By Feng Ye

This ebook intends to teach that radical naturalism (or physicalism), nominalism and strict finitism account for the purposes of classical arithmetic in present clinical theories. The utilized mathematical theories constructed within the publication comprise the fundamentals of calculus, metric house idea, complicated research, Lebesgue integration, Hilbert areas, and semi-Riemann geometry (sufficient for the functions in classical quantum mechanics and normal relativity). the truth that lots utilized arithmetic should be built inside this sort of susceptible, strictly finitistic procedure, is unbelievable in itself. It additionally exhibits that the functions of these classical theories to the finite actual international may be translated into the functions of strict finitism, which demonstrates the applicability of these classical theories with no assuming the literal fact of these theories or the truth of infinity.

Both expert researchers and scholars of philosophy of arithmetic will gain vastly from studying this book.

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Extra resources for Strict Finitism and the Logic of Mathematical Applications

Sample text

This should not be very surprising. After all, from the point of view of a naturalistic observer, humans are actually using their brains, assisted by paper-and-pencils or computers, to simulate other physical entities when they apply classical mathematics to those physical entities. The only puzzle for logicians is that when humans use classical mathematical concepts and thoughts that appear committed to infinity, the logic of how those concepts and thoughts simulate finite physical entities is not very clear.

That is why the finitistic version of (M-2) can be derived from (R-2) and (B-1). We will see that in general, conditions such as continuity and differentiability in strict finitism are not committed to any idealization to infinity. They can be interpreted into literally true conditions about discrete physical quantities, as long as those discrete quantities are ‘smooth at the macro-scale’. Note that this is the intuitive reason why we can simulate a population growth curve by a differentiable curve.

These imply that we can translate the derivation from (M-1), (M-2) to (M-3) in strict finitism rather straightforwardly into a derivation from (R-1), (R-2) to (R-3). In Chap. 3 I will explain how a continuity or differentiability condition in strict finitism can be satisfied by discrete quantities, and I will come back to this example in the last section of Chap. 3, to add more details on the finitistic version of (M-2) and the finitistic derivation of (M-3) from (M-1) and (M-2). Chapter 2 Strict Finitism This chapter presents the logical framework for strict finitism.

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