Ultrafinitism and Infinity: A Physicalistic Examination

The thesis Physicalism about Cognitive Subjects (PCS) says that human cognitive subjects and processes are themselves physical things and processes, namely, human brain neural networks, neural processes, and their physical interactions with human environments. This paper will first introduce reasons supporting PCS and introduce my previous researches exploring implications of PCS, which include a nominalist, fictionalist and finitist account of human practices of classical mathematics. After these introductions, this paper will argue that PCS implies liberalism for mathematical practices, which can even legitimize logically inconsistent practices in some situations. Several philosophical questions about ultrafinitism, experimental mathematics and the set-theoretic multiverse view will then be answered in this context. Finally, since PCS has answered the ontological and epistemological questions about infinity in mathematics, this paper will offer some analyses on two other questions concerning infinity that are still open.