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"Schlimm, Dirk"
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Calculus as method or calculus as rules? Boole and Frege on the aims of a logical calculus
By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the other hand, is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems.
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Extended mathematical cognition: external representations with non-derived content
Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that ‘non-derived’, or ‘original’, content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper we develop one aspect of Menary’s vehicle externalist theory of cognitive integration—the process of enculturation—to respond to this longstanding objection. We offer examples of how expert mathematicians introduce new symbols to represent new mathematical possibilities that are not yet understood, and we argue that these new symbols have genuine non-derived content, that is, content that is not dependent on an act of interpretation by a cognitive agent and that does not derive from conventional associations, as many linguistic representations do.
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On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others
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.
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PASCH’S PHILOSOPHY OF MATHEMATICS
Moritz Pasch (1843–1930) gave the first rigorous axiomatization of projective geometry in his Vorlesungen über neuere Geometrie (1882), in which he also clearly formulated the view that deductions must be independent from the meanings of the nonlogical terms involved. Pasch also presented in these lectures the main tenets of his philosophy of mathematics, which he continued to elaborate on throughout the rest of his life. This philosophy is quite unique in combining a deductivist methodology with a radically empiricist epistemology for mathematics. By taking into consideration publications from the entire span of Pasch’s career, the latter decades of which he devoted primarily to careful reflections on the nature of mathematics and of mathematical knowledge, Pasch’s highly original, but virtually unknown, philosophy of mathematics is presented.
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Two Ways of Analogy: Extending the Study of Analogies to Mathematical Domains
Thestructure‐mapping theoryhas 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 theaxiomaticapproach 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, for example, for the discovery of new analogies.
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Learning from the Existence of Models: On Psychic Machines, Tortoises, and Computer Simulations
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.
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Methodological Reflections on Typologies for Numerical Notations
Past and present societies world-wide have employed well over 100 distinct notational systems for representing natural numbers, some of which continue to play a crucial role in intellectual and cultural development today. The diversity of these notations has prompted the need for classificatory schemes, or typologies, to provide a systematic starting point for their discussion and appraisal. The present paper provides a general framework for assessing the efficacy of these typologies relative to certain desiderata, and it uses this framework to discuss the two influential typologies of Zhang & Norman and Chrisomalis. Following this, a new typology is presented that takes as its starting point the principles by which numerical notations represent multipliers (the principles of cumulation and cipherization), and bases (those of integration, parsing, and positionality). Many different examples show that this new typology provides a more refined classification of numerical notations than the ones put forward previously. In addition, the framework provided here can be used to assess typologies not only of numerical notations, but also of many other domains.
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On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan Have Proved the Jordan-Hölder Theorem?
In 1870 Jordan proved that the composition factors of two composition series of a group are the same. Almost 20 years later Holder (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.
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History and philosophy of infinity: Selected papers from the conference \Foundations of the Formal Sciences VIII\ held at Corpus Christi College, Cambridge, England, 20–23 September 2013
Issue Title: Special Section on History and philosophy of infinity
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