Online research in philosophy

Leonhard Euler (1707–1783) treated in his study „De motu vibratorio tympanorum [6], published in 1766, the theory of the vibrating rectangular and circular membrane mathematically in such a comprehensive way that little more had to be added, considering today's standards. However, he omitted to interprete his results physically. Therefore his uncomparable work found little recognition, especially in the field of musical accoustics.

Euler’s wave theory of light developed from a mere description of this notion based on an analogy between sound and light to a more and more mathematical elaboration on that notion. He was very successful in predicting the shape of achromatic lenses based on a new dispersion law that we now know is wrong. Most of his mathematical arguments were, however, guesswork without any solid physical reasoning. Guesswork is not always a bad thing in physics if it leads (...) to new experiments or makes the theory coherent with other theories. And Euler tried to find such experiments. He saw the construction of achromatic lenses, the explanation of colors of thin plates and of the opaque bodies as proof of his theory. When it came to the fundamental issues, the correctness of his dispersion law and the prediction of frequencies of light he was not at all successful. His wave theory degenerated, and it was not until Augustin Fresnel introduced transverse waves and an elaborate notion of interference that the wave theory again progressed. (shrink)

Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions (...) as in the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem. (shrink)

Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables (...) us to formalize and analyze free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs. (shrink)

MATHEMATICS is a wonderful, mad subject, full of imagination, fantasy and creativity that is not limited by the petty details of the physical world, but only by the strength of our inner light. Does this sound familiar? Probably not from the mathematics classes you may have attended. But consider the work of three famous earlier mathematicians: Leonhard Euler (18th century), Georg Cantor (19th century) and Srinivasa Ramanujan (20th century).

Newton rested his theory of mechanics on distinct metaphysical and epistemological foundations. After Leibniz's death in 1716, the Principia ran into sharp philosophical opposition from Christian Wolff and his disciples, who sought to subvert Newton's foundations or replace them with Leibnizian ideas. In what follows, I chronicle some of the Wolffians' reactions to Newton's notion of absolute space, his dynamical laws of motion, and his general theory of gravitation. I also touch on arguments advanced by Newton's Continental followers, such as (...) Leonhard Euler, who made novel attempts to defend his mechanical foundations against the pro-Leibnizian attack. This examination grants us deeper insight into the fate of Newton's mechanics on the Continent during the early eighteenth century and, more specifically, sheds needed light on the conflicts and tensions that characterized the reception of Newton's philosophy of mechanics among the Leibnizians. (shrink)

This article is a spiritual interpretation of Leonhard Euler’s famous equation linking the most important entities in mathematics: e (the base of natural logarithms), π (the ratio of the diameter to the circumference of a circle), i ( d -1),1 , and 0. The equation itself (e π i+1 = 0>) can be understood in terms of a traditional mathematical proof, but that does not give one a sense of what it might mean. While one might intuit, given (...) the significance of the elements of the equation, that there is a deeper meaning, one is not in a position to get at that meaning within the discipline of mathematics itself. It is only by going outside of mathematics and adopting the perspective of theology that any kind of understanding of the equation might be gained, the significant implication here being that the whole mathematical field might be a vast treasure house of insights into the mind of God. In this regard, the article is a response to the monograph by George Lakoff and Rafael Núñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (2000), which attempts to approach mathematics in general and the Euler equation in particular in terms of some basic principles of cognitive psychology. It is my position that while there may be an external basis for understanding mathematics, the results are somewhat disappointing and fail to reveal the full measure of meaning buried within that equation. (shrink)

The evolution of Euler diagrams is examined from Euler's original system through the modifications made by Venn and Peirce. It is shown that these modifications were motivated by an attempt to increase the expressivity of the diagrams, but that a side effect of these modifications was a loss of the visual clarity of Euler's original system. Euler's original system is reconstructed from a modern, logical point of view. Formal semantics and rules of inference are provided for (...) this reconstruction of Euler's system, and basic logical properties are proved. (shrink)

