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Dirk Schlimm
McGill University
  1. Dedekind’s Analysis of Number: Systems and Axioms.Wilfried Sieg & Dirk Schlimm - 2005 - Synthese 147 (1):121-170.
    Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.
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  2.  53
    The Cultural Challenge in Mathematical Cognition.Andrea Bender, Dirk Schlimm, Stephen Crisomalis, Fiona M. Jordan, Karenleigh A. Overmann & Geoffrey B. Saxe - 2018 - Journal of Numerical Cognition 2 (4):448–463.
    In their recent paper on “Challenges in mathematical cognition”, Alcock and colleagues (Alcock et al. [2016]. Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41) defined a research agenda through 26 specific research questions. An important dimension of mathematical cognition almost completely absent from their discussion is the cultural constitution of mathematical cognition. Spanning work from a broad range of disciplines – including anthropology, archaeology, cognitive science, history of science, linguistics, philosophy, and psychology – we (...)
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  3.  89
    Pasch’s Philosophy of Mathematics.Dirk Schlimm - 2010 - Review of Symbolic Logic 3 (1):93-118.
    Moritz Pasch (1843ber 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 Paschs (...)
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  4. Extended Mathematical Cognition: External Representations with Non-Derived Content.Karina Vold & Dirk Schlimm - 2019 - Synthese 1:1-21.
    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 (...)
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  5.  41
    The Cognitive Advantages of Counting Specifically: A Representational Analysis of Verbal Numeration Systems in Oceanic Languages.Andrea Bender, Dirk Schlimm & Sieghard Beller - 2015 - Topics in Cognitive Science 7 (4):552-569.
    The domain of numbers provides a paradigmatic case for investigating interactions of culture, language, and cognition: Numerical competencies are considered a core domain of knowledge, and yet the development of specifically human abilities presupposes cultural and linguistic input by way of counting sequences. These sequences constitute systems with distinct structural properties, the cross-linguistic variability of which has implications for number representation and processing. Such representational effects are scrutinized for two types of verbal numeration systems—general and object-specific ones—that were in parallel (...)
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  6.  72
    Axioms in Mathematical Practice.Dirk Schlimm - 2013 - Philosophia Mathematica 21 (1):37-92.
    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 (...)
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  7.  78
    Modeling Ancient and Modern Arithmetic Practices: Addition and Multiplication with Arabic and Roman Numerals.Dirk Schlimm & Hansjörg Neth - 2008 - In B. C. Love, K. McRae & V. M. Sloutsky (eds.), Proceedings of the 30th Annual Conference of the Cognitive Science Society. Cognitive Science Society. pp. 2097--2102.
    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. (...)
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  8.  35
    On Frege’s Begriffsschrift Notation for Propositional Logic: Design Principles and Trade-Offs.Dirk Schlimm - 2018 - History and Philosophy of Logic 39 (1):53-79.
    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 (...)
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  9.  9
    Numbers Through Numerals. The Constitutive Role of External Representations.Dirk Schlimm - 2018 - In Sorin Bangu (ed.), Naturalizing Logico-Mathematical Knowledge: Approaches from Psychology and Cognitive Science. New York, NY, USA: pp. 195–217.
    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 (...)
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  10.  95
    On the Creative Role of Axiomatics. The Discovery of Lattices by Schröder, Dedekind, Birkhoff, and Others.Dirk Schlimm - 2011 - Synthese 183 (1):47-68.
    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 (...)
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  11. Learning From the Existence of Models: On Psychic Machines, Tortoises, and Computer Simulations.Dirk Schlimm - 2009 - Synthese 169 (3):521 - 538.
    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 (...)
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  12.  33
    Methodological Reflections on Typologies for Numerical Notations.Theodore Reed Widom & Dirk Schlimm - 2012 - Science in Context 25 (2):155-195.
    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 (...)
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  13. Basic Mathematical Cognition.David Gaber & Dirk Schlimm - 2015 - WIREs Cognitive Science 4 (6):355-369.
    Mathematics is a powerful tool for describing and developing our knowledge of the physical world. It informs our understanding of subjects as diverse as music, games, science, economics, communications protocols, and visual arts. Mathematical thinking has its roots in the adaptive behavior of living creatures: animals must employ judgments about quantities and magnitudes in the assessment of both threats (how many foes) and opportunities (how much food) in order to make effective decisions, and use geometric information in the environment for (...)
     
