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Single Idea 17440

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle ]

Full Idea

Wright intends the claim that Hume's Principle (HP) embodies an explanation of the concept of number to imply that it is analytic of the concept of cardinal number - so it is an analytic or conceptual truth, much as a definition would be.

Clarification

HP says 1-1 correspondence means same number

Gist of Idea

Wright says Hume's Principle is analytic of cardinal numbers, like a definition

Source

report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1

Book Ref

-: 'Notre Dame Journal of Formal Logic' [-], p.189

A Reaction

Boolos is quoted as disagreeing. Wright is claiming a fundamental truth. Boolos says something can fix the character of something (as yellow fixes bananas), but that doesn't make it 'fundamental'. I want to defend 'fundamental'.

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]