Periodic Reporting for period 1 - MathematicsAnalogies (Mathematics Analogies)
Période du rapport: 2019-10-01 au 2021-09-30
This project investigates how mathematics impacts metaphysical realism-antirealism debates. Its goal is to develop a systematic account of the potentials and limits of using mathematics as a model for other a priori domains, thus responding to the rapidly increasing literature exploiting structural parallels between mathematics and domains like ethics, metaphysics, and religion.
While early Mesopotamian and Egyptian mathematics developed in response to the practical needs of settled civilizations, such as taxation, the measurement of plots of land, or the calculation of lunar calendars, philosophers soon assigned mathematics the additional role of paradigm for human reasoning and knowledge. Plato, for example, considered mathematics the highest form of knowledge, sharply distinguished from our uncertain beliefs about the empirical world. He thought that mathematical knowledge consists in knowledge of eternal ‘Forms’, perfect ideas that constitute ultimate reality, and he considered proficiency in mathematics essential for the acquisition of any knowledge at all. Similarly, Galilei saw mathematics as the ‘language of the Book of Nature’, Kant argued that mathematics provides essential insights into an entire class of human judgements, Cantor identified God with a particular class of transfinite numbers... In short: the idea that mathematics holds the key for understanding the fundamental structures of reality has, in different ways, dominated philosophy for centuries and also today many are convinced that the heuristic force of mathematics goes beyond its mere application in the natural sciences.
In current ontological, semantic, and epistemic debates that address these questions, mathematics frequently functions as a model for other a priori domains. Metaethicists in particular have started to use local structural parallels between mathematics and morality in order to corroborate particular metaethical views, but mathematics has also been argued to share relevant features with the domains of logic, modality, and religion. The mathematical analogies employed in those arguments share a common form: a local analogy between mathematics and another domain is identified, from which a global conclusion is drawn about one or both of the domains.
What is completely missing in these debates, however, is a discussion of
1. the plausibility of mathematics analogies in light of their mathematical background assumptions
2. how to develop an adequate methodology for analogical reasoning about a priori domains
3. the prospects of developing a unified framework for all a priori domains.
Only by answering these questions will it be possible to fully understand the evidential role mathematics has played for philosophers for centuries. This is the gap this project fills.
While early Mesopotamian and Egyptian mathematics developed in response to the practical needs of settled civilizations, such as taxation, the measurement of plots of land, or the calculation of lunar calendars, philosophers soon assigned mathematics the additional role of paradigm for human reasoning and knowledge. Plato, for example, considered mathematics the highest form of knowledge, sharply distinguished from our uncertain beliefs about the empirical world. He thought that mathematical knowledge consists in knowledge of eternal ‘Forms’, perfect ideas that constitute ultimate reality, and he considered proficiency in mathematics essential for the acquisition of any knowledge at all. Similarly, Galilei saw mathematics as the ‘language of the Book of Nature’, Kant argued that mathematics provides essential insights into an entire class of human judgements, Cantor identified God with a particular class of transfinite numbers... In short: the idea that mathematics holds the key for understanding the fundamental structures of reality has, in different ways, dominated philosophy for centuries and also today many are convinced that the heuristic force of mathematics goes beyond its mere application in the natural sciences.
In current ontological, semantic, and epistemic debates that address these questions, mathematics frequently functions as a model for other a priori domains. Metaethicists in particular have started to use local structural parallels between mathematics and morality in order to corroborate particular metaethical views, but mathematics has also been argued to share relevant features with the domains of logic, modality, and religion. The mathematical analogies employed in those arguments share a common form: a local analogy between mathematics and another domain is identified, from which a global conclusion is drawn about one or both of the domains.
What is completely missing in these debates, however, is a discussion of
1. the plausibility of mathematics analogies in light of their mathematical background assumptions
2. how to develop an adequate methodology for analogical reasoning about a priori domains
3. the prospects of developing a unified framework for all a priori domains.
Only by answering these questions will it be possible to fully understand the evidential role mathematics has played for philosophers for centuries. This is the gap this project fills.
(Please note that, due to the Covid-pandemic and my maternity leave, project dissemination activities such as conference presentations were severely limited.)
A personal career development plan was established; a project website created (https://silviajonas.com/mathematical-analogies/); the project was announced on PHILOS; I attended a course on project management and leadership by the LMU Mentoring program. Together with three colleagues, I discussed several new publications on the philosophy of set theory (reading group). Results of my research on mathematical analogies were presented at several conferences at the following universities: Bremen, Regensburg, Vienna, Linz Bayreuth, Birmingham. I deepened my knowledge of logic in an intensive course with Dr. Marta Sznajder (University of Amsterdam). A two-day conference on mathematical analogies was planned for June 2020 but had to be rescheduled twice due to the pandemic; it finally took place in June 2022 (conference website: https://silviajonas.wixsite.com/mathematicsanalogies).
Papers:
Published: 'Mathematical Indispensability and Arguments from Design' (Philosophia, 2020)
Revise and resubmit: 'Mathematical Pluralism and Indispensability'
In progress: 'What good are mathematical analogies?', 'Varieties of non-empirical confirmation', 'Mathematics as a philosophical heuristic', 'Mathematics and the limits of language'
Monograph (started) on the historical and contemporary role of mathematics in metaphysical debates.
A personal career development plan was established; a project website created (https://silviajonas.com/mathematical-analogies/); the project was announced on PHILOS; I attended a course on project management and leadership by the LMU Mentoring program. Together with three colleagues, I discussed several new publications on the philosophy of set theory (reading group). Results of my research on mathematical analogies were presented at several conferences at the following universities: Bremen, Regensburg, Vienna, Linz Bayreuth, Birmingham. I deepened my knowledge of logic in an intensive course with Dr. Marta Sznajder (University of Amsterdam). A two-day conference on mathematical analogies was planned for June 2020 but had to be rescheduled twice due to the pandemic; it finally took place in June 2022 (conference website: https://silviajonas.wixsite.com/mathematicsanalogies).
Papers:
Published: 'Mathematical Indispensability and Arguments from Design' (Philosophia, 2020)
Revise and resubmit: 'Mathematical Pluralism and Indispensability'
In progress: 'What good are mathematical analogies?', 'Varieties of non-empirical confirmation', 'Mathematics as a philosophical heuristic', 'Mathematics and the limits of language'
Monograph (started) on the historical and contemporary role of mathematics in metaphysical debates.
The most significant result of the project so far is that mathematical analogies in philosophy are crucially dependent on one's underlying picture of the mathematical cosmos and the notion of truth inherent in that picture. Much of my work during the project was invested into gaining a better understanding of mathematical pluralism and how to square this view with a realist commitment to mathematical ontology. As it turns out that there is not only one way of conceiving mathematical pluralism but several, developing a clear philosophical understanding of mathematical pluralism is expedient and must have priority over any further development of particular mathematical analogies. However, the expected result of the project will still be an account of the evidential role of mathematics analogies in current realism-antirealism debates, as well as a unified framework for tackling ontological, semantic, and epistemic questions about a priori domains. Yet these results will be based on a deeper understanding of the mathematical realm. These results will have a deep impact on realism-antirealism debates in the philosophy of mathematics, metaethics, philosophy of religion, and metaphysics, and may also affect neighbouring fields such as cognitive science and history of science. Moreover, it will impact a larger, non-scientific audience by showing how mathematics shapes our conceptualization of the fundamental structures of the abstract realms of reality.