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Single Idea 13863
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
]
Full Idea
Most would cite Russell's paradox, the non-logical character of the axioms which Russell and Whitehead's reconstruction of Frege's enterprise was constrained to employ, and the incompleteness theorems of Gödel, as decisive for logicism's failure.
Gist of Idea
Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms
Source
Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
Book Ref
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.-1
Related Idea
Idea 13895
The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
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]
|
13865
|
'Sortal' concepts show kinds, use indefinite articles, and require grasping identities
[Wright,C]
|
13866
|
A concept is only a sortal if it gives genuine identity
[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]
|