The vicissitudes of mathematical reason in the 20th century Content Type Journal Article Pages 1-6 DOI 10.1007/s11016-011-9556-y Authors ThomasMormann, Department of Logic and Philosophy of Science, University of the Basque Country UPV/EPU, Donostia-San Sebastian, Spain, Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
Abstract. In Dynamics of Reason Michael Friedman proposes a kind of synthesis between the neokantianism of Ernst Cassirer, the logical empiricism of Rudolf Carnap, and the historicism of Thomas Kuhn. Cassirer and Carnap are to take care of the Kantian legacy of modern philosophy of science, encapsulated in the concept of a relativized a priori and the globally rational or continuous evolution of scientific knowledge,while Kuhn´s role is to ensure that the historicist character of scientific knowledge is taken seriously. (...) More precisely, Carnapian linguistic frameworks, guarantee that the evolution of science procedes in a rational manner locally,while Cassirer’s concept of an internally defined conceptual convergence of empirical theories provides the means to maintain the global continuity of scientific reason. In this paper it is argued that Friedman’s neokantian account of scientific reason based on the concept of the relativized a priori underestimates the pragmatic aspects of the dynamics of scientific reason. To overcome this short-coming, I propose to reconsider C.I. Lewis’s account of a pragmatic the priori, recently modernized and elaborated by Hasok Chang. This may be<br><br><br><br><br><br><br><br><br><br&g t;<br><br><br><br><br><br>Keywords: Dynamics of reason, Paradigms, Logical Empiricism,Neokantianism, Pragmatism, Mathematics, Communicative Rationality. (shrink)
We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)
The aim of this paper is to present a general method for constructing natural tessellations of conceptual spaces that is based on their topological structure. This method works for a class of spaces that was defined some 80 years ago by the Russian mathematician Pavel Alexandroff. Alexandroff spaces, as they are called today, are distinguished from other topological spaces by the fact that they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, Ian Rumfitt (apparently not (...) being aware of Alexandroff’s work) used a very special case of Alexandroff’s method to elucidate the logic of vague concepts in a new way. Elaborating his approach, the color circle’s conceptual space can be shown to define an atomistic Boolean algebra of regular open concepts. In a similar way Gärdenfors’ geometrical discretization of conceptual spaces by Voronoi tessellations also can be shown to be a kind of geometrical version of Alexandroff’s topological construction. More precisely, a discretization à la Gärdenfors is extensionally equivalent to a topological discretization constructed by Alexandroff’s method. Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works much more generally, namely, for Alexandroff spaces. For these spaces (X, OX) the Boolean algebras O*X of regular open sets are still atomistic and yield natural tessellations of X. (shrink)
The aim of this paper is to show that a comprehensive account of the role of representations in science should reconsider some neglected theses of the classical philosophy of science proposed in the first decades of the 20th century. More precisely, it is argued that the accounts of Helmholtz and Hertz may be taken as prototypes of representational accounts in which structure preservation plays an essential role. Following Reichenbach, structure-preserving representations provide a useful device for formulating an up-to-date version of (...) a (relativized) Kantian a priori. An essential feature of modern scientific representations is their mathematical character. That is, representations can be conceived as (partially) structure-preserving maps or functions. This observation suggests an interesting but neglected perspective on the history and philosophy of this concept, namely, that structure-preserving representations are closely related to a priori elements of scientific knowledge. Reichenbach’s early theory of a relativized constitutive but non-apodictic a priori component of scientific knowledge provides a further elaboration of Kantian aspects of scientific representation. To cope with the dynamic aspects of the evolution of scientific knowledge, Cassirer proposed a re-interpretation of the concept of representation that conceived of a particular representation as only one phase in a continuous process determined by pragmatic considerations. Pragmatic aspects of representations are further elaborated in the classical account of C.I. Lewis and the more modern of Hasok Chang. (shrink)
Abstract. The main aim of this paper is to show that the concept of vagueness based on an S4 modal operator C of clearness (CA to be read as “It is clear that A”, “It is definitely the case that A”, or similarly) leads to a concept of columnar higher-order vagueness. More precisely, the class of topological models of the operator C that satisfy the requirements an S4 operator has to satisfy can be identified with the class of weakly scattered (...) topological spaces and leads to a concept of higher-order vagueness that is columnar. Polar spaces, recently introduced by Rumfitt to cope with the Sorites paradox and other problems of vague concepts, provide a philosophically particularly interesting class of weakly scattered spaces. More generally, all topological spaces support a concept of stably columnar vagueness that is only slightly weaker than proper columnar vagueness. Further, for all topological spaces, the boundaries of sets that satisfy the McKinsey axiom are columnar. Finally, it is proved that higher-order vagueness is (stably) columnar higher-order vagueness not only for S4-operators but also for operators that satisfy the axioms of Williamson’s “logic of clarity” (which is not S4 but characterized by Brouwer’s axiom (B)). Thus, (stably) columnar vagueness may be said to be almost ubiquitous in modal accounts of vagueness. (shrink)
