Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set of (...) theorems. The concept of an accessible domain comes from Wilfried Sieg's analysis of proof-theoretic practices, starting with his dissertation. In particular, he noticed the special epistemological character of elements of an accessible domain: they can always be uniquely identified with their build-up. Generally, the unique build-up of elements justifies the principles of induction and recursion. -/- I use category theory to give an abstract characterization of accessible domains. I claim that accessible domains are all instances of initial algebras for endofunctors. Grounded in the historical roots of Sieg's discussions, this dissertation shows how the properties of initial algebras for endofunctors and accessible domains coincide in a satisfying and natural way. -/- Filling out this characterization, I show how important examples of accessible domains fit into this broad characterization. I first characterize some accessible domains by relatively simple functors (e.g. finite, polynomial, those that preserve certain colimits) where we can see how iterating the functor can produce the accessible domain. Then I describe accessible domains that result from more involved specifications (e.g. ordinals and segments of the cumulative hierarchies associated with CZF, IZF, and ZF) by relying heavily on algebraic set theory. I end with a discussion of some of the methodological features of category theory in particular that helped characterize accessible domains. (shrink)

The main algebraic foundations of quantum mechanics are quickly reviewed. They have been suggested since the birth of this theory till up to last years. They are the following ones: Heisenberg-Born- Jordan’s (1925), Weyl’s (1928), Dirac’s (1930), von Neumann’s (1936), Segal’s (1947), T.F. Jordan’s (1986), Morchio and Strocchi’s (2009) and Buchholz and Fregenhagen’s (2019). Four cases are stressed: 1) the misinterpretation of Dirac’s algebraic foundation; 2) von Neumann’s ‘conversion’ from the analytic approach of Hilbert space to the algebraic approach (...) of the rings of operators; 3) Morchio and Strocchi’s improving Dirac’s analogy between commutators and Poisson Brackets into an exact equivalence; 4) the recent foundation of quantum mechanics upon the algebra of perturbations. Some considerations on alternating theoretical importance of the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices to group theory and C*-algebras but has not led to a definition of the foundations of physics; in particular, an algebraic formulation of QM organized as a problem-based theory and an exclusive use of constructive mathematics is still to be discovered. (shrink)

Der vorliegende Beitrag ist als ein Versuch zu verstehen, dem polnischen Leser eine recht seltsame Position aus der Reihe der deutschsprachigen Lodzaica, nämlich die 1913 im Verlag August Grüning in Pabianice herausgegebene Sammlung von Witzen und Anekdoten mit dem Titel 'Ne Mütze voll Witze aus Lodz und Pabianice’, näherzubringen. Der Verfasser des schmalen Bändchens, der sich hinter dem Pseudonym Wilfried Spectator verbirgt, ist Reinhold Piel, ein Deutscher aus Pabianice. Als Satelitenstadt von Łódź, der sich damals sehr rasch entwickelnden Textilmetropole, (...) bildete auch Pabianice in jenen Jahren zweifelsohne ein Teil des in ganz Europa einmaligen und eigenartigen multikulturellen und multiethnischen Tigcls. Reinhold Piel stellte sich mit seinem Büchlein die Aufgabe, ein Sammelporträt der Mentalität des typischen Vertreters jenes Tigels von Kulturen und Sprachen zu zeichnen. In der mehrere disparate Elemente integrierenden Gemeinschaft funktionierte der Humor sicherlich als eine Art Sicherheitsventil, mit dessen Hilfe man bestrebt war, Fremdes zu „zähmen“ und Konflikte zu entschärfen. Scheinbar abstrakte, „auslachende“ Beispiele haben sich wahrscheinlich ebenfalls vortrefflich als Methode bewährt, andere diplomatisch zu sensibilisieren, aber auch eigene Nachteile und Unzulänglichkeiten mit Abstand zu sehen. Der universale Charakter der Anekdote als literarisches Genre beruht u.a. auf ihrer Ahistorizität und Konkretheit zugleich, ihr narratives Werkzeug hingegen - die individualisierte Sprache des multiethnischen Milieus - liefert hervorragenden StolT für sprachwissenschaftliche Forschung. Ein besonders interessanter Aspekt ist dabei vor allem die Frage nach der Übersetzbarkeit bzw. Unübersetzbarkeit von Kulturidiomatik. Um dem Leser entgegenzukommen und ihm den historischen und informativen Wert der besprochenen Sammlung bewusst zu machen, wird im Anhang - zum ersten Mal in der polnischen Sprache - eine Auswahl von besonders charakteristischen Witzen der genannten Sammlung von R. Piel präsentiert. (shrink)

