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CEFISES Seminar: Stéphanie Ponsar, “Foundations of mathematics in terms of fundamentality and of grounding?”
Speaker: Stéphanie Ponsar
Title: “Foundations of mathematics in terms of fundamentality and of grounding?”
Although set theory is universally accepted as a theory of foundations of mathematics, there are several conceptions of foundations of mathematics. Starting from their main characteristics, we will discuss foundations of mathematics as metaphysical fundamentality and as grounding.
Metaphysical fundamentality can be thought of as something basic or primitive. Several varieties of metaphysical fundamentality exist. In this talk, we will only focus on the ones we have identified as relevant for our purpose. In particular, metaphysical fundamentality can be interpreted as providing a complete minimal basis for the description of reality. In this case, fundamental entities must include all and only the fundamental entities which determine everything else. Another standpoint on fundamentality is primitivism which states that fundamentality cannot be defined but can be characterised. Another task of fundamentality is to capture the idea that there is a foundation of being and that everything else depends on these fundamental entities and properties.
Grounding can be thought of as realist metaphysics. The objective of the study of grounding is to better understand the relation “what grounds what”. Grounding claims can be formulated in different manners. An important one distinguishes between explanation-based grounding and determination-based grounding. Each of these two types can be further separated between full and partial grounding. Another central feature of grounding is well-foundedness which ensures that any grounded fact is ultimately grounded by ungrounded facts.
We examine how the relationship “to be a foundation of mathematics” can be interpreted in terms of different accounts of metaphysical fundamentality (complete minimal basis, primitivism) and in terms of grounding. We also show that set theory and category theory have characteristics from both metaphysical fundamentality and grounding.