In this paper, we explore Peirce's work for insights into a theory of learning and cognition for education. Our focus for this exploration is Peirce's paper The Fixation of Belief (FOB), originally published in 1877 in Popular Science Monthly. We begin by examining Peirce's assertion that the study of logic is essential for understanding thought and reasoning. We explicate Peirce's view of the nature of reasoning itself—the characteristic guiding principles or ‘habits of mind’ that underlie acts of inference, the dimensions (...) of and interaction between doubt and belief, and his four methods of resolving or ‘fixing’ belief (i.e., tenacity, authority, a priori, and experimentation). The four methods are then juxtaposed against current models of teaching and learning such as constructivism, schema theory, situated cognition, and inquiry learning. Finally, we discuss Peirce's modes of inference as they relate educationally to the resolution of doubt and beliefs and offer an example of belief resolution from an experienced teacher in a professional development environment. (shrink)
On the occasion of the 150th birthday of Georg Cantor (1845â1918), the founder of the theory of sets, the development of the logical foundations of this theory is described as a sequence of catastrophes and of trials to save it. Presently, most mathematicians agree that the set theory exactly defines the subject of mathematics, i.e., any subject is a mathematical one if it may be defined in the language (i.e., in the notions) of set theory. Hence the nature of formal (...) definitions plays an important role within the logical foundations of mathematics. Its study is also helpful to answer the question of how it is possible that the set theory as a universal new ontology for the subject of mathematics (as people hoped around 1900) totally failed but nevertheless the language of set theory is successful in all the mathematical practice. (shrink)
Zusammenfassung Die folgenden AusfÃ¼hrungen enthalten einen Beitrag zum Problem der Induktion. Den bisher bekannt gewordenen Interpretationen induktiver Regeln wird eine weitere zur Seite gestellt. Sie beruht auf der von Popper propagierten Idee der AnnÃ¤herung an die Wahrheit, genauer: auf einer geeigneten topologischen PrÃ¤zisierung dieser AnnÃ¤herungsidee. Wie sich dabei herausstellt, ist die approximationstheoretische Deutung der allgemeinen Induktion zwar stringent, aber nicht induktivistisch, d.h. sie liefert keine irgendwie geartete Rechtfertigung induktiver Regeln.
The paper explores the relation between Kierkegaard’s concept of a “life-view,” understood as a certain quality of a person’s character, and his early account of Christian faith. To claim the need for such an exploration is motivated by two observations: First, defining a “life-view” as “an unshakable certainty in oneself won from all experience” (Kierkegaard’s formula in his debut book From the Papers of One Still Living ) essentially conforms with his characterization of faith as an “a priori certainty.” Second, (...) the relation between Kierkegaard’s notion of “life-view” and his concept of faith has been interpreted in different, at times in-compatible ways by Kierkegaard-scholars. Thus, after outlining the overall argument in Kierkegaard’s debut book, I will compare in detail the notions of “life-view” and faith, and this by using as a vantage point and paradigmatic example the opposing accounts of Emanuel Hirsch and Ulrich Klenke. (shrink)
Starting from the thesis that the history of mathematics, for saving its duration and support as a scientific discipline, has to look for the needs and problems of contemporary mathematics, six main problems of contemporary mathematics are listed (concerning partly its social state) and subsequently, 16 questions about the history of mathematics. Moreover, some analogues between the work-sharing of the historians of mathematics and that of the scientists are stated.
We explore a network architecture introduced by Elman (1988) for predicting successive elements of a sequence. The network uses the pattern of activation over a set of hidden units from time-step 25-1, together with element t, to predict element t + 1. When the network is trained with strings from a particular finite-state grammar, it can learn to be a perfect finite-state recognizer for the grammar. When the network has a minimal number of hidden units, patterns on the hidden units (...) come to correspond to the nodes of the grammar, although this correspondence is not necessary for the network to act as a perfect finite-state recognizer. We explore the conditions under which the network can carry information about distant sequential contingencies across intervening elements. Such information is maintained with relative ease if it is.. (shrink)