Changes of Setting and the History of Mathematics: A New Study of Frege
Thesis DisciplineHistory and Philosophy of Science
Degree GrantorUniversity of Canterbury
Degree NameMaster of Arts
This thesis addresses an issue in the philosophy of Mathematics which is little discussed, and indeed little recognised. This issue is the phenomenon of a ‘change of setting’. Changes of setting are events which involve a change in a scientific framework which is fruitful for answering questions which were, under an old framework, intractable. The formulation of the new setting usually involves a conceptual re-orientation to the subject matter. In the natural sciences, such re-orientations are arguably unremarkable, inasmuch as it is possible that within the former setting for one’s thinking one was merely in error, and under the new orientation one is merely getting closer to the truth of the matter. However, when the subject matter is pure mathematics, a problem arises in that mathematical truth is (in appearance) timelessly immutable. The conceptions that had been settled upon previously seem not the sort of thing that could be vitiated. Yet within a change of setting that is just what seems to happen. Changes of setting, in particular in their effects on the truth of individual propositions, pose a problem for how to understand mathematical truth.
Thus this thesis aims to give a philosophical analysis of the phenomenon of changes of setting, in the spirit of the investigations performed in Wilson (1992) and Manders (1987) and (1989). It does so in three stages, each of which occupies a chapter of the thesis:
1. An analysis of the relationship between mathematical truth and settingchanges, in terms of how the former must be viewed to allow for the latter. This results in a conception of truth in the mathematical sciences which gives a large role to the notion that a mathematical setting must ‘explain itself’ in terms of the problems it is intended to address.
2. In light of (1), I begin an analysis of the change of setting engendered in mathematical logic by Gottlob Frege. In particular, this chapter will address the question of whether Frege’s innovation constitutes a change of setting, by asking the question of whether he is seeking to answer questions which were present in the frameworks which preceded his innovations. I argue that the answer is yes, in that he is addressing the Kantian question of whether alternative systems of arithmetic are possible. This question arises because it had been shown earlier in the 19th century that Kant’s conclusion, that Euclid’s is the only possible description of space, was incorrect.
3. I conclude with an in-depth look at a specific aspect of the logical system constructed in Frege’s Grundgesetze der Arithmetik. The purpose of this chapter is to find evidence for the conclusions of chapter two in Frege’s technical work (as opposed to the philosophical). This is necessitated by chapter one’s conclusions regarding the epistemic interdependence of formal systems and informal views of those frameworks.
The overall goal is to give a contemporary account of the possibility of setting-changes; it will turn out that an epistemic grasp of a mathematical system requires that one understand it within a broader, somewhat historical context.