● Sergio Cremaschi, The non-existing Island. I discuss the way in which the cleavage between the Continental and the Anglo-American philosophies originated, the (self-)images of both philosophical worlds, the converging rediscoveries from the Seventies, as well as recent ecumenic or anti-ecumenic strategies. I argue that pragmatism provides an important counter-instance to both the familiar self-images and to the fashionable ecumenic or anti-ecumenic strategies. My conclusions are: (i) the only place where Continental philosophy exists (as Euro-Communism one decade ago) is America; (...) (ii) less obviously, also analytic philosophy does not exist, or does no more exist as a current or a paradigm; what does exist is, on the one hand, philosophy of language and, on the other, philosophy of mind, that is, two disciplines; (iii) the dissolution of analytic philosophy as a school has been extremely fruitful, precisely in so far as it has left room for disciplines and research programmes; (iv) what is left, of the Anglo-American/Continental cleavage is primarily differences in styles, depending partly on intellectual traditions, partly owing to sociology, history, institutional frameworks; these differences should not be blurred by rash ecumenism; besides, theoretical differences are alive as ever, but within both camps; finally, there is indeed a lag (not a difference) in the appropriation of intellectual techniques by most schools of 'Continental' philosophy, and this should be overcome through appropriation of what the best 'analytic' philosophers have produced. ● Michael Strauss, Language and sense-perception: an aspect of analytic philosophy. To test an assertion about one fact by comparing it with perceived reality seems quite unproblematic. But the very possibility of such a procedure is incompatible with the intellectualistic basis of logical positivism and atomism (as it is for example to be found in Russell's Analysis of Mind). According to the intellectualistic approach pure sensation is meaningless. Sensation receives its meaning and order from the intellect through interpretation, which is performed with the help of linguistic tools, i.e. words and sentences. Before being interpreted, sensation is not a picture or a representation, it is neither true nor false, neither an illusion nor knowledge; it does not tell us anything; it is a lifeless and order-less matter. But how can a thought (or a proposition) be compared with such a lifeless matter? This difficulty confronts the intellectualist, if on the one hand he admits the necessity of comparing thought with sense-perception, and on the other hand presupposes that we possess only intellectual and no immediate perceptual understanding of what we see and hear. In this paper I give a critical exposition of three attempts, made by Russell, Neurath and Wittgenstein, to solve this problem. The first attempt adheres to strict conventionalism, the second tends to naturalism and the third leads to an amended, very moderate version of conventionalism. This amended conventionalism looks at sense impressions as being a peculiar language, which includes primary symbols, i.e. symbols not founded on convention and not being in need of interpretation. ● Ernst Tugendhat, Phenomenology and language analysis. The paper, first published in German in 1970, by which Tugendhat gave a start to the German rediscovery of analytic philosophy. The author stages a confrontation between phenomenology and language analysis. He argues that language analysis does not differ from phenomenology as far as the topics dealt with are concerned; instead, both currents are quite different in method. The author argues that language-analytic philosophy does not simply lay out of the mainstream of transcendental philosophy, but that instead it challenges this tradition on the very level of foundations. The author criticizes the linguistic-analytic approach centred on the subject as well as any object-centred approach, while proposing inter-subjective understanding through language as the new universal framework. This is, when construed in so general terms, the same program of hermeneutics, though in a more basic version. ● Jürgen Habermas, Language game, intention and meaning. On a few suggestions by Sellars and Wittgenstein. The paper, first published in German in 1975, in which Habermas announces his own linguistic turn through a discovery of speech acts. In this essay the author wants to work out a categorical framework for a communicative theory of society; he takes Wittgenstein's concept of language game as a Leitfade and, besides, he takes advantage also of Wilfried Sellars's quasi-transcendental account of the genesis of intentionality. His goal is to single out the problems connected with a theory of consciousness oriented in a logical-linguistic sense. ● Zvie Bar-On, Isomorphism of speech acts and intentional states. This essay presents the problem of the formal relationship between speech acts and intentional states as an essential part of the perennial philosophical question of the relation between language and thought. I attempt to show how this problem had been dealt with by two prominent philosophers of different camps in our century, Edmund Husserl and John Searle. Both of them wrote extensively about the theory of intentionality. I point out an interesting, as it were unintended, continuity of their work on that theory. Searle started where Husserl left off 80 years earlier. Their meeting point could be used as the first clue in our search. They both adopted in effect the same distinction between two basic aspects of