We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...) main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)
In this paper it is shown that Heyting and Co-Heyting mereological systems provide a convenient conceptual framework for spatial reasoning, in which spatial concepts such as connectedness, interior parts, (exterior) contact, and boundary can be defined in a natural and intuitively appealing way. This fact refutes the wide-spread contention that mereology cannot deal with the more advanced aspects of spatial reasoning and therefore has to be enhanced by further non-mereological concepts to overcome its congenital limitations. The allegedly unmereological concept of (...) boundary is treated in detail and shown to be essentially affected by mereological considerations. More precisely, the concept of boundary turns out to be realizable in a variety of different mereologically grounded versions. In particular, every part K of a Heyting algebra H gives rise to a well-behaved K-relative boundary operator. (shrink)
Abstract: One of the institutional highlights of the encounter between Austrian “wissen¬schaftliche Philosophie” and French “philosophie scientifique” in the first half of the 20th century was the “First International Congress for Unity of Science” that took place 1935 in Paris. In my contribution I deal with an episode of the philosophical mega-event whose protagonist was the American philosopher and semiotician Charles William Morris. At the Paris congress he presented his programme of a comprehensive, practice-oriented scientific philosophy and, in a more (...) elaborated version he published it two years later in Logical Positivism, Pragmatism and Scientific Empiricism (Morris 1937). Morris aimed at a synthesis of formalism, pragmatism, and traditional empiricism that combined the virtues of these accounts while avoided their shortocmings. The core of approach was a comprehensive theory of the concept of meaning. Through an analysis of the concept of meaning he sought to sort out the existing differences and the options for a possible future rapprochment between logical empiricism and pragmatism. Against the overly narrow logical empiricist understanding of philosophy as the syntax of the language of science Morris argued for a “scientific pragmatism” that comprised four levels: (1) Philosophy as Logic of Science, (2) Philosophy as Clarification of Meaning (Peirce), (3) Philosophy as Empirical Axiology (Dewey), and (4) Philosophy as Empirical Cosmology (Whitehead). (shrink)
A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that Natorp's (...) metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)
The aim of this paper is to discuss the “Austro-American” logical empiricism proposed by physicist and philosopher Philipp Frank, particularly his interpretation of Carnap’s Aufbau, which he considered the charter of logical empiricism as a scientific world conception. According to Frank, the Aufbau was to be read as an integration of the ideas of Mach and Poincaré, leading eventually to a pragmatism quite similar to that of the American pragmatist William James. Relying on this peculiar interpretation, Frank intended to bring (...) about a rapprochement between the logical empiricism of the Vienna Circle in exile and American pragmatism. In the course of this project, in the last years of his career, Frank outlined a comprehensive, socially engaged philosophy of science that could serve as a “link between science and philosophy”. (shrink)
Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.
Nach gängiger Auffassung nahm das Thema „Werte” in Carnaps Philosophie nur einen geringen Stellenwert ein. In dieser Arbeit soll gezeigt werden, daß diese Einschätzung der Korrektur bedarf: So wird der im „Aufbau“ vorgetragene Entwurf eines Konstitutionssystems mit Werten als der höchsten Schicht des Konstitutionssystems abgeschlossen. Auch die Quasianalyse als allgemeine Konstitutionsmethode steht in enger Beziehung zur Unterscheidung zwischen „Sein“ und „Gelten“, die für den werttheoretisch orientierten Neukantianismus der südwest-deutschen Schule charakteristisch war. Allgemein erlaubt die Wertthematik einen Blick auf Unterströmungen des (...) carnapschen Denkens, die in den offiziellen logisch-empiristischen Darstellungen seines Denkens meist vernachlässigt werden. Die Wertthematik, so die These dieser Arbeit, eröffnet überdies eine neue Perspektive auf die spezifisch Jenaer Konstellation von Neukantianismus und Lebensphilosophie, die Carnaps Philosophie wesentlich geprägt hat. (shrink)
A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)
In this paper I want to show that topology has a bearing on the theory of tropes. More precisely, I propose a topological ontology of tropes. This is to be understood as follows: trope ontology is a „one-category”-ontology countenancing only one kind of basic entities, to wit, tropes. 1 Hence, individuals, properties, relations, etc. are to be constructed from tropes.
