Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Mathematicians writing for mathematicians


We present a case study of how mathematicians write for mathematicians. We have conducted interviews with two research mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research paper they wrote together. Over the course of 2 years, Adam and Thomas revised Adam’s very detailed first draft. At the beginning of this collaboration, Adam was very knowledgeable about the subject of the paper and had good presentational skills but, as a new PhD student, did not yet have experience writing research papers for mathematicians. Thus, one main purpose of revising the paper was to make it take into account the intended audience. For this reason, the changes made to the initial draft and the authors’ purpose in making them provide a window for viewing how mathematicians write for mathematicians. We examined how their paper attracts the interest of the reader and prepares their proofs for validation by the reader. Among other findings, we found that their paper prepares the proofs for two types of validation that the reader can easily switch between.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    For an account of the relevance of this work to the philosophy of mathematics, see Dufour (2013).

  2. 2.

    We were asking leading questions in the sense of asking questions about the aims of the changes they made to the paper while suggesting what these aims may be. Lewis Anthony Dexter wrote that leading questions “are helpful in interviewing experts; most experts are predisposed to argue about professional matters and set people right, and few of them are so malleable as to fall tout court for a leading question” (1970/2006, p. 79).

  3. 3.

    We should stress that we used the term ‘narrative structure’ in the question Thomas is responding to here. Since in using the term Thomas is merely copying our choice of words for the structural template in question, we ascribe no significance to his use of this term.

  4. 4.

    From comparing the two paragraphs of the answer, it seems that Thomas uses the term ‘local narrative’ to refer to the structure of the proof. We should not put too much emphasis on his use of the word ‘narrative’ here, since he may just use it because we do, even if we do not use it in this particular sense.

  5. 5.

    He added that,

    This approach works well for me and is quite effective in weeding out errors. However, I could imagine that other mathematicians might have other approaches, some of which are much more detail-oriented and less narrative or pattern based.

  6. 6.

    We take Thomas’ response to imply that a proof typically cannot be validated using only higher-level proof validation. Even when higher-level proof validation indicates that the proof is correct, “some additional checking of the more oblique or technical parts” is typically called for, although the patterns of the proof, including the patterns of these parts, have been validated. It is not clear how “the more oblique or technical parts” of the proof are identified.

  7. 7.

    0.10 and 2.3 indicate the number of the theorem in the first and the last version of the paper, respectively.

  8. 8.

    Consequently, when a reader does not check a proof line-by-line this does not imply that she is just trusting the author to have done so. She may have validated parts of the proof line-by-line and validated the other parts by independently validating parts of the Level I argument.


  1. Aberdein, A., & Dove, I. J. (Eds.). (2013). The argument of mathematics. Dordrecht: Springer.

  2. Andersen, L. E. (2018). Acceptable gaps in mathematical proofs. Synthese.

  3. Dahn, B. I. (1998). Robbins algebras are Boolean: A revision of McCune’s computer-generated solution of Robbins problem. Journal of Algebra, 208(2), 526–532.

  4. Dexter, L. A. (1970/2006). Elite and specialized interviewing. Colchester: ECPR Press.

  5. Dufour, M. (2013). Arguing around mathematical proofs. In A. Aberdein & I. J. Dove (Eds.), The argument of mathematics (pp. 61–76). Dordrecht: Springer.

  6. Fallis, D. (2003). Intentional gaps in mathematical proofs. Synthese, 134, 45–69.

  7. Hersh, R. (1979). Some proposals for reviving the philosophy of mathematics. Advances in Mathematics, 31, 31–50.

  8. Hilbert, D. (1925/1967). On the infinite. In J. Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 18791931 (pp. 369-392). Cambridge, MA: Harvard University Press.

  9. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43, 358–390.

  10. Johansen, M. W., & 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). Cham: Springer.

  11. Johnson, R. H. (2000). Manifest rationality: A pragmatic theory of argument. Mahwah, NJ: Lawrence Erlbaum Associates.

  12. Krabbe, E. C. W. (2013 [1997]). Arguments, proofs, and dialogues. In A. Aberdein & I. J. Dove (Eds.), The argument of mathematics (pp. 31–45). Dordrecht: Springer.

  13. Krabbe, E. C. W. (2013 [2008]). Strategic maneuvering in mathematical proofs. In A. Aberdein & I. J. Dove (Eds.), The argument of mathematics (pp. 181–193). Dordrecht: Springer.

  14. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. New York: Cambridge University Press.

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

  16. Misfeldt, M., & Johansen, M. W. (2015). Research mathematicians’ practices in selecting mathematical problems. Educational Studies in Mathematics, 89, 357–373.

  17. Olson, R. (2015). Houston, we have a narrative: Why science needs story. Chicago: University of Chicago Press.

  18. Paseau, A. C. (2016). What’s the point of complete rigour? Mind, 125, 177–207.

  19. Perelman, C., & Olbrechts-Tyteca, L. (1969). The new rhetoric: A treatise on argumentation. Notre Dame: The University of Notre Dame Press.

  20. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7, 5–41.

  21. Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica, 15, 291–320.

  22. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30, 161–177.

Download references


We are deeply indebted to Adam and Thomas for letting us into their space of collaboration and supervision. We have presented our research in local groups and at the Oxford conference, and we are very grateful for the feedback and discussion, we have received, in particular from Alan Bundy and from colleagues. We are also very grateful for the challenging and constructive feedback we received from two anonymous referees. Part of the research for this paper was conducted while the first author was a postdoctoral researcher at the Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussels, Belgium. At Aarhus University, she is supported by K. Brad Wray’s Grant, AUFF-E-2017-FLS-7-3.

Author information

Correspondence to Line Edslev Andersen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Verify currency and authenticity via CrossMark

Cite this article

Andersen, L.E., Johansen, M.W. & Sørensen, H.K. Mathematicians writing for mathematicians. Synthese (2019).

Download citation


  • Mathematical publication
  • Mathematical audience
  • Mathematical argument
  • The nature of proof
  • Contextualization