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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
36 ideas
from 'Frege's Concept of Numbers as Objects'
10142
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The attempt to define numbers by contextual definition has been revived
[Wright,C, by Fine,K]
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9868
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An expression refers if it is a singular term in some true sentences
[Wright,C, by Dummett]
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7804
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Wright has revived Frege's discredited logicism
[Wright,C, by Benardete,JA]
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17441
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Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are
[Wright,C, by Heck]
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10140
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We derive Hume's Law from Law V, then discard the latter in deriving arithmetic
[Wright,C, by Fine,K]
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8692
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Frege has a good system if his 'number principle' replaces his basic law V
[Wright,C, by Friend]
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17440
|
Wright says Hume's Principle is analytic of cardinal numbers, like a definition
[Wright,C, by Heck]
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9878
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Contextually defined abstract terms genuinely refer to objects
[Wright,C, by Dummett]
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13863
|
Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms
[Wright,C]
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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]
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13861
|
Number theory aims at the essence of natural numbers, giving their nature, and the epistemology
[Wright,C]
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13860
|
We can only learn from philosophers of the past if we accept the risk of major misrepresentation
[Wright,C]
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13867
|
Instances of a non-sortal concept can only be counted relative to a sortal concept
[Wright,C]
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13869
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Number platonism says that natural number is a sortal concept
[Wright,C]
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13870
|
We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism
[Wright,C]
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13868
|
Sortal concepts cannot require that things don't survive their loss, because of phase sortals
[Wright,C]
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13865
|
'Sortal' concepts show kinds, use indefinite articles, and require grasping identities
[Wright,C]
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13866
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A concept is only a sortal if it gives genuine identity
[Wright,C]
|
13873
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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
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We can accept Frege's idea of object without assuming that predicates have a reference
[Wright,C]
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13882
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A milder claim is that understanding requires some evidence of that understanding
[Wright,C]
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13883
|
The best way to understand a philosophical idea is to defend it
[Wright,C]
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13884
|
The idea that 'exist' has multiple senses is not coherent
[Wright,C]
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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]
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13888
|
If numbers are extensions, Frege must first solve the Caesar problem for extensions
[Wright,C]
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13893
|
It is 1-1 correlation of concepts, and not progression, which distinguishes natural number
[Wright,C]
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13892
|
One could grasp numbers, and name sizes with them, without grasping ordering
[Wright,C]
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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]
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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]
|