The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come. Contributors Paul Benacerraf, George Boolos, John P. Burgess, Charles S. Chihara, Michael Detlefsen, Michael Dummett, Hartry Field, Kit Fine, (...) Bob Hale, Richard G. Heck, Jnr., Geoffrey Hellman, Penelope Maddy, Karl-Georg Niebergall, Charles D. Parsons, Michael D. Resnik, Matthias Schirn, Stewart Shapiro, Peter Simons, W.W. Tait, Crispin Wright. (shrink)

The main claim of this paper is that notions of implementation based on an isomorphic correspondence between physical and computational states are not tenable. Rather, ``implementation'' has to be based on the notion of ``bisimulation'' in order to be able to block unwanted implementation results and incorporate intuitions from computational practice. A formal definition of implementation is suggested, which satisfies theoretical and practical requirements and may also be used to make the functionalist notion of ``physical realization'' precise. The upshot of (...) this new definition of implementation is that implementation cannot distinguish isomorphic bisimilar from non-isomporphic bisimilar systems anymore, thus driving a wedge between the notions of causal and computational complexity. While computationalism does not seem to be affected by this result, the consequences for functionalism are not clear and need further investigations. (shrink)

In both the public and the business world, in academe as well as in practice, the ideas of Adam Smith are regarded as the bedrock of modern economics. When present economic conditions and management practices are criticised, Adam Smith is referred to by defenders and detractors of the current status quo alike. Smith, it is believed, defined the essential terms of reference of these debates, such as the rational pursuit of self-interest on part of the individual and the resultant optimal (...) allocation of goods in free markets thanks to the workings of an “invisible hand.” In this article, we question whether this standard view of Smith, the economist, is tenable. We provide an extensive review of the extant secondary literature from economists, business ethicists, and philosophers, comparing their assessments to crucial elements of Smith’s theoretical system. As a result, we show that Smith, far from being an advocate of a value-free or even value-averse conception of economic transactions, stood for a virtue-based and values-oriented model of business. Accordingly, we argue current management education and the pedagogy of business ethics ought to be changed, and certain strategic conclusions drawn for business practice. (shrink)

Adam Smith’s is often falsely portrayed as having argued that radical selfishness is a force for the good and that this “invisible hand’ is his market mechanism. This paper argues that Smith’s real market mechanism, the sympathy manoeuvre, is a viable alternative to Schumpeterian and mainstream models of innovation in economics and also could help build a firmer theoretical basis for other approaches such as Responsible Innovation. To Smith all human activity was social and must be understood and explained in (...) terms of the sentiments involved. Discovery, for instance, is driven by three sentiments (wonder, surprise, admiration), economic activities by sympathetic imagination, the need for exchange, the need to better one’s position in life, and the need for gratitude. Through sympathetic imagination, his famous model of the impartial spectator, Smith elegantly connects the individual and society. Smith’s innovation process is thus an exercise in social construction and not a destructive process based on radical selfishness. The paper argues that this social innovation process is a viable alternative to the extant approaches that are essentially asocial and amoral (economism) or ideologically normative (Responsible Innovation). (shrink)

Das Gerucht, Gott sei tot, scheint ubertrieben gewesen zu sein. An dessen Verbreitung haben die Naturwissenschaften einen wesentlichen Anteil gehabt. Gegenwartig fuhren dagegen gerade sie Gott wieder im Munde. Massgeblichen Anteil daran hat die Entdeckung des Phanomens der Feinabstimmung Wenn Naturkonstanten, die die Gestalt unseres Universums bestimmen, nur wenig andere Werte angenommen hatten, hatte es bewusstes Leben nie gegeben. Matthias Schleiff entwickelt daraus das teleologische Feinabstimmungsargument. In der Nachfolge traditioneller Gottesbeweise, aber mit den wissenschaftlichen Einsichten unserer Zeit, argumentiert er (...) dafur, dass sich die Annahme eines kosmischen Schopfers auf rationale Grunde stutzen kann. Er setzt sich dabei intensiv mit alternativen Erklarungshypothesen auseinander - etwa der Erklarung durch den Zufall, das Anthropische Prinzip oder die Theorie des Multiversums - und bringt so Naturwissenschaften, Philosophie und Theologie wieder in ein Gesprach uber den Anfang der Welt. (shrink)

Setting out the 'state of the art' in proportionality doctrine, this book combines theoretical reconstruction with case-law examples, defending and developing the dominant model of proportionality.

