We demonstrate that the quantum-mechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finite-dimensional 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, permutation-invariant categorical relations, i.e. relations independent of the quantum-mechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all (...) their admissible states; but their categorical discernibility turns out to be a state-dependent matter. In all demonstrated cases of discernibility, the fermions and the bosons are discerned (i) with only minimal assumptions on the interpretation of quantum mechanics; (ii) without appealing to metaphysical notions, such as Scotusian haecceitas, Lockean substrata, Postian transcendental individuality or Adamsian primitive thisness; and (iii) without revising the general framework of classical elementary predicate logic and standard set theory, thus without revising standard mathematics. This confutes: (a) the currently dominant view that, provided (i) and (ii), the quantum-mechanical description of such composite physical systems always conflicts with PII; and (b) that if PII can be saved at all, the only way to do it is by adopting one or other of the thick metaphysical notions mentioned above. Among the most general and influential arguments for the currently dominant view are those due to Schrodinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Giuntini, Mittelstaedt, Castellani, Krause and Huggett. We review them succinctly and critically as well as related arguments by van Fraassen and Massimi. (shrink)

A variety of ideas arising in decoherence theory, and in the ongoing debate over Everett's relative-state theory, can be linked to issues in relativity theory and the philosophy of time, specifically the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995; 1996a); in what follows we shall consider the same circle of ideas, but specifically in relation to the interpretation of probability, and its identification with relations in the Hilbert Space (...) norm. The familiar objection that Everett's approach yields probabilities different from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as defined by Everett's theory, and that this can be transparently interpreted in terms of the ordinary notion of the chance occurrence of an event, as witnessed in the present. It is in this sense that the theory has empirical content. (shrink)

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 (...) the most defensible form of relativistic endurantism. A recurring theme of this paper is that even those philosophers who do worry about relativity have not taken it seriously enough. (shrink)

This paper examines in detail two paths that lead to Structural Realism, viz. a substantive philosophical position which asserts that only the structure of the world is knowable. The upward path is any attempt to begin with empiricist premises and reach a sustainable realist position. The downward path is any attempt to start from realist premises and construct a weaker realist position. This paper unravels and criticizes the metaphysical presuppositions of both paths to SR. It questions its very possibility as (...) a substantive—and viable—realist thesis. (shrink)

A theory of temporal passage is put forward which is "objective" in the sense that time flow characterizes the universe independently of the existence of conscious beings. The theory differs from Grunbaum's "mind-dependence" theory, and is designed to avoid Grunbaum's criticisms of an earlier theory of Reichenbach's. The representation of temporal becoming is accomplished by the introduction of indeterministic universe-models; each model representing the universe at a time. The models depict the past as a single four-dimensional manifold, and the future (...) as a branched structure of such manifolds. Time flow is relativistic in that it manifests itself in a frame-dependent (but not observer-dependent) way. The indeterministic character of the universe-models is mirrored in a "temporal" theory of truth which rejects the principle of bivalence, and suitable semantics are provided for this theory. Finally, an account of physical law is given which defines it in terms of physical possibility, rather than vice versa. (shrink)

We analyze the geometric foundations of classical Yang-Mills 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 Yang-Mills 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 (...) local gauge transformations. (shrink)

An ontology of Leibnizian relationalism, consisting in distance relations among sparse matter points and their change only, is well recognized as a serious option in the context of classical mechanics. In this paper, we investigate how this ontology fares when it comes to general relativistic physics. Using a Humean strategy, we regard the gravitational field as a means to represent the overall change in the distance relations among point particles in a way that achieves the best combination of being simple (...) and being informative. (shrink)

The dependence on history of both present and future dynamics of life is a common intuition in biology and in humanities. Historicity will be understood in terms of changes of the space of possibilities as well as by the role of diversity in life’s structural stability and of rare events in history formation. We hint to a rigorous analysis of “path dependence” in terms of invariants and invariance preserving transformations, as it may be found also in physics, while departing from (...) the physico-mathematical analyses. The idea is that the invariant traces of the past under organismal or ecosystemic transformations contribute to the understanding of present and future states of affairs. This yields a peculiar form of unpredictability in biology, at the core of novelty formation: the changes of observables and pertinent parameters may depend also on past events. In particular, in relation to the properties of synchronic measurement in physics, the relevance of diachronic measurement in biology is highlighted. This analysis may a fortiori apply to cognitive and historical human dynamics, while allowing to investigate some general properties of historicity in biology. (shrink)

