Ioannis G. Ntagkas

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about.

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical.

Since the time of Aristotle, metaphysics has been an ill-defined term. This paper defines it as a logically idempotent metalinguistic identity of reality which couples the two initial ingredients of awareness: perceptual reality (the basis of physics), and cognitive-perceptual syntax, a formalization of mind. The explanation has been reduced to a few very simple, clearly explained mathematical ingredients. This paper contains no assumptions or arguable assertions, and is therefore presented as an advanced formulation of logic which has been updated for meaningful reference to the structure of reality at large. This structure, called the Cognitive-Theoretic Model of the Universe or CTMU, resolves the problems attending Cartesian dualism by replacing dualism with the mathematical property of self-duality, meaning (for reality-theoretic purposes) the quantum-level invariance of identity under permutation of objective and spatiotemporal data types. The CTMU takes the form of a global coupling or superposition of mind and physical reality in a self-dual metaphysical identity M:L<-->U, which can be intrinsically developed into a logico-geometrically self-dual, ontologically self- contained language incorporating its own medium of existence and comprising its own model therein.

The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some causal structure. The difficult case is when the causal structure includes latent variables. We here introduce the inflation technique for tackling this problem. An inflation of a causal structure is a new causal structure that can contain multiple copies of each of the original variables, but where the ancestry of each copy mirrors that of the original. To every distribution of the observed variables that is compatible with the original causal structure, we assign a family of marginal distributions on certain subsets of the copies that are compatible with the inflated causal structure. It follows that compatibility constraints for the inflation can be translated into compatibility constraints for the original causal structure. Even if the constraints at the level of inflation are weak, such as observable statistical independences implied by disjoint causal ancestry, the translated constraints can be strong. We apply this method to derive new inequalities whose violation by a distribution witnesses that distribution’s incompatibility with the causal structure. We describe an algorithm for deriving all such inequalities for the original causal structure that follow from ancestral independences in the inflation. For three observed binary variables with pairwise common causes, it yields inequalities that are stronger in at least some aspects than those obtainable by existing methods. We also describe an algorithm that derives a weaker set of inequalities but is more efficient. Finally, we discuss which inflations are such that the inequalities one obtains from them remain valid even for quantum generalizations of the notion of a causal model.

In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitrary experimental procedures, it is shown that the outcome of a measurement depends deterministically on the ontic state of the system being measured if and only if the measurement is sharp. Hence for every unsharp measurement, its outcome necessarily has an indeterministic dependence on the ontic state. We defend this proposal against alternatives. In particular, we demonstrate why considerations parallel to Fine’s theorem do not challenge this conclusion

Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. We show that representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the status of something real, the failure of locality can be established through an argument considerably more straightforward than Bell’s theorem. The historical significance of this result becomes evident when one recognizes that the same reasoning is present in Einstein’s preferred argument for incompleteness, which dates back to 1935. This fact suggests that Einstein was seeking not just any completion of quantum theory, but one wherein quantum states are solely representative of our knowledge. Our hypothesis is supported by an analysis of Einstein’s attempts to clarify his views on quantum theory and the circumstance of his otherwise puzzling abandonment of an even simpler argument for incompleteness from 1927

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropie quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of 'conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.

A Gedanken experiment is presented where an excited and a ground-state atom are positioned such that, within the former’s half-life time, they exchange a photon with 50% probability. A measurement of their energy state will therefore indicate in 50% of the cases that no photon was exchanged. Yet other measurements would reveal that, by the mere possibility of exchange, the two atoms have become entangled. Consequently, the “no exchange” result, apparently precluding entanglement, is non-locally established between the atoms by this very entanglement. This quantum-mechanical version of the ancient Liar Paradox can be realized with already existing transmission schemes, with the addition of Bell’s theorem applied to the no-exchange cases. Under appropriate probabilities, the initially-excited atom, still excited, can be entangled with additional atoms time and again, or alternatively, exert multipartite nonlocal correlations in an interaction free manner. When densely repeated several times, this result also gives rise to the Quantum Zeno effect, again exerted between distant atoms without photon exchange. We discuss these experiments as variants of interaction-free-measurement, now generalized for both spatial and temporal uncertainties. We next employ weak measurements for elucidating the paradox. Interpretational issues are discussed in the conclusion, and a resolution is offered within the Two-State Vector Formalism and its new Heisenberg framework.

