Articles (refereed)

43. Who's afraid of mathematical platonism? An historical perspective. In: K. Chemla, J. Ferreirós, L. Li, E. Scholz, C. Wang (eds.), The Richness of the History of Mathematics, pp. 595–615, Springer, Cham, 2024. (Published online, December 2023.) Platonism Cantor Frege (online)

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.

42. Mathematical experiments on paper and computer (with Juan Fernández González). In: B. Sriraman, (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer, New York, 2024. (Published online, November 2023.) Discovery (online)

We propose a characterization of mathematical experiments in terms of a setup, a process with an outcome, and an interpretation. Using a broad notion of process, this allows us to consider arithmetic calculations and geometric constructions as components of mathematical experiments. Moreover, we argue that mathematical experiments should be considered within a broader context of an experimental research project. Finally, we present a particular case study of the genesis of a geometric construction to illustrate the experimental use of hand drawings and computers in mathematical practice.

41. From a doodle to a theorem: A case study in mathematical discovery (with Juan Fernández González). Journal of Humanistic Mathematics, 13(1):4–35, January 2023. Discovery (online)

We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical practice.

40. Tables as powerful representational tools. In: V. Giardino, S. Linker, R. Burns, F. Bellucci, J.-M. Boucheix, and P. Viana (eds.), Diagrammatic Representation and Inference. 13th International Conference, Diagrams 2022, Rome, Italy, September 14–16, 2022, Proceedings, pp. 185–201, Springer, New York, 2022. Cognition Notation (online)

Tables are widely used for storing, retrieving, communicating, and processing information, but in the literature on the study of representations they are still somewhat neglected. The strong structural constraints on tables allow for a clear identification of their characteristic features and the roles these play in the use of tables as representational and cognitive tools. After introducing syntactic, spatial, and semantic features of tables, we give an account of how these affect our perception and cognition on the basis of fundamental principles of Gestalt psychology. Next are discussed the ways in which these features of tables support their uses in providing a global access to information, retrieving information, and visualizing relational structure and patterns. The latter is particularly important, because it shows how tables can contribute to the generation of new knowledge. In addition, tables also provide efficient means for manipulating information in general and in structured notations. In sum, tables are powerful and efficient representational tools.

39. How can numerals be iconic? More varieties of iconicity. In: A. Basu, G. Stapleton, S. Linker, C. Legg, E. Manalo, and P. Viana (eds.), Diagrammatic Representation and Inference. 12th International Conference, Diagrams 2021, Virtual, September 28–30, 2021, Proceedings, pp. 520–528, Springer, New York, 2021. Notation Numerals (online)

The standard notion of iconicity, which is based on degrees of similarity or resemblance, does not provide a satisfactory account of the iconic character of some representations of abstract entities when those entities do not exhibit any imitable internal structure. Individual numbers are paradigmatic examples of such structureless entities. Nevertheless, numerals are frequently described as iconic or symbolic; for example, we say that the number three is represented symbolically by '3', but iconically by '|||'. To address this difficulty, I discuss various alternative notions of iconicity that have been presented in the literature, and I propose two novel accounts.

38. Calculus as method or calculus as rules? Boole and Frege on the aims of a logical calculus (with David Waszek). Synthese, 199:11913–11943, 2021. Notation Boole Frege (view only) (online)

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 goaldirected 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.

37. Babbage's guidelines for the design of mathematical notations (with Jonah Dutz). Studies in History and Philosophy of Science, 88:92–101, 2021. Notation Babbage (accepted manuscript) (online)

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.

36. Peano on symbolization, design principles for notations, and the dot notation. Philosophia Scientiæ, 25(1):95–126, 2021. Notation Peano (online)

Peano was one of the driving forces behind the development of the current mathematical formalism. In this paper, we study his particular approach to notational design and present some original features of his notations. To explain the motivations underlying Peano's approach, we first present his view of logic as a method of analysis and his desire for a rigorous and concise symbolism to represent mathematical ideas. On the basis of both his practice and his explicit reflections on notations, we discuss the principles that guided Peano's introduction of new symbols, the choice of characters, and the layout of formulas. Finally, we take a closer look, from a systematic and historical perspective, at one of Peano's most striking innovations, his use of dots for the grouping of subformulas.

