Abstract
The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970s, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by Tymoczko and some wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem.
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Notes
- 1.
Mentions to the 4CT proof can be found in Avigad (2008) and McEvoy (2013). For more detailed approaches considering new philosophical questions on the role of computers in mathematical practices, see MacKenzie (2001, 2005), and for old questions revisited see Bassler (2006), McEvoy (2008, 2013) and Prawitz (2008).
- 2.
A normal planar map is a map on the plane in which there is no region completely surrounded by other and there are no more than three regions connecting in the same point. The topological version of the 4CT affirms: “For every map there exists an admissible 4-colouring”; the combinatorial one, without any reference to geometry or topology: “Every planar graph has an admissible vertex 4-coloring” (Fritsch and Fritsch 1998).
- 3.
The guess was made by a Geography teacher whose brother was a student of Augustus de Morgan. It was De Morgan who first brought mathematical attention to the problem in letters to Hamilton (from 1852 to 1853) and in a review of William Whewell’s The Philosophy of Discovery published in 1860. For details about the history of the different attempts to solve the problem, besides Fritsch and Fritsch 1998, see Saaty and Kainen 1986, MacKenzie 2001 and Wilson 2002.
- 4.
- 5.
Appel said (see MacKenzie 2001: 138) that in one of the first public briefings of the result the audience had been clearly divided into two groups: people with more than 40 years that “could not be convinced that a proof by computer could be correct” and “people under forty [who] could not be convinced that a proof that took 700 pages of hand calculations could be correct”. Appel was referring to a talk given by Haken’s son at Berkeley, who at that time (1977) was a graduate student at the UCB and gave this talk to announce and describe the proof in that institution.
- 6.
A topic not explored here is the connection between the alleged lack of structure of the computational part of the proof and the absence of explanatory virtues in it (a sample of this kind of criticism can be found in Ian Stewart’s Concepts of Modern Mathematics (Stewart 1995: 304)). Nevertheless, Swart argues that if the proof of the 4CT is not explanatory, then no proof by exhaustion is. They can be divided into three parts: “(i) Establishing the fact that the theorem is true provided a certain set of graphs, configurations, or – in general – cases possesses (or do not possess, as the case may be) a stated property; (ii) Obtaining an exhaustive listing of these cases; (iii) Confirming that all the members of this set do possess the required property. The finite set of cases concerned may, at one extreme, be so small and so simple that the case testing can be done in our heads, or it may, at the other extreme, be so large and/or so complicated that it is impossible to carry out without the help of a computer” (Swart 1980: p. 699).
- 7.
In fact, during the 1980s, a student of electric engineering, Ulrich Schmidt, found an error that could be considered relevant in the proof – being “relevant” an error that could interfere in the final result of the derivation. Nevertheless, Appel and Haken not only developed an “error-correcting routine” (Appel and Haken 1986: 19–20), but also presented a typology of errors according to which the main error found by Schmidt wasn’t relevant. The authors attribute the rumours surrounding the correctness of the proof to “a misinterpretation of the results of the independent check of details of the proof by U. Schmidt” (Appel and Haken 1986: 10).
- 8.
- 9.
Georg Kreisel (Kreisel 1977) and Hao Wang (Wang 1981) were the first members of the philosophical community that mentioned the 4CT proof in 1977. They knew about the result, respectively, by means of an article published in Scientific American (Appel Haken 1977b) and an expository paper published in New Scientist (Appel 1976). Wang (1981) is the register of a series of talks given by the author in the Chinese Academy of Science in 1977.
- 10.
- 11.
- 12.
- 13.
Prawitz deals with cases of proofs produced by computers programs – the 4CT proof being his main example – and proofs that verify programs. Investigating the sociology of mathematical proofs MacKenzie (2005) also considers two categories: proofs using computers and proofs about computers. Each category contains different dimensions: in the first one, automated theorem provers and model checkers, mathematical proofs of immense complication of detail and, in the universe of artificial intelligence, the question about the possibility of a computer be ‘an artificial mathematician’ (MacKenzie 2005: 2336); the second category deals with key aspects of microprocessors and computer systems on which lives and national security depend.
- 14.
