In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of rest (...) and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises. (shrink)

A critical survey of the main philosophical theories about events and event talk, organized in three main sections: (i) Events and Other Categories (Events vs. Objects; Events vs. Facts; Events vs. Properties; Events vs. Times); (ii) Types of Events (Activities, Accomplishments, Achievements, and States; Static and Dynamic Events; Actions and Bodily Movements; Mental and Physical Events; Negative Events); (iii) Existence, Identity, and Indeterminacy.

There are numerous occasions on which we need to reason about a finite number of events. And we often need to consider only those events which are given or which we perceive. These give rise to the Criteria of Finiteness and Closedness. Allen's logic provides a way of reasoning about events. In this paper I examine Allen and Hayes' axiomatisation of this logic, and develop two other axiomatisations based on the work by Russell and Thomason. I shall show that these (...) three axiomatisations are weakly equivalent, and that only the last two meet the Criteria of Finiteness and Closedness (to different degrees). I shall then examine two ways of constructing instants of time in a finite and closed world, i.e. the Russell construction and the Thomason construction. I shall prove that these two constructions are equivalent under certain conditions. (shrink)

The idea that temporal propositions are vague predicates is examined with attention to the nature of the objects over which the predicates range. These objects should not, it is argued, be identified once and for all with points or intervals in the real line (or any fixed linear order). Context has an important role to play not only in sidestepping the Sorites paradox (Gaifman 2002) but also in shaping temporal moments/extent (Landman 1991). The Russell-Wiener construction of time from events (Kamp (...) 1979) is related to a notion of context given by a string of observations, the vagueness in which is brought out by grounding the observations in the real line. With this notion of context, the context dependency functions in Gaifman 2002 are adapted to interpret temporal propositions. (shrink)

Events are center stage in several fields of psychological research. There is a long tradition in the study of event perception, event recognition, event memory, event conceptualization and segmentation. There are studies devoted to the description of events in language and to their representation in the brain. There are also metapsychological studies aimed at assessing the nature of mental events or the grounding of intentional action. Outside psychology, the notion of an event plays a prominent role in various areas of (...) philosophy as well as in such diverse disciplines as linguistics, probability theory, artifical intelligence, physics, and�of course� history. This plethora of concerns and applications is indicative of the prima facie centrality of the notion of an event in our conceptual scheme, but it also gives rise to some important methodological questions. Can we identify a core notion that is preserved across disciplines? Does this notion, or some such notion, correspond to the pre-theoretical conception countenanced by common sense? Does it correspond to a genuine metaphysical category? (shrink)

In many of his writings Russell developed a theory of time,highly interesting bothfrom a philosophical and from a logical point of view.Strangely enough, this has not acquired generalattention. The most important relational propertiesof the duration and points of time will be presented.In addition, Russell's considerations on the existenceand density of time points will becritically analysed and systematically reconstructed.A. G. Walker's explication of the concept of timepoint is unlike that of Russell. His theory oftime reflects the Dedekindian concept of cut.Walker's constructions (...) will be modified in a suitableway and supplemented so that it becomes possileto prove the continuity of the temporal series. (shrink)

This essay attempts to develop a psychologically informed semantics of perception reports, whose predictions match with the linguistic data. As suggested by the quotation from Miller and Johnson-Laird, we take a hallmark of perception to be its fallible nature; the resulting semantics thus necessarily differs from situation semantics. On the psychological side, our main inspiration is Marr's (1982) theory of vision, which can easily accomodate fallible perception. In Marr's theory, vision is a multi-layered process. The different layers have filters of (...) different gradation, which makes vision at each of them approximate. On the logical side, our task is therefore twofold - to formalise the layers and the ways in which they may refine each other, and. (shrink)

In this paper we provide a mathematical model of Kant’s temporal continuum that yields formal correlates for Kant’s informal treatment of this concept in theCritique of Pure Reasonand in other works of his critical period. We show that the formal model satisfies Kant’s synthetic a priori principles for time and that it even illuminates what “faculties and functions” must be in place, as “conditions for the possibility of experience”, for time to satisfy such principles. We then present a mathematically precise (...) account of Kant’s transcendental theory of time—the most precise account to date.Moreover, we show that the Kantian continuum which we obtain has some affinities with the Brouwerian continuum but that it also has “infinitesimal intervals” consisting of nilpotent infinitesimals; these allow us to capture Kant’s theory of rest and motion in theMetaphysical Foundations of Natural Science.While our focus is on Kant’s theory of time the material in this paper is more generally relevant for the problem of developing a rigorous theory of the phenomenological continuum, in the tradition of Whitehead, Russell, and Weyl among others. (shrink)

Finite-state methods are applied to the Russell-Wiener notion of time (based on events) and developed into an account of interval relations and temporal propositions. Strings are formed and collected in regular languages and regular relations that are argued to embody temporal relations in their various underspecified guises. The regular relations include retractions that reduce computations by projecting strings down to an appropriate level of granularity, and non-deterministic relations defining notions of partiality within and across such levels.

