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This book intends to show that radical naturalism (or physicalism),
nominalism and strict finitism account for the applications of
classical mathematics in current scientific theories. The applied
mathematical theories developed in the book include the basics of
calculus, metric space theory, complex analysis, Lebesgue
integration, Hilbert spaces, and semi-Riemann geometry (sufficient
for the applications in classical quantum mechanics and general
relativity). The fact that so much applied mathematics can be
developed within such a weak, strictly finitistic system, is
surprising in itself. It also shows that the applications of those
classical theories to the finite physical world can be translated
into the applications of strict finitism, which demonstrates the
applicability of those classical theories without assuming the
literal truth of those theories or the reality of infinity. Both
professional researchers and students of philosophy of mathematics
will benefit greatly from reading this book.
This book demonstrates how a radical version of physicalism
('No-Self Physicalism') can offer an internally coherent and
comprehensive philosophical worldview. It first argues that a
coherent physicalist should explicitly treat a cognitive subject
merely as a physical thing and should not vaguely assume an
amorphous or even soul-like subject or self. This approach forces
the physicalist to re-examine traditional core philosophical
notions such as truth, analyticity, modality, apriority because our
traditional understandings of them appear to be predicated on a
cognitive subject that is not literally just a physical thing. In
turn, working on the assumption that a cognitive subject is itself
completely physical, namely a neural network-based robot programmed
by evolution (hence the term 'No-Self'), the book proposes
physicalistic theories on conceptual representation, truth,
analyticity, modality, the nature of mathematics, epistemic
justification, knowledge, apriority and intuition, as well as a
physicalistic ontology. These are meant to show that this No-Self
Physicalism, perhaps the most minimalistic and radical version of
physicalism proposed to date, can accommodate many aspects that
have traditionally interested philosophers. Given its refreshingly
radical approach and painstakingly developed content, the book is
of interest to anyone who is seeking a coherent philosophical
worldview in this age of science.
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