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Full Idea
The Peano Axioms are logical consequences of a statement constituting the core of an explanation of the notion of cardinal number. The infinity of cardinal numbers emerges as a consequence of the way cardinal number is explained.
Gist of Idea
The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals
Source
Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xix)
Book Ref
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.168
A Reaction
This, along with Idea 13896, nicely summarises the neo-logicist project. I tend to favour a strategy which starts from ordering, rather than identities (1-1), but an attraction is that this approach is closer to counting objects in its basics.
Related Ideas
Idea 13896 The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes [Wright,C]
Idea 17312 It is more explanatory if you show how a number is constructed from basic entities and relations [Koslicki]
| 21718 | Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B] |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| 7804 | Wright has revived Frege's discredited logicism [Wright,C, by Benardete,JA] |
| 13899 | The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C] |
| 13896 | The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes [Wright,C] |
| 10622 | The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright] |
| 8783 | Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright] |
| 12225 | Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright] |
| 10580 | Mathematics is both necessary and a priori because it really consists of logical truths [Yablo] |
| 9224 | Proceduralism offers a version of logicism with no axioms, or objects, or ontological commitment [Fine,K] |
| 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
| 18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
| 19295 | Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers [Hale] |
| 21648 | Neo-Fregeans are dazzled by a technical result, and ignore practicalities [Hofweber] |