•  1
    The cognitive basis of arithmetic
    with Helen3 De Cruz and Hansjörg Neth
    In Benedikt Löwe & Thomas Müller (eds.), PhiMSAMP. Philosophy of mathematics: Sociological aspects and mathematical practice, . pp. 59-106. 2010.
    status: published.
  • Basic mathematical cognition
    with David Gaber
    WIREs Cognitive Science 4 (6): 355-369. 2015.
    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 …Read more
  •  59
    The cultural challenge in mathematical cognition
    with Andrea Bender, Stephen Crisomalis, Fiona M. Jordan, Karenleigh A. Overmann, and Geoffrey B. Saxe
    Journal of Numerical Cognition 2 (4). 2018.
    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 –…Read more
  •  9
    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 r…Read more
  •  129
    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…Read more
  •  10
    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 …Read more
  •  14
    Proceedings of the Canadian Society for History and Philosophy of Mathematics.
  •  35
    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 idiosy…Read more
  •  74
    Axioms in Mathematical Practice
    Philosophia Mathematica 21 (1): 37-92. 2013.
    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 clar…Read more
  •  41
    The Cognitive Advantages of Counting Specifically: A Representational Analysis of Verbal Numeration Systems in Oceanic Languages
    with Andrea Bender and Sieghard Beller
    Topics in Cognitive Science 7 (4): 552-569. 2015.
    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 represent…Read more
  •  33
    Methodological Reflections on Typologies for Numerical Notations
    with Theodore Reed Widom
    Science in Context 25 (2): 155-195. 2012.
    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…Read more
  •  36
    Metaphors for Mathematics from Pasch to Hilbert
    Philosophia Mathematica 24 (3): 308-329. 2016.
    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 taking these metap…Read more
  •  22
    Formal Languages in Logic. A Philosophical and Cognitive Analysis (review)
    History and Philosophy of Logic 35 (1): 1-3. 2014.
    History and Philosophy of Logic, Volume 35, Issue 1, Page 108-110, February 2014
  •  71
    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 …Read more
  •  72
    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
  •  1
    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-suite…Read more
  •  53
    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…Read more
  •  28
    History and Philosophy of Logic
    Bulletin of Symbolic Logic 11 (2): 247-248. 2005.
  •  112
    Against Against Intuitionism
    Synthese 147 (1): 171-188. 2005.
    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.
  •  60
    Dedekind's Abstract Concepts: Models and Mappings
    with Wilfried Sieg
    Philosophia Mathematica. 2014.
    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.
  •  78
    Modeling ancient and modern arithmetic practices: Addition and multiplication with Arabic and Roman numerals
    with Hansjörg Neth
    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. 2008.
    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 s…Read more
  •  89
    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 c…Read more
  • Of the association for symbolic logic
    with Janet Folina, Douglas Jesseph, Emily Grosholz, Kenneth Manders, Sun-Joo Shin, Saul Kripke, and William Ewald
    Bulletin of Symbolic Logic 15 (2): 229. 2009.
  •  95
    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…Read more
  •  42
    Axiomatics and progress in the light of 20th century philosophy of science and mathematics
    In Benedikt Löwe, Volker Peckhaus & T. Rasch (eds.), Foundations of the Formal Sciences IV, College Publications. 2006.
    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 surpris…Read more