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the (...) ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings. (shrink)

We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle $WPHP_{n}^{2n}$ : two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A. B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate (...) Euler characteristic must satisfy $$WPHP_{n}^{2n}$ . Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle $WPHP_{n}^{2n}$ : for no definable set A with more than one element can $A^2$ definably embed into A. (shrink)

Behind a succinct account about Euler and his connection with the philosophy, on show forth his arguments for to prove the effective existence of the absolute space in the Mechanica sive motus scientia (1736) and in the Reflexions sur l’espace et le temps (1748). These works constitute the Euler’s first approximation in defence of the doctrine of space held by Newton, and against: the Metaphysicians(Leibniz, Berkeley). This paper point at the possible ascendancy about Kant, especially in Von de (...) m ersten Grunde des Unterschiedes der Gegenden im Raume (1768). (shrink)

draftt of review of new edition of K. L. Reinhold's Versuch (1789), ed. E.-O. Onnasch.

This contribution presents for the first time in critical edition two early speeches written by Reinhold. Reinhold wrote them in 1783 to be delivered during meetings of the Viennese Masonic Lodge “Zur wahren Eintracht” (To True Harmony) of which he was a member. The first, “Über die Kunst des Lebens zu genüssen” (On the art of enjoying life), discusses the best way for Masons to wisely deal with the joys and pains of life. In the second, “Der Werth einer Gesellschaft (...) hängt von der Beschaffenheit ihrer Glieder ab” (The worth of a society depends on the disposition of its members), Reinhold discusses the nature of the Masonic society from an Illuminatist point of view. This second speech is especially relevant with regard to Reinhold’s views on the Enlightenment and his later reception of Kant. (shrink)

Studies of Reinhold have not paid sufficient attention to the systematic connection of the early Elementarphilosophie with the history of philosophy. Reinhold understands his own system as the last historical step of a purposive philosophizing activity of reason that ends the history of philosophy and enables the accomplishment of the true Copernican revolution. Reinhold discusses different aspects of this self-understanding in the writings of 1789–1791. Reinhold develops the core of this approach in a neglected and not republished essay from 1791: (...) “Ueber den Begrif der Geschichte der Philosophie: Eine akademische Vorlesung.” The complete picture of Reinhold’s approach emerges only after the respective arguments of the Versuchschrift, Beiträge vol. 1, Ueber das Fundament, and “Ueber den Begrif ” are methodically integrated. In addition, “Ueber den Begrif ” fulfils another unnoticed function; it reveals the role that Reinhold’s theory of representation plays in the systematic construction of the rational history of philosophy. (shrink)

Karl Leonhard Reinhold<br>Versuch einer neuen Theorie des Vorstellungsvermögens, Teilband 1<br>Einleitung, Vorrede, Erstes Buch<br><br>Mit einer Einleitung und Anmerkungen herausgegeben von Ernst-Otto Onnasch.<br>PhB 599a. 2010. CLVII, 210 Seiten.<br>978-3-7873-1934-3. Leinen 68.00<br><br>Karl Leonhard Reinholds Versuch einer neuen Theorie des menschlichen Vorstellungsvermögens (1789) ist aufgegliedert in eine lange Vorrede und drei Bücher. In der Vorrede und im ersten Buch stellt der Autor die epochale Bedeutung der kritischen Philosophie heraus. Im zweiten Buch folgt die eigentliche Theorie des Vorstellungsvermögens, von der aus im dritten Buch (...) Kants wichtigste Entdeckungen in der Kritik der reinen Vernunft, nämlich die Unterscheidung von Sinnlichkeit, Verstand und Vernunft, neu dargestellt werden. Hier liefert Reinhold eine eigene und höchst originelle Ableitung der Kategorien und der Ideen.<br><br>In seiner Einleitung beschreibt der Herausgeber Reinholds philosophische Entwicklung und erweist ihn als einen eigenständigen Denker mit einer ganz eigenen philosophischen Agenda, die er allerdings auf eine sehr geschickte Weise mit dem philosophischen Anliegen Kants zu verbinden vermochte: Reinholds Philosophie war, entgegen der überkommenen Einschätzung, alles andere als epigonal und von enormer Bedeutung für die Ausprägung und Genese der Philosophie des deutschen Idealismus.<br><br>Bereits mit seinen populären Briefen über die Kantische Philosophie (1786/87) traf Reinhold den Nerv der Zeit und setzte damit die kritische Philosophie Kants für ein breiteres Publikum auf die philosophische Agenda (nur wenige der Zeitgenossen lasen Kant im Original, die meisten bezogen ihr Urteil über Kant aus den Briefen). Der Versuch einer neuen Theorie des menschlichen Vorstellungsvermögens ist dann sein erstes großes theoretisches Werk mit eigenem Anspruch. Reinhold präsentiert es als einen Versuch, die kritische Philosophie auf der Grundlage des Vorstellungsvermögens allgemein verständlich zu machen.<br>. (shrink)