the intentional experience: its content or matter, and its quality or mode. Husserl did not yet have the concept of a speech act as contradistinguished from an intentional state. The working hypothesis, however, which he suggested, could be used as a second clue for the further elaboration of the theory. The relationship of the two levels, the mental and the linguistic, which remained for Husserl in the background only, became the cornerstone of Searle' s inquiry. He employed the speech act as the model and analysed the intentional experience by means of the conceptual apparatus of his own theory of speech acts. This procedure enabled him to mark out a number of parallelisms and correlations between the two levels. This procedure explains the phenomenon of the partial isomorphism of speech acts and intentional states. ● Roberta de Monticelli, Ontology. A dialogue among the linguistic philosopher, the naturalist, and the phenomenological philosopher. This paper proposes a comparison between two main ways of conceiving the role and scope of that fundamental part of philosophy (or of "first" philosophy) which is traditionally called "ontology". One way, originated within the analytic tradition, consists of two main streams, namely philosophy of language and (contemporary) philosophy of mind, the former yielding "reduced ontology" and the latter "neo-Aristotelian ontology". The other way of conceiving ontology is exemplified by "phenomenological ontology" (more precisely, the Husserlian, not the Heideggerian version). Ontology as a theory of reference ("reduced" ontology, or ontology as depending on semantics) is presented and justified on the basis of some classical thesis of traditional philosophy of language (from Frege to Quine). "Reduced ontology" is shown to be identifiable with one level of a traditional, Aristotelian ontology, namely the one which corresponds to one of the four "senses of being" listed in Aristotle's Metaphysics: "being" as "being true". This identification is justified on the basis of Franz Brentano's "rules for translation" of the Aristotelian table of judgements in terms of (positive and negative) existential judgments such as are easily translatable into sentences of first order predicate logic. The second part of the paper is concerned with "neo-Aristotelian ontology", i.e. with naturalism and physicalism as the main ontological options underlying most of contemporary discussion in the philosophy of mind. The qualification of such options as "neo-Aristotelian" is justified; the relationships between "neo-Aristotelian ontology" and "reduced ontology" are discussed. In the third part the fundamental tenet of "phenomenological ontology" is identified by the thesis that a logical theory of existence and being does capture a sense of "existing" and "being" which, even though not the basic one, is grounded in the basic one. An attempt is done of further clarifying this "more basic" sense of "being". An argument making use of this supposedly "more basic" sense is advanced in favour of a "phenomenological ontology". ● Kuno Lorenz, Analytic Roots in Dialogic Constructivism. Both in the Vienna Circle ad in Russell's early philosophy the division of knowledge into two kinds (or two levels), perceptual and conceptual, plays a vital role. Constructivism in philosophy, in trying to provide a pragmatic foundation - a knowing-how - to perceptual as well as conceptual competences, discovered that this is dependent on semiotic tools. Therefore, the "principle of method" had to be amended by the "principle of dialogue". Analytic philosophy being an heir of classical empiricism, conceptually grasping the "given", and constructive philosophy being an heir of classical rationalism, perceptually providing the "constructed", merge into dialogical constructivism, a contemporary development of ideas derived especially from the works of Charles S. Peirce (his pragmatic maxim as a means of giving meaning to signs) and of Ludwig Wittgenstein (his language games as tools of comparison for understanding ways of life). 7. Albrecht Wellmer, "Autonomy of meaning" and "principle of charity" from the viewpoint of the pragmatics of language. In this essay I present an interpretation of the principle of the autonomy of meaning and of the principle of charity, the two main principles of Davidson's semantic view of truth, showing how both principles may fit in a perspective dictated by the pragmatics of language. I argue that (I) the principle of the autonomy of meaning may be thoroughly reformulated in terms of the pragmatics of language, (ii) the principle of charity needs a supplement in terms of pragmatics of language in order to become really enlightening as a principle of interpretation. Besides, I argue that: (i) on the one hand, the fundamental thesis of Habermas on the pragmatic theory of meaning ("we understand a speech act when we know what makes it admissible") is correlated with the seemingly intentionalist thesis according to which we understand a speech act when we know what a speaker means; (ii) on the other hand, to say that the meaning competence of a competent speaker is basically a competence about a potential of