As is well known, the late Husserl warned against the dangers of reifying and objectifying the mathematical models that operate at the heart of our physical theories. Although Husserl’s worries were mainly directed at Galilean physics, the first aim of our paper is to show that many of his critical arguments are no less relevant today. By addressing the formalism and current interpretations of quantum theory, we illustrate how topics surrounding the mathematization of nature come to the fore naturally. Our (...) second aim is to consider the program of reconstructing quantum theory, a program that currently enjoys popularity in the field of quantum foundations. We will conclude by arguing that, seen from this vantage point, certain insights delivered by phenomenology and quantum theory regarding perspectivity are remarkably concordant. Our overall hope with this paper is to show that there is much room for mutual learning between phenomenology and modern physics. (shrink)

Husserl argues in the Crisis that the prevalent tradition of positive science in his time had a philosophical core, called by him "Galilean science", that mistook the quest for objective theory with the quest for truth. Husserl is here referring to Gottingen science of the Golden Years. For Husserl, theory "grows" out of the "soil" of the prescientific, that is, pretheoretical, life-world. Scientific truth finally is to be sought not in theory but rather in the pragmatic-perceptual praxes of measurement. Husserl (...) is faulted for taking measuring processes to be "infinitely perfectible". The dependence of new scientific phenomena on the existence of prior "prescientific" inductive praxis is analyzed, also Husserl's residual objectivism and failure to appreciate the hermeneutic character of measurement. Though not a scientific (theory-)realist, neither was he an instrumentalist, but he was a scientific (phenomena-)realist. (shrink)

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as (...) Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...) all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

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.

A scientific theory offers models for the phenomena in its domain; these models involve theoretical quantities, and a model's structure is the set of relations it imposes on these quantities. A fundamental demand in scientific practice is for those quantities to be clearly and feasibly related to measurement. This demand for empirical grounding can be articulated by displaying the theory-dependent criteria for a procedure to count as a measurement and for identifying the quantity it measures.

By inserting the dialogue between Einstein, Schlick and Reichenbach into a wider network of debates about the epistemology of geometry, this paper shows that not only did Einstein and Logical Empiricists come to disagree about the role, principled or provisional, played by rods and clocks in General Relativity, but also that in their lifelong interchange, they never clearly identified the problem they were discussing. Einstein’s reflections on geometry can be understood only in the context of his ”measuring rod objection” against (...) Weyl. On the contrary, Logical Empiricists, though carefully analyzing the Einstein–Weyl debate, tried to interpret Einstein’s epistemology of geometry as a continuation of the Helmholtz–Poincaré debate by other means. The origin of the misunderstanding, it is argued, should be found in the failed appreciation of the difference between a “Helmholtzian” and a “Riemannian” tradition. The epistemological problems raised by General Relativity are extraneous to the first tradition and can only be understood in the context of the latter, the philosophical significance of which, however, still needs to be fully explored. (shrink)

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.

I argue for a four dimensional, non-dynamical view of space-time, where becoming is not an intrinsic property of reality. This view has many features in common with the Parmenidean conception of the universe. I discuss some recent objections to this position and I offer a comparison of the Parmenidean space-time with an interpretation of Heraclitus’ thought that presents no major antagonism.

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 (...) as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)

Standard quantum mechanics unquestionably violates the separability principle that classical physics (be it point-like analytic, statistical, or field-theoretic) accustomed us to consider as valid. In this paper, quantum nonseparability is viewed as a consequence of the Hilbert-space quantum mechanical formalism, avoiding thus any direct recourse to the ramifications of Kochen-Specker’s argument or Bell’s inequality. Depending on the mode of assignment of states to physical systems – unit state vectors versus non-idempotent density operators – we distinguish between strong/relational and weak/deconstructional forms (...) of quantum nonseparability. The origin of the latter is traced down and discussed at length, whereas its relation to the all important concept of potentiality in forming a coherent picture of the puzzling entangled interconnections among spatially separated systems is also considered. Finally, certain philosophical consequences of quantum non-separability concerning the nature of quantum objects, the question of realism in quantum mechanics, and possible limitations in revealing the actual character of physical reality in its entirety are explored. (shrink)