This book considers Robert Brandom's philosophy and presents an original and exciting analysis of the semantic argument in legal argumentation.

All conceptions of equal opportunity draw on some distinction between morally justified and unjustified inequalities. We discuss how this distinction varies across a range of philosophical positions. We find that these positions often advance equality of opportunity in tandem with distributive principles based on merit, desert, consequentialist criteria or individuals' responsibility for outcomes. The result of this amalgam of principles is a festering controversy that unnecessarily diminishes the widespread acceptability of opportunity concerns. We therefore propose to restore the conceptual separation (...) of opportunity principles concerning unjustified inequalities from distributive principles concerning justifiable inequalities. On this view, equal opportunity implies that that morally irrelevant factors should engender no differences in individuals' attainment, while remaining silent on inequalities due to morally relevant factors. We examine this idea by introducing the principle of ‘opportunity dominance' and explore in a simple application to what extent this principle may help us arbitrate between opposing distributive principles. We also compare this principle to the selection rules developed by John Roemer and Dirk Van de Gaer. (shrink)

Metrology is a discipline of expunging impurities. The mid-nineteenth century French physicist Henri-Victor Regnault created a whole new way of doing experiments, attempting to produce standards physically by the "direct method." His immodest ambition to control all disturbing parameters represents a relict in the physical sciences of Romantic hopes for an all-embracing, artistic and aesthetic approach to nature, expressed in the absolute, eternal determination of nature's constants and their numerical relationships. The novelist Gustave Flaubert, whose rejection of metaphysics, love for (...) finest detail, concern for impartial treatment of the human psyche, and perfectionism, made him Regnault's literary complement, likewise hoped for this utopian union of science and art. There is a romantic and a modern ring to their ideals which were not to be achieved by optimism, but by introducing systematic doubt, by thoroughly studying impurities and expunging them. Nature did not speak clearly, but hid behind a multitude of disturbing effects—revealed only by the experimenter's or writer's voluntary restraint and self-denial, a disinterestedness Regnault and Flaubert celebrated as the highest and holiest morality. For them, attention to secondary matters became primary, because they expected them to lead the way to the grail. (shrink)

In this article, I try to shed some new light onGrundgesetze§10, §29–§31 with special emphasis on Frege’s criteria and proof of referentiality and his treatment of the semantics of canonical value-range names. I begin by arguing against the claim, recently defended by several Frege scholars, that the first-order domain inGrundgesetzeis restricted to value-ranges, but conclude that there is an irresolvable tension in Frege’s view. The tension has a direct impact on the semantics of the concept-script, not least on the semantics (...) of value-range names. I further argue that despite first appearances truth-value names play a distinguished role as semantic “target names” for “test names” in the criteria of referentiality and do not figure themselves as “test names” regarding referentiality. Accordingly, I show in detail that Frege’s attempt to demonstrate that by virtue of his stipulations “regular” value-range names have indeed been endowed with a unique reference, can plausibly be regarded as a direct application of the context principle. In a subsequent section, I turn to some special issues involved in §10. §10 is closely intertwined with §31 and in my and Richard Heck’s view would have been better positioned between §30 and §31. In a first step, I discuss the piecemeal strategy which Frege applies when he attempts to bestow a unique reference on value-range names in §3, §10–§12. In a second step, I critically analyze his tentative, but predictably unsuccessful proposal to identify all objects whatsoever, including those already clad in the garb of value-ranges, with their unit classes. In conclusion, I present two arguments for my claim that Frege’s identification of the True and the False with their unit classes in §10 is illicit even if both the permutation argument and the identifiability thesis that he states in §10 are regarded as formally sound. The first argument is set out from the point of view of the syntax of his formal language. It suggests though that a reorganization of the exposition of the concept-script would have solved at least one of the problems to which the twin stipulations in §10 give rise. The second argument rests on semantic considerations. If it is sound, it may call into question, if not undermine the legitimacy of the twin stipulations. (shrink)

Freges Kontextprinzip "Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas" hat auch nach der von ihm vollzogenen Angleichung von Behauptungssätzen an Eigennamen Gültigkeit für die formale Sprache der "Grundgesetze". Der Bedeutungsvollständigkcitsbeweis, den er für sein Logiksystem anstrebt, schließt eine unmittelbare Anwendung dieses Prinzips nicht nur auf die unvollständigen Funktionsausdrücke, sondern auch auf die leerstellenfreien Wertverlaufsnamen ein. Wahrheitsnamen (Sätze) zeichnen sich vor anderen symbolsprachlichen Eigennamen in mehrfacher Hinsicht, insbesondere durch ihre semantische Selbständigkeit aus. Wertverlaufsnamen haben nur im Zusammenhang eines Wahrheitswertnamens (...) eine Bedeutung. Ihre Bedeutung besteht in ihrem Beitrag zur Bestimmung der Bedeutung von Wahrheitswertnamen, in denen sie vorkommen. (shrink)

Phänomenologie der Normativität ist der Versuch zu zeigen, wie Normativität in der menschlichen Natur gründet. Im Anschluss an die phänomenologische Anthropologie Max Schelers und Helmuth Plessners stellt der Autor die Strukturen menschlichen Lebens heraus, die alle menschlichen Lebensformen fundieren. Menschen sind Lebewesen, die einen Leib haben, und dieses Phänomen ist ursprünglicher als die Unterscheidung von Körper und Geist. Hält man sich dies vor Augen, werden bestimmte Formen menschlicher Normativität verständlich: Die menschliche Lebensweise ist auf ein Gleichgewicht verschiedener Formen des Miteinanders (...) angelegt, das nur möglich ist, weil es leiblich vermitteltes Ausdrucksverstehen gibt. Mitfühlen mit dem anderen und Liebe zu ihm fundiert das Erkennen, und was menschliche Würde ist, verstehen wir, weil wir wissen, wie die Integrität des Leibes geschützt, aber auch verletzt werden kann. (shrink)

Freges Kontextprinzip "Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas" hat auch nach der von ihm vollzogenen Angleichung von Behauptungssätzen an Eigennamen Gültigkeit für die formale Sprache der "Grundgesetze". Der Bedeutungsvollständigkcitsbeweis, den er für sein Logiksystem anstrebt, schließt eine unmittelbare Anwendung dieses Prinzips nicht nur auf die unvollständigen Funktionsausdrücke, sondern auch auf die leerstellenfreien Wertverlaufsnamen ein. Wahrheitsnamen (Sätze) zeichnen sich vor anderen symbolsprachlichen Eigennamen in mehrfacher Hinsicht, insbesondere durch ihre semantische Selbständigkeit aus. Wertverlaufsnamen haben nur im Zusammenhang eines Wahrheitswertnamens (...) eine Bedeutung. Ihre Bedeutung besteht in ihrem Beitrag zur Bestimmung der Bedeutung von Wahrheitswertnamen, in denen sie vorkommen. (shrink)

1. Logik und Philosophie der Mathematik.--2. Logik und Sprachphilosophie.--3. Logik und Semantik.

Closed systems are mental artefacts whose structure can neither be explained in terms of innate dispositions nor in terms of the mechanism of natural selection and which stands in contradiction to the conditions of the possibility of information transfer. Kant takes the formal structure of pure mathematics as a guarantor for his reconstruction of a closed system of the subject's structures of knowledge a priori, thereby giving rise to the problem of a normative interface between interior system and exterior information (...) – a problem which is, in this form, not to be solved. By tracing the structure of generative grammar and formal logics back to innate modules, respectively, Chomsky and Fodor are creating the same interface problem, whose basic structure has not yet received the necessary attention. (shrink)

This chapter contains sections titled: The Pure Subject of Cognition The Artist as Pure Subject of Cognition Notes References Further Reading.

