Fictionalist Mathematical Structuralism

Project Information

FICTSTRUCT

Grant agreement ID: 890376

DOI

10.3030/890376

Project closed

EC signature date 22 April 2020

Start date 1 July 2020

End date 29 May 2024

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 224 933,76

EU contribution

€ 224 933,76

Coordinated by

UNIVERSITY OF YORK
United Kingdom

Periodic Reporting for period 1 - FICTSTRUCT (Fictionalist Mathematical Structuralism)

Reporting period: 2020-07-01 to 2022-06-30

Summary of the context and overall objectives of the project

This project addresses the abstract nature of mathematics and the philosophical questions surrounding the existence of mathematical entities such as numbers, functions, and sets. Traditionally, the philosophical perspectives of structuralism and fictionalism have offered competing views: realist, or ‘ante rem’, structuralism views mathematical theories as descriptions of abstract structures, while fictionalism denies the existence of these entities, treating mathematical discourse as fictional. This project has been implemented to develop a comprehensive theory of mathematical structure, based on the fictionalist approach, which can advance long-standing debates in the philosophy of mathematics and the philosophy of science. Understanding the nature of mathematical entities and the applicability of mathematics is crucial not only for philosophical inquiry but also for enhancing our comprehension of mathematical practice and its applications in science.

Core objectives:
1. To develop a comprehensive theory of mathematical structure by integrating structuralism and fictionalism.
2. To advance debates in the philosophy of mathematics and the philosophy of science.
3. To promote interdisciplinary research and collaboration among philosophers, mathematicians, and scientists.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Overall, the project has made significant strides in bridging the gap between structuralism and fictionalism within the philosophy of mathematics. By developing a comprehensive theory of mathematical structure, the project not only advances philosophical debates but also enhances our understanding of mathematical practice and its applications in science. The interdisciplinary nature of the project has fostered collaboration and knowledge transfer, laying the groundwork for future research and innovation across multiple fields.

Summary of Results:
- The project successfully developed an axiomatic approach to fictionalist mathematical structuralism. This novel approach allows for the description and application of mathematical structures without commitment to the existence of abstract mathematical entities.
- By offering an understanding of structural ‘entities’ as fictional, the project offered an explanation of some puzzling elements of structural entities.
- Significant progress was made in understanding how mathematical abstraction operates within both structuralist and fictionalist frameworks.
- The project generated multiple high-impact academic outputs, including research articles, conference presentations, and a planned special issue in a leading journal.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The project has made significant progress beyond the state of the art by developing a novel axiomatic approach that integrates structuralism and fictionalism, allowing for the description and application of mathematical structures without the need to commit to the existence of abstract entities. Expected results of the project include the publication of high-impact research articles, a special journal issue, and continued dissemination of findings through conferences and workshops. The research addresses fundamental questions in the philosophy of mathematics, such as the nature of mathematical entities and the applicability of mathematics in science. By influencing and advancing debates in these areas, the project contributes to the theoretical foundations that underpin many scientific and mathematical practices, which can have long-term benefits for various scientific fields. The project also promotes interdisciplinary collaboration between philosophy, mathematics, physics, linguistics, and cognitive psychology. It fosters knowledge transfer and the integration of philosophical thinking into scientific practices, potentially leading to new research directions and innovations in these fields.

fictionalist mathematical structuralism