This edited collection is the first of its kind to explore the view called perspectivism in philosophy of science. The book brings together an array of essays that reflect on the methodological promises and scientific challenges of perspectivism in a variety of fields such as physics, biology, cognitive neuroscience, and cancer research, just as a few examples. What are the advantages of using a plurality of perspectives in a given scientific field and for interdisciplinary research? Can different perspectives be integrated? What is the relation between perspectivism, pluralism, and pragmatism? These ten new essays by top scholars in the field offer a polyphonic journey towards understanding the view called ‘perspectivism’ and its relevance to science.

Since Leibniz's time, Cartesian mental causation has been criticized for violating the conservation of energy and momentum. Many dualist responses clearly fail. But conservation laws have important neglected features generally undermining the objection. Conservation is _local_, holding first not for the universe, but for everywhere separately. The energy in any volume changes only due to what flows through the boundaries. Constant total energy holds if the global summing-up of local conservation laws converges; it probably doesn't in reality. Energy conservation holds if there is symmetry, the sameness of the laws over time. Thus, if there are time-places where symmetries fail due to nonphysical influence, conservation laws fail there and then, while holding elsewhere, such as refrigerators and stars. Noether's converse first theorem shows that conservation laws imply symmetries. Thus conservation trivially nearly entails the causal closure of the physical. But expecting conservation to hold in the brain simply assumes the falsehood of Cartesianism. Hence Leibniz's objection begs the question. Empirical neuroscience is another matter. So is Einstein's General Relativity: far from providing a loophole, General Relativity makes mental causation _harder_.

The principle of energy conservation is widely taken to be a se- rious difficulty for interactionist dualism (whether property or sub- stance). Interactionists often have therefore tried to make it satisfy energy conservation. This paper examines several such attempts, especially including E. J. Lowe’s varying constants proposal, show- ing how they all miss their goal due to lack of engagement with the physico-mathematical roots of energy conservation physics: the first Noether theorem (that symmetries imply conservation laws), its converse (that conservation laws imply symmetries), and the locality of continuum/field physics. Thus the “conditionality re- sponse”, which sees conservation as (bi)conditional upon symme- tries and simply accepts energy non-conservation as an aspect of interactionist dualism, is seen to be, perhaps surprisingly, the one most in accord with contemporary physics (apart from quantum mechanics) by not conflicting with mathematical theorems basic to physics. A decent objection to interactionism should be a posteri- ori, based on empirically studying the brain.

Russellian monism—an influential doctrine proposed by Russell (The analysis of matter, Routledge, London, 1927/1992)—is roughly the view that physics can only ever tell us about the causal, dispositional, and structural properties of physical entities and not their categorical (or intrinsic) properties, whereas our qualia are constituted by those categorical properties. In this paper, I will discuss the relation between Russellian monism and a seminal paradox facing epiphenomenalism, the paradox of phenomenal judgment: if epiphenomenalism is true—qualia are causally inefficacious—then any judgment concerning qualia, including epiphenomenalism itself, cannot be caused by qualia. For many writers, including Hawthorne (Philos Perspect 15:361–378, 2001), Smart (J Conscious Stud 11(2):41–50, 2004), and Braddon-Mitchell and Jackson (The philosophy of mind and cognition, Blackwell, Malden, 2007), Russellian monism faces the same paradox as epiphenomenalism does. I will assess Chalmers’s (The conscious mind: in search of a fundamental theory. Oxford University Press, New York, 1996) and Seager’s (in: Beckermann A, McLaughlin BP (eds) The Oxford handbook of philosophy of mind. Oxford University Press, New York, 2009) defences of Russellian monism against the paradox, and will put forward a novel argument against those defences.

