Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent (...) formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other. (shrink)

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)

To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals. This demonstrates that Roman numerals are not only informationally equivalent to Arabic ones but also computationally similar—a claim that is widely disputed. An analysis of our models' elementary processing steps reveals intricate tradeoffs between problem representation, algorithm, and interactive resources. Our simulations allow for a more nuanced view of the received wisdom on Roman numerals. (...) While symbolic computation with Roman numerals requires fewer internal resources than with Arabic ones, the large number of needed symbols inflates the number of external processing steps. (shrink)

The structure-mapping theory has become the de-facto standard account of analogies in cognitive science and philosophy of science. In this paper I propose a distinction between two kinds of domains and I show how the account of analogies based on structure-preserving mappings fails in certain (object-rich) domains, which are very common in mathematics, and how the axiomatic approach to analogies, which is based on a common linguistic description of the analogs in terms of laws or axioms, can be used successfully (...) to explicate analogies of this kind. Thus, the two accounts of analogies should be regarded as complementary, since each of them is adequate for explicating analogies that are drawn between different kinds of domains. In addition, I illustrate how the account of analogies based on axioms has also considerable practical advantages, e. g., for the discovery of new analogies. (shrink)

Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various ways in which lattices have been introduced in mathematics by Schröder et al. These historical episodes reveal that the axiomatic method is not only a way of systematizing our knowledge, but that it can also be used as a fruitful tool for discovering and introducing new mathematical notions. Looked at it from this perspective, the creative (...) aspect of axiomatics for mathematical practice is brought to the fore. (shrink)

Using four examples of models and computer simulations from the history of psychology, I discuss some of the methodological aspects involved in their construction and use, and I illustrate how the existence of a model can demonstrate the viability of a hypothesis that had previously been deemed impossible on a priori grounds. This shows a new way in which scientists can learn from models that extends the analysis of Morgan (1999), who has identified the construction and manipulation of models as (...) those phases in which learning from models takes place. (shrink)

In 1870 Jordan proved that the composition factors of two composition series of a group are the same. Almost 20 years later Hölder (1889) was able to extend this result by showing that the factor groups, which are quotient groups corresponding to the composition factors, are isomorphic. This result, nowadays called the Jordan-Hölder Theorem, is one of the fundamental theorems in the theory of groups. The fact that Jordan, who was working in the framework of substitution groups, was able to (...) prove only a part of this theorem is often used to emphasize the importance and even the necessity of the abstract conception of groups, which was employed by Hölder. However, as a little-known paper from 1873 reveals, Jordan had all the necessary ingredients to prove the Jordan-Hölder Theorem at his disposal (namely, composition series, quotient groups, and isomorphisms), and he also noted a connection between composition factors and corresponding quotient groups. Thus, I argue that the answer to the question posed in the title is “Yes.” It was not the lack of the abstract notion of groups which prevented Jordan from proving the Jordan-Hölder Theorem, but the fact that he did not ask the right research question that would have led him to this result. In addition, I suggest some reasons why this has been overlooked in the historiography of algebra, and I argue that, by hiding computational and cognitive complexities, abstraction has important pragmatic advantages. (shrink)

The main ideas behind Brouwer’s philosophy of Intuitionism are presented. Then some critical remarks against Intuitionism made by William Tait in “Against Intuitionism” [Journal of Philosophical Logic, 12, 173–195] are answered.

This article argues against the term and concept of ‘hate speech’ and in favour of using the concept and term ‘discriminatory speech’. ‘Hate speech’ is a misnomer; we should name the harmful speech in question by what it in fact does: it discriminates. The article argues for this conceptual replacement claim by identifying a number of functions the concept ‘hate speech’ has been meant to serve and by arguing that extant concepts of hate speech have not served this function well. (...) Roughly, they do not serve the functions well because of five properties of hate speech: hate centricity, perpetrator centricity, intention centricity, emotion centricity, and individual centricity. The article then proposes a definition of discriminatory speech and argues that it fulfils the conceptual functions better. (shrink)

The design of good notation is a cause that was dear to Charles Babbage's heart throughout his career. He was convinced of the "immense power of signs" (1864, 364), both to rigorously express complex ideas and to facilitate the discovery of new ones. As a young man, he promoted the Leibnizian notation for the calculus in England, and later he developed a Mechanical Notation for designing his computational engines. In addition, he reflected on the principles that underlie the design of (...) good mathematical notations. In this paper, we discuss these reflections, which can be found somewhat scattered in Babbage's writings, for the first time in a systematic way. Babbage's desiderata for mathematical notations are presented as ten guidelines pertinent to notational design and its application to both individual symbols and complex expressions. To illustrate the applicability of these guidelines in non-mathematical domains, some aspects of his Mechanical Notation are also discussed. (shrink)

