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)

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...) Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (shrink)

What can be more fascinating than experimental metaphysics, to quote one of Abner Shimony’s enlightening expressions? Bell inequalities are at the heart of the study of nonlocality. I present a list of open questions, organised in three categories: fundamental; linked to experiments; and exploring nonlocality as a resource. New families of inequalities for binary outcomes are presented.

Kant argues in the Critique of the Teleological Power of Judgment that the first stage in resolving the problem of teleology is conceiving it correctly. He explains that the conflict between mechanism and teleology, properly conceived, is an antinomy of the power of judgment in its reflective use regarding regulative maxims, and not an antinomy of the power of judgment in its determining use regarding constitutive principles. The matter in hand does not concern objective propositions regarding the possibility of objects (...) or actual features of certain objects, namely, organisms. It is rather a methodological issue as to the appropriate way to explain the generation, development, and function of organisms. Taken in this manner as subjective maxims guiding the explanation and inquiry of organisms, the principles of mechanism and teleology need not necessarily be seen as contradictorily opposed but instead can be combined in the study of organisms. This, however, is not enough to complete the analysis of the antinomy of the teleological power of judgment. In order to show that there is an antinomy in this case, Kant has to establish that both seemingly conflicting maxims are necessary and natural to the human mind. He does it by grounding them in the ‘special character’ or peculiarity (Eigentümlichkeit) of the human understanding. However, it is not entirely clear just what exactly this peculiarity of the human understanding is. Paul Guyer argues that Kant suggests two different accounts of the peculiarity of the human intellect. According to one account, this peculiarity consists in the fact that our understanding forms general concepts and according to another, in its propensity to proceed from the parts to the whole. I will argue in this paper that Kant puts forward a single account, in which the combination of these two features demonstrate the peculiarity of the human understanding manifested in the encounter with organisms. This account explains the necessity of the regulative maxims of mechanism and teleology, and thus completes Kant’s analysis of the antinomy of the teleological power of judgment. (shrink)

Locke and Leibniz are often classified as proponents of compatibilist theories of human freedom, since both maintain that freedom is consistent with determinism and that the difference between being and not being free turns on how one is determined. However, we will argue in this paper that their versions of compatibilism are essentially different and that they have significantly distinct commitments to compatibilism. To this end, we will first analyze the definitions and examples for freedom and necessity that Locke and (...) Leibniz present in sections 8-13 of chapter 21 of the Essay on Human Understanding and the Nouveaux essais respectively, and then conjecture how Locke and Leibniz would have continued the discussion, if they had had the opportunity to engage in an exchange of opinions. In this way, we believe, one will be in a position to understand why Leibniz thinks that Locke’s discussion of freedom “est un des plus prolixes et des plus subtils de son ouvrage.”. (shrink)

SPA: En este artículo me pregunto qué posición ocupa el lenguaje en la estructura metafísica realista propuesta por Xavier Zubiri, asumiendo las consecuencias de su crítica radical contra el pensamiento representacional. Para ello exploro los bordes de la tesis según la cual sólo hay una realidad, a la vez que me remito principalmente a las consideraciones del filósofo vasco acerca del lenguaje en Sobre el hombre, así como a las relaciones conceptuales que elabora entre lenguaje, mentalidad y esquemas de intelección (...) en esta obra y en Sobre la esencia. Concluyo tentativamente que el lenguaje es la realización del contenido en el despliegue del logos a partir de los esquemas de intelección. ENG: In this article I ask what position language occupies in the realist metaphysical structure proposed by Xavier Zubiri, assuming the consequences of his radical critique of representational thinking. To this end I explore the boundaries of the thesis that states that there is only one reality, while circling back primarily to the Basque philosopher’s considerations about language in On Man, as well as the conceptual relations that he posits between language, mentality and intellection schemes in this work and in On Essence. I tentatively conclude that language is the realization of content in the deployment of logos by means of intellection schemes. (shrink)