This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwrizm, Cardano, and Descartes to Euler, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form (in the sense of Wittgenstein's Tractatus) of the symbolic language (...) of algebra. Thus the paper develops further the method of reconstruction which the author introduced for the analysis of the development of geometry. (shrink)

The main point of this paper is to identify a set of interlocking views that became (and still are!) very influential within philosophy in the wake of Newton’s success. These views use the authority of natural philosophy/mechanics to settle debates within philosophy. I label these “Newton’s Challenge.” Newton had some hand in promoting them, but he is not responsible for all of them. My paper, thus, revisits an old theme articulated by A.E. Burt, but I offer new arguments and evidence. (...) The heart of the paper (sections II-III) identifies the core set of related views that constitute “Newton’s Challenge.” In section IIA I draw on two eighteenth century figures (Euler and Musschenbroek) to introduce arguments that give evidence for the existence of “Newton’s Challenge” and distinguish four strands within it. In section IIB I use Berkeley as evidence that something like “Newton’s Challenge” was recognized by philosophical opponents to Newton and I identify in Berkeley’s work five counter-strategies. In section IIC, I identify Newton’s contribution to Newton’s Challenge. In section III, I use the writings by MacLaurin, ’s Gravesande, and Musschenbroek to identify eight arguments that constitute the way “Newton’s Challenge” was articulated and developed in practice by eighteenth century Newtonians. (shrink)

The discussions which follow rest on a distinction, first expounded by Husserl, between formal logic and formal ontology. The former concerns itself with (formal) meaning-structures; the latter with formal structures amongst objects and their parts. The paper attempts to show how, when formal ontological considerations are brought into play, contemporary extensionalist theories of part and whole, and above all the mereology of Leniewski, can be generalised to embrace not only relations between concrete objects and object-pieces, but also relations between what (...) we shall call dependent parts or moments. A two-dimensional formal language is canvassed for the resultant ontological theory, a language which owes more to the tradition of Euler, Boole and Venn than to the quantifier-centred languages which have predominated amongst analytic philosophers since the time of Frege and Russell. Analytic philosophical arguments against moments, and against the entire project of a formal ontology, are considered and rejected. The paper concludes with a brief account of some applications of the theory presented. (shrink)

Must space be a unity? This question, which exercised Aristotle, Descartes and Kant, is a specific instance of a more general one; namely, can the topology of physical space change with time? In this paper we show how the discussion of the unity of space has been altered but survives in contemporary research in theoretical physics. With a pedagogical review of the role played by the Euler characteristic in the mathematics of relativistic spacetimes, we explain how classical general relativity (...) (modulo considerations about energy conditions) allows virtually unrestrained spatial topology change in four dimensions. We also survey the situation in many other dimensions of interest. However, topology change comes with a cost: a famous theorem by Robert Geroch shows that, for many interesting types of such change, transitions of spatial topology imply the existence of closed timelike curves or temporal non-orientability. Ways of living with this theorem and of evading it are discussed. (shrink)

