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Single Idea 14103
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
]
Full Idea
Pure mathematics is the class of all propositions of the form 'p implies q', where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants.
Gist of Idea
Pure mathematics is the class of propositions of the form 'p implies q'
Source
Bertrand Russell (The Principles of Mathematics [1903], §001)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.3
A Reaction
Linnebo calls Russell's view here 'deductive structuralism'. Russell gives (§5) as an example that Euclid is just whatever is deduced from his axioms.
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]
|
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]
|
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]
|
8655
|
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
|
Frege aimed to discover the logical foundations which justify arithmetical judgements
[Frege, by Burge]
|
8689
|
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
|
We tried to define all of pure maths using logical premisses and concepts
[Russell]
|
10037
|
'Principia' lacks a precise statement of the syntax
[Gödel on Russell/Whitehead]
|
8683
|
Russell and Whitehead were not realists, but embraced nearly all of maths in logic
[Russell/Whitehead, by Friend]
|
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
|
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
|
Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology
[Hale/Wright]
|
10568
|
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
|
The task of logicism was to define by logic the concepts 'number', 'successor' and '0'
[Linsky,B]
|
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]
|