|24.||Metaphors for mathematics from Pasch to Hilbert. Philosophia Mathematica (accepted for publication).
How mathematicians conceptualize the nature of mathematics is reflected in the metaphors they use. In this paper I investigate a change in the use of these metaphors in the early development of modern mathematics. 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 attitude towards mathematics. By taking these metaphors seriously, I contend, we can learn something about the different ways in which mathematics was understood in the late 19th and early 20th centuries.
|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.
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.
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, Advance Access. September 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 (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.
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.
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.
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.
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.
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.
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.
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. (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. (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. (online)|
|10.||Pasch's philosophy of mathematics. Review of Symbolic Logic,
3(1): 93–118, March 2010.
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,
Cognitive Science Society, Austin, TX, 2009.
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.
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. (info, preprint)|
|6.||Two ways of analogy: Extending the study of analogies to
mathematical domains. Philosophy of Science,
75(2):178–200. April 2008.
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
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.
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.
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.
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.
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.
– Reprinted in: W. Sieg, Hilbert's Programs and Beyond, pp. 35–72, Oxford University Press, Oxford, 2013.
- 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, amazon.ca)
- 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)
|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)|
- 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.
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
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, Erick 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).