Machine generated contents note: Introduction Andrew Janiak and Eric Schliesser; Part I. Newton and his Contemporaries: 1. Newton's law-constitutive approach to bodies: a response to Descartes Katherine Brading; 2. Leibniz, Newton and force Daniel Garber; 3. Locke's qualified embrace of Newton's Principia Mary Domski; 4. What geometry postulates: Newton and Barrow on the relationship of mathematics to nature Katherine Dunlop; Part II. Philosophical Themes in Newton: 5. Cotes' queries: Newton's Empiricism and Conceptions of Matter Zvi Biener and Chris Smeenk; 6. (...) Newton's Scientific Method and the Universal Law of Gravitation Ori Belkind; 7. Measurement and method: some remarks on Newton, Huygens and Euler on natural philosophy William Harper; 8. What did Newton mean by 'Absolute Motion'? Nick Huggett; 9. From velocities to fluxions Marco Panza; Part III. The Reception of Newton: 10. Newton, Locke, and Hume Graciela de Pierris; 11. Maupertuis on attraction as an inherent property of matter Lisa Downing; 12. The Newtonian refutation of Spinoza: Newton's Challenge and the Socratic Problem Eric Schliesser; 13. Dispositional explanations: Boyle's problem, Newton's solution, Hume's response Lynn Joy; 14. Newton and Kant on Absolute Space: from theology to transcendental philosophy Michael Friedman; 15. How Newton's Principia changed physics George Smith; Bibliography. (shrink)

Throughout its long history, mathematics has involved the use ofsystems of written signs, most notably, diagrams in Euclidean geometry and formulae in the symbolic language of arithmetic and algebra in the mathematics of Descartes, Euler, and others. Such systems of signs, I argue, enable one to embody chains of mathematical reasoning. I then show that, properly understood, Frege’s Begriffsschrift or concept-script similarly enables one to write mathematical reasoning. Much as a demonstration in Euclid or in early modern algebra does, (...) a proof in Frege’s concept-script shows how it goes. (shrink)

It is controversial whether Wittgenstein's philosophy of mathematics is of critical importance for mathematical proofs, or is only concerned with the adequate philosophical interpretation of mathematics. Wittgenstein's remarks on the infinity of prime numbers provide a helpful example which will be used to clarify this question. His antiplatonistic view of mathematics contradicts the widespread understanding of proofs as logical derivations from a set of axioms or assumptions. Wittgenstein's critique of traditional proofs of the infinity of prime numbers, specifically those of (...) Euler and Euclid, not only offers philosophical insight but also suggests substantive improvements. A careful examination of his comments leads to a deeper understanding of what proves the infinity of primes. (shrink)

On September 6, 2004, using the Isabelle proof assistant, I verified the following statement: (%x. pi x * ln (real x) / (real x)) ----> 1 The system thereby confirmed that the prime number theorem is a consequence of the axioms of higher-order logic together with an axiom asserting the existence of an infinite set. All told, our number theory session, including the proof of the prime number theorem and supporting libraries, constitutes 673 pages of proof scripts, or roughly 30,000 (...) lines. This count includes about 65 pages of elementary number theory that we had at the outset, developed by Larry Paulson and others; also about 50 pages devoted to a proof of the law of quadratic reciprocity and properties of Euler’s ϕ function, neither of which are used in the proof of the prime number theorem. The page count does not include the basic HOL library, or properties of the real numbers that we obtained from the HOL-Complex library. (shrink)

There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are "the other way around"). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood (...) as involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators— Posts and Kleenes—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)

