This article deals with six aspects of analogical thinking in mathematics: 1. Platonism and continuity principle or the “geometric voices of analogy” (as Kepler put it), 2. analogies and the surpassing of limits, 3. analogies and rule stretching, 4. analogies and concept stretching, 5. language and the art of inventing, 6. translation, or constructions instead of discovery. It takes especially into account the works of Kepler, Wallis, Leibniz, Euler, and Laplace who all underlined the importance of analogy in finding out (...) new mathematical truth. But the meaning of analogy varies with the different authors. Isomorphic structures are interpreted as an outcome of analogical thinking. (shrink)

Die Wissenschaft hat - von der Wissenschaftstheorie des 20. Jahrhunderts gründlich untersucht - bestimmte Strukturen ihres Wissens, ihrer Methodik der Wissensgewinnung und Wissensabsicherung entwickelt, die sie von anderen Arten des Umgangs mit Wissen - z.B. der Alltagserfahrung - signifikant unterscheidet. Für die Technikwissenschaften haben solche wissenschaftstheoretischen Untersuchungen jedoch bisher kaum stattgefunden. Was etwa die analytische Wissenschaftstheorie für die Naturwissenschaften geleistet hat, steht daher für die Technikwissenschaften noch aus. Denn das technische Wissen unterscheidet sich vom wissenschaftlichen Wissen in seiner inneren (...) Struktur. Diese Struktur aufzuklären ist eine lohnende Aufgabe, weil deren Kenntnis hilft, besser mit Technik umzugehen. Die in diesem Band vorgelegten analytischen Studien zur Wissenschaftstheorie der Technikwissenschaften behandeln - ausgehend vom Problem der Formalisierung und der Wahrheitsfrage in der Philosophie - die Grundeinheit des technischen Wissens, die technologische Regel, und den Aufbau und Zusammenhang technologischer Theorien. Daraus ergeben sich Hinweise auf die innere, d.h. auch logische Struktur des technischen Wissens. (shrink)

In der modernen Physik spielen Symmetrien eine herausragende Rolle zur Identifikation und Klassifizierung der fundamentalen Theorien und Entitäten. Symmetrien dienen der Darstellung invarianter Strukturen, das geeignete mathematische Werkzeug hierfür ist die Gruppentheorie. Eine Struktur lässt sich als eine Menge von Relationen verstehen, die einer Menge von Objekten aufgeprägt sind. Strukturell charakterisierte Objekte sind daher wesentlich über ihre relationalen Eigenschaften charakterisiert. Sieht man die theoretischen Entitäten wissenschaftlicher Theorien vornehmlich in dieser strukturellen Weise an, vertritt man eine moderate Variante eines wissenschaftlichen (...) Realismus, die als Strukturenrealismus bekannt ist. Im Umfeld dieser drei Konzepte – Symmetrien, Strukturen, Realismus – bewegt sich der folgende Aufsatz. (shrink)

We study the partial orderings of the form ⟨P,⊂⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P}, \subset\rangle}$$\end{document}, where X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} is a binary relational structure with the connectivity components isomorphic to a strongly connected structure Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Y}}$$\end{document} and P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} }$$\end{document} is the set of substructures of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {X}}$$\end{document} isomorphic to X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document}. We show that, for example, for a countable X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document}, the poset ⟨P,⊂⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb {P}, \subset\rangle}$$\end{document} is either isomorphic to a finite power of P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} }$$\end{document} or forcing equivalent to a separative atomless σ-closed poset and, consistently, to P/fin. In particular, this holds for each ultrahomogeneous structure X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} such that X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} or Xc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}^{c}}$$\end{document} is a disconnected structure and in this case Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Y}}$$\end{document} can be replaced by an ultrahomogeneous connected digraph. (shrink)

Wo Meinungen aufeinanderprallen, um Verständnis geworben und Überzeugungsarbeit geleistet wird, sind Begründungen nicht weit. Für jede Überzeugung gibt es immer ein, zwei Gründe, die mit Gegengründen konfrontiert und, im Gegenzug, mit weiteren Überlegungen verteidigt werden usw. usf. Schnell sind wir verwirrt und drohen, ohne uns der "Grammatikregeln" vernünftigen Argumentierens zu besinnen, nicht mehr durchzublicken. Die Theorie dialektischer Strukturen leistet einen Beitrag zur Grammatik vernünftigen Argumentierens. Sie stellt Begriffe und Verfahren bereit, um Fragen, die sich angesichts einer komplexen Argumentation stellen (...) können, zu beantworten: Wie beziehen sich die einzelnen Argumente der Debatte aufeinander? Welche Positionen vertreten die verschiedenen Proponenten, und wie lassen sich diese bewerten? Wie gut ist die zentrale These der Kontroverse alles in allem begründet? Ist die und die Argumentationshandlung angesichts des Debattenstandes tatsächlich zweckmäßig und rational? Liegt hier eine zirkuläre - und damit fehlerhafte - Argumentation vor? Die klassische Logik, die sogenannte informal logic und argumentationstheoretische Ansätze der Künstlichen Intelligenz bilden Anknüpfungspunkte der Theorie dialektischer Strukturen. Exemplarisch werden die entwickelten argumentationstheoretischen Methoden zur Analyse umfangreicher philosophischer Kontroversen eingesetzt. (shrink)

Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.