Die dritte Abteilung der Kritischen Gesamtausgabe bietet in vierzehn Bänden sämtliche erhaltenen Predigten Friedrich Schleiermachers (1768-1834). Der vorliegende Band 13 enthält alle 59 Predigten des Jahres 1832, die durch Drucktexte oder Nachschriften dokumentierbar sind. Die Erläuterungen des Bandherausgebers erschließen die zeitgeschichtlichen und zahlreichen biblischen Bezüge der Texte.

Keine ausführliche Beschreibung für "Religion und Christentum in Fichtes Spätphilosophie 1810-1813" verfügbar.

In lieu of an abstract, here is a brief excerpt of the content:The Idea of Progress and the Art of Grammar: Charisius Ars Grammatica 1.15Dirk M. SchenkeveldIIn studies on the history of the concept of progress many passages have been cited from Greek and Roman texts on the progress of mankind, culture and/or the arts, but no attention has ever been given to a passage which, thanks to Flavius Sosipater Charisius (ca. 360), 1 we may read in his Ars Grammatica. (...) It not only contains a definite statement on continuing growth and development of the arts but also discusses the effect this may produce when one considers such progress. 2 The latter aspect is found again in very few other texts, but our text is unique for its application of the concept of progress to the ars grammatica. For these reasons the passage in Charisius deserves a discussion of its own. It is not easily read because of its textual condition and its style, and this may well be the main reason why those who have studied it have until now (with one exception) 3 overlooked the underlying idea of progress. 4Charisius’ grammar is a huge collection of earlier material, which he has compiled and ordered after the plan of the Schulgrammatik. Within this framework he has inserted huge blocks of material to accommodate the form. 5 Thus the fifteenth chapter of his first book starts [End Page 443] with a long portion of text, apparently an introduction to some book, but what follows thereafter has no clear connection to it. Accordingly, the first portion is an independent piece of writing and should be discussed as such. The text takes up pages 61.16–63.20 in Barwick’s edition (5 Keil GL I 50.8–I 51.20) and consists of five sections:a. 61.16–62.2, on the progress of arts in generalb. 62.2–14, on Latin language originally being unruly and later being put in order by grammarc. 62.14–63.9, on the four criteria of correct usage of Latin, namely natura, analogia, consuetudo, and auctoritas [End Page 444]d. 63.9–16, more on the relationship between analogy and consuetudoe. 63.16–20, announcement of the subjects to be treated hereafter.As we are concerned here with the aspect of progress, only the first two sections are relevant. 6 I have supplied my own text (above), based on Barwick’s [with his line numbers], 7 and add a short critical apparatus and a translation. [End Page 445]The first sentence (ne ipsa... satiata) shows stylistic traits that we shall meet with throughout the passage: an involuted style with much variatio, 8 daring imagery, and difficult periphrases. Ancient authors liked to turn their prefaces into rhetorical showpieces, and our author has done so to a high degree. Instead of simply saying, “the arts have no limits,” he introduces an a fortiori argument: if nature, though finite, is so wide as not to allot to us her farthest boundary, 9 the more is this true for the arts. No argument is given for the idea that nature is finite. 10 At any rate, our author is more interested in the status of nature’s usual complement, the arts. At first—and this would be in line with his a fortiori argument—he seems to accept as feasible the idea that at some time the arts will be complete and finished, and he explains that this point of perfection has not yet been reached because of human weakness: we humans are too weak to make the arts perfect. This is so either because the work to be done is of extreme difficulty, 11 or because we are already satisfied by having founded an art and are disinclined to continue our job. The reader is left with the choice between these two possibilities as the author proceeds to a more basic explanation. Now he opts for the idea that in principle the arts are incomplete and remain so: every artisan always adds something to his art, adds, as it were, extensions to the building that his art forms; 12 consequently the arts will never be finished. The author next anticipates... (shrink)

Jonathan Wolff is Professor of Philosophy at University College London. He is the author of Robert Nozick (1991), An Introduction to Political Philosophy (1996) and Why Read Marx Today (2002). He is currently working on a number of topics at the intersection of political philosophy and public policy.

Places the development of Anaximander's thought within social, political, cosmological, astronomical, and technological contexts.