This paper is a contribution to a book symposium on my book Experiencing Time. I reply to comments on the book by Natalja Deng, Geoffrey Lee and Bradford Skow. Although several chapters of the book are discussed, the main focus of my reply is on Chapters 2 and 6. In Chapter 2 I argue that the putative mind-independent passage of time could not be experienced, and from this I develop an argument against the A-theory of time. In Chapter 6 I (...) offer one part of an explanation of why we are disposed to think that time passes, relating to the supposedly ‘dynamic’ quality of experienced change. Deng, Lee, and Skow’s comments help me to clarify several issues, add some new thoughts, and make a new distinction that was needed, and I acknowledge, as I did in the book, that certain arguments in Chapter 6 are not conclusive; but I otherwise concede very little regarding the main claims and arguments defended in the book. (shrink)

It has been suggested that puzzles in the interpretation of quantum mechanics motivate consideration of entities that are numerically distinct but do not stand in a relation of identity with themselves or non-identity with others. Quite apart from metaphysical concerns, I argue that talk about such entities is either meaningless or not about such entities. It is meaningless insofar as we attempt to take the foregoing characterization literally. It is meaningful, however, if talk about entities without identity is taken as (...) elliptical for either nominal or predicative use of a special class of mass terms. (shrink)

The purpose of this paper is to show that the mathematics of quantum mechanics is the mathematics of set partitions linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite (...) states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac’s Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being “definite all the way down”. (shrink)

Ebbhinghaus, H., J. Flum, and W. Thomas. 1984. Mathematical Logic. New York, NY: Springer-Verlag. Forster, T. Typescript. The significance of Yablo’s paradox without self-reference. 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.

An objective and relational theory of local time is expounded and its philosophical implications are discussed in Sect. 2. In Sect. 3 certain physical and metaphysical questions concerning time are taken up in the light of that theory. The basic concepts of the theory are those of event, reference frame, chronometric scale, and time function. These are subject to four axioms: existence of events, frames and scales; time is a real valued function; the set of events is compact; and any (...) duration can be subdivided into two contiguous durations. Several theorems are derived, among them the one of the asymmetry of time. And a number of concepts are defined, such as those of time order, instant, and time coordinate. It is argued that the theory, though untestable, belongs to the background of a number of scientific theories. It is also shown that it includes all relational theories of time. The usual confusion between the asymmetry of time and the direction of irreversible processes is clarified. Time reversal is interpreted either as a purely formal operation or as a convenient way of describing motion reversed processes. Time orders are shown to be both relative and objective, apart from the choice of the positive direction, which is conventional. The various attempts to define the direction of time in terms of irreversible processes are shown to be logically untenable and methodologically undesirable. A number of metaphysical questions, such as the one of the reality and the fundamental character of time, are tackled. Finally the occasion is seized to extoll the advantages of systematization over both ordinary language discussions and open context analyses. (shrink)

And the first step of the Peripatetick argument is that, where Aristotle proveth the integrity and perfection of the World, telling us, that it is not a simple line, nor a bare superficies, but a body adorned with Longitude, Latitude and Profundity; and because there are no more dimensions but these three; the World having them, hath all, and having all, is to be concluded perfect. And again, that by simple length, that magnitude is constituted, which is called a line, (...) to which adding breadth, there is formed a Superficies, and yet further adding the altitude or profundity, there results the Body, and after these three the dimensions there is no passing farther, so that in these three the integrity, and to so speak, totality is terminated, which I might but with justice have required Aristotle to have proved to me by necessary consequences, the rather in regard he was able to do it very plainly and speedily. *Received 9. iii. 54. (shrink)

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 (...) a deductive-nomological explanation, their main contribution lies in their explication rather than their explanation of inertial motion. (shrink)

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...) central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

Hempel's high probability requirement asserts that any rationally acceptable answer to the question 'Why did event X occur?' must offer information which shows that X was to be expected at least with reasonable probability. Salmon rejected this requirement in his S-R model. This led to a series of paradoxical consequences, such as the assertion that an explanation of an event can both lower its probability and make it arbitrarily low, and the assertion that the explanation of an outcome would have (...) qualified as an explanation of its non-occurrence as well. We argue that if inductive explanations are to be seen as generalizations of the causal-deterministic model, or if they are to be seen as satisfying the requirement--fulfilled by the D-N model--that explanations ought to identify certain features of the universe that are nomically responsible for the explanadum event, then the high probability requirement seems to be unacceptable. If this is so, a realistically inspired theory of inductive explanation will be committed to the paradoxes that follow from Salmon's model. (shrink)