35. Dedekind et la crèation du continu arithmétique (with Emmylou Haffner). In: E. Haffner and D. Rabouin (eds.), L'Épistemologie du dedans. Mélanges en l'honneur de Hourya Benis-Sinaceur, pp. 341–378. Classiques Garnier, Paris, 2021. Dedekind (online)

Cet article étudie la conception de la continuité dans les travaux de Richard Dedekind et les méthodes qu'il a adoptées pour définir des domaines continus. Il analyse une sélection de textes (publiés par Dedekind et extraits de son Nachlass) traitant de domaines continus, et montre que l'approche de Dedekind est profondément arithmétique. Il suggère que le point de vue ensembliste est, dans les travaux de Dedekind, subordonné au point de vue arithmétique.

34. Dedekind on continuity (with Emmylou Haffner). In: S. Shapiro and G. Hellman (eds.), The History of Continua. Philosophical and mathematical perspectives, pp. 255–282, Oxford University Press, New York, 2021. Dedekind (online)

In this chapter, we will provide an overview of Richard Dedekind's work on continuity, both foundational and mathematical. His seminal contribution to the foundations of analysis is the well-known 1872 booklet Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), which is based on Dedekind's insight into the essence of continuity that he arrived at in the fall of 1858. After analysing the intuitive understanding of the continuity of the geometric line, Dedekind characterized the property of continuity for the real numbers in terms of what are nowadays called 'Dedekind cuts' on the rational numbers. This treatment, which can be characterized as being 'arithmetical' as well as 'axiomatic', will be presented in detail. To better position Dedekind's contributions in their historical context, we will also consider some of his more mathematical treatments of continuity in addition to his foundational work. Of particular interest is the definition of the Riemann surface in his joint work with Heinrich Weber (1882). Moreover, Dedekind's reflections on space and continuity in his unpublished papers 'Allgemeine Sätze über Räume' (General theorems about spaces; before 1870) and 'Beweis und Anwendungen eines allgemeinen Satzes über mehrfach ausgedehnte stetige Gebiete' (Proof and applications of a general theorem about multiply extended continuous domains; 1892) illustrate the wide range and general coherency of his thoughts. By discussing Dedekind's works in which the notion of continuity plays a central role, we will show how Dedekind's approaches became increasingly abstract, while at the same time retaining a common methodology.

33. Multiple readability in principle and practice: Existential Graphs and complex symbols (with David Waszek). Logique & Analyse, 251:231–260, 2020. Notation Peirce (publisher) (McGill eScholarship)

Since Sun-Joo Shin's groundbreaking study (2002), Peirce's existential graphs have attracted much attention as a way of writing logic that seems profoundly different from our usual logical calculi. In particular, Shin argued that existential graphs enjoy a distinctive property that marks them out as "diagrammatic": they are "multiply readable," in the sense that there are several di erent, equally legitimate ways to translate one and the same graph into a standard logical language. Stenning (2000) and Bellucci and Pietarinen (2016) have retorted that similar phenomena of multiple readability can arise for sentential notations as well. Focusing on the simplest kinds of existential graphs, called alpha graphs (AGs), this paper argues that multiple readability does point to important features of AGs, but that both Shin and her critics have misdiagnosed its source. As a preliminary, and because the existing literature often glosses over such issues, we show that despite their non-linearity, AGs are uniquely parsable and allow for inductive de nitions. Extending earlier discussions, we then show that that in principle, all propositional calculi are multiply readable, just like AGs: contrary to what has been suggested in the literature, multiple readability is linked neither to non-linearity nor to AGs' dearth of connectives. However, we argue that in practice, AGs are more amenable to multiple readability than our usual notations, because the patterns that one needs to recognize to multiply translate an AG form what we call complex symbols, whose structural properties make it easy to perceive and process them as units. Nevertheless, we show that such complex symbols, though largely absent from our usual notations, are not inherently diagrammatic and can be found in seemingly sentential languages. Hence, while ultimately vindicating Shin's idea of multiple readability, our analysis traces it to a di erent source and thus severs its link with diagrammaticity.

32. Iconicity in mathematical notation: commutativity and symmetry (with Theresa Wege, Sophie Batchelor, Matthew Inglis, and Honali Mistry). Journal of Numerical Cognition, 6(3):378–392, 2020. Cognition Notation (online)

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd's hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance.

