Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

336 jeremy avigadmathematics was viewed as the science of magnitude, and judgments as to therelative magnitude of various types of quantities still play a key role in thesubject. In a sense, methods of deriving equalities are better developed, andcomputer algebra systems carry out symbolic calculations quite effectively. Thisis not to say that issues regarding equality are trivial; the task of determiningwhether two terms are equal, in an axiomatic theory or in an intendedinterpretation, is often difficult or even algorithmically undecidable. Onestrategy for determining whether an equality holds is to find ways of puttingterms into canonical ‘normal forms’, so that an assertion s = t is then valid ifand only if s and t have the same normal form. There is an elaborate theory of‘rewrite systems’ for simplifying terms and verifying equalities in this way, butwe will not consider this here.Let Ɣ be a set of equalities and inequalities between, say, real or integervaluedexpressions, involving variables x 1 , ... , x n . Asking whether an inequalityof the form s < t is a consequence of Ɣ is the same as asking whether Ɣ togetherwith the hypothesis t ≤ s is unsatisfiable. Here, too, there is a well-developedbody of research, which provides methods of determining whether such asystem of equations is satisfiable, and finding a solution if it is. This researchfalls under the heading ‘constraint programming’, or, more specifically, ‘linearprogramming’, ‘nonlinear programming’, ‘integer programming’, and so on.In automated reasoning, such tasks typically arise in connection to schedulingand planning problems, and heuristic methods have been developed to dealwith complex systems involving hundreds of constraints.But the task of verifying the entailments that arise in ordinary mathematicalreasoning has a different character. To start with, the emphasis is on findinga proof that an entailment is valid, rather than finding a counterexample.Often the inequality is tight, which means that conservative methods ofapproximation will not work; and the structure of the terms is often moreelaborate than those that arise in industrial applications. On the other hand,the inference may involve only a few hypotheses, in the presence of suitablebackground knowledge. So the problems are generally smaller, if structurallymore complex.Let us consider two examples of proofs involving inequalities. The first comesfrom a branch of combinatorics known as Ramsey theory. An (undirected) graphconsists of a set of vertices and a set of edges between them, barring ‘loops’,i.e. edges from a vertex to itself. The complete graph on n vertices is the graph inwhich between any two vertices there is an edge. Imagine coloring every edgeof a complete graph either red or blue. A collection of k points is said to behomogeneous for the coloring if either all the edges between the points are red,or all of them are blue. A remarkable result due to F. P. Ramsey is that for any