The purpose and place of formal systems in the development of science

Abstract
The aim of this paper is to re-emphasise that the purpose of formal systems is to provide something to map into and to stem the tide of unjustified formal systems. I start by arguing that expressiveness alone is not a sufficient justification for a new formal system but that it must be justified on pragmatic grounds. I then deal with a possible objection as might be raised by a pure mathematician and after that to the objection that theory can be later used by more specific models. I go on to compare two different methods of developing new formal systems: by a priori principles and intuitions; and by post hoc generalisation from data and examples. I briefly describe the phenomena of “social embedding” and use it to explain the social processes that underpin “normal” and “revolutionary” science. This suggests social grounds for the popularity of normal science. I characterise the “foundational” and “empirical” approaches to the use of formal systems and situate these with respect to “normal” and “revolutionary” modes of science. I suggest that successful sciences (in the sense of developing relevant mappings to formal systems) bare either more tolerant of revolutionary ideas or this tolerance is part of the reason they are successful. I finish by enumerating a number of ‘tell-tale’ signs that a paper is presenting an unjustified formal system.
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