Skip to main content
SpringerLink
Log in

Entering the valley of formalism: trends and changes in mathematicians’ publication practice—1885 to 2015

  • Original Research
  • Published:
Synthese Aims and scope Submit manuscript

Abstract

Over the last century, there have been considerable variations in the frequency of use and types of diagrams used in mathematical publications. In order to track these changes, we developed a method enabling large-scale quantitative analysis of mathematical publications to investigate the number and types of diagrams published in three leading mathematical journals in the period from 1885 to 2015. The results show that diagrams were relatively common at the beginning of the period under investigation. However, beginning in 1910, they were almost completely unused for about four decades before reappearing in the 1950s. The diagrams from the 1950s, however, were of a different type than those used earlier in the century. We see this change in publication practice as a clear indication that the formalist ideology has influenced mathematicians’ choice of representations. Although this could be seen as a minor stylistic aspect of mathematics, we argue that mathematicians’ representational practice is deeply connected to their cognitive practice and to the contentual development of the discipline. These changes in publication style therefore indicate more fundamental changes in the period under consideration.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. After—and in part building on—the empirical part of this study machine learning tools for detecting, but not classifying diagrams have been build. For reports on their performance see (Sørensen & Johansen, 2020; Sørensen, 2021).

  2. In Johansen et al. (2018) we named this category of diagrams “Cartesian diagrams”, but we have chosen to change the name to stress that it is a general category and not a particular type of diagrams.

  3. The ‘in doubt’ code is included in the raw data set available at https://doi.org/10.17894/ucph.6e18e1c7-7eef-4445-8d9c-3b4d38949079. The reader can thus see which concrete papers we were in doubt about and how we chose to code them.

  4. Although there are exceptions. For instance, we consider knot diagrams to be resemblance diagrams due to the clear topological resemblance with the objects they model, but it can be claimed that such diagrams can be used for rigorous, semi-syntactic manipulations (e.g. (Toffoli & Giardino, 2014)).

  5. We are indebted to one of the anonymous reviewers for pointing out the connection between representations and known results and proof tecniques.

References

  • Barany, M. J., & MacKenzie, D. (2014). Chalk: Materials and concepts in mathematics research. In C. Coopman, J. Vertesi, M. Lynch, & S. Woolgar (Eds.), Representation in scientific practice revisited, Inside technology (pp. 107–129). The MIT Press.

  • Bishop, C. J. (2015). Constructing entire functions by quasiconformal folding. Acta Mathematica, 214(1), 1–60.

    Article  Google Scholar 

  • Buch, A. S. (2015). Mutations of puzzles and equivariant cohomology of two-step flag varieties. Annals of Mathematics, 182(1), 173–220.

    Article  Google Scholar 

  • Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1), 1–14.

    Article  Google Scholar 

  • Cruz, H. D., & Smedt, J. D. (2013). Mathematical symbols as epistemic actions. Synthese, 190(1), 3–19.

    Article  Google Scholar 

  • De Toffoli, S. (2017). ‘Chasing’ the diagram—The use of visualizations in algebraic reasoning. The review of symbolic logic, 10(1), 158–186.

    Article  Google Scholar 

  • Dolgushev, V. A., Rogers, C. L., & Willwacher, T. H. (2015). Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields. Annals of Mathematics, 182(3), 855–943.

    Article  Google Scholar 

  • Drasin, D., & Pankka, P. (2015). Sharpness of Rickman’s Picard theorem in all dimensions. Acta Mathematica, 214(2), 209–306.

    Article  Google Scholar 

  • Giaquinto, M. M. (2007). Visual thinking in mathematics an epistemological study. Clarendon.

  • Giaquinto, M. (2020). The epistemology of visual thinking in mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Metaphysics Research Lab, Stanford University.

  • Giardino, V. (2017). Diagrammatic reasoning in mathematics. In L. Magnani & T. Bertolotti (Eds.), Springer handbook of model-based science (2017th ed., pp. 499–522). Springer Handbooks, Springer International Publishing.

  • Hersh, R. (1991). Mathematics has a front and a back. Synthese, 88(2), 127–133.

    Article  Google Scholar 

  • Hilbert, D. (1902). Mathematical problems. Bulletin (new series) of the American Mathematical Society, 8(10), 437–479.

    Article  Google Scholar 

  • Hilbert, D. (1902/2004). Grundlagen der geometrie. In: David Hilbert’s lectures on the foundations of geometry, 1891–1902, chapter 6. Springer, Berlin. Republished 2004.

  • Hilbert, D. (1925/1983). On the infinite. In Benacerraf, P., & Putnam, H., (Eds.), Philosophy of mathematics: Selected readings, (pp. 183–201). Cambridge University Press.

  • Hoffman, D., & Meeks, W. (1990). Embedded minimal surfaces of finite topology. Annals of Mathematics, 131(1), 1–34.

    Article  Google Scholar 

  • Inglis, M., & Aberdein, A. (2015). Beauty is not simplicity: An analysis of mathematicians’ proof appraisals. Philosophia Mathematica, 23(1), 87–109.

    Article  Google Scholar 

  • Johansen, M. (2014). What’s in a diagram? On the classification of symbols, figures and diagrams. In L. Magnani (Ed.), Model-based reasoning in science and technology. Theoretical and cognitive issues, sapere (pp. 89–108). Springer.

  • Johansen, M., & Misfeldt, M. (2016). An empirical approach to the mathematical values of problem choice and argumentation. In B. Larvor (Ed.), Mathematical cultures: The London meetings 2012–2014 (pp. 259–269). Switzerland: Springer.

  • Johansen, M. W., & Misfeldt, M. (2018). Material representations in mathematical research practice. Synthese, 197(9), 3721–3741.

    Article  Google Scholar 

  • Johansen, M., Misfeldt, M., & Pallavicini, J. (2018). A typology of mathematical diagrams. In Stapleton, G., Bellucci, F., Moktefi, A., Chapman, P., & Perez-Kriz, S., (Eds.), Diagrammatic Representation and Inference - 10th International Conference, Diagrams 2018, Proceedings, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), (pp. 105–119). Springer. 10th International Conference, Diagrams, 2018, Edinburgh.

  • Jungerman, M., & Ringel, G. (1980). Minimal triangulations on orientable surfaces. Acta Mathematica, 145, 121–154.

    Article  Google Scholar 

  • Karoubi, M. (1980). Theorie de quillen et homologie du groupe orthogonal. Annals of Mathematics, 112(1), 207–257.

    Article  Google Scholar 

  • Kjelsen, T. (2009). Egg-forms and measure-bodies: Different mathematical practices in the early history of the modern theory of convexity. Science in Context, 22(1), 85–113.

    Article  Google Scholar 

  • Knuth, D. E. (1979). Mathematical typography. Bulletin (new series) of the American Mathematical Society, 1(2), 337–372.

    Article  Google Scholar 

  • Krömer, R. (2007). Tool and object: A history and philosophy of category theory, Science networks: Historical studies. Birkhäuser Verlag.

  • Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11(1), 65–100.

    Article  Google Scholar 

  • Mancosu, P. (2005). Visualization in logic and mathematics. In P. Mancosu, S. A. Pedersen, & K. F. Jørgensen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 13–30). Springer.

  • Mandelbaum, R. (1980). Four-dimensional topology: An introduction. Bulletin (new Series) of the American Mathematical Society, 2(1), 1–160.

    Article  Google Scholar 

  • Manders, K. (2008). The euclidean diagram (1995). In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 80–133). Oxford University Press.

  • Morgan, F. (1990). A sharp counterexample on the regularity of \(\phi \)-minimizing hypersurfaces. Bulletin of the American Mathematical Society, 22(2), 297.