ABSTRACT. The objectivity of physics has been called into question by social theorists, Kuhnian relativists, and by anomalous aspects of quantum mechanics. Here we focus on one neglected background issue, the categorical structure of the language of classical physics. The first half is an historical overview of the formation of the language of classical physics (LCP), beginning with Aristotle's Categories and the novel idea of the quantity of a quality introduced by medieval Aristotelians. Descartes and Newton at-tempted to put the (...) new mechanics on an ontological foundation of atomism. Euler was the pivotal figure in basing mechanics on a macroscopic concept of matter. The second scientific revolution, led by Laplace, took mechanics as foundational and attempted to fit the Baconian sciences into a framework of atomistic mechanism. This protracted effort had the unintended effect of supplying an informal unification of physics in a mixture of ordinary language and mechanistic terms. The second half treats LCP as a linguistic para-site that can attach itself to any language and effect mutations in the host without chang-ing its essential form. This puts LCP in the context of a language of discourse and sug-gests that philosophers should concentrate more on the dialog between experimenters and theoreticians and less on analyses of theories. This orientation supplies a basis for treating objectivity. (shrink)

Though the seminal importance of Karl Leonhard Reinhold for the development of German philosophy in the immediate aftermath of the Kantian revolution has never been in question, his actual writings have generally remained out of print and unread. Recently, however, this situation has begun to change dramatically, first, with the publication of new Felix Meiner “Philosphische Bibliothek” editions of the first and second volumes of Beiträge zur Berichtigung bisheriger Mißverständnisse der Philosophen (1790/1794), expertly edited by Faustino Fabianelli, and then (...) with the first installment of a new multi-volume edition of Versuch einer neuen Theorie des menschlichen Vorstellungsvermögens (1789),edited and with a .. (shrink)

In this paper, I would like to show that considering technological models as they arise in engineering disciplines can greatly enrich the philosophical perspective on models. In fluid mechanics, (at least) three types of models are distinguished: mathematical, computer and physical models. Very often, the choice of a particular mathematical, computer or physical model highly affects the type of solutions and the computational time needed for it. Technological models not only aim at a correct description of the physical phenomena, but (...) also for an efficient and accurate simulation. The problem arises how heterogeneous models of an engineering problem can be brought together and be compared to each other as regards their function and technological efficiency. There are two developments in the history of fluid mechanics that have greatly influenced the use of models in the field: The introduction of the concept of the boundary layer by Ludwig Prandtl in 1904 made it possible to apply ideal analytical solutions, which at the time were almost entirely based on Euler’s equations for inviscid fluids, to interesting real cases and to approximate the theoretical Navier-Stokes equations to practical engineering problems, i.e. to cases at high Reynolds numbers. This made it possible to link the empirical tradition of hydraulics with the theoretical tradition of analytical mechanics and therefore lead to a kind of equilibrium in the use of mathematical and physical models. In the 1970s the introduction of the computer has greatly pushed back the importance of both physical and mathematical (analytic) models alike without making them superfluous. There remain, however, three different ways to conceive of physical models in fluid mechanics, and thus of the experimental ingredient, depending on whether they are devised from an analytical, computational or measurement theoretical point of view. Yet even inside the tradition of computer simulation, different practices have formed according to the programming methods used. The choice of method either depends on considerations of efficiency in terms of costs and time, or on historically contingent factors, like availability of instruments and programming packages or the arbitrary choice of a forerunner. Seen from a technological point of view the factors that make models “autonomous agents” and thus (relatively) independent from theory depend on efficiency constraints. Models are means to solve problems in a certain practical perspective by the most efficient means available. To develop a model is a “fast and frugal way” to get to grips with a certain region of reality, whereas the theoretical approach stresses the importance of universal features. (shrink)

