*Eric Snyder and Stewart Shapiro*

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198712084
- eISBN:
- 9780191780240
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198712084.003.0014
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory ...
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This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.Less

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece *Grundgesetze der Arithmetik*. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.

*Rafael E. Núñez*

- Published in print:
- 2010
- Published Online:
- August 2013
- ISBN:
- 9780262014601
- eISBN:
- 9780262289795
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262014601.003.0012
- Subject:
- Philosophy, Philosophy of Mind

This chapter argues that cases wherein an empirically observable physical reality is lacking—when there is no pregiven world to be mentally re-presented—are provided by mathematics. In particular, it ...
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This chapter argues that cases wherein an empirically observable physical reality is lacking—when there is no pregiven world to be mentally re-presented—are provided by mathematics. In particular, it argues that mathematical infinity, as a form of cognition which by definition is not directly available to experience due to the finite nature of living systems, is an excellent candidate for fully exploring the power of enaction as a paradigm for cognitive science. The chapter focuses on a particular form of actual infinity—or infinity as a complete entity—namely, the transfinite cardinal numbers as they were conceived by one of the most controversial characters in the history of mathematics, the nineteenth-century mathematician Georg Cantor.Less

This chapter argues that cases wherein an empirically observable physical reality is lacking—when there is no pregiven world to be mentally re-presented—are provided by mathematics. In particular, it argues that mathematical infinity, as a form of cognition which by definition is not directly available to experience due to the finite nature of living systems, is an excellent candidate for fully exploring the power of enaction as a paradigm for cognitive science. The chapter focuses on a particular form of actual infinity—or infinity as a complete entity—namely, the transfinite cardinal numbers as they were conceived by one of the most controversial characters in the history of mathematics, the nineteenth-century mathematician Georg Cantor.

*Ian Rumfitt*

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198792161
- eISBN:
- 9780191866876
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198792161.003.0010
- Subject:
- Philosophy, Moral Philosophy

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the ...
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This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the Burali-Forti Paradox) when combined with impredicative second-order logic. It continues by arguing that the fault lies in the use of impredicative logic rather than in the abstraction principle per se. As the focus is on a form of predicative logic which reflects a philosophical diagnosis of the source of the paradox, the chapter considers how far Hale and Wright’s neo-logicist programme in cardinal arithmetic can be carried out in that logic.Less

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the Burali-Forti Paradox) when combined with impredicative second-order logic. It continues by arguing that the fault lies in the use of impredicative logic rather than in the abstraction principle *per se*. As the focus is on a form of predicative logic which reflects a philosophical diagnosis of the source of the paradox, the chapter considers how far Hale and Wright’s neo-logicist programme in cardinal arithmetic can be carried out in that logic.

- Published in print:
- 2008
- Published Online:
- May 2014
- ISBN:
- 9781846311314
- eISBN:
- 9781781380680
- Item type:
- chapter

- Publisher:
- Liverpool University Press
- DOI:
- 10.5949/UPO9781846315596.009
- Subject:
- Society and Culture, Cultural Studies

Manx cardinal and ordinal numerals can occur with or without a noun. This chapter discusses cardinal numbers; cardinal numbers with noun; duals; nouns of measure after numeral; numerals preceding ...
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Manx cardinal and ordinal numerals can occur with or without a noun. This chapter discusses cardinal numbers; cardinal numbers with noun; duals; nouns of measure after numeral; numerals preceding nouns; nouns embedded in compounds; numerals with dy; numerals used as nouns; ordinal numbers; fractions; telling the time; and mathematical terms.Less

Manx cardinal and ordinal numerals can occur with or without a noun. This chapter discusses cardinal numbers; cardinal numbers with noun; duals; nouns of measure after numeral; numerals preceding nouns; nouns embedded in compounds; numerals with *dy*; numerals used as nouns; ordinal numbers; fractions; telling the time; and mathematical terms.