31. Pasch's empiricism as methodological structuralism. In: E. Reck and G. Schiemer (eds.), The Pre-History of Mathematical Structuralism, pp. 80–105. Oxford University Press, New York, 2020. Pasch (online)
30. Über Freges Begriffsschrift für Aussagenlogik: Gestaltungsprinzipien, Vor- und Nachteile. In: M. Wille (ed.), Fregesche Variationen. Essays zu Ehren von Christian Thiel, pp. 129–167. Mentis, Paderborn, 2019. Notation Frege (online)
29. Extended mathematical cognition: External representations with non-derived content (with Karina Vold). Synthese, 197(9): 3757-3777, August 2020. (Published online, January 2019.) Cognition Notation (online)

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.

28. The cultural challenge in mathematical cognition (with Sieghard Beller, Andrea Bender, Stephen Chrisomalis, Fiona M. Jordan, Karenleigh A. Overmann, Geoffrey B. Saxe). Journal of Numerical Cognition, 4(2): 448–463, 2018. Cognition (online) 

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 argue that for any research agenda on mathematical cognition the cultural dimension is indispensable, and we propose a set of exemplary research questions related to it.

27. Numbers through numerals. The constitutive role of external representations. In: S. Bangu (ed.), Naturalizing Logico-Mathematical Knowledge: Approaches from Psychology and Cognitive Science, pp. 195–217. Routledge, New York, February 2018. Cognition Numerals (online) 

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.

26. On Frege's Begriffsschrift notation for propositional logic: Design principles and trade-offs. History and Philosophy of Logic, 39(1): 53–79, January 2018. (Published online, May 2017.) Notation Frege (online) 

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.

25. On Dedekind's axiomatic approach to the foundations of mathematics. In: K. Scheel, T. Sonar, and P. Ullrich (eds.), In Memoriam Richard Dedekind (1831–1916), pp. 75–82. WTM Verlag für wissenschaftliche Texte und Medien, Münster. 2017. Axiomatics Dedekind (prefinal draft)

This paper brings together some points made previously in (Sieg and Schlimm 2005) and (Sieg and Schlimm 2014) focusing on some aspects of Dedekind's axiomatic approach to the foundations of mathematics. In particular the terminology for his axiomatic definitions of concepts and the development of the notion of mapping are discussed.

24. Metaphors for mathematics from Pasch to Hilbert. Philosophia Mathematica, 24(3): 308–329. October 2016. Metaphors Pasch (accepted manuscript) (online)

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 metaphors seriously we attain a new vantage point for understanding historical changes in conceptions of mathematics.

23. 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, October 2015. Cognition Numerals (online)

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 counting sequences-general and object-specific ones-that were in parallel use in several Oceanic languages (English with its general system is included for comparison). The analysis reveals that the object-specific systems outperform the general systems with respect to counting and mental arithmetic, largely due to their regular and more compact representation. What these findings reveal on cognitive diversity, how the conjectures involved speak to more general issues in cognitive science, and how the approach taken here might help to bridge the gap between anthropology and other cognitive sciences is discussed in the conclusion.

22. Basic mathematical cognition (with David Gaber). WIREs Cognitive Science, 6(4):355–369, July/August 2015. Cognition (online)

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 recognizing landmarks and navigating environments. Correspondingly, cognitive systems that are dedicated to the processing of distinctly mathematical information have developed. In particular, there is evidence that certain core systems for understanding different aspects of arithmetic as well as geometry are employed by humans and many other animals. They become active early in life and, particularly in the case of humans, develop through maturation. While these core systems individually seem to be quite limited in application, in combination they allow for the recognition of mathematical properties and the formation of appropriate inferences based upon those properties. In this overview, the core systems, their roles, their limitations, and their interaction with external representations are discussed, as well as possibilities for how they can be employed together to allow us to reason about more complex mathematical domains.

21. Dedekind's abstract concepts: models and mappings (with Wilfried Sieg). Philosophia Mathematica, 25(3):292–317, October 2017. (Advance Access online: September 2014.) Dedekind (online)

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 (for systems of mathematical objects), models (systems satisfying such concepts), and mappings (connecting models in a structure-preserving way).

20. A subjective comparison between a historical and a contemporary textbook on geometry (with Douglas Roland Campbell). Journal of Humanistic Mathematics, 4(2):58–66, July 2014. Pasch (online)

In order to investigate how a 19th century mathematical textbook (in contrast to a contemporary one) would be experienced by a novice reader, we embarked on the following project: In the summer of 2013, a student with no previous training in college-level mathematics (the first author) set out to learn projective geometry from Pasch's 1882 textbook Lectures on Modern Geometry. Afterwards, he studied the same material from Coxeter's 1994 popular undergraduate textbook Projective Geometry. We report here some of his experiences and impressions contextualizing them along the way.