To be precise, in the first paper the authors refer to results obtained by others for “more detailed descriptions of the algorithms” (Appel Haken 1977a: 431), offering “a general description of the method of defining the discharging procedure” (loc. cit.) in the fifth section of the paper; in the second paper they illustrate “the reducer-choosing algorithms” (Appel, Haken Koch 1977: 493) for the case of configurations with a 12-ring size (a ring size being the number of regions that circumvents a reducible configuration). We are using indifferently “program” and “algorithm” even knowing that the distinction between them is one of the main problems in the philosophy of computer science, as it can be seen in Turner Eden 2009.
- 15.
See Tractatus 4.1252, 5.232, 6.2 ssq.
- 16.
For this point see the second chapter of Marion (1998).
- 17.
From the Tractatus onwards, and in a multitude of formulations, Wittgenstein permanently insisted in the distinction between the descriptive nature of the language, predominantly in experimental activities, and the normative character of language at stake in mathematical activities such as calculations and proofs. “Calculation is not an experiment”, this sort of slogan stated in the final phrase of a couple of tractarian passages (6.233 and 6.2331), appears in a context in which Wittgenstein is dealing with the question about the need of some kind of intuition in the resolution of mathematical problems – to which he responds that it is the language itself, manipulated in the process of calculating, that brings about this intuition.
- 18.
We are emphasizing the idea that this distinction appears late in more recent discussions about the 4CT to stress that its proper wittgensteinian character only began with Shanker’s approach to the problem. We can perfectly recognize that Tymoczko’s uses the distinction, but given that he did not affiliate himself to Wittgenstein’s ideas and, as we considered above, his paper treats the notion of surveyability (the main criterion for the distinction “proof vs. experiment” in its wittgensteinian use) in an ambiguous way, we prefer not to attribute to Tymoczko any strong association to Wittgenstein’s perspective.
- 19.
Some textual support for each one of these comparison will be given in the footnotes. We will refer to Wittgenstein’s Remarks on the Foundations of Mathematics as RFM, indicating as usual the sections and paragraphs, not the pages.
- 20.
RFM I, Appendix III, 17; III, 29, 31–2, 41; VII, 12, 18.
- 21.
RFM, III, 55, 65–76.
- 22.
RFM, I, 98; IV, 4, 6.
- 23.
RFM III, 68–69.
- 24.
Stillwell probably inherit this idea from Shanker, who was very much concerned with the elimination of any reference to “medical limitations” in the analysis of Wittgenstein’s notion of surveyability, such as not being able to grasp a proof or calculation in one coup d’oeil. This is a point of controversy in the wittgensteinian scholarship milieu. For specific discussions on the notion in Wittgenstein’s philosophy of mathematics see Wright (1980), Shanker (1987) and Frascolla (1994). More recent approaches can be found in Mühlhölzer (2006) and Marion (2011).
- 25.
“A clear proof is reproducible ‘as a whole’ in that it can be copied, if carefully, without error. But this does not imply that copying a proof entails copying its result. That is, perspicuity does not imply holism. On the other hand, holism does not guarantee perspicuity. The lack of it might make it impossible for us to reproduce a proof, but does not entail that any proof is not holistic. Put otherwise, from the fact that (copying) any proof must include (copying) its result, nothing follows about whether or not that proof is laid clear to view” (Stillwell 1992: 122).
- 26.
We would like to add: (P6) The description of a proof (with some details and in relation with some audiences) is a proof; (E6) The description of an experiment never replaces the experiment.
- 27.
We find in Esquisabel’s approach of Leibniz conception of symbolic thought an elegant articulation between local and global surveyability given that amongst the functions of this kind of thought one finds a computational function (identifiable with local surveyability) and a cognitive function (related to the idea that in this kind of thought we grasp the in the syntax of a symbolic system the structures being worked with. This grasping can be associated with global surveyability).
- 28.
Some discussions Wittgenstein had with his students, including Alan Turing, about the empirical status of calculations during 1939 explores this point (see Wittgenstein 1989).
- 29.
- 30.
We would like to thank Marcos Silva for organizing the Colloquium in which this paper was first presented and also to CNPq for funding the research projects that resulted in the paper.
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Secco, G.D., Pereira, L.C. (2017). Proofs Versus Experiments: Wittgensteinian Themes Surrounding the Four-Color Theorem. In: Silva, M. (eds) How Colours Matter to Philosophy. Synthese Library, vol 388. Springer, Cham. https://doi.org/10.1007/978-3-319-67398-1_17
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