Russell and Walker proposed different ways of constructing instants from events. For an explanation of "time as a continuum," Thomason favored Walker's construction. The present article shows that Russell's construction fares as well. To this end, a mathematical characterization problem is solved which corresponds to the characterization problem that Thomason solved with regard to Walker's construction. It is shown how to characterize those event structures (formally, interval orders) which, through Russell's construction of instants, become linear orders isomorphic to a given (...) (or, deriving, to some—nontrivial ordered) real interval. As tools, separate characterizations for each of resulting (i) Dedekind completeness, (ii) separability, (iii) plurality of elements, (iv) existence of certain endpoints are provided. Denseness is characterized to replace Russell's erroneous attempt. Somewhat minimal nonconstructive principles needed are exhibited, and some alternative approaches are surveyed. (shrink)

Situations serving as partial worlds as well as events in natural language semantics are constructed from a type-theoretic interpretation of firstorder formulae and (after a type reduction) temporal formulae. Limitations of the Russell-Wiener-Kamp derivation of time from events are discussed and overcome to give a more widely applicable account of temporal granularity. Finite situations are formulated as strings of observations, conceptualized to persist inertially (in the absence of forces).

We classify two of Bertrand Russell's theories of events within the point-free ontology. The first of such approaches was presented informally by Russell in ‘The World of Physics and the World of Sense’ (Lecture IV in Our Knowledge of the External World of 1914). Based on this theory, Russell sketched ways to construct instants as collections of events. This paper formalizes Russell's approach from 1914. We will also show that in such a reconstructed theory, we obtain all axioms of Russell's (...) second theory from 1936 and all axioms of Thomason's theory of events from 1989. Russell's work certainly influenced the works of Stanisław Leśniewski, his student Alfred Tarski, and Czesław Lejewski – prominent members of the Lvov-Warsaw School (LWS). We see our work in the tradition of the research of Leśniewski and Tarski. Building on the technical tools developed in this environment and in the spirit of the traditional research of the LWS, we engage here, in particular, with two classic works by Russell on fundamental ontology. (shrink)

Obwohl es "die Zeit" nicht gibt, ordnet sich doch alles, was wir erleben, zeitlich. Auch die großen Schlagworte unserer Tage betreffen allesamt von zeitlichen Herausforderungen: Nachhaltigkeit, Resilienz, Transformation, Zeitenwende. Dieses Buch handelt davon, was Zeitliches ausmacht, warum sich die Wirklichkeit zeitlich ordnet und was das mit der wechselseitigen Taktung von Ereignissen und Autonomieerfahrungen zu tun hat. Es werden Missverständnisse aufgelöst, indem aufgezeigt wird, inwiefern es "die Zeit" nicht gibt, es oftmals sogar leidvolle bis hin zu pathologischen Konsequenzen mit sich bringt, (...) wenn Zeit als Gegenstand oder Ressource betrachtet wird. Es ist ein Buch über den Vergleich von Abläufen und darüber, was es bedeutet, wenn Ereignisse miteinander im Takt sind, und wie sie durch Wiederholung und Variation Bedeutung gewinnen und damit "intakt" wirken. Musik und vor allem das Hören dienen als Modelle, um diese allgemeinen zeitlichen Struktureigenschaften der Wirklichkeit begrifflich anklingen zu lassen. Darüber hinaus regen Hyperlinks zu konkreten Hörbeispielen dazu an, sich direkt auch auf besondere Erfahrungen von Zeit einzulassen. - Ein Buch für all diejenigen, die Grundlegendes über Zeit nicht nur lesen wollen, sondern sich im Idealfall auch für einen taktvollen Umgang mit Zeit motivieren lassen möchten. (shrink)

Each phenomenon contains variable components, which are conservative. Because of their conservation, they accumulate. Present phenomena contain constituents of phenomena, belonging to the past which form the present and the future, and their dependence on time is an exponential one - S = Sₒe^t-tᵖ. We assume that before and after tₒ = t-tᵖ = 0 the change pertains to phenomena of one type. The dependency is for each defined phenomenon of one and the same type. The concrete aspect of the (...) change S will depend on the type of the phenomenon. We show in our study how in some cosmological phenomena, the exponential dependence on time is present. The processes of radioactive disintegration of atomic nuclei, are also phenomena of this type. We present the real phenomena as a sum of exponents. Each phenomenon originates, develops and is destroyed. In reality most phenomena are formed as a composition of exponential dependencies of the change. (shrink)