Outsiders enabling scientific change: learning from the sociohistory of a mathematical proof

Abstract

It has been a common belief among scientists, including mathematicians, that young scientists are especially good at bringing about scientific change. A number of studies suggest, however, that older scientists are not more resistant to change than young scientists are. It is nonetheless worth examining why a scientist’s or mathematician’s outsider status – due to age, educational background, or something else – can sometimes be effective in enabling scientific change. This paper focuses on the case of the solving of the Four Color Problem by Wolfgang Haken and Kenneth Appel. Building on Donald MacKenzie’s paper ‘Slaying the kraken: The sociohistory of a mathematical proof’ (1999), I argue that Haken’s outsider status is central to understanding his success with the problem. On this basis, I offer an argument against Margaret Gilbert’s account of why a scientist’s outsider status can be effective in enabling scientific change.

Keywords:

• Scientific change
• outsiders
• Margaret Gilbert
• collective beliefs
• Thomas Kuhn

Acknowledgments

I am particularly indebted to Henrik Kragh Sørensen for extensive and valuable comments on earlier versions of this draft. This paper has also benefited from comments from two anonymous referees, Dania Achermann, Hanne Andersen, Mads Goddiksen, Gabriel Henderson, Matthias Heymann, Søren Harnow Klausen, Eva Lykkegaard Poulsen.

Notes

1. He had in mind questions like, What are the fundamental entities constituting the universe; what questions may legitimately be asked about them; and what techniques used in trying to solve them.

2. While Kuhn was thinking of the natural sciences here, it appears that something similar can be said of mathematics. His point may even apply especially well to mathematics since it would seem that mathematics education is more homogeneous across countries than any natural science education (Robitaille and Travers 1992).

3. There has been some debate as to whether the type of collective attitude referred to by Gilbert is collective acceptance rather than collective belief, a debate started by Wray (2001). But it is enough for my purposes to speak in terms of a group’s joint commitment to the view that p (as do Mathiesen 2006 and Rolin 2010). The question of whether this amounts to a proper collective belief that p is thus set aside.

4. Haken indicated this division of labor when he described the final stretch of their work on the Four Color Problem to Tony Dale, who conducted the interviews for MacKenzie’s study. Haken said that “50 configurations remained to be proven reducible. And so over the 4th of July weekend Ken […] recomputed those 50 configurations, and only 12 of them did not work. And then I worked on those for one day, and I found that they could be replaced […] by something like 20 others, and two of them did not work, and then again it took a day” (MacKenzie 1999, 39).

5. It seems that Haken himself was very aware of this, for, when interviewed by Dale, he noted, referring to his mathematical status in this period of time, that he “was an outsider and engineer then” (MacKenzie 1999, 32). In the interview he only spoke directly of himself as an outsider in this sense (not by virtue of his unusual education) and when he did, it was with the aim of explaining the skepticism that met his mathematical work in those years. I aim to explain his success with solving the Four Color Problem.

6. Haken himself linked his elementary style to his unusual education, described by MacKenzie as an education “from one teacher, essentially without books, and without in-depth study of many important areas of modern ‘higher’ mathematics” (MacKenzie 1999, 49).

7. He was editor in chief of the prestigious Journal of Combinatorial Theory from 1967 to 1985. Graph theorists Arthur M. Hobbs and James G. Oxley write that he was the leading graph and matroid theorist of his generation and that “he advanced graph theory from a subject with one text [König, 1936] toward its present extremely active state” (Hobbs and Oxley 2004, 320).

8. Let me give an example of Tutte’s authority being called upon to support the validity of the Appel–Haken proof when presenting it to the broader audience. Science journalist Gina Bari Kolata began her piece on it in the Research News section of Science by admitting that “countless erroneous ‘proofs’ [of the conjecture] have been advanced”, but then stressed that “the most recent proof […] differs from its predecessors in that it is accepted by experts in the field of combinatorics, who are intimately involved with ramifications of the four color conjecture. Moreover, these experts include some of the greatest skeptics of previously acclaimed ‘proofs”’ (Kolata 1976). She was primarily thinking of Tutte of whom she wrote that he “is an example of a mathematician who has seriously studied various aspects of the four color conjecture,” and she proceeded to say that he, “who was skeptical of and found an error in a previous well-publicized ‘proof’ of the four color conjecture, believes that Appel and Haken have really proved this conjecture”.