19. The correspondence between Moritz Pasch and Felix Klein. Historia Mathematica, 40(2):183–202, May 2013. Pasch (online)

The extant correspondence, consisting of ten letters from the period from 1882 to 1902, from Moritz Pasch to Felix Klein is presented together with an English translation and a short introduction. These letters provide insights into the views of Pasch and Klein regarding the role of intuition and axioms in mathematics, and also into the hiring practices of mathematics professors in the 1880s.

– Selected as one of 16 articles from Historia Mathematica that were reprinted in the 40th year anniversary issue Four decades of excellence in the history of mathematics.
18. Conceptual metaphors and mathematical practice: On cognitive studies of historical developments in mathematics. Topics in Cognitive Science, 5(2): 283–298, April 2013. Cognition (online)  

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 historical developments of mathematics has been primarily on the former, even if they claim to be about the latter.

17. Axioms in mathematical practice. Philosophia Mathematica, 21(1):37–92, February 2013. Axiomatics (accepted manuscript) (online)

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.

16. Mathematical concepts and investigative practice. In: U. Feest and F. Steinle (eds.), Scientific Concepts and Investigative Practice, pp. 127–147. De Gruyter, Berlin, 2012. Concepts Discovery (prefinal draft)

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 capture important aspects of mathematical and scientific reasoning, and, consequently, that a pluralistic approach that allows for representing both of these aspects is most useful for an adequate account of investigative practices.

15. Methodological reflections on typologies for numerical notations (with Theodore R. Widom). Science in Context, 25(2): 155–195, June 2012. Numerals (online)

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.

14. On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others. Synthese, 183(1): 47–68, November 2011. Dedekind Axiomatics (online)

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.

13. Learning and understanding numeral systems: Semantic aspects of number representations from an educational perspective (with Katja Lengnink). In: B. Löwe and T. Müller (eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice, pp. 235–264. College Publications, London, 2010. Numerals (online)
12. Loss of vision: How mathematics turned blind while it learned to see more clearly (with Bernd Buldt). In: B. Löwe and T. Müller (eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice, pp. 87–106. College Publications, London, 2010. Concepts (online)
11. The cognitive basis of arithmetic (with Helen de Cruz and Hansjörg Neth). In: B. Löwe and T. Müller (eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice, pp. 39–86. College Publications, London, 2010. Cognition (online)
10. Pasch's philosophy of mathematics. Review of Symbolic Logic, 3(1): 93–118, March 2010. Pasch (online)

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.

9. Learning the structure of abstract groups (with Thomas R. Shultz). In: N.A. Taatgen & H. van Rijn (eds.), Proceedings of the 31th Annual Conference of the Cognitive Science Society, pp. 2950–2955. Cognitive Science Society, Austin, TX, 2009. Cognition (online)

It has recently been shown that neural networks can learn particular mathematical groups, for example, the Klein 4-group (Jamrozik & Shultz, 2007). However, there are groups with any number of elements, all of which are said to instantiate the abstract group structure. Learning to differentiate groups from other structures that are not groups is a very difficult task. Contrary to some views, we show that neural networks can learn to recognize finite groups consisting of up to 4 elements. We present this problem as a case study that exhibits the advantages of knowledge-based learning over knowledge-free learning. In addition, we also show the surprising result that the way in which the KBCC algorithm recruits previous knowledge reflects some deep structural properties of the patterns that are learned, namely, the structure of the subgroups of a given group.

8. Learning from the existence of models. On psychic machines, tortoises, and computer simulations. Synthese, 169(3):521–538, August 2009. Discovery (online).

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.

7. Bridging theories with axioms: Boole, Stone, and Tarski. In: B. van Kerkhove (ed.), New Perspectives on Mathematical Practices, pp. 222–235. World Scientific, 2009. Axiomatics Discovery (info, preprint)
6. Two ways of analogy: Extending the study of analogies to mathematical domains. Philosophy of Science, 75(2):178–200. April 2008. Analogy (online)

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, for example, for the discovery of new analogies.

5. Modeling ancient and modern arithmetic practices: Addition and multiplication with Arabic and Roman numerals (with Hansjörg Neth). In: V. Sloutsky, B. Love, and K. McRae (eds.), Proceedings of the 30th Annual Meeting of the Cognitive Science Society, pp. 2007–2012. Cognitive Science Society, Austin, TX, 2008. Numerals Cognition (online)

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.