To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As they stand, these results appear to elude constructivization. On the philosophical side, it (...) is argued that any mentalist-based radical constructivism suffers from a kind of neo-Kantian apriorism. It would be at best a lucky accident if objective spacetime structure mirrored mentalist mathematics. the latter would seem implicitly committed to a Leibnizian relationist view of spacetime, but is it doubtful if implementation of such a view would overcome the objection. As a result, an anti-realist view of physics seems forced on the radical constructivist. (shrink)

The concept of indiscernibility in a structure is analysed with the aim of emphasizing that in asserting that two objects are indiscernible, it is useful to consider these objects as members of (the domain of) a structure. A case for this usefulness is presented by examining the consequences of this view to the philosophical discussion on identity and indiscernibility in quantum theory.

This paper explicates Husserl’s usage of what he calls “radical Besinnung” in Formale und transzendentale Logik. Husserl introduces radical Besinnung as his method in the introduction to FTL. Radical Besinnung aims at criticizing the practice of formal sciences by means of transcendental phenomenological clarification of its aims and presuppositions. By showing how Husserl applies this method to the history of formal sciences down to mathematicians’ work in his time, the paper explains in detail the relationship between historical critical Besinnung and (...) transcendental phenomenology. Ultimately the paper suggests that radical Besinnung should be viewed as a general methodological framework within which transcendental phenomenological descriptions are used to criticize historically given goal-directed practices. (shrink)

The Timaeus is the dialogue that was for many centuries the most influential of Plato’s works. Among its readers we find Descartes, Boyle, Kepler and Heisenberg. In the first division of Timaeus Plato deals with the theory of celestial motion, in the second he presents us with the first mathematical theory of the structure of matter. Here, in a gigantic step forward with respect to the preceding Democritean atomistic theory with its unalterable micro-entities, he introduces the intertransformability of elementary corpuscles (...) and with that the first “chemical” reactions in history. Plato’s geometrical interpretation of Empedocles’ elements and his geometrical atomism is described at the molecular, atomic and sub-atomic level. The “chemical” reactions reported in Timaeus are written in the enlightening chemical symbolism, analogies with modern chemistry are noticed throughout. (shrink)

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 so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what (...) is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo—Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect. (shrink)

The article focuses on the unifying and explanatory power of the selective realism defended by Anjan Chakravartty. Our main aim is twofold. First, we critically analyse the purported synthesis between entity realism and structural realism offered by the author. We give reasons to think that this unification is an inconvenient marriage. In the second step, we deal with certain controversial aspects of the intended unification among three metaphysical concepts: causation, laws of nature and natural kinds. After pointing out that Chakravartty’s (...) conception of laws is a plausible view that a scientific realist might endorse, we contend, on the contrary, that the concept of natural kind is dispensable in the framework of Chakravartty’s realism. (shrink)

SummaryLeśniewski presented his logical systems in a way which conformed to his nominalism, so the question arises whether Leśniewski's logic can be given a natural formal semantics which, unlike current versions, avoids commitment to abstract entities. Building on hints in Wittgenstein's Tractatus, I develop the idea of a way of meaning which is the basis for what I call combinatorial semantics. I then consider whether this commits us to abstract objects or an intensional metalogic.

The present paper introduces a simple framework for modeling the relationship between environmental states, perceptual states, and action. The framework represents situations where an agent’s perceptual state forms the basis for choosing an action, and what action the agent performs determines the agent’s payoff, as a function of the environmental conditions in which the action is performed. The framework is used as the basis for a simulation study of the sorts of correspondence between perceptual and environmental states that are important (...) for successful navigation of the world. Some of the results are surprising and conflict with long held views about the kind of perception-to-environment correspondence that is important for knowledge of the world. The results also raise doubts concerning the view that our perceptual states provide a basis for knowledge of the real structure of the external world. (shrink)

This paper develops a novel interpretation of Kant's argument from incongruent counterparts to the effect that the representations of space and time are intuitions rather than concepts. When properly understood, the argument anticipates the contemporary position whereby the meaning of indexicals cannot be captured by descriptive contents.

