The twofold role of diagrams in Euclid’s plane geometry

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid??s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid??s plane geometry (EPG). Euclid??s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: (i) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; (ii) EPG objects inherit some properties and relations from these diagrams.     

The mathematical form of measurement and the argument for Proposition I in Newton’s Principia

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton??s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition??the putting-together in space??of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton??s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler??s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry.     

Interrogative Belief Revision Based on Epistemic Strategies

I develop a dynamic logic for reasoning about ??interrogative belief revision??, a new branch of belief revision theory that has been developed in a small number of papers, beginning with E. J. Olsson and D. Westlund??s paper ??On the role of the research agenda in epistemic change?? [12]. In interrogative belief revision, epistemic states are taken to include a research agenda, consisting of questions the agent seeks to answer. I present a logic for revision of such epistemic states based on the notion of an epistemic strategy, a stable plan of action that determines changes in the agent??s research agenda. This idea is a further development of an idea put forward in [6], that changes in the research agenda of an agent should be determined by stable, ??long term?? research interests. I provide complete axioms and a decidability result for the logic.     

Big toy models

We pursue a model-oriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we aim for a ??big toy model??, in which both quantum and classical systems can be faithfully represented??as well as, possibly, more exotic kinds of systems. To this end, we show how Chu spaces can be used to represent physical systems of various kinds. In particular, we show how quantum systems can be represented as Chu spaces over the unit interval in such a way that the Chu morphisms correspond exactly to the physically meaningful symmetries of the systems??the unitaries and antiunitaries. In this way we obtain a full and faithful functor from the groupoid of Hilbert spaces and their symmetries to Chu spaces. We also consider whether it is possible to use a finite value set rather than the unit interval; we show that three values suffice, while the two standard possibilistic reductions to two values both fail to preserve fullness.     

Diagrams as sketches

This article puts forward the notion of ??evolving diagram?? as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter ??diagrams*??) in the context of ??sketch theory,?? a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning. Thus, the aim of the paper is twofold. First, it claims that diagrams* provide a clear example of evolving diagrams, and shed light on them as a general phenomenon. Second, in return, it uses sketches, understood as evolving diagrams, to show how diagrams* in general should be re-evaluated positively.     

Hume on space, geometry, and diagrammatic reasoning

Hume??s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume??s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume??s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume??s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume??s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume??s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume??s most original contributions. First, Hume??s epistemological model invokes the ??confounding?? of indivisible minima to explain the appearance of spatial continuity. Second, Hume??s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the ??loose judgements?? of the vulgar.     

FORMALIZATION OF HILBERT'S GEOMETRY OF INCIDENCE AND PARALLELISM

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.     

Notes on Mally’s deontic logic and the collapse of {\varvec{Seinsollen}} and {\varvec{Sein}}

This paper analyzes Mally’s system of deontic logic, introduced in his The Basic Laws of Ought: Elements of the Logic of Willing (1926). We discuss Mally’s text against the background of some contributions in the literature which show that Mally’s axiomatic system for deontic logic is flawed, in so far as it derives, for an arbitrary A, the theorem “A ought to be the case if and only if A is the case”, which represents a collapse of obligation. We then try to sort out and understand which axioms are responsible for the collapse and consider two ways of amending Mally’s system: (i) by changing its original underlying logical basis, that is classical logic, and (ii) by modifying Mally’s axioms.     

The ‘theory theory’ of mind and the aims of Sellars’ original myth of Jones

Recent proponents of the ??theory theory?? of mind often trace its roots back to Wilfrid Sellars?? famous ??myth of Jones?? in his 1956 article, ??Empiricism and the Philosophy of Mind??. Sellars developed an account of the intersubjective basis of our knowledge of the inner mental states of both self and others, an account which included the claim that such knowledge is in some sense theoretical knowledge. This paper examines the nature of this claim in Sellars?? original account and its relationship to more recent debates concerning ??theory of mind??, in particular the theory theory. A close look reveals that Sellars?? original view embodied several distinctions that would enable more recent theory theorists to accommodate certain phenomenological objections that have been raised against that outlook. At the heart of the philosophical issue is an overlooked complexity involved in Sellars?? account of the ??theory/observation?? distinction, involving a conception of the distinction that is both independently plausible and a key to the issue in dispute.     