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  14.  69
    Two Ways of Analogy: Extending the Study of Analogies to Mathematical Domains.Dirk Schlimm - 2008 - Philosophy of Science 75 (2):178-200.
    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 (...)
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  15.  1
    The Cognitive Basis of Arithmetic.Helen De Cruz, Hansjörg Neth & Dirk Schlimm - 2010 - In Benedikt Löwe & Thomas Müller (eds.), PhiMSAMP. Philosophy of mathematics: Sociological aspects and mathematical practice. pp. 59-106.
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  16.  57
    Dedekind's Abstract Concepts: Models and Mappings.Wilfried Sieg & Dirk Schlimm - 2014 - Philosophia Mathematica:nku021.
    Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his ‘axiomatic standpoint’: abstract concepts, models, and mappings.
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  17.  42
    Axiomatics and Progress in the Light of 20th Century Philosophy of Science and Mathematics.Dirk Schlimm - 2006 - In Benedikt Löwe, Volker Peckhaus & T. Rasch (eds.), Foundations of the Formal Sciences IV. College Publications. pp. 233–253.
    This paper is a contribution to the question of how aspects of science have been perceived through history. In particular, I will discuss how the contribution of axiomatics to the development of science and mathematics was viewed in 20th century philosophy of science and philosophy of mathematics. It will turn out that in connection with scientific methodology, in particular regarding its use in the context of discovery, axiomatics has received only very little attention. This is a rather surprising result, since (...)
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  18. Against Against Intuitionism.Dirk Schlimm - 2005 - Synthese 147 (1):171-188.
    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.
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  19.  36
    Metaphors for Mathematics From Pasch to Hilbert.Dirk Schlimm - 2016 - Philosophia Mathematica 24 (3):308-329.
    How mathematicians conceive of the nature of mathematics is reflected in the metaphors they use to talk about it. In this paper I investigate a change in the use of metaphors in the late nineteenth and early twentieth centuries. In particular, I argue that the metaphor of mathematics as a tree was used systematically by Pasch and some of his contemporaries, while that of mathematics as a building was deliberately chosen by Hilbert to reflect a different view of mathematics. By (...)
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  20.  89
    Conceptual Metaphors and Mathematical Practice: On Cognitive Studies of Historical Developments in Mathematics.Dirk Schlimm - 2013 - Topics in Cognitive Science 5 (2):283-298.
    This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive analyses of (...)
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  21.  53
    On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan Have Proved the Jordan-Hölder Theorem?Dirk Schlimm - 2008 - Erkenntnis 68 (3):409-420.
    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 (...)
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  22.  22
    Formal Languages in Logic. A Philosophical and Cognitive Analysis. [REVIEW]Dirk Schlimm - 2014 - History and Philosophy of Logic 35 (1):1-3.
    History and Philosophy of Logic, Volume 35, Issue 1, Page 108-110, February 2014.
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  23.  40
    Mathematical Concepts and Investigative Practice.Dirk Schlimm - 2012 - In Uljana Feest & Friedrich Steinle (eds.), Scientific Concepts and Investigative Practice. De Gruyter. pp. 3--127.
    In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both notions of concepts (...)
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  24.  30
    Richard Zach, Hilbert's 'Verunglückter Beweis', the First Epsilon Theorem, and Consistency Proofs. [REVIEW]Dirk Schlimm - 2005 - Bulletin of Symbolic Logic 11 (2):247-248.
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  25.  41
    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.Brendan P. Larvor, Benedikt Löwe & Dirk Schlimm - 2015 - Synthese 192 (8):2339-2344.
  26. Loss of Vision: How Mathematics Turned Blind While It Learned to See More Clearly.Bernd Buldt & Dirk Schlimm - 2010 - In Benedikt Löwe & Thomas Müller (eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. London: College Publications. pp. 87-106.
    To discuss the developments of mathematics that have to do with the introduction of new objects, we distinguish between ‘Aristotelian’ and ‘non-Aristotelian’ accounts of abstraction and mathematical ‘top-down’ and ‘bottom-up’ approaches. The development of mathematics from the 19th to the 20th century is then characterized as a move from a ‘bottom-up’ to a ‘top-down’ approach. Since the latter also leads to more abstract objects for which the Aristotelian account of abstraction is not well-suited, this development has also lead to a (...)
     
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  27.  72
    A New Look at Analogical Reasoning: Paul F. A. Bartha: By Parallel Reasoning: The Construction and Evaluation of Analogical Arguments. New York: Oxford University Press, 2010, X+354pp, US$74.00 HB.Dirk Schlimm - 2012 - Metascience 21 (1):197-201.
    A new look at analogical reasoning Content Type Journal Article Pages 1-5 DOI 10.1007/s11016-011-9563-z Authors Dirk Schlimm, Department of Philosophy, McGill University, Montreal, QC H3A 2T7, Canada Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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  28.  28
    José Ferreirós. Mathematical Knowledge and the Interplay of Practices. [REVIEW]Dirk Schlimm - 2017 - Philosophia Mathematica 25 (1):139-143.
  29.  28
    History and Philosophy of Logic.Dirk Schlimm - 2005 - Bulletin of Symbolic Logic 11 (2):247-248.
  30. Learning the Structure of Abstract Groups.Dirk Schlimm & Thomas R. Shultz - 2009 - In N. A. Taatgen & H. van Rijn (eds.), Proceedings of the 31st Annual Conference of the Cognitive Science Society. pp. 2100--5.
     
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  31.  31
    Critical Thinking.Dirk Schlimm - 2003 - Teaching Philosophy 26 (3):305-307.
  32.  26
    The Marriott Hotel Philadelphia, Pennsylvania December 27–30, 2008.Janet Folina, Douglas Jesseph, Dirk Schlimm, Emily Grosholz, Kenneth Manders, Sun-Joo Shin, Saul Kripke & William Ewald - 2009 - Bulletin of Symbolic Logic 15 (2).
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  33. Of the Association for Symbolic Logic.Janet Folina, Douglas Jesseph, Dirk Schlimm, Emily Grosholz, Kenneth Manders, Sun-Joo Shin, Saul Kripke & William Ewald - 2009 - Bulletin of Symbolic Logic 15 (2):229.
  34.  12
    Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta.Maria Zack & Dirk Schlimm (eds.) - 2017 - New York: Birkhäuser.
    Proceedings of the Canadian Society for History and Philosophy of Mathematics.
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  35.  7
    Research in History and Philosophy of Mathematics The CSHPM 2017 Annual Meeting in Toronto, Ontario.Maria Zack & Dirk Schlimm (eds.) - 2018 - New York: Birkhäuser.
    This volume contains thirteen papers that were presented at the 2017 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques, which was held at Ryerson University in Toronto. It showcases rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics from Ancient Greece to the twentieth century. -/- A series of chapters all set in the eighteenth century consider topics such as John Marsh’s (...)
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  36.  14
    Research in History and Philosophy of Mathematics: The CSHPM 2018 Volume.Maria Zack & Dirk Schlimm (eds.) - 2020 - New York, USA: Springer Verlag.