Line Andersen, line.edslev.andersen@gmail.com, postdoc at the Centre for Science Studies, Aarhus University

The talk will be based on joint work with Henrik Kragh Sørensen and Mikkel Willum Johansen from the University of Copenhagen.

Forty years ago, mathematician and philosopher Reuben Hersh suggested that we reject formalism as a philosophy of mathematics; that we give up the picture of mathematics as formal derivations from some given set of formulas. Hersh wanted to replace formalism with “a philosophy that is true to the reality of mathematical experience.” (1979, 40) Since then many philosophers have examined how ordinary mathematical proofs, as they occur in mathematical practice, differ from formal derivations (e.g., Lakatos 1974; Rav 1999; Manders 2008). In this talk, we continue this general line of work by examining mathematical proofs in the context of their intended audience.

As mathematician William Thurston has pointed out, mathematicians “prove things in a certain context and address them to a certain audience.” (1994, 175) Ordinary mathematical proofs appeal to the intuitions and background knowledge of the intended reader. Hence, what such proofs look like depends on their intended audience, and to understand the nature of mathematical proofs as presented, we need to examine how they are made to address their audience. Some studies take into account the intended audience of mathematical proofs when discussing the level of granularity of proofs; they point out that the appropriate level of granularity of a proof depends on the audience (e.g., Fallis 2003; Larvor 2012; and Paseau 2016). For example, the level of granularity of a textbook proof written for high school students will often be higher than the level of granularity of research proofs. However, these studies do not go into detail with how the level of granularity of a proof is made to fit the audience.

In this talk, we focus on how the level of granularity of research proofs are made to fit the intended expert audience and, more generally, on how mathematicians frame their proofs when writing for mathematicians. We have conducted interviews with two mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research article they wrote together. Over the course of two years, Thomas and Adam revised Adam’s very detailed first draft. At the beginning of this collaboration, Adam was a new PhD student and, for this reason, did not really know the intended audience of the article, but he was very knowledgeable about the subject of the article. Thus, one main purpose of revising the article 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 to how mathematicians write for mathematicians. We examined how their article prepares their proofs for validation by the reader and found that it prepares the proofs for two types of validation that the reader can easily switch between: line-by-line validation and a higher-level validation, which we will describe in some detail in the talk. The two types of validation do not require the same level of granularity of the proofs, and the proofs are thus made to present themselves to the reader at two different levels of granularity.

References

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

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

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

Larvor, B. (2012). How to think about informal proofs. Synthese, 187, 715–730.

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

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

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

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