Auf dem Hintergrund der kritischen Theorie der Gesellschaft und der Sprachphilosophie versucht der Autor, die geschichtsphilosophischen Strukturen im Werk Nietzsches sichtbar zu machen. Diese werden erkannt als soziale Strukturen, vermittelt durch sprachliche Kommunikation.

We show that, under PFA, a coherent Suslin tree forces that every two Aronszajn trees are club-isomorphic.

Name der Zeitschrift: Deutsche Zeitschrift für Philosophie Jahrgang: 65 Heft: 1 Seiten: 1-15.

The spectrum of a relation on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between and any other computable structure . The relation is intrinsically computably enumerable if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of (...) the minimum element of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family. (shrink)

In diesem Band deckt Diego D'Angelo semiotische Strukturen in der Husserl’schen Phänomenologie der Wahrnehmung auf. Ist es der Phänomenologie darum zu tun, die Erfahrung von Dingen in unserer Umwelt zu beschreiben, so ist dabei der Begriff des Horizontes von zentraler Bedeutung: Was wir unmittelbar wahrnehmen, verweist immer schon auf anderes, was nur „mitgegeben“ ist. Wenn wir Dinge wahrnehmen, haben wir nur eine bestimmte Perspektive, d.h. wir sehen lediglich einen Aspekt. Aber wir nehmen immer ganze Gegenstände wahr (wir sehen Tische (...) und Stühle und andere Menschen). Jeder dieser Gegenstände erscheint in einem Feld weiterer Gegenstände, und es ist der Horizontbegriff, der es erlaubt, das Verhältnis zwischen Selbstgegebenheit und Mitgegebenheit zu explizieren. Dieses Buch stellt den ersten detaillierten Versuch dar, die Ursprünge solcher horizontaler Felder in semiotischen Strukturen zu suchen. Aus der Verbindung zwischen Husserls eigener Semiotik und seiner Phänomenologie der Wahrnehmung ergibt sich, dass das wahrgenommene Phänomen als Zeichen verstanden werden muss. Das Zeichen wiederum bezeichnet etwas, was in leiblicher Bewegung eingeholt werden kann. Mit der Verbindung von Leiblichkeit, Semiotik und Wahrnehmung thematisiert diese Monographie das Verhältnis zwischen folgenden phänomenologischen Forschungsgebieten: • Husserls Semiotik der Wahrnehmung in den Logischen Untersuchungen • Phänomenologische Raumanalyse – kinästhetische Indikation • Horizont und Noema • Passive Anzeige • Zeichen und Leiblichkeit als Grundlagen der Fremderfahrung • Genetische Phänomenologie und Semiotik der Erfahrung • Protentionen und teleologische Semiose • Induktion und Ursprung des menschlichen Ichs Das Buch eröffnet die Möglichkeit, Husserls Phänomenologie jenseits einer Metaphysik der Präsenz zu verstehen. Zudem leisten D'Angelos Einzeluntersuchungen einen Beitrag zu aktuellen Diskussionen in der Philosophie der leiblichen Kognition. – Eine hilfreiche Leseempfehlung für • Interessierte Themenneulinge • Bachelor- und Masterstudenten der Geisteswissenschaften • Hochschulabsolventen sowie Forschungswissenschaftler. (shrink)

We investigate the second order bounded arithmetical systems which is isomorphic to TAC i , TNC i or TLS.

Politische Strukturen im Guptareich. By Fred Virkus. Asien- und Afrika-Studien der Humboldt-Universität zu Berlin, vol. 18. Wiesbaden: Harrassowitz Verlag, 2004. Pp. x + 319. €78. The Archaeology of Hindu Ritual: Temples and the Establishment of the Gods. By Michael Willis. Pp. xiv + 375. Cambridge: caMBridge university Press, 2009. £50.

This paper belongs to cylindric-algebraic model theory understood in the sense of algebraic logic. We show the existence of isomorphic but not lower base-isomorphic cylindric set algebras. These algebras are regular and locally finite. This solves a problem raised in [N 83] which was implicitly present also in [HMTAN 81]. This result implies that a theorem of Vaught for prime models of countable languages does not continue to hold for languages of any greater power.

Palmer describes a “subjective barrier” that limits knowledge of others' experience. We discuss how this barrier extends to all knowledge, becoming less distinct as theoretical constructs are strengthened. We provide evidence for isomorphic experience, among individuals with similar physiologies, by showing that perceived relations between colors are as similar when viewing pigments as when viewing subjective colors caused by flickering bars.