On A- and B-theoretic elements of branching spacetimes

This paper assesses branching spacetime theories in light of metaphysical considerations concerning time. I present the A, B, and C series in terms of the temporal structure they impose on sets of events, and raise problems for two elements of extant branching spacetime theories??McCall??s ??branch attrition??, and the ??no backward branching?? feature of Belnap??s ??branching space?Ctime????in terms of their respective A- and B-theoretic nature. I argue that McCall??s presentation of branch attrition can only be coherently formulated on a model with at least two temporal dimensions, and that this results in severing the link between branch attrition and the flow of time. I argue that ??no backward branching?? prohibits Belnap??s theory from capturing the modal content of indeterministic physical theories, and results in it ascribing to the world a time-asymmetric modal structure that lacks physical justification.     

Comparative concepts

Comparative concepts such as greener than or higher than are ways of ordering objects. They are fundamental to our grasp of gradable concepts, that is, the type of meanings expressed by gradable general terms, such as ??is green?? or ??is high??, which are embeddable in comparative constructions in natural language. Some comparative concepts seem natural, whereas others seem gerrymandered. The aim of this paper is to outline a theoretical approach to comparative concepts that bears both on the account of naturalness for comparative concepts and on the theory of gradable concepts. The approach is novel in that it carries some basic assumptions from Peter G?rdenfors?? conceptual spaces account of categorical concepts over to comparative concepts. The offered approach is more general than G?rdenfors?? account in that it supplies a framework of graded categorisation that includes his categorisation rule as a limiting case. Importantly, it provides also a new argument for adopting G?rdenfors?? particular model of categorisation.     

Looking for Arguments

How do people find arguments while engaged in a discussion? Following an analogy with visual search, a mechanism that performs this task is described. It is a metarepresentational device that examines representations in a mostly serial manner until it finds a good enough argument supporting one??s position. It is argued that the mechanism described in dual process theories as ??system 2??, or analytic reasoning fulfills these requirements. This provides support for the hypothesis that reasoning serves an argumentative function.     

The Pedagogy of Self-Fashioning: A Foucaultian Study of Montaigne’s “On Educating Children”

In this paper I interpret Montaigne??s essay, ??On Educating Children??, as a pedagogical text through its performance of a distinct epistolary function, one that addresses the letter-recipient for the purpose of shaping the ideas, actions, and beliefs of that individual. At the same time, I also read ??On Educating Children?? within the context of the wider project of Montaigne??s Essays, which, as I suggest, is an ethical-aesthetic project of self-fashioning and self-cultivation. The net result is an interpretation of teaching as an ethical-aesthetic practice of the self, one that is in concert with the interpretation of Montaigne??s writing of the Essays as a similar practice of the self. In order to build this case, I employ Michel Foucault??s fourfold schema of ethical subjectivity, mapping that schema onto ??On Educating Children??, so as to reveal a unique pedagogy of self-formation??a pedagogy that works as much upon the self of the teacher as it does the self of the student.     

Looking Through Ulanowicz’s “Third Window”

After a ??very personal?? introduction, and a reference to how accurate indeed is the use of the ??new window?? metaphor by Ulanowicz and about what ??can be seen through it??, the article dwells into the evolution of our understanding about the most general sources??material and/or non-material??of change and transformation; in order to examine further the item about the ways through which ??information?? can be a source of change and transformation also in pre-biotic processes, where commonly it is not taken into account. Thus, the article argues the convergence between Ulanowicz??s and the Author??s treatments of??non only biotic??networks of configuration of processes (a central theme in Ulanowicz??s book). Finally, the article pays attention to how Ulanowicz??s vision through his ??new Third Window??, can indeed help us transcend the non desirable division of contemporary culture into two isolated compartments (science and technology versus the humanities and ethics), which has contributed not only to the globalization of communications, transactions, knowledge and so on, but also to the globalization of multiple crisis of different sort that need to be solved if we want ??a better world to be possible??.     

