Strict Finitism and the Logic of Mathematical Applications

1. Front Matter

   Pages i-xii

   PDF

2. Introduction

   • Feng Ye

   Pages 1-33

3. Strict Finitism

   • Feng Ye

   Pages 35-78

4. Calculus

   • Feng Ye

   Pages 79-112

5. Metric Space

   • Feng Ye

   Pages 113-123

6. Complex Analysis

   • Feng Ye

   Pages 125-147

7. Integration

   • Feng Ye

   Pages 149-169

8. Hilbert Space

   • Feng Ye

   Pages 171-215

9. Semi-Riemannian Geometry

   • Feng Ye

   Pages 217-266

10. Back Matter

    Pages 267-272

    PDF

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.

• Constructive Mathematics
• Constructivism
• Elementary Recursive Arithmetic
• Finitism
• Foundations of Mathematics
• Naturalism
• Nominalism
• Philosophy of Mathematics
• Physicalism
• Strict Finitism

"Strict finitism is a very attractive view that has generally suffered just from the sense that it couldn't reproduce enough mathematics. This book takes strides toward removing that worry and making the view a viable alternative." James Tappenden, University of Michigan, Ann Arbor, U.S.A.

• , Department of Philosophy, Peking University, Beijing, China, People's Republic

  Feng Ye

Feng Ye is a professor of philosophy at Peking University, China. He has a B.S. degree in mathematics from Xiamen University, China, and a Ph.D. degree in philosophy from Princeton University, U.S.A.. His research areas include constructive and finitistic mathematics, philosophy of mathematics, and philosophy of mind and language. He used to prove, for the first time, a constructive version of the spectral theorem and Stone’s theorem for unbounded linear operators on Hilbert spaces. He is currently developing a radically naturalistic, nominalistic, and strictly finitistic philosophy of mathematics, a naturalistic theory of content, and a naturalistic interpretation of modality. His research articles have been published in The Journal of Symbolic Logic, Philosophia Mathematica, and Synthese, among others. He is also the author of the book Philosophy of Mathematics in the 20th Century: a Naturalistic Commentary (in Chinese, Peking University Press, 2010). His philosophical interests revolve around naturalism.

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