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Single Idea 8782

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism ]

Full Idea

Since Frege's defence of his thesis that the laws of arithmetic are analytic depended upon a realm of independently existing objects - the finite cardinal numbers and the real numbers - his view amounted to a Platonist version of logicism.

Gist of Idea

Frege offered a Platonist version of logicism, committed to cardinal and real numbers

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by B Hale / C Wright - Logicism in the 21st Century 1

Book Ref

'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.166

A Reaction

Nice to have this spelled out. Along with Gödel, Frege is the most distinguished Platonist since the great man. Frege has lots of modern fans, but I would have thought that this makes his position a non-starter. Alternatives are needed.

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]
7739 Arithmetic is analytic [Frege, by Weiner]
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]
16905 Arithmetic must be based on logic, because of its total generality [Frege, by Jeshion]
5658 Numbers are definable in terms of mapping items which fall under concepts [Frege, by Scruton]
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]
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]
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]