We discuss external and internal graphical and linguistic representational systems. We argue that a cognitive theory of peoples' reasoning performance must account for (a) the logical equivalence of inferences expressed in graphical and linguistic form; and (b) the implementational differences that affect facility of inference. Our theory proposes that graphical representations limit abstraction and thereby aid processibility. We discuss the ideas of specificity and abstraction, and their cognitive relevance. Empirical support comes from tasks (i) involving and (ii) not involving the (...) manipulation of external graphics. For (i), we take Euler's Circles, provide a novel computational reconstruction, show how it captures abstractions, and contrast it with earlier construals, and with Mental Models' representations. We demonstrate equivalence of the graphical Euler system, and the non-graphical Mental Models system. For (ii), we discuss text comprehension, and the mental performance of syllogisms. By positing an internal system with the same specificity as Euler's Circles we cover the Mental Models data, and generate new empirical predictions. Finally, we consider how the architecture of working memory explains why such specific representations are relatively easy to store. (shrink)

We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (Proofs and Refutations, 1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, in which we use (...) work by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods. (shrink)

EM Music Education /EM is a collection of thematically organized essays that present an historical background of the picture of education first in Greece and Rome, the Middle Ages, then Early-Modern Europe. The bulk of the book focuses on American education up to the present. This third edition includes readings by Orff, Kodály, Sinichi Suzuki, William Channing Woodbridge, Allan Britton, and Charles Leonhard. In addition, essays include timely topics on feminism, diversity, cognitive psych, testing (the Praxis exam) and the (...) No Child Left Behind Act. (shrink)

In the course of his long development, Kant's concept of matter changed somewhat, while his concept of scientific explanation changed considerably. Both developments achieved a coherent integration in Kant's Metaphysical Foundations of Natural Science. Using this developmental background, the present paper argues that the Foundations should be interpreted as an attempted rational reconstruction of the mechanics of Newton and Euler. Kant attempted to do this by constructing a concept of matter that would confer a Leibnizian intelligibility on Newtonian mechanics, (...) and also accord with Kant's theories on the nature of concepts and their role in scientific explanation. (shrink)

Friedrich Albert Lange (1828?1875) author of a famous History of Materialism and Critique of Its Present Significance(1866, English transi.I?III 1877?79, repr.1925 with introduction by Bertrand Russell), was also interested in the epistemological foundations of formal logic.Part I of his intended two?volume Logische Studienwas published posthumously in 1877 by Hermann Cohen?head?of the Marburg school of neo?Kantianism.Lange, departing from Kant, claims that spatial intuition is the source of the apodeictic character not only of the truths of mathematics, but also of the truths (...) of logic.He aims at showing this by basing validity and invalidity of syllogistic inferences on an interpretation of the standard forms (of proposition in assertoric syllogistic) with the help of the five kinds of possible relations (in fact what is known today as the Gergonne?Euler relations) between extensions of concepts given to us as areas in a plane, i.e.in space.Generality is achieved by considering all possible variations within each type of spatial relation, exhibiting a connection between concept and intuition reminding Lange of the Kantian ?schema?. Lange is well aware of the contemporary English ?algebraic? logic, but he considers its approach as the appropriate one for a logic of content (Inhaltslogik)and not for a logic of extension (Umfangslogik)Lange did not live to enjoy the recognition by some leading logicians (amongst them John Venn, to whose reference in 1881 to Lange?s ?admirable Logische Studien?the present paper owes it title), nor could he respond to the many critics of his proposed foundation of logic.Its radicality as well as its broad reception (and discussion up to at least 1959) seem to entitle Lange?s Logische Studiento an, if modest, place in the history of logic in the 19th century. (shrink)

Reinhold's Letters on the Kantian Philosophy is arguably the most influential book ever written concerning Kant. It propelled Kant's Critical Philosophy, which had previously enjoyed an equivocal reception, into the central position which it has held to this day. It also brought fame to Reinhold, who became a professor at Jena and later developed his own "Elementary Philosophy". This volume presents the first English translation of the work, together with an introduction that sets it in its philosophical and historical contexts.

