Abstract
The present paper provides an analysis of Euler’s solutions to the Königsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present case. I also propose computational complexity and logical strength as measures of relevant information.
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Notes
See Molinini (2012) for a successful example of analyzing Euler’s work in the context of mathematical explanations.
Euler does not specify how to carry out the Brute Force Method in detail. One way of implementing it is to write down all (finitely many) paths of length seven starting from one of the areas A,B,C,D, and see whether any one of these paths includes just seven different bridges.
A multigraph is a graph where two vertices can be connected by more than one edge. Consequently, the multiset of edges may contain a pair of vertices more than one time.
Arguably, the fact that Euler’s paper stands at the beginnings of graph theory is its most important innovation. This supports an observation by Rav (1999) that one of the most important roles of proving theorems lies in the novel insights generated by proof methods beyond determining the truth of particular theorems.
Euler does not employ the notation d(X) for the degree of X; his argument hardly uses any algebraic expressions. The reconstruction given here follows Euler closely, but transforms some of his reasoning into algebra in order to make it more accessible.
Euler’s Theorem is different from modern formulations of the theorem in several respects; see, e.g., Diestel (2006, pp. 21) for a modern account. First, Euler asks if it is possible to cross every bridge in Königsberg exactly once, without the assumption that the Euler path should be closed. Second, Euler does not assume that graphs are connected. This is a necessary assumption for the theorem. Third, Euler only proved one direction of the modern version of the theorem, viz. the statement that a closed Euler path exists if, and only if, every area has even number of edges. He did not prove that if a closed Euler path exists, every vertex has an even number of edges; this was only proved 135 years later; see Wilson (1986, p. 270).
To use an imperfect analogy with the deductive-nomological model of explanation, we need both a general law as well as initial conditions to deduce the explanandum.
Brute Force Methods can be applied whenever a problem is decidable. For example, we may want to find out whether a mathematical structure has a certain property or not, and this can be determined by going through all (finitely many) possible cases. A Brute Force Method then systematically goes through all possible cases, and answers “Yes” if at least one case comes out positive, and “No” otherwise.
We could interpret the Brute Force Method as not being explanatory at all. However, this is, strictly speaking, not true. There are proofs, so-called zero-knowledge proofs, that only show that a result is true, without giving us any knowledge as to why this is so, in a strict, cryptographic sense of knowledge; see Aaronson (2013, Sec. 9.1). However, these are non-classical, probabilistic proofs, and deductive analogues will always convey some information.
This does not speak against all kinds of algorithms that compile lists of paths; the redundancy could be overcome, to a certain degree, by using a clever search algorithm.
Note that while the Intermediate Method and Euler’s Theorm fare better than the Brute Force Method in terms of time complexity, the Brute Force Method is simpler to state than the other methods, i.e., the program-size complexity of the algorithms may increase, and thus be uncorrelated with, explanatory power.
Note that the complexity may depend on the size of both E and V; see, e.g., Gibbons (1985, Ch. 6) for more on issues of complexity.
Note that there may be other fruitful ways of measuring relevant information. To give an example, Mark Colyvan (2012, pp. 83) briefly discusses the idea of using so-called relevant logic to distinguish between explanatory and non-explanatory proofs. I think that this idea is compatible with what I have proposed here.
I thank an anonymous referee for suggesting this perspective of the relation between ontic and epistemic conceptions of explanation.
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Acknowledgements
Thanks to Alan Baker, Claus Beisbart, Matthias Egg, Michael Esfeld, Marion Hämmerli, Marc Lange, Hannes Leitgeb, Philip Mills, Thomas Müller, Christopher Pincock, Antje Rumberg, Tilman Sauer, Raphael Scholl, various anonymous referees, audiences in Bern, Lausanne, and Konstanz for comments on earlier drafts of this paper, to Andreas Verdun for help with the historical literature, to Scott Aaronson for correspondence, and to Dan Ward for proofreading. The usual disclaimer applies.
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This work was partially supported by the Swiss National Science Foundation, grant numbers 100011-124462/1 and 100018-140201/1, as well as by the Templeton World Charity Foundation through grant TWCF0078/AB46
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Räz, T. Euler’s Königsberg: the explanatory power of mathematics. Euro Jnl Phil Sci 8, 331–346 (2018). https://doi.org/10.1007/s13194-017-0189-x
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DOI: https://doi.org/10.1007/s13194-017-0189-x