reasons (or also of possible justifications) which is inherently connected with the meaning of statements, or with their use in utterances. ● Rüdiger Bubner, The convergence of analytic and hermeneutic philosophy This paper argues that the analytic philosophy does not exist, at least as understood by its original programs. Differences in the analytic camp have always been bigger than they were believed to be. Now these differences are coming to the fore thanks to a process of dissolution of dogmatism. Philosophical analysis is led by its own inner logic towards questions that may be fairly qualified as hermeneutic. Recent developments in analytic philosophy, e.g. Davidson, seem to indicate a growing convergence of themes between philosophical analysis and hermeneutics; thus, the familiar opposition of Anglo-Saxon and Continental philosophy might soon belong to history. The fact of an ongoing appropriation of analytical techniques by present-day German philosophers may provide a basis for a powerful argument for the unity of philosophizing, beyond its strained images privileging one technique of thinking and rejecting the remainder. Actual philosophical practice should take the dialogue between the two camps more seriously; in fact, the processes described so far are no danger to philosophical work. They may be a danger for parochial approaches to philosophizing; indeed, contrary to what happens in the natural sciences, Thomas Kuhn's "normal science" developing within the framework of one fixed paradigm is not typical for philosophical thinking. And in philosophy innovating revolutions are symptoms more of vitality than of crisis. ● Karl-Otto Apel, The impact of analytic philosophy on my intellectual biography. In my paper I try to reconstruct the history of my Auseinandersetzung mit - as I called it - "language-analytical" philosophy (including even Peircean semiotics) since the late Fifties. The heuristics of my study was predetermined by two main motives of my beginnings: the hermeneutic turn of phenomenology and the transformation of "transcendental philosophy" in the light of the "language a priori". Thus, I took issue with the early and the later Wittgenstein, logical positivism, and post-Wittgensteinian and post-empiricist philosophy of science (i.e. G.H. von Wright and the renewal of the "explanation vs understanding controversy" as well as the debate between Th. Kuhn and Popper/Lakatos); besides, with speech act theory and the debate about "transcendental arguments" since Strawson. The "pragmatic turn", started already by C.L. Morris and the later Carnap, led me to study also the relationship between Wittgensteinian "use" theory of meaning and of truth. This resulted on my side in something like a program of "transcendental semiotics", i.e. "transcendental pragmatics" and "transcendental hermeneutics". ● Ben-Ami Scharfstein, A doubt on both their houses: the blindness to non-western philosophies. The burden of my criticism is that contemporary European philosophers of all kinds have continued to think as if there were no true philosophy but that of the West. For the most part, the existentialists have been oblivious of their Eastern congeners; the hermeneuticians have yet to stretch their horizons beyond the most familiar ones; and the analysts remain unaware of the analyses and linguistic sensitivities of the ancient non-European philosophers. Briefly, ignorance still blinds almost all contemporary Western philosophers to the rich, variegated philosophical traditions outside of their familiar orbit. Both Continental and Anglo-Americans have lost the breadth of view that once characterized such thinkers as Herder and the Humboldts. The blindness that has resulted is not simply that of individual Western philosophers but of our whole, still parochial philosophical culture. (shrink)

Gödel's incompleteness theorems establish the stunning result that mathematics cannot be fully formalized and, further, that any formal system containing a modicum of number or set theory cannot establish its own consistency. Wilfried Sieg and Clinton Field, in their paper Automated Search for Gödel's Proofs, presented automated proofs of Gödel's theorems at an abstract axiomatic level; they used an appropriate expansion of the strategic considerations that guide the search of the automated theorem prover AProS. The representability conditions that allow (...) the syntactic notions of the metalanguage to be represented inside the object language were taken as axioms in the automated proofs. The concrete task I am taking on in this project is to extend the search by formally verifying these conditions. Using a formal metatheory defined in the language of binary trees, the syntactic objects of the metatheory lend themselves naturally to a direct encoding in Zermelo's theory of sets. The metatheoretic notions can then be inductively defined and shown to be representable in the object-theory using appropriate inductive arguments. Formal verification of the representability conditions is the first step towards an automated proof thereof which, in turn, brings the automated verification of Gödel's theorems one step closer to completion. (shrink)