This paper consists roughly of three parts. In the first part, an attempt has been made to find some tenable interpretation of Hamilton's logic. This results in accepting that Hamilton's logic can be "saved" if it is understood as being an everday language version of Euler's relations, i.e., extensional relations between terms (classes). In the second part, the propositions of Euler and the propositions of Aristotle are compared and found to be interdefinable: every proposition of Aristotle can be (...) defined by a disjunction of Euler's propositions, and every proposition of Euler can be defined by a conjunction of Aristotle's propositions. In the third part, extensional interpretation is applied to the traditional logic. As a result the 19 traditional syllogisms are reduced to 11. (shrink)

HTA and TA institutions at national parliaments (PTA) both share the same origin and of course have objectives and some of their methods in common. Nevertheless both TA branches developed in some distance during the 1970s and 1980s. Drawing on the case of biomedicine this paper outlines the differences between HTA and PTA, highlighting the “clinical perspective” of HTA and the “societal perspective” of PTA. It is shown that biomedicine which has developed rapidly during the last decade has hardly been (...) dealt with by HTA, whereas it ranked quite prominent on the agendas of PTA institutions. Biomedical technologies became a subject of policy making beyond the boundaries of health care politics since biomedicine is perceived as an ethical challenge to society and not only as a medical innovation that has to be assessed by clinical experts. It is argued that there may however be good reasons to integrate the HTA and the PTA perspective in future TA on biomedical technologies. (shrink)

One of the ways in which the artificial languages of mathematics are “generous”, that is, in which they assists the advance of thought, is through its establishment of advanced operatory structures that permit an even further advance of intuition. However, this generosity may be delusive, suggest ideas which in the longer run turn out to be untenable. The paper analyses two cases of “honest generosity”, namely a “proof” of the sign rule “less times less makes plus” from the 1340s and (...) a result in partition theory obtained by Euler by means of rash manipulations of infinite series and products, case-Cantor’s introduction of transfinite numbers from 1895-and (in modern terms) a failed attempt to extend the semi-group of algebraic powers into a complete group, also from c. 1340. Gewöhnlich glaubt der Mensch, wenn er nur Worte hört es müsse sich dabei wohl auch was denken lassen Goethe, Faust I, 2565-2566 He gives the kids free samples because he knows full well that today’s young innocent faces will be tomorrow’s clientele Tom Lehrer, “The Old Dope Peddler”. (shrink)

The problem of an Euler–Bernoulli cantilever beam whose free end impacts with a point constraint is revisited from the point of view of modal analysis. It is shown that there is non-uniqueness of consistent impact laws for a given modal truncation. Moreover, taking an N-mode compliant, bilinear formulation and passing to the rigid limit leads to a sequence of impact models that does not converge as . The dynamics of such truncated models are studied numerically and found to give (...) rise to quite different dynamics depending on the number of degrees of freedom taken. The simulations are compared with results from simple experiments that show a propensity for multiple-tap dynamics, in which higher-order modes lead to rapidly cycling intermittent contact. The conclusion reached is that, to derive an accurate model, one needs to avoid the impact limit altogether, and take sufficiently many modes in the formulation to match the actual stiffness of the constraining stop. mechanical engineering, applied mathematics. (shrink)

Abstract Instrumental variables estimation is widely applied in econometrics. To implement the method, it is necessary to specify a vector of instruments. In this paper, it is argued that there are compelling reasons to use the data for instrument selection, but that it is desirable to ensure the resulting estimator still behaves in the way predicted by standard textbook theory. These arguments lead one to propose three criteria for data based instrument selection. The remainder of the paper assesses the extent (...) to which these criteria are met by two algorithms for data based instrument selection. The first algorithm is the method of structurally ordered instrumental variables proposed in the context of economy-wide linear simultaneous equation models. The second algorithm is proposed in the context of the method of generalized instrumental variables, which is commonly used to estimate the parameters of Euler equation models. (shrink)