Our epistemic access to mathematical objects, like numbers, is mediated through our external representations of them, like numerals. Nevertheless, the role of formal notations and, in particular, of the internal structure of these notations has not received much attention in philosophy of mathematics and cognitive science. While systems of number words and of numerals are often treated alike, I argue that they have crucial structural differences, and that one has to understand how the external representation works in order to form (...) an advanced conception of numbers. The relation between external representations and mathematical cognition will be discussed by drawing on experimental results from cognitive science and mathematics education, and I argue for the constitutive role of external representations for the development of an advanced conception of numbers. (shrink)

Prototype theory makes a crucial distinction between central and peripheral sense of words. Geeraerts explores the implications of this model for a theory of semantic change, in the first full-scale treatment of the impact of the most recent developments in lexicological theory on the study of meaning change. He identifies structural features of the development of word meanings which follow from a prototype-theoretical model of semantic structure, and incorporates these diachronic prototypicality effects into a theory of meaning change.

Dirk Paul Bogner widmet sich der Frage, welche Relevanz Kurt Lewins Feldtheorie im Bereich der pädagogischen Diagnostik haben könnte. Er untermauert die These, dass die Lewin’sche Feldtheorie eine völlig neue Sicht auf Lern-, Entwicklungs- und Erziehungsprozesse eröffnen würde, und legt dar, inwiefern die Einbeziehung der Dimension des Raumes die Analyse dieser erziehungswissenschaftlich relevanten Topoi verfeinern könnte. Er erörtert mittels hermeneutischer Textauslegung und vergleichender Dokumentenanalyse, warum Lewin keine erziehungswissenschaftliche Resonanz erzeugen konnte, obwohl in der Erziehungswissenschaft auch heute noch intensiv Begriffe, (...) die entweder direkt aus der Feldtheorie stammen oder aber von dieser konsequent definiert wurden, verwendet werden. (shrink)

There has recently been a wave of attempts to make sense of the role of de se thoughts in linguistic communication. A majority of the attempts assume a Perryan or a Lewisian view of de se thought. Views with these assumptions, I suggest, come in four varieties: uncentering (Egan 2007, Kölbel 2013, Moss 2012), recentering (Heim 2004, Weber 2012), multicentering (Kindermann 2014, Ninan 2010, Torre 2009), and no centering (Kaplan 1989, Perry 1979). I argue first that all four varieties of (...) centering are committed to what I call a shifting operation on the hearer's part. I argue second that, against common assumption, there is no real choice to make between the views. By showing that attempts to establish an advantage for some view over the others fail across the board, I make the case for neutralism regarding the varieties of centering – the claim that coverage of the empirical data is exactly the same for each view, and that the views are broadly equal in simplicity and elegance. (shrink)

In Miletus, about 550 B.C., together with our world-picture cosmology was born. This book tells the story. In Part One the reader is introduced in the archaic world-picture of a flat earth with the cupola of the celestial vault onto which the celestial bodies are attached. One of the subjects treated in that context is the riddle of the tilted celestial axis. This part also contains an extensive chapter on archaic astronomical instruments. Part Two shows how Anaximander (610-547 B.C.) blew (...) up this archaic world-picture and replaced it by a new one that is essentially still ours. He taught that the celestial bodies orbit at different distances and that the earth floats unsupported in space. This makes him the founding father of cosmology. Part Three discusses topics that completed the new picture described by Anaximander. Special attention is paid to the confrontation between Anaxagoras and Aristotle on the question whether the earth is flat or spherical, and on the battle between Aristotle and Heraclides Ponticus on the question whether the universe is finite or infinite. (shrink)

Gesprekken tussen filosoof en ecoloog Johan Braeckman en moraalfilosoof Dirk Verhofstadt over de evolutietheorie van Darwin, religie, irrationalieit, humanisme en atheïsme."--Publisher information.

In "Plato's Ghost" Jeremy Gray presented many connections between mathematical practices in the nineteenth century and the rise of mathematical platonism in the context of more general developments, which he refers to as modernism. In this paper, I take up this theme and present a condensed discussion of some arguments put forward in favor of and against the view of mathematical platonism. In particular, I highlight some pressures that arose in the work of Frege, Cantor, and Gödel, which support adopting (...) a platonist position. The aim of this discussion is to provide an historically informed introduction to the philosophical position of mathematical platonism and to point at some of its mathematical and philosophical roots in the nineteenth century. (shrink)