An Axiomatisation of a Pure Calculus of Names

A calculus of names is a logical theory describing relations between names. By a pure calculus of names we mean a quantifier-free formulation of such a theory, based on classical propositional calculus. An axiomatisation of a pure calculus of names is presented and its completeness is discussed. It is shown that the axiomatisation is complete in three different ways: with respect to a set theoretical model, with respect to Le?niewski??s Ontology and in a sense defined with the use of axiomatic rejection. The independence of axioms is proved. A decision procedure based on syntactic transformations and models defined in the domain of only two members is defined.     

Hilbert,logicism, and mathematical existence

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).     

Zeitlosigkeit und Verg?nglichkeit (in) der Psychoanalyse

The author shows that the accessible psychoanalytic unconscious is not timeless. Freud??s ??timeless unconscious??, which was ontologized by himself and his successors, proves to be fiction with negative consequences. Assuming ??timeless?? primary processes psychoanalysis as a therapy would bite the hand which feeds it. Instead of contrasting linear time to the cyclic personal experience of time, the author suggests to speak about the ??emergence of lived time?? and the ??new moments?? of encounter in the therapeutic situation, to emphasize the importance of the opportune moment for change in contrast to the cyclic repetition of transference. In order to understand and approach disorders of time perception knowledge of unconscious conflicts and defense processes remains crucial. Even dreams are not timeless. While our conscious mind works according to chronological patterns, the pictorial representation in the particular form of thinking asleep, dreaming, makes it possible to map different memory traces simultaneously. Freud??s speculations about the ??timeless unconscious?? and Nirvana may originate from a longing for redemption. In contrast the author??s final chapter deals with the potential life span and transience of some central psychoanalytic concepts. Not even those who prove to be very close to observation can be attributed with ??eternal life??.     

A System of Axioms for Minkowski Spacetime

Journal of Philosophical Logic - We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the...     

A ��non-aligned�� intelligentsia: Timur Novikov��s neo-avantgarde and the afterlife of Leningrad non-conformism

This article describes a logic of distinction and succession within the late-twentieth-century Leningrad-St. Petersburg cultural field, whereby consecutive intelligentsia mainstreams were replaced by their avant-garde peripheries. In this dynamic picture of socio-cultural transformations, I propose a working hypothesis of a repeated stratification of the field into an ??official??, an ??unofficial??, and a third ??non-aligned?? intelligentsia. This hypothesis is tested in reference to the ??non-aligned?? groups founded by the avant-garde artist and ideologue Timur Novikov (1958?C2002). Three major shifts are described: from the politicized late-Brezhnevite early 1980s to the apolitical radicalism of Novikov??s New Artists; from this anarchistic underground, through the perestroika era, to the playful ??classicism?? of the New Academy of Fine Arts in the 1990s; and from this postmodern international orientation to an arch-reactionary, neo-imperial posturing at the turn of the 2000s. Lastly, this ??non-aligned?? intelligentsia is suggested as a possible precedent, or, indeed, a model for understanding other historically significant avant-garde peripheries, which commonly seek to distinguish themselves from (often mutually-exclusive) centres.     

Trust and the doxastic family

This article examines Keith Lehrer??s distinction between belief and acceptance and how it differs from other accounts of belief and of the family of doxastic attitudes. I sketch a different taxonomy of doxastic attitudes. Lehrer??s notion of acceptance is mostly epistemic and at the service of his account of the ??loop of reason??, whereas for other writers acceptance is mostly a pragmatic attitude. I argue, however, that his account of acceptance underdetermines the role that the attitude of trust plays in his analysis of reason.