The paper contributes to the current discussion on the role of participatory methods in the context of technology assessment (TA) and science and technology (S&T) governance. It is argued that TA has to be understood as a form of democratic policy consulting in the sense of the Habermasian model of a “pragmatist” relation of science and politics. This notion implies that public participation is an indispensable element of TA in the context of policy advice. Against this background, participatory TA (pTA) (...) is defended against recent criticism of procedures of lay participation which states that pTA is lacking impact on S&T decision making, that pTA instead of opening S&T policies to new perspectives is used as a means to support mainstream S&T policy and that in pTA procedure the authentic lay perspective is systematically contorted by dominant expert knowledge. (shrink)

What good is logic? -- Seventeen ways this book is different -- The two logics -- All of logic in two pages : an overview -- The three acts of the mind -- I. The first act of the mind : understanding -- Understanding : the thing that distinguishes man from both beast and computer -- Concepts, terms and words -- The problem of universals -- The comprehension and extension of terms -- II. Terms -- Classifying terms -- Categories -- (...) Predicables -- Division -- III. Material fallacies -- Fallacies of language -- Fallacies of diversion -- Fallacies of oversimplification -- Fallacies of argumentation -- Inductive fallacies -- Procedural fallacies -- Metaphysical fallacies -- Short story : love is a fallacy -- IV. Definition -- The nature of definition -- The rules of definition -- The kinds of definition -- The limits of definition -- V. Second act of the mind : judgment -- Judgments, propositions, and sentences -- What is truth? -- The four kinds of categorical propositions -- Logical form -- Euler's circles -- Tricky propositions -- The distribution of terms -- VI. Changing propositions -- Immediate inference -- Conversion -- Obversion -- Contraposition -- VII. Contradiction -- What is contradiction? -- The square of opposition -- Existential import -- Tricky propositions on the square -- Some practical uses of the square of opposition -- VIII. The third act of the mind : reasoning -- What does reason mean? -- The ultimate foundations of the syllogism -- How to detect arguments -- Arguments vs. explanations -- Truth and validity -- IX. Different kinds of arguments -- Three meanings of because -- The four causes -- A classification of arguments -- Simple argument maps -- Deductive and inductive arguments -- Combining deduction and induction : socratic method -- X. Syllogisms -- The structure and strategy of the syllogism -- The skeptics objection to the syllogism -- The empiricist's objection to the syllogism -- Demonstrative syllogisms -- How to construct convincing syllogisms -- XI. Checking syllogisms for validity -- By Euler's circles -- By Aristotle's six rules -- Barbara Celarent : mood and figure -- Venn diagrams -- XII. More difficult syllogisms -- Enthymemes : abbreviated syllogisms -- Sorites : chain syllogisms -- Epicheiremas : multiple syllogisms -- Complex argument maps -- XIII. Compound syllogisms -- Hypothetical syllogisms -- Reductio ad absurdum arguments -- The practical syllogism : arguing about means and ends -- Disjunctive syllogisms -- Conjunctive syllogisms -- Dilemmas -- XIV. Induction -- What is induction? -- Generalization -- Causal induction : Mill's methods -- Scientific hypotheses -- Statistical probability -- Arguments from analogy -- A fortiori and a minore arguments -- XV. Some practical applications of logic -- How to write a logical essay -- How to write a socratic dialogue -- How to have a socratic debate -- How to use socratic method on difficult people -- How to read a book socratically -- XVI. Some philosophical applications of logic -- Logic and theology -- Logic and metaphysics -- Logic and cosmology -- Logic and philosophical anthropology -- Logic and epistemology -- Logic and ethics. (shrink)

Between the Orthodox Russian Church and modern science were not any serious conflicts. For examples, in XVII century students of Academy of Kiev-Mohyla studied heliocentric system of Copernic and doctrine of Galileo. In 1724 According to the project of Leibniz Tsar Peter I founded Russian Academy of Science in St-Petersburg. There worked D. Bernoulli and L. Euler. The Russian philosophical though presents an attempt of accord of science and theology.