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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism

[first developments of the logicist idea]

31 ideas
Mathematics is close to logic, but is even more abstract [Peirce]
I hold that algebra and number are developments of logic [Jevons]
Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege]
My Basic Law V is a law of pure logic [Frege]
Logicism shows that no empirical truths are needed to justify arithmetic [George/Velleman on Frege]
Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Hale/Wright on Frege]
Numbers are definable in terms of mapping items which fall under concepts [Scruton on Frege]
Arithmetic is analytic [Weiner on Frege]
Arithmetic must be based on logic, because of its total generality [Jeshion on Frege]
Arithmetic is analytic and a priori, and thus it is part of logic [Frege]
The loss of my Rule V seems to make foundations for arithmetic impossible [Frege]
Frege aimed to discover the logical foundations which justify arithmetical judgements [Burge on Frege]
Eventually Frege tried to found arithmetic in geometry instead of in logic [Friend on Frege]
For Russell, numbers are sets of equivalent sets [Benacerraf on Russell]
Maths can be deduced from logical axioms and the logic of relations [Russell]
We tried to define all of pure maths using logical premisses and concepts [Russell]
Russell and Whitehead took arithmetic to be higher-order logic [Hodes on Russell/Whitehead]
'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead]
Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Friend on Russell/Whitehead]
Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
Arithmetic can have even simpler logical premises than the Peano Axioms [Russell]
Logical positivists incorporated geometry into logicism, saying axioms are just definitions [Shapiro on Carnap]
Questions about numbers are answered by analysis, and are analytic, and hence logically true [Carnap]
Two and one making three has the necessity of logical inference [Wittgenstein]
Maths and logic are true universally because they are analytic or tautological [Ayer]
If mathematics follows from definitions, then it is conventional, and part of logic [Quine]
Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Bostock]
Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright]
Logicists say mathematics can be derived from definitions, and can be known that way [Fine,K]
Russell needed three extra axioms to reduce maths to logic: infinity, choice and reducibility [Grayling]
The logicists held that is-a-member-of is a logical constant, making set theory part of logic [Orenstein]