The paper has three parts. In the first part ExtOSR, an extended version of Ontic Structural Realism, will be introduced. ExtOSR considers structural properties as ontological primitives, where structural properties are understood as comprising both relational and structurally derived intrinsic properties or structure invariants. It is argued that ExtOSR is best suited to accommodate gauge symmetry invariants and zero value properties. In the second part, ExtOSR will be given a Humean shape by considering structures as categorical and global. It will be laid out how such structures serve to reconstruct non-essential structural kinds and laws. In the third part Humean structural realism will be defended against the threat of quidditism.

When it is spoken of scientific representations it is often understood that science can offer "only" representations but does not enable us to know reality. This tenet is the inheritance of a gratuitous and inconsistent presupposition that affected modern philosophy during almost two centuries, according to which we know our representations and not things, and we have to find warranties in order to believe that such representations correspond to reality. The present paper analyzes this presupposition, shows its inconsistency and, through a discourse regarding the relations between thought and ontology, between sense and reference of the intellectual constructions, between abstract encoding of properties and concrete exemplification of the same by means of operational criteria of reference, justifies the cognitive purport of scientific representations, including the mathematical representations of physical phenomena.

Henri Poincaré’s views on the foundations of mechanics and the nature of mechanical explanation were influenced by the work of two of the most renowned nineteenth century scientists, James Clerk Maxwell and Heinrich Hertz. In order then to unravel Poincaré’s views and own contribution to the subject it is important to see the connection between Maxwell ’s and Hertz’s researches on the one hand and Poincaré’s on the other. Consequently, I start this paper with a brief account of Poincaré’s encounter with Maxwell ’s work in electromagnetism. Then, in section 2, I move on to show how Hertz’s work on the foundations of mechanics shaped Poincaré own views. In sections 3 and 4, I formulate Poincaré’s own conventionalist philosophy of mechanics and show how several methodological considerations, especially the search for unity, mitigated his conventionalism. Having thus examined Poincaré’s views on the foundations of mechanics, in section 5 I turn my attention to his notion of mechanical explanation and his proof that a mechanical explanation of a set of phenomena is possible if the principle of conservation of energy is satisfied. I then go on to show how Poincaré secured the possibility of a mechanical explanation of electromagnetic phenomena, and also how, having done so, he ended up with an unlimited number of configurations of matter in motion that could underpin electromagnetic phenomena. The upshot of this paper will be that Poincaré departed from the traditional conceptions on mechanical explanation and defended a purely structural conception, the strong point of which was that it promoted — as the motto of this paper says — the true and only aim of science, namely unity

Many contemporary thinkers seeking to integrate theistic belief and scientific thought reject what they regard as two extremes. They disavow deism in which God is understood simply to uphold the existence of the physical universe, and they exclude any view of divine influence that suggests the performance of physical work through an immaterial cause. Deism is viewed as theologically inadequate, and acceptance of direct immaterial causation of physical events is viewed as scientifically illegitimate. This desire to avoid both deism and any positing of God as directly intervening in the physical order has led to models of divine agency that seek to defend the reality of divine causal power yet affirm the causal closure of the physical. I argue, negatively, that such models are unsuccessful in their attempts to affirm both the reality of divine causal power acting in the created world and the causal closure of the physical and, positively, that the assumption that underlies these models, namely that any genuine integration of theistic and scientific belief must posit the causal closure of the physical on pain of violating well-established conservation principles, is mistaken.