4. On abstraction and the importance of asking the right research questions: Could Jordan have proved the Jordan-Hölder Theorem? Erkenntnis, 68(3):409–420, May 2008. Discovery (online)

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.

3. Axiomatics and progress in the light of 20th century philosophy of science and mathematics. In: B. Löwe, V. Peckhaus, and T. Rasch (eds.), Foundations of the Formal Sciences IV, pp. 233–253. Studies in Logic Series, College Publications, London, 2006. Axiomatics Discovery (preprint)

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 axiomatizations have been employed extensively in mathematics, science, and also by the philosophers themselves.

2. Against against Intuitionism. Synthese 147(1):171–188, October 2005. Brouwer (online)

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.

1. Dedekind's analysis of number: systems and axioms (with Wilfried Sieg). Synthese 147(1):121–170, October 2005. Axiomatics Dedekind (online)
– Reprinted in: W. Sieg, Hilbert's Programs and Beyond, pp. 35–72, Oxford University Press, Oxford, 2013.

Edited books

  • Research in History and Philosophy of Mathematics: The CSHPM 2019–2020 Volume. Edited by M. Zack and D. Schlimm. Annals of the Canadian Society for History and Philosophy of Mathematics. Birkhäuser, New York, 2022. (springer)
  • Research in History and Philosophy of Mathematics: The CSHPM 2018 Volume. Edited by M. Zack and D. Schlimm. Proceedings of the Canadian Society for History and Philosophy of Mathematics. Birkhäuser, New York, 2020. (springer)
  • Research in History and Philosophy of Mathematics: The CSHPM 2017 Annual Meeting in Toronto, Ontario. Edited by M. Zack and D. Schlimm. Proceedings of the Canadian Society for History and Philosophy of Mathematics. Birkhäuser, New York, 2018. (springer)
  • Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Edited by M. Zack and D. Schlimm. Proceedings of the Canadian Society for History and Philosophy of Mathematics. Birkhäuser, New York, 2017. (springer)
Assistant editor:
  • The Collected Works of Rudolf Carnap, vol. 1, Early Writings. R. Creath, general editor. Edited by A. W. Carus, M. Friedman, W. Kienzler, A. Richardson, S. Schlotter, with editorial assistance from S. Awodey, D. Schlimm, R. Zach. Oxford University Press, London, 2019.
  • David Hilbert's Lectures on the Foundations of Logic and Arithmetic, 1917–1933. Edited by W. Ewald and W. Sieg, in association with M. Hallett, in collaboration with U. Majer and D. Schlimm. Vol. 3 of Hilbert's Lectures on the Foundations of Mathematics and Physics, Springer, Berlin, 2013. (springer,

Edited journal issue

  • History and Philosophy of Infinity, special section of Synthese, 192(8): 2339–2511. Edited by B. Larvor, B. Löwe, and D. Schlimm. August, 2015. (online)


  8. Book review of José Ferreirós, "Mathematical Knowledge and the Interplay of Practices," Philosophia Mathematica, 25(1):139–143, February 2017. (online)
  7. Book review of Catarina Dutilh Novaes, "Formal Languages in Logic. A Philosophical and Cognitive Analysis," History and Philosophy of Logic, 35(1):108–110, 2014. (online)
  6. A new look at analogical reasoning. (Book review of Paul F. A. Bartha, "By Parallel Reasoning. The construction and evaluation of analogical arguments"), Metascience, 21(1):197–201, March 2012. (online)
  5. Book review of Torkel Franzén, "Gödel's Theorem," Review of Modern Logic, 10(3/4):257–261, March 2005–May 2007. (online)
4. Book review of Volker Peckhaus (ed.), "Oskar Becker und die Philosophie der Mathematik," History and Philosophy of Logic, 27(2):198–200, May 2006. (online)
3. Review of Richard Zach, "Hilbert's 'Verunglückter Beweis' the first epsilon theorem, and consistency proofs," Bulletin of Symbolic Logic, 11(2):247–248, June 2005. (online)
2. Book review of Marcus Giaquinto, "The Search for Certainty," Review of Modern Logic, 10(1/2):187–190, September 2004–February 2005. (online)
1. Book review of Kevin Possin, "Critical Thinking," Teaching Philosophy, 26(3):305–307, September 2003. (online)

