The growth of mathematical knowledge - introduction of convex bodies
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The article addresses the topic of the growth of mathematicalknowledge with a special focus on the question: How are mathematical objects introduced to mathematical practice? It takes as starting point a proposal made in a previous paper which is based on a case study on the introduction of Riemann surfaces. The claim is that (i) a new object first refers to previously accepted objects, and that (ii) reasoning is possible via a correspondence to the objects with reference to which it is introduced. In addition Riemann surfaces are geometrical objects, i.e., they are placed in a geometrical context, which makes new definitions possible. This proposal is tested on a case study on Minkowski’s introduction of convexbodies. The conclusion is that the proposal holds also for this example. In both cases we notice that in a first stage is a close connection between the new object and the objects it is introduced with reference to, and that in a later stage, the new object is given an independent definition. Even though the two cases display similarity in these respects, we also point to certain differences between the cases in the process of the first stage. Overall we notice the fruitfulness of representing problems in different contexts.
|Studies in History and Philosophy of Science Part A
|Published - 2012