Abstract. Let REL(O*E) be the relation algebra of binary relations defined on the Boolean algebra O*E of regular open regions of the Euclidean plane E. The aim of this paper is to prove that the canonical contact relation C of O*E generates a subalgebra REL(O*E, C) of REL(O*E) that has infinitely many elements. More precisely, REL(O*,C) contains an infinite family {SPPn, n ≥ 1} of relations generated by the relation SPP (Separable Proper Part). This relation can be used to define (...) point-free concept of connectedness that for the regular open regions of E coincides with the standard topological notion of connectedness, i.e., a region of the plane E is connected in the sense of topology if and only if it has no separable proper part. Moreover, it is shown that the contact relation algebra REL(O*E, C) and the relation algebra REL(O*E, NTPP) generated by the non-tangential proper parthood relation NTPP, coincide. This entails that the allegedly purely topological notion of connectedness can be defined in mereological terms. (shrink)
In this paper some classical representational ideas of Hertz and Duhem are used to show how the dichotomy between representation and intervention can be overcome. More precisely, scientific theories are reconstructed as complex networks of intervening representations (or representational interventions). The formal apparatus developed is applied to elucidate various theoretical and practical aspects of the in vivo/in vitro problem of biochemistry. Moreover, adjoint situations (Galois connections) are used to explain the relation berween empirical facts and theoretical laws in a new (...) way. (shrink)
Carnap and Twentieth-Century Thought: Explication as En lighten ment is the first book in the English language that seeks to place Carnap's philosophy in a broad cultural, political and intellectual context. According to the author, Carnap synthesized many different cur rents of thought and thereby arrived at a novel philosophical perspective that remains strik ing ly relevant today. Whether the reader agrees with Carus's bold theses on Carnap's place in the landscape of twentieth-century philosophy, and his even bolder claims concerning (...) the role that philosophy in Carnap's style should play in the thought of our century, does not matter so much as the excellent opportunity Carus's book offers to thoroughly rethink one's ideas about Carnap's philosophy. One reason why Carnap and Twentieth-Century Thought might change one's ideas is that Carus has unearthed much hitherto unknown material from the archives that sheds new light on Carnap's early life and thought. Indeed, the many archival findings presented in CTT for the first time suffice to make the book re warding reading for philosophers and historians of philosophy alike. CTT exhibits a high standard of historical scholarship, and the book itself is a beautiful example of high-quality academic publishing. Up to now, Carnap has remained a controversial figure on the philo sophical scene. On the one hand, he has a solid reputation as a leading figure of logical positivism . According to conventional wisdom, this was a school of thought characterized by its formal and technical philosophy, as well as being rather dismissive of other ways of doing philosophy, dogmatically sticking to its own theses. As a typical example of this arrogant logical empiricist attitude, one usually refers to Carnap's notorious Overcoming Metaphysics by Logical Analysis of Language , written when the Vienna Circle's Logical Empiricism had entered its most radical phase. Self-proclaimed postpositivist philosophers of science dismissed logical positivism, in particular Carnap's, as the dogmatic and orthodox “received view.” The tendency to portray logical empiricism as an obsolete doctrine centering around certain “dogmas” started with Quine's Two Dogmas of Empiricism and reached its somewhat ridiculous culmination in the early 1980s when allegedly “six or seven dogmas” were discovered . Thereby an allegedly un brid geable gap between classical “dogmatic” logical em pi ricism and its modern “enlightened” suc ces sors was construed. (shrink)
Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, from the central position it (...) used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)
The aim of this paper is to elucidate the mereological structure of complex states of affairs without relying on the problematic notion of structural universals. For this task tools from graph theory, lattice theory, and the theory of relational systems are employed. Our starting point is the mereology of similarity structures. Since similarity structures are structured sets, their mereology can be considered as a generalization of the mereology of sets..
