divide themselves into the two major camps of Purdurantists and Endurantists. What is required according to each for a given object at a given time to be identified with a given object at another time is held to be dramatically different, even while both often look to the same general sort of indicators for their (...) 

When dealing with a certain class of physical systems, the mathematical characterization of a generic system aims to describe the phase portrait of all its possible states. Because they are defined only up to isomorphism, the mathematical objects involved are “schematic structures”. If one imposes the condition that these mathematical definitions completely capture the physical information of a given system, one is led to a strong requirement of individuation for physical states. However, we show there are not enough qualitatively distinct (...) 

During his whole scientific life Hermann Weyl was fascinated by the interrelation of physical and mathematical theories. From the mid 1920s onward he reflected also on the typical difference between the two epistemic fields and tried to identify it by comparing their respective automorphism structures. In a talk given at the end of the 1940s he gave the most detailed and coherent discussion of his thoughts on this topic. This paper presents his arguments in the talk and puts it in (...) 

Ebbhinghaus, H., J. Flum, and W. Thomas. 1984. Mathematical Logic. New York, NY: SpringerVerlag. Forster, T. Typescript. The significance of Yablo’s paradox without selfreference. Available from http://www.dpmms.cam.ac.uk. Gold, M. 1965. Limiting recursion. Journal of Symbolic Logic 30: 28–47. Karp, C. 1964. Languages with Expressions of Infinite Length. Amsterdam. 

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics (...) 

En el presente trabajo analizamos críticamente el modo en que el Estructuralismo Empirista de van Fraassen caracteriza la relación de representación entre las teorías y los fenómenos. Nuestro objetivo es ofrecer argumentos que destaquen el papel del objeto en la construcción de modelos de datos. Asimismo, nos proponemos mostrar que la opción metodológica sugerida en su obra reciente resulta insuficiente para recuperar un vínculo plausible entre los modelos de datos y los fenómenos. 

I examine two claims that arise in Brown’s account of inertial motion. Brown claims there is something objectionable about the way in which the motions of free particles in Newtonian theory and special relativity are coordinated. Brown also claims that since a geodesic principle can be derived in Einsteinian gravitation, the objectionable feature is explained away. I argue that there is nothing objectionable about inertia and that while the theorems that motivate Brown’s second claim can be said to figure in (...) 

The concept of time is examined using the second law of thermodynamics that was recently formulated as an equation of motion. According to the statistical notion of increasing entropy, flows of energy diminish differences between energy densities that form space. The flow of energy is identified with the flow of time. The nonEuclidean energy landscape, i.e. the curved space–time, is in evolution when energy is flowing down along gradients and levelling the density differences. The flows along the steepest descents, i.e. (...) 

A crucial question for a process view of life is how to identify a process and how to follow it through time. The genidentity view can contribute decisively to this project. It says that the identity through time of an entity X is given by a wellidentified series of continuous states of affairs. Genidentity helps address the problem of diachronic identity in the living world. This chapter describes the centrality of the concept of genidentity for David Hull and proposes an (...) 

Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the socalled “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasicombinatorialism or combinatorial maximality. After explaining what (...) 



Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series (...) 

The main thrust of the argument of this thesis is to show the possibility of articulating a method of construction or of synthesisas against the most common method of analysis or divisionwhich has always been (so we shall argue) a necessary component of scientific theorization. This method will be shown to be based on a fundamental synthetic logical relation of thought, that we shall call inversionto be understood as a species of logical opposition, and as one of the basic monadic (...) 

In his 1927 Analysis of Matter and elsewhere, Russell argued that we can successfully infer the structure of the external world from that of our explanatory schemes. While nothing guarantees that the intrinsic qualities of experiences are shared by their objects, he held that the relations tying together those relata perforce mirror relations that actually obtain (these being expressible in the formal idiom of the Principia Mathematica). This claim was subsequently criticized by the Cambridge mathematician Max Newman as true but (...) 

Brentano’s theory of continuity is based on his account of boundaries. The core idea of the theory is that boundaries and coincidences thereof belong to the essence of continua. Brentano is confident that he developed a fullfledged, boundarybased, theory of continuity1; and scholars often concur: whether or not they accept Brentano’s take on continua they consider it a clear contender. My impression, on the contrary, is that, although it is infused with invaluable insights, several aspects of Brentano’s account of continuity (...) 