    Article  Google Scholar 

  • Mumma, J., & Panza, M. (2012). Diagrams in mathematics: History and philosophy [special issue]. Synthese, 186(1), 1.

    Article  Google Scholar 

  • Pasch, M., & Dehn, M. (1926/1882). Vorlesungen über neuere geometrie. Die Grundlehren der mathematischen Wissenschaften, 23.

  • Payne, S. (2015). Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry. Bulletin (new Series) of the American Mathematical Society, 22(2), 223–247.

    Article  Google Scholar 

  • Russell, B. (1917/1901). Mathematics and the metaphysicians. In Mysticism and Logic and other Essays, (pp. 74–96). George Allen and Unwin, London. (First published as “Recent Work on the Principles of Mathematics” in International Monthly, 4: 83–101).

  • Sattinger, D. (1980). Bifurcation and symmetry breaking in applied mathematics. Bulletin of the American Mathematical Society, 3(2), 779–819.

    Article  Google Scholar 

  • ScienceWatch. (2008). Sci-bytes—What’s new in research. Internet resource. Retrieved from http://archive.sciencewatch.com/dr/sci/08/jan20-08_22/.

  • Smith, I. (2015). A symplectic prolegomenon. Bulletin (new Series) of the American Mathematical Society, 52(3), 415–464.

    Article  Google Scholar 

  • Sørensen, H. (2021). Beyond counting: Measuring diagram intensity in mathematical research papers. In: Basu, A., Stapleton, G., Linker, S., Legg, C., Manalo, E., & Viana, P., (Eds.), Diagrammatic Representation and Inference, volume 12909 of Lecture Notes in Computer Science, (pp. 63–70). Springer International Publishing, Chambridge. 12th International Conference, Diagrams 2021.

  • Sørensen, H., & Johansen, M. (2020). Counting mathematical diagrams with machine learning. In: Pietarinen, A.-V., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., & Linker, S., (Eds.), Diagrammatic Representation and Inference, volume 12169 of Lecture Notes in Computer Science, (pp. 26–33). Springer. 11th International Conference, Diagrams 2020.

  • Steensen, A., & Johansen, M. (2016). The role of diagram materiality in mathematics. Cognitive Semiotics, 9(2), 183–201.

    Article  Google Scholar 

  • Steensen & Johansen. (Accepted). Textual materiality and abstraction in mathematics. Science in Context.

  • Stjernfelt, F. (2007). Diagrammatology, an Investigation on the Borderlines of Phenomenology, Ontology, and Semiotics. Synthese Library v. 336. Springer.

  • Toffoli, S. D., & Giardino, V. (2014). Forms and roles of diagrams in knot theory. Erkenntnis, 79(4), 829–842.

    Article  Google Scholar 

  • Vold, K., & Schlimm, D. (2020). Extended mathematical cognition: External representations with non-derived content. Synthese, 197(9), 3757–3777.

    Article  Google Scholar 

  • Wishart, D. (2003). The printing of mathematics. In: Type & Typography: Highlights from Matrix, the review for printers and bibliophiles, (pp. 285–295). Mark Batty Publ. LLC., New York, 1. ed. edition.

Download references

Acknowledgements

We wish to thank professor Morten Misfeldt, University of Copenhagen, for his contribution to the original idea and design of the study. We also wish to thank the editors of Annals of Mathematics for kindly letting us reproduce diagrams from the journal.

Author information

Authors and Affiliations

  1. Department of Science Education, University of Copenhagen, Copenhagen, Denmark

    Mikkel Willum Johansen & Josefine Lomholt Pallavicini

Authors
  1. Mikkel Willum Johansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josefine Lomholt Pallavicini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikkel Willum Johansen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Johansen, M.W., Pallavicini, J.L. Entering the valley of formalism: trends and changes in mathematicians’ publication practice—1885 to 2015. Synthese 200, 239 (2022). https://doi.org/10.1007/s11229-022-03741-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11229-022-03741-8

Keywords

Search

Navigation