In A Parting of the Ways Michael Friedman proposed to conceive the contemporary divide between analytic philosophy (AP) and continental philosophy (CP) as the outcome of the bifurcation between the Neokantians of Heidelbarg and Marburg. According to Friedman, Carnap can be characterized as the executor of the Marburg school, while Heidegger is to be considered as the heir of the Southwest Neokantianism. In this paper it is argued that Carnap was much closer to the Southwest Neokantianism than usually recognized. To (...) some extent, Carnap’s project in the Aufbau may be described as an attempt to modernize the Neokantian approach of the Southwest school with the help of modern relational logic. This entails that Carnap cannot be lined so easily with the Marburg school as Friedman assumes. (shrink)
Ernst Cassirer, 2011, Symbolische Prägnanz, Ausdrucksphänomen und „Wiener Kreis“, Nachgelassene Manuskripte und Texte, vol. 4, ed. Christian Möckel, 478pp., Hamburg, Felix Meiner Verlag.
The aim of this paper is make a contribution to the ongoing search for an adequate concept of the a priori element in scientific knowledge. The point of departure is C.I. Lewis’s account of a pragmatic a priori put forward in his "Mind and the World Order" (1929). Recently, Hasok Chang in "Contingent Transcendental Arguments for Metaphysical Principles" (2008) reconsidered Lewis’s pragmatic a priori and proposed to conceive it as the basic ingredient of the dynamics of an embodied scientific reason. (...) The present paper intends to further elaborate Chang’s account by relating it with some conceptual tools from cognitive semantics and certain ideas that first emerged in the context of the category-theoretical foundations of mathematics. (shrink)
In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous lattices and “pointless” (...) topology.
The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures.
Quine’s classical classic interpretation succinctly characterized characterizes Carnap’s Aufbau as an attempt “to account for the external world as a logical construct of sense-data....” Consequently, “Russell” was characterized as the most important influence on the Aufbau. Those times have passed. Formulating a comprehensive and balanced interpretation of the Aufbau has turned out to be a difficult task and one that must take into account several disjointed sources. My thesis is that the core of the Aufbau rested on a problem that (...) had haunted German philosophy since the end of the 19th century. In terms fashionable at the time, this problem may be expressed as the polarity between Leben and Geist that characterized German philosophy during the years of the Weimar Republic. At that time, many philosophers, including Cassirer, Rickert and Vaihinger, were engaged in overcoming this polarity. As I will show, Carnap’s Aufbau joined the ranks of these projects. This suggests that Lebensphilosophie and Rickert’s System der Philosophie exerted a strong influence on Carnap’s projects, an influence that is particularly conspicuous in his unpublished manuscript Vom Chaos zur Wirklichkeit. Carnap himself asserted that this manuscript could be considered “the germ of the constitution theory” of the Aufbau. Reading Chaos also reveals another strong but neglected influence on the Aufbau, namely a specific version of neutral monism put forward by the philosopher and psychologist Theodor Ziehen before World War I. Ziehen’s work contributed much to the invention of the constitutional method of quasi-analysis. -/-. (shrink)
The concept of similarity has had a rather mixed reputation in philosophy and the sciences. On the one hand, philosophers such as Goodman and Quine emphasized the „logically repugnant“ and „insidious“ character of the concept of similarity that allegedly renders it inaccessible for a proper logical analysis. On the other hand, a philosopher such as Carnap assigned a central role to similarity in his constitutional theory. Moreover, the importance and perhaps even indispensibility of the concept of similarity for many empirical (...) sciences can hardly be denied. The aim of this paper is to show that Quine’s and Goodman’s harsh verdicts about this notion are mistaken. The concept of similarity is susceptible to a precise logico-mathematical analysis through which its place in the conceptual landscape of modern mathematical theories such as order theory, topology, and graph theory becomes visible. Thereby it can be shown that a quasi-analysis of a similarity structure S can be conceived of as a sheaf (etale space) over S. (shrink)
In this paper we argue for the thesis that theories are to be considered as representations. The term "representation" is used in a sense inspired by its mathematical meaning. Our main thesis asserts that theories of empirical theories can be conceived as geometrical representations. This idea may be traced back to Galileo. The geometric format of empirical theories should not be simply considered as a clever device for displaying a theory. Rather, the geometrical character deeply influences the theory s ontology. (...) We argue that it would be desastrous for philosophy if it followed Rorty s "neo-pragmatic" proposal to discard the concept of representation from philosophical discourse. (shrink)