In this paper we investigate computable isomorphisms of Boolean algebras with operators (BAOs). We prove that there are examples of polymodal Boolean algebras with finitely many computable isomorphism types. We provide an example of a polymodal BAO such that it has exactly one computable isomorphism type but whose expansions by a constant have more than one computable isomorphism type. We also prove a general result showing that BAOs are complete with respect to the degree spectra of structures, computable dimensions, expansions (...) by constants, and the degree spectra of relations. (shrink)

It is proved that all the isomorphisms in the cartesian category freely generated by a set of objects can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free cartesian category is an isomorphism.

This study assesses the impact had by institutional isomorphic pressures in the organisational fields of 185 businesses operating within the United Kingdom. The emphasis throughout is on how external institutions affect the socially and environmentally responsible aspects of an organization’s purchasing practice. Factor analyses and a linear regression model are employed to test the influence of these pressures. Initial findings suggest that what other industry participants are doing in this area is not as important in affecting the procurement practice of (...) the focal organisation as is the managers’ perception of how legitimacy is awarded by stakeholders and, indeed, if competitors with well-developed social and environmental supply chain management programs are perceived favourably in the industry. (shrink)

Name der Zeitschrift: Deutsche Zeitschrift für Philosophie Jahrgang: 64 Heft: 1 Seiten: 155-161.

Previous research into the formalisation of statute law identified a number of uses of language which posed problems for formalisation. A previous paper argued that these uses establish the requirement that a formalisation be isomorphic, but noted that this has odd consequences. This paper expands on what these consequences are and argues that they undermine the very idea of formalisation. Therefore, the whole argument constitutes a reductio ad absurdum of the idea of formalising statute law. The paper provides reasons why (...) this activity is impossible and places the conclusion in the context of the development of modem logic, in which context it appears uncontroversial. (shrink)

In this paper it will be shown that HEO and HRO E are isomorphic with respect to extensional equality. This answers a question of Troelstra [T, 2.4.12, p. 128]. The main problem is to extend effective operations to a larger domain. This will be achieved by a modification of the proof of the continuity of effective operations. Following a suggestion of A.S. Troelstra, similar results were obtained for ECF(U) and ICF E (U), where U is any universe of functions closed (...) under "recursive in". (shrink)

In this paper it will be shown that HEO and $HRO^E$ are isomorphic with respect to extensional equality. This answers a question of Troelstra [T, 2.4.12, p. 128]. The main problem is to extend effective operations to a larger domain. This will be achieved by a modification of the proof of the continuity of effective operations. Following a suggestion of A.S. Troelstra, similar results were obtained for ECF(U) and $ICF^E(U)$, where $U$ is any universe of functions closed under "recursive in".

In this paper the existence or nonexistence of isomorphic mappings between graph models for the untyped lambda calculus is studied. It is shown that Engeler's D A is completely determined, up to isomorphism, by the cardinality of its `atom-set' A. A similar characterization is given for a collection of graph models of the Pω-type; from this some propositions regarding automorphisms are obtained. Also we give an indication of the complexity of the first-order theory of graph models by showing that the (...) second-order theory of first-order definable elements of a graph model is first-order expressable in the model. (shrink)

We show that if A and $\widehat{A}$ are automorphic via Φ then the structures $S_{R}(A)$ and $S_{R}(\widehat{A})$ are $\Delta_{3}^{0}-isomorphic$ via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ℝ2 and ℝ. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non-measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non-measurable.

In this paper we prove that, in the Cohen extension of a model M of ZFC+CH containing a simplified -morass, η1-orderings without endpoints having cardinality of the continuum, and satisfying specified technical conditions, are order-isomorphic. Furthermore, any order-isomorphism in M between countable subsets of the η1-orderings can be extended to an order-isomorphism between the η1-orderings in the Cohen extension of M. We use the simplified -morass, and commutativity conditions with morass maps on terms in the forcing language, to extend countable (...) partial functions on terms in the forcing language that are forced in all generic extensions to be order-preserving injections. This technique provides for the construction of functions in Cohen extensions adding 2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω1 in models of ZFC+CH. The specific example presented in this paper is an extension of Tarski’s classic result that in models of ZFC+CH, η1-orderings are order-isomorphic. (shrink)

An oft-defended claim of a close relationship between Gentzen inference rules and the meaning of the connectives they introduce and eliminate has given rise to a whole domain called proof-theoretic semantics, see Schroeder- Heister (1991); Prawitz (2006). A branch of proof-theoretic semantics, mainly developed by Dosen (2019); Dosen and Petric (2011), isolates in a precise mathematical manner formulas (of a logic L) that have the same meaning. These isomorphic formulas are defined to be those that behave identically in inferences. The (...) aim of this paper is to investigate another type of recently discussed rules in the literature, namely grounding rules, and their link to the meaning of the connectives they provide the grounds for. In particular, by using grounding rules, we will refine the notion of isomorphic formulas through the notion of hyper-isomorphic formulas. We will argue that it is actually the notion of hyper-isomorphic formulas that identify those formulas that have the same meaning. (shrink)

It is proved that all the isomorphisms in the cartesian category freely generated by a set of objects (i.e., a graph without arrows) can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free cartesian category is an isomorphism.