Whitehead’s cosmology centers on the selfcreation of actual occasions that perish as they come to be, but somehow do combine to constitute societies that are persistent agents and/or patients. “Instance Ontology” developed by D.W. Mertz concerns unification of relata into facts of relatedness by specific intensions. These two conceptual systems are similar in that they both avoid the substanceproperty distinction: they differ in their understanding of how basic units combine to constitute complex unities. “Process Structural Realism” (PSR) draws from both (...) 

Standard quantum mechanics unquestionably violates the separability principle that classical physics (be it pointlike analytic, statistical, or fieldtheoretic) accustomed us to consider as valid. In this paper, quantum nonseparability is viewed as a consequence of the Hilbertspace quantum mechanical formalism, avoiding thus any direct recourse to the ramifications of KochenSpecker’s argument or Bell’s inequality. Depending on the mode of assignment of states to physical systems – unit state vectors versus nonidempotent density operators – we distinguish between strong/relational and weak/deconstructional forms (...) 

Deux choses sont en contact s'il n'y a rien entre elles (ni volume, ni ligne, ni point) et qu'elles ne se chevauchent pas (en un volume, un ligne ou un point). Le contact est la limite de proximité des choses : si deux choses sont en contact, deux autres choses ne peuvent être pas être plus près l'une de l'autre sans se pénétrer. 

This contribution explores Wolfgang Pauli's idea that mind and matter are complementary aspects of the same reality. We adopt the working hypothesis that there is an undivided timeless primordial reality (the primordial "one world''). Breaking its symmetry, we obtain a contextual description of the holistic reality in terms of two categorically different domains, one tensed and the other tenseless. The tensed domain includes, in addition to tensed time, nonmaterial processes and mental events. The tenseless domain refers to matter and physical (...) 

We demonstrate that the quantummechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finitedimensional Hilbert spaces, is not in conflict with Leibniz's Principle of the Identity of Indiscernibles (PII). We discern the fermions by means of physically meaningful, permutationinvariant categorical relations, i.e. relations independent of the quantummechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all (...) 

Classical physics and quantum physics suggest two metaphysical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets which are categorytheoretically dual to one another (...) 

In this paper I focus on the impact on structuralism of the quantum treatment of objects in terms of symmetry groups and, in particular, on the question as to how we might eliminate, or better, reconceptualise such objects in structural terms. With regard to the former, both Cassirer and Eddington not only explicitly and famously tied their structuralism to the development of group theory but also drew on the quantum treatment in order to further their structuralist aims and here I (...) 



Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector spacewhich is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a categorytheoretic sense) to (...) 

The ManyWorlds Interpretation (MWI) is an approach to quantum mechanics according to which, in addition to the world we are aware of directly, there are many other similar worlds which exist in parallel at the same space and time. The existence of the other worlds makes it possible to remove randomness and action at a distance from quantum theory and thus from all physics. 

I use modal logic and transfinite settheory to define metaphysical foundations for a general theory of computation. A possible universe is a certain kind of situation; a situation is a set of facts. An algorithm is a certain kind of inductively defined property. A machine is a series of situations that instantiates an algorithm in a certain way. There are finite as well as transfinite algorithms and machines of any degree of complexity (e.g., Turing and superTuring machines and more). There (...) 

One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject. 

We analyze the geometric foundations of classical YangMills theory by studying the relationships between internal relativity, locality, global/local invariance, and background independence. We argue that internal relativity and background independence are the two independent defining principles of YangMills theory. We show that local gauge invariance heuristically implemented by means of the gauge argument is a direct consequence of internal relativity. Finally, we analyze the conceptual meaning of BRST symmetry in terms of the invariance of the gauge fixed theory under general (...) 

It is often thought that there is little that seems more obvious from experience than that time objectively passes, and that time is, in this respect, quite unlike space. Yet nothing in the physical picture of the world seems to correspond to the idea of such an objective passage of time. In this paper, I discuss some attempts to explain this apparent conflict between appearance and reality. I argue that existing attempts to explain the conflict as the result of a (...) 