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Philosophy of Mathematics/Mathematical Practice
In recent years, there has been growing attention on foundational theories of mathematics which are not set-theoretic. These alternative theories have conceptually different values to set-theory which I am interested in. One example is Homotopy Type Theory (HoTT). HoTT builds on Martin-Loef Type Theory, instead of classical first-order logic, and applies homotopical interpretation of types. The practical value of HoTT is understood to be its application on a computer for proof verification. However, HoTT offers a conceptually distinct foundation of mathematics. Although these theories can usually be logically interpreted by each other (with some additional axioms), the conceptual values of each theory are lost in the process. I am interested in understanding the differences in foundational theories in terms of their conceptual values found in mathematical practice.
Phenomenology
Phenomenology can be seen as the study of consciousness, or as a philosophical methodology. I am interested in both aspects of phenomenology. In the former sense of phenomenology, I explore the structure of intentionality and the relationship between the subject and the intended object of experience. Some questions that interest me are how do we experience other conscious beings? Can an AI be conscious? Fundamental to answering these questions is the concept of empathy. In the latter of sense phenomenology, I am interested in developing a phenomenological method for studying scientific practice. Husserl, himself, viewed the scientific practice as teleological (also having the structure of intentionality). A well-developed phenomenological method, I believe, could provide a phenomenological account of group identity, group knowledge and group consciousness.
Descartes/Early Modern Rationalism
Descartes’s contributions to both mathematics and philosophy are well-known to mathematicians and philosophers, respectively. However, the connection between the two is less developed in the current philosophical literature. My ongoing research considers how Descartes’s philosophical views on mathematics influenced his development of algebraic geometry, and the status of mathematics during his time affected his epistemology.
Mathematical Structuralism
There are two kinds of structuralisms in mathematics: ontological and methodological. The former is a philosophical view concerning the nature of mathematics and mathematical structures. The latter concerns properties or methods within mathematical practice which might be considered structural. I am interested in the relationship between these two kinds of structuralism.
Formal Theories of Truth
First-order Peano Arithmetic can be consistently extended with various axiom schemes of the truth predicate T. I am particularly interested in philosophical questions arising from the models of PA +T.