Book contents
- Philosophy of Mathematics after Wittgenstein
- Philosophy of Mathematics after Wittgenstein
- Copyright page
- Dedication
- Epigraph
- Contents
- Figures
- Acknowledgments
- Abbreviations
- Citation of Wittgenstein’s Works
- Introduction
- Chapter 1Wittgenstein’s Normativism about Mathematics
- Chapter 2Normativism as Eliminativism
- Chapter 3Basic Mathematical Rules (I)
- Chapter 4Basic Mathematical Rules (II)
- Chapter 5Advanced Rules
- Chapter 6Cantor’s Proof (I)
- Chapter 7Cantor’s Proof (II)
- Chapter 8Logicism, Number, and the Infinite
- Chapter 9Conclusion
- References
- Index
Chapter 3 - Basic Mathematical Rules (I)
Hardening
Published online by Cambridge University Press: 14 May 2026
Book contents
- Philosophy of Mathematics after Wittgenstein
- Philosophy of Mathematics after Wittgenstein
- Copyright page
- Dedication
- Epigraph
- Contents
- Figures
- Acknowledgments
- Abbreviations
- Citation of Wittgenstein’s Works
- Introduction
- Chapter 1Wittgenstein’s Normativism about Mathematics
- Chapter 2Normativism as Eliminativism
- Chapter 3Basic Mathematical Rules (I)
- Chapter 4Basic Mathematical Rules (II)
- Chapter 5Advanced Rules
- Chapter 6Cantor’s Proof (I)
- Chapter 7Cantor’s Proof (II)
- Chapter 8Logicism, Number, and the Infinite
- Chapter 9Conclusion
- References
- Index
Summary
In Chapter 3, we return to several questions asked at the end of Chapter 1: what, if any, are the grounds for the introduction of the rules we currently have in our mathematics books? Are they adopted arbitrarily? Why these rules, and not others? How does the introduction of such rules proceed? And so on. However, the chapter looks only at the ‘basic’ rules, such as the simplest addition rules (e.g., 2 + 2 = 4), the rules in the multiplication table (e.g., 3 × 2 = 6), and other simple multiplications, e.g., 21 × 36 = 756. These rules are ‘basic’ insofar as they emerge by ‘hardening’ certain empirical regularities displayed by people’s behavior in arranging and counting manageable physical objects; so, this is in essence a genealogical story. This chapter also serves as preparation for the discussion in the next chapter, where we tackle the possible challenges to these rules (by ‘skeptics’), with the aim of dissolving the problem of mathematical skepticism: what if we have always been wrong about our basic arithmetic?
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- Philosophy of Mathematics after Wittgenstein , pp. 85 - 110Publisher: Cambridge University PressPrint publication year: 2026
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