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Single Idea 13896

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

Full Idea

We shall endeavour to see whether it is possible to follow through the strategy adumbrated in 'Grundlagen' for establishing the Peano Axioms without at any stage invoking classes.

Gist of Idea

The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes

Source

Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi)

Book Ref

Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.131

A Reaction

The key idea of neo-logicism. If you can avoid classes entirely, then set theory paradoxes become irrelevant, and classes aren't logic. Philosophers now try to derive the Peano Axioms from all sorts of things. Wright admits infinity is a problem.

Related Idea

Idea 13899 The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C]

The 36 ideas from 'Frege's Concept of Numbers as Objects'

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]