In his paper "Finitism", W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by him, (...) is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA. /// En su articulo "Finitism", W.W. Tait sostiene que la dificultad principal para quien quiere comprender la concepción hilbertiana de la matemática finitista es ésta: especificar el sentido de la demostrabilidad de enunciados generales sobre los números naturales sin presuponer totalidades infinitas. Además, Tait argumenta que todo razonamiento finitista es esencialmente primitivo recursivo. En este artículo tratamos de mostrar que su tesis "Las funciones finitistas son precisamente las funciones primitivas recursivas" es discutible y que otra, también defendida por él, resulta insostenible. La segunda tesis es que los teoremas finitistas son precisamente las clausuras universales de las ecuaciones que pueden demostrarse en PRA. (shrink)

The question whether God should be thought of as personal or a-personal is closely linked to the issue of an appropriate model of God-world relation on the one hand and the question how to conceive divine action on the other hand. Starting with a discussion of the scientific character of theology, this article critically examines the univocal-personal concept of God. Traditional Christian conceptions of God have, however, always acknowledged a radical asymmetry between the personal existence of created beings and the (...) ground of being itself. In a second step, the ontological truth claim associated with this way of speaking about God is being related to its methodological consequences. In final step, attention is given to the relation of immanence and transcendence as it is defended in different versions of panentheism: As an alternative to divine interventionism, panentheism can be shown to explicate divine providence as formal and final causation. (shrink)

Frege’s method of introducing abstract singular terms by transforming an equivalence statement into an identity statement suffers from one major defect: it is haunted by a pervasive indeterminacy of putative reference. In this paper, I. discuss mainly Frege’s introduction of courses-of-values in his magnum opus Grundgesetze der Arithmetik (Volume I, 1893, Volume 11, 1903). More specifically, I want to assesscritically, with respect to course-of-values names, what I call Frege’s indeterminacy problem. In the first part, I sketch the nature of this (...) problem in connection with the introduction of numerical singular terms in his book Die Grundlagen der Arithmetik of 1884. In the second part, I try to shed light on the analogies and differences concerning the introduction of numerical terms on the one hand and of course-of-values terms on the other. (shrink)

The essay deals with the enormous success of the Russian television series The Boy’s Word. Blood on the Asphalt (Slovo patsana. Krov’ na asfalte, 2023), directed by Zhora Kryzhovnikov, on both sides of the trenches during the Russo-Ukrainian war. It discusses the role of social media and the public reception in Ukraine and Russia. In particular, the way the series depicts the so-called ‘Kazan phenomenon’, when the Tatar capital was shocked by criminal youth gangs in the late Soviet Union, makes (...) it attractive and topical in the current situation. By analyzing this, the essay also attempts to gain a better understanding of how popular cultural products take up and process social problems and conflicts in a complex way. (shrink)

The basic concept of Type-2 Theory of Effectivity to define computability on topological spaces or limit spaces are representations, i. e. surjection functions from the Baire space onto X. Representations having the topological property of admissibility are known to provide a reasonable computability theory. In this article, we investigate several additional properties of representations which guarantee that such representations induce a reasonable Type-2 Complexity Theory on the represented spaces. For each of these properties, we give a nice characterization of the (...) class of spaces that are equipped with a representation having the respective property. (shrink)

Theories of epistemology make reference—via the perspective of an observer—to the structure of information transfer, which generates reality, of which the observer himself forms a part. It can be shown that any epistemological approach which implies the participation of tautological structural elements in the information transfer necessarily leads to an antinomy. Nevertheless, since the time of Aristotle the paradigm of mathematics—and thus tautological structure—has always been a hidden ingredient in the various concepts of knowledge acquisition or general theories of information (...) transfer. We hold that Darwin’s Evolutionary Theory is the first scientific theory which consistently presupposes a non-tautological structure for the information transfer and, at the same time, keeps it strictly distinct from the tautological metric of scientific observation. The consequences of this technique—namely the dissociation of information from intentionality—have not yet been fully drawn. (shrink)

What is the core of legal scholarship? How can we understand its relation to other disciplines, such as moral and political philosophy, sociology, and economics? I explore these questions by analysing the impact of the dual nature thesis. Criticising established theories of legal scholarship, I defend the ideal of an integrative jurisprudence. Integrative jurisprudence combines the two dimensions of law by employing analytical, empirical, and normative methods. I then discuss three objections and address the problem of how to bridge is (...) and ought. In the course of my analysis of three different bridge theories, I ultimately further develop the dual nature thesis into a triadic nature thesis. (shrink)