Der Antirepräsentationalismus behauptet, es sei an der Zeit, das auf dem Begriff der Repräsentation basierende Paradigma der Philosophie zu verabschieden. Die von Rorty und anderen propagierte Unterscheidung von Repräsentationalismus und Antirepräsentationalismus beruht jedoch auf einem verkürzten Repräsentationsbegriff als Spiegelung. Der Begriff der Repräsentation, so wie er in der Philosophie und Wissenschaft gebraucht wird, hat nur wenig mit Spiegelung zu tun. Statt den Begriff der Repräsentation mit Hilfe der Spiegelungsmetapher zu explizieren, sollte man Repräsentation als strukturerhaltende Abbildung konzipieren. Dadurch lassen sich (...) die für den Repräsentationsbegriff wesentlichen pragmatischen Aspekte der Reduktion und Induktion von Komplexität explizieren. Der Zusammenhang der gegenläufigen Aspekte Reduktion und Induktion wird an Beispielen aus der Konstitution extensiver Größen sowie an Cassirers Theorie der Konstitution von Erkenntnis und Wirklichkeit erörtert. Man sollte daher den Begriff der Repräsentation nicht aus dem philosophischen Diskurs eliminieren, sondern versuchen, die Ansätze einer allgemeinen pragmatischen Repräsentationstheorie auszubauen. (shrink)
These days, a number of philosophers of science indulge in lamenting about a crisis of their discipline. They complain about its loss of relevance, and bemoan the mar gi na lization of their dis cipline in the philosophical community and in the wider academia , Hardcastle and Richardson ). The Munich take on the philosophy of science does not succumb to this temptation. According to it, philosophy of science is well and alive. In Carlos Ulises Moulines’s Die Entwicklung der modernen (...) Wissen schaftstheorie Eine historische Einführung the word “crisis” is used only in reference to the 1940s when clas sical logical positivism encountered some dif fi culties in dealing with problems concerning veri fi cation, the ana ly tic/synthetic distinction, and similar conundrums. For Moulines, “crisis” is not a word that applies to contemporary philosophy of science. (shrink)
In this paper we discuss three examples of the appropriation of Kuhn’s ideas in philosophy of science. First we deal with classical logical empiricism. Perhaps somewhat surprisingly, the arch-logical empiricist Carnap considered Kuhn’s socio-historical account as a useful complementation, and not as a threat of the philosophy of science of logical empiricism. As a second example we consider the attempt of the so-called struc- turalist philosophy of science to provide a “rational reconstruction” of Kuhn’s approach. Finally, we will deal with (...) Friedman’s proposal to apply Kuhn’s ideas to the formulation of a modernized, historically enlightened Kantian approach that is based on the concept of a non-apodictic constitutive and historically moving a priori. (shrink)
The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to (...) prove a completeness theorem for S4.1. Further, it is shown that the McKinsey algebra MKX of a space X endoewed with an alpha-topologiy satisfies Esakia's GRZ axiom. (shrink)
The aim of this paper is to show that global scientific promises aka “scientific world-conceptions” have an interesting history that should be taken into account also for contemporary debates. I argue that the prototypes of many contemporary philosophical positions concerning the role of science in society can already be found in the philosophy of science of the 1920s and 1930s. First to be mentioned in this respect is the Scientific World-Conception of the Vienna Circle (The Manifesto) that promised to contribute (...) to the realization of an enlightened, rational and science-oriented society and culture. The Manifesto was not the only „scientific world-conception“ that philosophers and scientists put forward in the 1920s. Also the scientific world-conception of the philosopher and physicist Moritz Schlick, and the Weltanschauung of Sigmund Freud deserve to be mentioned. Still other examples of are Carnap’s Scientific Humanism and the project of The International Encyclopedia of Unified Science which was related to American pragmatism as well, as is shown by Charles W. Morris and others. Forgotten for a long time, since the beginning of the 21rst century, at least some of the Viennese projects are reconsidered in a new wave of „socially engaged philosophy of science”. (shrink)
The general aim of this paper is to introduce some ideas of the theory of infinite topological games into the philosophical debate on supertasks. First, we discuss the elementary aspects of some infinite topological games, among them the Banach-Mazur game.Then it is shown that the Banach-Mazur game may be conceived as a Newtonian supertask.In section 4 we propose to conceive physical experiments as infinite games. This leads to the distinction between determined and undetermined experiments and the problem of how it (...) is related to that between determinism and indeter-minism. Finally the role of the Axiom of Choice as a source of indetermi-nacy of supertasks is discussed. (shrink)
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