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Single Idea 10025
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
]
Full Idea
Russell and Whitehead took arithmetic to be higher-order logic, ..and came close to identifying numbers with numerical quantifiers.
Gist of Idea
Russell and Whitehead took arithmetic to be higher-order logic
Source
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.148
Book Ref
-: 'Journal of Philosophy' [-], p.148
A Reaction
The point here is 'higher-order'.
The
33 ideas
with the same theme
[first developments of the logicist idea]:
14788
|
Mathematics is close to logic, but is even more abstract
[Peirce]
|
8628
|
I hold that algebra and number are developments of logic
[Jevons]
|
8487
|
Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism
[Frege]
|
18165
|
My Basic Law V is a law of pure logic
[Frege]
|
13608
|
Mathematics has no special axioms of its own, but follows from principles of logic (with definitions)
[Frege, by Bostock]
|
5658
|
Numbers are definable in terms of mapping items which fall under concepts
[Frege, by Scruton]
|
16905
|
Arithmetic must be based on logic, because of its total generality
[Frege, by Jeshion]
|
7739
|
Arithmetic is analytic
[Frege, by Weiner]
|
9945
|
Logicism shows that no empirical truths are needed to justify arithmetic
[Frege, by George/Velleman]
|
8782
|
Frege offered a Platonist version of logicism, committed to cardinal and real numbers
[Frege, by Hale/Wright]
|
8655
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Arithmetic is analytic and a priori, and thus it is part of logic
[Frege]
|
18166
|
The loss of my Rule V seems to make foundations for arithmetic impossible
[Frege]
|
16880
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Frege aimed to discover the logical foundations which justify arithmetical judgements
[Frege, by Burge]
|
8689
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Eventually Frege tried to found arithmetic in geometry instead of in logic
[Frege, by Friend]
|
17635
|
Arithmetic can have even simpler logical premises than the Peano Axioms
[Russell on Peano]
|
13414
|
For Russell, numbers are sets of equivalent sets
[Russell, by Benacerraf]
|
6108
|
Maths can be deduced from logical axioms and the logic of relations
[Russell]
|
6423
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We tried to define all of pure maths using logical premisses and concepts
[Russell]
|
8683
|
Russell and Whitehead were not realists, but embraced nearly all of maths in logic
[Russell/Whitehead, by Friend]
|
10037
|
'Principia' lacks a precise statement of the syntax
[Gödel on Russell/Whitehead]
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10025
|
Russell and Whitehead took arithmetic to be higher-order logic
[Russell/Whitehead, by Hodes]
|
14103
|
Pure mathematics is the class of propositions of the form 'p implies q'
[Russell]
|
8748
|
Logical positivists incorporated geometry into logicism, saying axioms are just definitions
[Carnap, by Shapiro]
|
13936
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Questions about numbers are answered by analysis, and are analytic, and hence logically true
[Carnap]
|
11073
|
Two and one making three has the necessity of logical inference
[Wittgenstein]
|
5202
|
Maths and logic are true universally because they are analytic or tautological
[Ayer]
|
8993
|
If mathematics follows from definitions, then it is conventional, and part of logic
[Quine]
|
8788
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Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology
[Hale/Wright]
|
10568
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Logicists say mathematics can be derived from definitions, and can be known that way
[Fine,K]
|
6408
|
Russell needed three extra axioms to reduce maths to logic: infinity, choice and reducibility
[Grayling]
|
21723
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The task of logicism was to define by logic the concepts 'number', 'successor' and '0'
[Linsky,B]
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8473
|
The logicists held that is-a-member-of is a logical constant, making set theory part of logic
[Orenstein]
|
21647
|
Logicism makes sense of our ability to know arithmetic just by thought
[Hofweber]
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