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

15The Boundary BetweenMathematics and PhysicsALASDAIR URQUHARTThe traditional or ‘received view’ of scientific explanation widely held in the1960s and1970s was that scientific theories are applied axiomatic systems,with explanations and predictions taking the form of logical derivations fromobservational statements. However, this model does not seem to describeaccurately some aspects of scientific practice, for example, the use of mean fieldmodels in condensed matter physics. Such models are more plausibly describedas being mathematical constructs in their own right that may be only looselyrelated to the phenomena they are designed to model.This latter view of models, however, has its own difficulties, rooted inthe fact that the methods of physicists are more often than not lacking inmathematical rigour. The fruitful tension resulting from this is the main topicof this chapter, together with the problems arising from the recently renewedinteraction between mathematics and physics.15.1 Mathematics and physicsIn the 1950s, when the logical positivist approach to the philosophy of sciencewas still a dominating force in North American philosophical circles, a commonview of scientific theories was that they were simply applied logical theories. Anabstract or purely logical theory was held to consist of a set of logical axioms andrules, in which no fixed interpretation was assigned to the primitive relationsand concepts of the theory. An applied theory was one in which observationalmeanings were assigned to certain primitive terms, the observation vocabulary,while the remaining terms (other than purely logical or mathematical notions),the theoretical terms, were to be explicitly defined using correspondence rules