Guillaume Massas
Welcome to my personal website! I am a sixth year PhD candidate in the Group in Logic and the Methodology of Science at UC Berkeley, working under the supervision of Wes Holliday and Paolo Mancosu. I defended my dissertation Duality and Infinity in October 2023. You can find a recording of the defense and a copy of the slides here.
My main research interests are in the application of categorical methods broadly construed to problems in logic, and in the history and philosophy of mathematics. I am particularly interested in possibility semantics, an approach to semantics that takes partiality to be a fundamental feature of semantic structures, and in its applications to the philosophy of the infinite and to a foundational framework called semiconstructive mathematics. Recently, I have been thinking about the following topics:
Constructive dualities in lattice theory: Dualities use the language of category theory to bring together algebraic and geometric structures, and often provide the robust mathematical machinery behind soundness and completeness proofs for many logics. I am particularly interested in dualities that can be established without the Axiom of Choice. I have developed such results in my papers Choice-free de Vries Duality and Duality for Fundamental Logic (in preparation).
Possibility semantics for non-classical logics: Possibility semantics is an alternative semantic framework in which the fundamental objects are partial possibilities rather than completely determined objects such as possible worlds. My paper B-frame Duality explores the potential of this approach for the semantics of intuitionistic logic, while my paper Orthologic and the Open Future (under review) discusses an application of possibility semantics to an old problem of Aristotle.
Infinitesimals: Quantities that are infinitely small yet non-zero have a long and troubled history. Although the modern foundations of analysis avoid any mention of them, infinitesimals were given a rigorous treatment in Robinson's nonstandard analysis, and have played a role in recent debates in the philosophy of probability theory. I have argued for a semiconstructive treatment of infinitesimals in my paper A Semiconstructive Approach to the Hyperreal Line, and my papers Totality, Regularity and Cardinality in Probability Theory (with Paolo Mancosu) and Infinitesimal Credences without the Axiom of Choice (in preparation) discuss the use of infinitesimals in probability theory.
Part-whole alternatives to the Cantorian transfinite: The modern notion of cardinality offers a rigorous extension of the concept of size from the finite to the infinite that is based on the intuition that two sets have the same size if and only if there is a one-to-one correspondence between them. Historically, there is however an alternative approach, based on the intuition that a whole is always strictly greater in size than any of its proper parts. My paper Bolzano's Mathematical Infinite (with Anna Bellomo) offers a rigorous formalization of such a view presented by 19th century mathematician Bernard Bolzano, while my paper Possibility Semantics and Galileo's Paradox (in preparation) offers a new alternative to the Cantorian notion of size that refines a recent proposal called the theory of numerosities.