In this paper, I argue that Heisenberg's mature philosophy of quantum mechanics must be understood in the context of his epistemological project to reinterpret and redefine Kant's notion of the a priori. After discussions with Weizsäcker and Hermann in Leipzig in the 1930s, Heisenberg attempted to ground his interpretation of quantum mechanics on what might be termed a 'practical' transformation of Kantian philosophy. Taking as his starting point, Bohr's doctrine of the indispensability of classical concepts, Heisenberg argued that concepts such (...) as space, time and causality can be regarded as 'practically a priori', in so far as they remain the conditions for the possibility of experience and even of 'objective reality', though they are not universal and necessary in a strictly Kantian sense. We cannot avoid using classical concepts in the description of experiments in quantum theory, despite the fact that there are limits to their applicability. Such concepts are, for Heisenberg, historically contingent, yet indispensable in our time, because we have no other language through which we can describe and conceive of the interaction between 'object' and 'measuring device'. (shrink)

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 (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)

One of the main criticisms of the theory of collections of indiscernible objects is that once we quantify over one of them, we are quantifying over all of them since they cannot be discerned from one another. In this way, we would call the collapse of quantifiers: ‘There exists one x such as P’ would entail ‘All x are P’. In this paper we argue that there are situations (quantum theory is the sample case) where we do refer to a (...) certain quantum entity, saying that it has a certain property, even without committing all other indistinguishable entities with the considered property. Mathematically, within the realm of the theory of quasi-sets $$\mathfrak {Q}$$, we can give sense to this claim. We show that the above-mentioned ‘collapse of quantifiers’ depends on the interpretation of the quantifiers and on the mathematical background where they are ranging. In this way, we hope to strengthen the idea that quantification over indiscernibles, in particular in the quantum domain, does not conform with quantification in the standard sense of classical logic. (shrink)

The question why natural processes tend to flow along a preferred direction has always been considered from within the perspective of the Second Law of Thermodynamics, especially its statistical formulation due to Maxwell and Boltzmann. In this article, we re-examine the subject from the perspective of a new historico-philosophical formulation based on the careful use of selected theoretical elements taken from three key modern thinkers: Hans Reichenbach, Ilya Prigogine, and Roger Penrose, who are seldom considered together in the literature. We (...) emphasize in our analysis how the entropy concept was introduced in response to the desire to extend the applicability of the Second Law to the cosmos at large (Reichenbach and Penrose), and to examine whether intrinsic irreversibility is a fundamental universal characteristics of nature (Prigogine). While the three thinkers operate with vastly different technical proposals and belong to quite distinct intellectual backgrounds, some similarities are detected in their thinking. We philosophically examine these similarities but also bring into focus the uniqueness of each approach. Our purpose is not providing an exhaustive derivations of logical concepts identified in one thinker in terms of ideas found in the others. Instead, the main objective of this work is to stimulate historico-philosophical investigations and inquiries into the problem of the direction of time in nature by way of crossdisciplinary examinations of previous theories commonly treated in literature as disparate domains. (shrink)

The Received View of particles in quantum mechanics is that they are indistinguishable entities within their kinds and that, as a consequence, they are not individuals in the metaphysical sense and self-identity does not meaningfully apply to them. Nevertheless cardinality does apply, in that one can have n> 1 such particles. A number of authors have recently argued that this cluster of claims is internally contradictory: roughly, that having more than one such particle requires that the concepts of distinctness and (...) identity must apply after all. A common thread here is that the notion of identity is too fundamental to forego in any metaphysical account. I argue that this argument fails. I then argue that the failure of individuality and identity applies also to macroscopic physical objects, that the problems cannot be constrained to apply only within the microscopic realm. (shrink)

If p(x 1 ,...,x n ) and q(x 1 ,...,x n ) are two logically equivalent propositions then p(π (x 1 ),...,π (x n )) and q(π (x 1 ),...,π (x n )) are also logically equivalent where π is an arbitrary permutation of the elementary constituents x 1 ,...,x n . In Quantum Logic the invariance of logical equivalences breaks down. It is proved that the distribution rules of classical logic are in fact equivalent to the meta-linguistic rule of (...) universal substitution and that the more restrictive structure of the substitution group of Quantum Logic prevents us from defining truth in a classical fashion. These observations lead to a more profound understanding of the Logic of Quantum Mechanics and of the role that symmetry principles play in that theory. (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical (...) geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. (shrink)