*Paolo Mancosu*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198746829
- eISBN:
- 9780191809095
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198746829.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, History of Philosophy

In this chapter Mancosu generalizes some worries, raised by Richard Heck, emerging from the theory of numerosities (discussed in Chapter 3) to a line of thought resulting in what he calls a ‘good ...
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In this chapter Mancosu generalizes some worries, raised by Richard Heck, emerging from the theory of numerosities (discussed in Chapter 3) to a line of thought resulting in what he calls a ‘good company’ objection to Hume’s Principle (HP). The chapter is centered around five main parts. The first takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where Mancosu shows that there are countably-infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which one can derive the axioms of second-order arithmetic. All the principles he presents agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section Mancosu offers a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, the fifth part makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.Less

In this chapter Mancosu generalizes some worries, raised by Richard Heck, emerging from the theory of numerosities (discussed in Chapter 3) to a line of thought resulting in what he calls a ‘good company’ objection to Hume’s Principle (HP). The chapter is centered around five main parts. The first takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where Mancosu shows that there are countably-infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which one can derive the axioms of second-order arithmetic. All the principles he presents agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section Mancosu offers a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, the fifth part makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

*Paolo Mancosu*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198746829
- eISBN:
- 9780191809095
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198746829.003.0002
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, History of Philosophy

The first chapter is devoted to showing that, contrary to what Frege intimates in the Grundlagender Arithmetik, abstraction principles were quite widespread in the mathematical practice that preceded ...
More

The first chapter is devoted to showing that, contrary to what Frege intimates in the Grundlagender Arithmetik, abstraction principles were quite widespread in the mathematical practice that preceded Frege’s discussion of them. Mancosu cites extensively from nineteenth century sources in number theory, geometry, algebra, vector theory, set theory and foundations of the number systems, in order to show the widespread use of abstraction principles in the mathematical practice of the time and to articulate the foundational problems such principles gave rise to.Less

The first chapter is devoted to showing that, contrary to what Frege intimates in the *Grundlagen**der Arithmetik*, abstraction principles were quite widespread in the mathematical practice that preceded Frege’s discussion of them. Mancosu cites extensively from nineteenth century sources in number theory, geometry, algebra, vector theory, set theory and foundations of the number systems, in order to show the widespread use of abstraction principles in the mathematical practice of the time and to articulate the foundational problems such principles gave rise to.