Kant’s theory of biology in the Critique of the Power of Judgment may be rejected as obsolete and attacked from two opposite perspectives. In light of recent advances in biology one can claim contra Kant, on the one hand, that biological phenomena, which Kant held could only be explicated with the help of teleological principles, can in fact be explained in an entirely mechanical manner, or on the other, that despite the irreducibility of biology to physico-mechanical explanations, it is nonetheless (...) proper science. I argue in response that Kant’s analysis of organisms is by no means obsolete. It reveals biology’s uniqueness in much the same way as several current theorists do. It brings to the fore the unique purposive characteristics of living phenomena, which are encapsulated in Kant’s concept of “natural end” and which must be explicated in natural terms in order for biology to become a science. I maintain that Kant’s reluctance to consider biology proper science is not a consequence of his critical philosophy but rather of his inability to complete this task. Kant lacked an appropriate theoretical framework, such as provided later by modern biology, which would enable the integration of the unique features of biology in an empirical system. Nevertheless, as I show in this paper, the conceptual problems with which Kant struggled attest more to the relevance and depth of his insights than to the shortcomings of his view. His contribution to the biological thought consists in insisting on an empirical approach to biology and in providing the essential philosophical underpinning of the autonomous status of biology. (shrink)

Leibniz’s main thesis regarding the nature of space is that space is relational. This means that space is not an independent object or existent in itself, but rather a set of relations between objects existing at the same time. The reality of space, therefore, is derived from objects and their relations. For Leibniz and his successors, this view of space was intimately connected with the understanding of the composite nature of material objects. The nature of the relation between space and (...) matter was crucial to the conceptualization of both space and matter. In this paper, I discuss Leibniz’s account of relational space and examine its novel elaborations by two of his successors, namely, the young Immanuel Kant and the Croat natural philosopher Roger Boscovich. Kant’s and Boscovich’s studies of Leibniz’s account lead them to original versions of the relational view of space. Thus, Leibniz’s relational space proved to be a philosophically fruitful notion, as it yielded bold and intriguing attempts to decipher the nature of space and was a key part in innovative scientific ideas. (shrink)

In this chapter we will deal with “mechanizing” induction, i.e. with ways in which theoretical computer science approaches inductive generalization. In the field of Machine Learning, algorithms for induction are developed. Depending on the form of the available data, the nature of these algorithms may be very different. Some of them combine geometric and statistical ideas, while others use classical reasoning based on logical formalism. However, we are not so much interested in the algorithms themselves, but more on the philosophical (...) and theoretical foundations they share. Thus in the first of two parts, we will examine different approaches and inductive assumptions in two particular learning settings. While many machine learning algorithms work well on a lot of tasks, the interpretation of the learned hypothesis is often difficult. Thus, while e.g. an algorithm surprisingly is able to determine the gender of the author of a given text with about 80 percent accuracy [Argamon and Shimoni, 2003], for a human it takes some extra effort to understand on the basis of which criteria the algorithm is able to do so. With that respect the advantage of approaches using logic are obvious: If the output hypothesis is a formula of predicate logic, it is easy to interpret. However, if decision trees or algorithms from the area of inductive logic programming are based purely on classical logic, they suffer from the fact that most universal statements do not hold for exceptional cases, and classical logic does not offer any convenient way of representing statements which are meant to hold in the “normal case”. Thus, in the second part we will focus on approaches for Nonmonotonic Reasoning that try to handle this problem. Both Machine Learning and Nonmonotonic Reasoning have been anticipated partially by work in philosophy of science and philosophical logic. At the same time, recent developments in theoretical computer science are expected to trig- ger further progress in philosophical theories of inference, confirmation, theory revision, learning, and the semantic and pragmatics of conditionals. We hope this survey will contribute to this kind of progress by building bridges between computational, logical, and philosophical accounts of induction. (shrink)