On the Effectiveness of Mathematics in the Natural Sciences: A Pragmatic Structuralist Perspective

Abstract We want to present a pragmatic conception about applied mathematics. We initiate our investigation stating such problem: If many mathematical theories were developed independently of theoretical physics, so why many of such theories have been so effective in scientific theories? We claim for an initial isomorphism between structures in mathematics and natural sciences. Then, we connect the heuristic searching for regularities and how it sheds light to our crucial question. Furthermore, it is noteworthy that this isomorphism is not always preserved when it is applied to scientific purposes. Thus, we present the operational attitude of scientists to use and manipulate mathematical tools to solve theoretical problems demanded by theoretic-empirical researches. Pure mathematics furnishes a plenty variety of structures in which, when applied to physics, they become relations between quantities or between observables. Scientists adapt and shape mathematical theories in different levels to make it useful tools to grasp regularities in natural world. Finally, we conclude that this isomorphic similarity and the plasticity of the operational use of mathematical tools in physics give us the possibility to make this hybrid language be so successful in applications. Consequences of this discussion to foundations of mathematics are presented as well.