Skip to main content
SpringerLink
Log in
Book cover

International Conference on Theory and Application of Diagrams

Diagrams 2018: Diagrammatic Representation and Inference pp 105–119Cite as

A Typology of Mathematical Diagrams

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10871))

Abstract

In this paper, we develop and discuss a classification scheme that allows us to distinguish between the types of diagrams used in mathematical research based on the cognitive support offered by diagrams. By cognitive support, we refer to the gain that research mathematicians get from using diagrams. This support transcends the specific mathematical topic and diagram type involved and arises from the cognitive strategies mathematicians tend to use. The overall goal of this classification scheme is to facilitate a large-scale quantitative investigation of the norms and values governing the publication style of mathematical research, as well as trends in the kinds of cognitive support used in mathematics. This paper, however, focuses only on the development of the classification scheme.

The classification scheme takes its point of departure from case studies known from the literature, but in this paper, we validate the scheme using examples from a preliminary investigation of developments in the use of diagrams. Building on these results, we discuss the potential and pitfalls in using one generic classification scheme, as done in this analysis. This approach is contrasted with attempts that respect and build on individual diagram types, and as part of this discussion, we report the problems we experienced when using that strategy. The paper ends with a description of possible next steps in using text corpora as an empirical approach to understanding the nature of mathematical diagrams and their relation to mathematical culture.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Johansen, M.W., Misfeldt, M.: An empirical approach to the mathematical values of problem choice and argumentation. In: Larvor, B. (ed.) Mathematical Cultures. THS, pp. 259–269. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-28582-5_15

    Chapter  Google Scholar 

  2. Hersh, R.: Mathematics has a front and a back. Synthese 80(2), 127–133 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Pasch, M., Dehn, M.: Vorlesungen über Neuere Geometrie. Die Grundlehren der Mathematischen Wissenschaften, vol. 23. Springer, Berlin (1926)

    MATH  Google Scholar 

  4. Brown, J.R.: Philosophy of Mathematics, an Introduction to a World of Proofs and Pictures. Philosophical Issues in Science. Routledge, London (1999)

    MATH  Google Scholar 

  5. Giaquinto, M.: Visual Thinking in Mathematics, An Epistemological Study. Oxford University Press, New York (2007)

    Book  MATH  Google Scholar 

  6. Livingston, E.: Cultures of proving. Soc. Stud. Sci. 29(6), 867–888 (1999)

    Article  Google Scholar 

  7. Krömer, R.: Tool and Object, a History and Philosophy of Category Theory. Science Networks Historical Studies, vol. 32. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  8. Carter, J.: Diagrams and proofs in analysis. Int. Stud. Philos. Sci. 24(1), 1–14 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kjeldsen, T.H.: From measuring tool to geometrical object: Minkowski’s development of the concept of convex bodies. Arch. Hist. Exact Sci. 62(1), 59–89 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Johansen, M.W., Misfeldt, M.: Semiotic scaffolding in mathematics. Biosemiotics 8, 325–340 (2015)

    Article  Google Scholar 

  11. Epstein, D., Levy, S., Llave, R.: About this journal. Exp. Math. 1(1), 1–3 (1992)

    Article  Google Scholar 

  12. Hatcher, A.E.: Topology of Numbers. https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf. Accessed 4 Dec 2017

  13. Johansen, M.W.: What’s in a diagram? On the classification of symbols, figures and diagrams. In: Magnani, L. (ed.) Model-Based Reasoning in Science and Technology. Theoretical and Cognitive Issues (SAPERE), vol. 8, pp. 89–108. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-37428-9_6

    Chapter  Google Scholar 

  14. Stjernfelt, F.: Diagrammatology, an Investigation on the Borderlines of Phenomenology, Ontology, and Semiotics. Synthese Library, vol. 336. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-5652-9

    Book  Google Scholar 

  15. Larkin, J.H., Simon, H.A.: Why a diagram is (sometimes) worth ten thousand words. Cogn. Sci. 11(1), 65–100 (1987)

    Article  Google Scholar 

  16. Steensen, A.K., Johansen, M.W.: The role of diagram materiality in mathematics. Cogn. Semiot. 9(2), 183–201 (2016)

    Article  Google Scholar 

  17. De Toffoli, S.: ‘Chasing’ the diagram—the use of visualizations in algebraic reasoning. Rev. Symb. Log. 10(1), 158–186 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Stenning, K.: Distinctions with differences: comparing criteria for distinguishing diagrammatic from sentential systems. In: Anderson, M., Cheng, P., Haarslev, V. (eds.) Diagrams 2000. LNCS (LNAI), vol. 1889, pp. 132–148. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44590-0_15

    Chapter  Google Scholar 

  19. Johansen, M.W., Misfeldt, M.: Material representations in mathematical research practice (in preparation)

    Google Scholar 

  20. Lakoff, G., Núñez, R.: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, New York (2000)

    MATH  Google Scholar 

  21. Candy, A.L.: A general theorem relating to transversals, and its consequences. Ann. Math. 10(1), 175–190 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  22. Serre, J.-P.: Faisceaux algebriques coherents. Ann. Math. 61(2), 197–278 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gehman, H.M.: Concerning the subsets of a plane continuous curve. Ann. Math. 27(1), 29–46 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moritz, R.E.: Some physical solutions of the general equation of the nth degree. Ann. Math. 6(3), 64–78 (1905)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guerra, F., Rosen, L., Simon, B.: The P(\( {\upvarphi } \))2 Euclidean quantum field theory as classical statistical mechanics. Ann. Math. 101(2), 191–259 (1975)

    Google Scholar 

  26. Hatcher, A.E.: Higher simple homotopy theory. Ann. Math. 102(1), 101–137 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sela, Z.: The isomorphism problem for hyperbolic groups I. Ann. Math. 141(2), 217–283 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Morse, M., Heins, M.: Topological methods in the theory of functions of a single complex variable I. Deformation types of locally simple plane curves. Ann. Math. 46(4), 600–624 (1945)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Copenhagen, Copenhagen, Denmark

    Mikkel Willum Johansen & Josefine Lomholt Pallavicini

  2. Aalborg University, Copenhagen, Denmark

    Morten Misfeldt

Authors
  1. Mikkel Willum Johansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Morten Misfeldt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Josefine Lomholt Pallavicini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikkel Willum Johansen .

Editor information

Editors and Affiliations

  1. Edinburgh Napier University, Edinburgh, UK

    Peter Chapman

  2. University of Brighton, Brighton, UK

    Gem Stapleton

  3. Tallinn University of Technology, Tallinn, Estonia

    Amirouche Moktefi

  4. Mitre Corporation, McLean, VA, USA

    Sarah Perez-Kriz

  5. University of Bologna, Bologna, Italy

    Francesco Bellucci

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Johansen, M.W., Misfeldt, M., Pallavicini, J.L. (2018). A Typology of Mathematical Diagrams. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-91376-6_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91375-9

  • Online ISBN: 978-3-319-91376-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation