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Single Idea 13899

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism ]

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]

The 38 ideas from Crispin Wright

10142 The attempt to define numbers by contextual definition has been revived [Wright,C, by Fine,K]
9868 An expression refers if it is a singular term in some true sentences [Wright,C, by Dummett]
7804 Wright has revived Frege's discredited logicism [Wright,C, by Benardete,JA]
17441 Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck]
10140 We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K]
8692 Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend]
17440 Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck]
9878 Contextually defined abstract terms genuinely refer to objects [Wright,C, by Dummett]
13863 Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C]
13862 There are five Peano axioms, which can be expressed informally [Wright,C]
17853 Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C]
17854 What facts underpin the truths of the Peano axioms? [Wright,C]
13861 Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
13860 We can only learn from philosophers of the past if we accept the risk of major misrepresentation [Wright,C]
13867 Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C]
13869 Number platonism says that natural number is a sortal concept [Wright,C]
13870 We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C]
13868 Sortal concepts cannot require that things don't survive their loss, because of phase sortals [Wright,C]
13866 A concept is only a sortal if it gives genuine identity [Wright,C]
13865 'Sortal' concepts show kinds, use indefinite articles, and require grasping identities [Wright,C]
13873 Treating numbers adjectivally is treating them as quantifiers [Wright,C]
13877 Singular terms in true sentences must refer to objects; there is no further question about their existence [Wright,C]
17857 We can accept Frege's idea of object without assuming that predicates have a reference [Wright,C]
13882 A milder claim is that understanding requires some evidence of that understanding [Wright,C]
13883 The best way to understand a philosophical idea is to defend it [Wright,C]
13884 The idea that 'exist' has multiple senses is not coherent [Wright,C]
13885 If apparent reference can mislead, then so can apparent lack of reference [Wright,C]
13890 Entities fall under a sortal concept if they can be used to explain identity statements concerning them [Wright,C]
13888 If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C]
13893 It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C]
13892 One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
13894 Sameness of number is fundamental, not counting, despite children learning that first [Wright,C]
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]
13895 The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
13898 If we can establish directions from lines and parallelism, we were already committed to directions [Wright,C]
7320 Holism cannot give a coherent account of scientific methodology [Wright,C, by Miller,A]
12189 Logical necessity involves a decision about usage, and is non-realist and non-cognitive [Wright,C, by McFetridge]