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Explore a philosophical lecture that examines the relationship between mathematical methodology and philosophical inquiry, focusing on how an 'as-if' interpretation of mathematics can illuminate both its practice and applicability. Starting with Platonic perspectives, learn how the conflation of mathematical and philosophical methods has led to persistent challenges in understanding mathematical foundations. Discover the concept of 'as-ifism' - treating mathematical hypotheses as if they were true first principles for solving mathematical rather than philosophical problems. Follow the evolution from as-if hypotheses to as-if axioms in modern mathematics, and understand the distinction between as-ifism and if-thenism in developing a structural as-ifist position. Examine how methodological considerations, rather than metaphysical ones, condition our assumptions about mathematical axioms and meta-mathematical theory. Conclude with a fresh perspective on the foundational debate between set theory and category theory through the lens of methodological as-ifism.
