Formalism, Formalization, Intuition and Understanding in Mathematics: From Informal Practice to Formal Systems and Back Again

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This project investigates the interplay between informal mathematical theories and their formalization, and argues that this dynamism generates three different forms of understanding:(I) Different kinds of formalizations fix the boundaries and conceptual dependences between concepts in different ways, thus contributing to our understanding of the content of an informal mathematical theory. We argue that this form of understanding of an informal theory is achieved by recasting it as a formal theory, i.e. by transforming its expressive means.(II) Once a formal theory is available, it becomes an object of understanding. An essential contribution to this understanding is made by our recognition of the theory in question as a formalization of a particular corpus of informal mathematics. This form of understanding will be clarified by studying both singular intended models, and classes of models that reveal the underlying conceptual commonalities between objects in different areas of mathematics.(III) The third level concerns how the study of different formalizations of the same area of mathematics can lead to a transformation of the content of those areas, and a change in the geography of informal mathematics itself. In investigating these forms of mathematical understanding, the project will draw on philosophical and logical analyses of case studies from the history of mathematical practice, in order to construct a compelling new picture of the relationship of formalization to informal mathematical practice. One of the main consequences of this investigation will be to show that the process of acquiring mathematical understanding is far more complex than current philosophical views allow us to account for.While formalization is often thought to be negligible in terms of its impact on mathematical practice, we will defend the view that formalization is an epistemic tool which not only enforces limits on the problems studied in the practice, but also produces new modes of reasoning that can augment the standard methods of proof in different areas of mathematics. Reflecting on the interplay between informal mathematical theories and their formalization means reflecting on mathematical practice and on what makes it rigorous, and how this dynamism generates different forms of understanding. We therefore also aim to investigate the connection between the three levels of understanding described above, and the notion of rigor in mathematics. The notion of formal rigor (in the proof theoretic sense) has been extensively investigated in philosophy and logic, though an account of the epistemic role of the process of formalization is currently missing. We argue that formal rigor is best understood as a dynamic abstraction from informally rigorous mathematical arguments. Such informally rigorous arguments will be studied by critically analyzing case studies from different subfields of mathematics, in order to identify patterns of rigorous reasoning.