Published abstracts

  • How to find axioms for finite domains: A computational exploration of mathematical discovery, with Gordon Krieger. In: A. K. Goel, C. M. Seifert, and C. Freksa (eds.), Proceedings of the 41st Annual Conference of the Cognitive Science Society, p. 3299, Cognitive Science Society, Montreal, QC, 2019.
  • General principles for the design of logical notations, abstract of invited talk given June 23, 2018, at UNILOG 2018, Vichy, France. In: J.-Y. Beziau, A. Buchsbaum, and C. Rey (eds.), Handbook of the 6th World Congress and School on Universal Logic, pp. 158–159, 2018.
  • On making mathematical inferences explicit: Pasch's thoughts on logic, abstract of invited talk given January 11, 2012, at the Mathematisches Forschungsinstitut Oberwolfach, Germany. In: T. Archibald, J. Pfeiffer, and N. Schappacher (eds.), Explicit versus Tacit Knowledge in Mathematics, pp. 39–41, Report No. 4/2012, 2012.
  • Empirico-deductivism: Moritz Pasch's philosophy of mathematics, abstract of invited talk given December 27, 2008, at the Winter meeting of the Association of Symbolic Logic (ASL), Philadelphia, PA. Bulletin of Symbolic Logic, 15(2):233, June 2009.
  • Moritz Pasch: A nineteenth century logical empiricist, abstract of invited talk given March 28, 2008, at the Annual meeting of the Association of Symbolic Logic (ASL), Irvine, CA. Bulletin of Symbolic Logic, 14(3):426, September 2008.

Articles and technical reports (not refereed)

  • Symbols for nothing: Different symbolic roles of zero and their gradual emergence in Mesopotamia (with Katherine Skosnik). In: A. Cupillari (ed.), Proceedings of the 2010 Meeting of the Canadian Society for History and Philosophy of Mathematics, Montreal, 29–31 May 2010, vol. 23, pp. 257–266, 2011. (online)

    Zero plays a number of different roles in our decimal place-value system. To allow for a nuanced discussion of the importance of zero, these roles should be distinguished carefully. We present such a differentiation of symbolic roles of zero and illustrate them by looking at the use ofsymbols for zero in ancient Mesopotamia. Old and Late Babylonian mathematicians used a place-value system (like ours, but with base sixty instead of ten), but did not use zeros in the way we do now. This shows that our current uses of zero are not a necessary consequence of the adoption of a place-value system and that the lack of a zero does not necessarily render a place-value system unusable.

  • Dedekind's analysis of number (Part I) – systems and axioms (with Wilfried Sieg). Technical Report CMU-PHIL-139, March 18 2003.
  • Towards axiomatic foundations of mathematics: The evolution of Richard Dedekind's treatment of numbers. In: D. Curtin, D. Kullmann, D. Otero (eds.), Proceedings of the Eighth Midwest History of Mathematics Conference, Northern Kentucky University, October 13–14, 2000.


  • Routes of Writing project, The McGill Writing Centre. July 2014. (online)
  • Introduction, Ampersand, Journal of the Bachelor in Arts and Science, McGill University. Vol. 2: viii–ix, December 2009. (online)
  • Two metaphors for teaching, Graduate Times, Carnegie Mellon Graduate Student Newsletter. Vol. V, No. 4, p. 2, Summer 2003. (online)


  • Axiomatics as Engine for Driving Discovery in Mathematics and Science,
    Ph.D. thesis in Logic, Computation, and Methodology, Department of Philosophy, Carnegie Mellon University, Pittsburgh, May 2005.
    Thesis committee: Clark Glymour (chair), Jeremy Avigad, John Earman (U Pittsburgh), Richard Scheines.
  • Richard Dedekind: Axiomatic Foundations of Mathematics,
    Master's thesis in Logic and Computation, Department of Philosophy, Carnegie Mellon University, Pittsburgh, May 2000.
    Thesis committee: Wilfried Sieg (chair), Steve Awodey, Erich Reck (UC Riverside).
  • Intuitionism and Computer Science. A historical and philosophical investigation of the logical foundations of computer science with special attention to L.E.J. Brouwer's Intuitionism,
    Diploma thesis, Department of Computer Science, Technical University Darmstadt, January 1997 (in German).
    Thesis committee: Christoph Kreitz (Computer Science, TU Darmstadt), Barbara Brüning (Philosophy, Frankfurt).