Formalismus, Formalisierung, Intuition und Verstehen in der Mathematik: Von der informellen Praxis zu formalen Systemen und zurück
Zusammenfassung der Projektergebnisse
The central theme of this project was how formalization contributes to our understanding of mathematics: firstly through transforming informal mathematics into a formalized form that clarifies the logical dependencies between different statements; secondly by allowing for the identification of natural models (and classes of models) of the formal theory; and thirdly through comparing the formal theory with the informal theory that gave rise to it. The first task of the project was to provide a scientific basis and an institutional context for extending the collaboration between the French and the German team. The second task was entitled: “Investigating the Interplay Between Formalization and Understanding in Mathematics: The forms of Understanding” (managers: G. Heinzmann and M. Panza, from the French ANR-funded team). It was divided into three subtasks: (2.1) Rethinking structuralism and investigating the relationship between structures and intuition; (2.2) Investigating the impact of formalization on mathematical practice; (2.3) Reverse Mathematics. Finally, the third task was entitled: “Formal and Informal rigor” (manager: H. Leitgeb, from the German DFG-funded team). It consisted of two subtasks: (3.1) Formal rigor in logical and mathematical reasoning; (3.2) informal rigor. The work carried out within the project contributed substantially to enlarge and enrich the debate on formalization and on its epistemic significance, casting new light on problems that were already the object of discussion in the philosophy of mathematics and logic, and the philosophy of mathematical practice, and developing new areas of debate. The main result of the project, consistent with the aims it set out to achieve, is to have provided an account of different levels of understanding produced by the process of formalizing informal domains of reasoning in mathematics and logic. In doing so, the project created a novel framework for posing new questions about the role and significance of formalization and addressing old ones in a new light. For example, the project did not simply extend the existing debate on limitative formal results following Gödel’s incompleteness or the debate concerning solutions to the paradoxes, but provided a new framework for understanding the significance of formalization in obtaining and comprehending the results in question. Interesting open questions generated by the project include the analysis of further case studies from historical and contemporary mathematics, logic, computer science, and mathematical philosophy that could be fruitfully illuminated by analyzing them within the framework proposed by the project. It also emerged from the work carried out within the project that while substantial progress was made on how formalization contributes to the understanding of logical and mathematical theories, the notion of understanding itself is still very elusive, and it remains hard to come close to a definition of it. An especially difficult hurdle to overcome in reaching a general characterization of the notion of understanding is the elusiveness of the notion of content of a mathematical or logical theory, as became apparent from the discussion at the closing event of the project. Another notion that remains hard to pin down is that of informal rigor: while we have a good grasp of the notion of formal rigor (as characterized for example in proof theory), the characterization of the notion of informal rigor, and its relation to formalization, depend in part on the characterization of the notions of mathematical (or logical) understanding and content. Clarifying the challenges implicit in characterizing the notions of mathematical understanding and content, and of informal rigor, constitutes progress in itself and lays the ground for achieving progress in the forthcoming years.
Projektbezogene Publikationen (Auswahl)
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What's Hot in Mathematical Philosophy, The Reasoner, 13(9), 60–61 (September 2019)
Antonutti Marfori, M.
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A More Unified Approach to Free Logics. Journal of Philosophical Logic, 50(1), 117-148.
Pavlović, Edi & Gratzl, Norbert
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Intuition, Understanding, and Proof: Tatiana Afanassjewa on Teaching and Learning Geometry. Women in the History of Philosophy and Sciences, 29-54. Springer International Publishing.
Antonutti Marfori, Marianna
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A Structural Justification of Probabilism: From Partition Invariance to Subjective Probability. Philosophy of Science, 88(2), 341-365.
Leitgeb, Hannes
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ABSTRACT FORMS OF QUANTIFICATION IN THE QUANTIFIED ARGUMENT CALCULUS. The Review of Symbolic Logic, 16(2), 449-479.
PAVLOVIĆ, EDI & GRATZL, NORBERT
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RAMSIFICATION AND SEMANTIC INDETERMINACY. The Review of Symbolic Logic, 16(3), 900-950.
LEITGEB, HANNES