In this work, we will try to state the opposition between two approaches to the problem of the overall reliability of human knowing capacities, and a possible solution to that conflict. On the one hand, as we will point out, there exist a number of approaches that fall under the broad term of “evolutionary reliabilism” and according to which the reasons that we have for believing in the reliability of human cognition are empirical in character. Namely, the adaptive success of our species in a biological environment characterised by the survival of the fittest provides us with a reason to believe that our belief-forming mechanisms are truth-tracking; if they were not, we would have gone extinct. On the other hand, nevertheless, we find analyses—of which we will focus on the case of Ernest Sosa—according to which the tenet that our knowing capacities are reliable cannot be empirically based, and thus contingent; on the contrary, they point out that we need to presuppose—without need of a proof—that such capacities are trustworthy. Given this conflict between an “empirical” defence and a defence “in principle” of our knowing capacities, we will consider two possible objections against Sosa’s proposal and try to answer them.

In a provocative, yet scarcely discussed, argument at the end of Knowing Full Well, Ernest Sosa has attempted to determine what kind of evidence we possess in support of the belief that our cognitive capacities as human beings are reliable. According to Sosa, we can appeal to considerations of coherence to prove that such capacities are reliable. However, Sosa also declares that such considerations are not “determinative, ultima facie” reasons−which is to say, they are to be regarded as defeasible. As we will try to point out, this overall strategy is ultimately incoherent. Furthermore, as we will argue, Sosa fails in attempting to provide us with an analogy between the case of doubting the reliability of the cognitive faculties of an individual and doubting such reliability in the case of the species.

According to what Jonathan Bennett calls the Kant–Frege view of existence, Frege gave solid logical foundations to Kant’s claim that existence is not a real predicate. In this article I will challenge Bennett’s claim by arguing that although Kant and Frege agree on what existence is not, they agree neither on what it is nor on the importance and justification of existential propositions. I identify three main differences: first, whereas for Frege existence is a property of a concept, for Kant it is a relational property pertaining between the concept and intuition of an object. Second, whereas for Frege truth about individuals presupposes their existence, for Kant truth is in many cases independent of the existence of objects. Third, whereas Frege binds logic to existence and removes modalities from logic, for Kant existence is a modal category that is emphatically removed from the domain of logic and set in the core of metaphysics. Due to these differences in Kant’s and Frege’s theories of existence, Frege cannot be seen as giving logical clarity to Kant’s view.

What is possible and why? What is the difference between the merely possible and the actual? In Kants Modal Metaphysics Nicholas Stang examines Kants lifelong engagement with these questions and their role in his philosophical development. This is the first book to trace Kants theory of possibility all theway from the so-called pre-Critical writings of the 1750s and 1760s to the Critical system of philosophy inaugurated by the Critique of Pure Reason in 1781. Stang argues that the key to understanding both the change and the continuity between Kants pre-Critical and Critical theory of possibility is his transformation of the ontological question about possibility-what is it for a being to be possible?-into a question in transcendental philosophy-what is it to represent an object as possible?

In this essay, I propose a novel way of thinking about Kant’s philosophical methodology during the critical period. According to this interpretation, the critical Kant can generally be understood as operating within a “capacities‐first” philosophical framework – that is, within a framework in which our basic rational or cognitive capacities play both an explanatorily and epistemically fundamental role in philosophy – or, at least, in the sort of philosophy that limited creatures like us are capable of. In discussing this idea, I consider the complicated relationship between the explanatory and epistemic roles that such capacities play in Kant. I also sketch how this way of thinking about Kant’s methodology can illuminate the foundations of both his theoretical and his practical philosophy, before discussing some of Kant’s reasons for finding this approach to philosophy attractive. I close with a brief discussion of the contemporary relevance of this approach.

In this paper I discuss the role of the nonsensical ‘statements’ of Wittgenstein’s Tractatus and the aims of the book, a topic which has in recent years been the subject of, at times heated, controversy among Wittgenstein’s readers.1 In this debate the so-called ineffability interpretation argues that the role of nonsense in the Tractatus is to make us grasp ineffable truths which ‘strictly speaking’ cannot be said or thought2. By contrast, the interpretation known as the resolute reading emphasises the incomprehensibility of the notion of ineffable truths. According to the latter, nonsense in the Tractatus serves a therapeutic purpose: that of curing us from attempts to put forward nonsensical philosophical doctrines3. By employing a method of juxtaposition I..