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Single Idea 5658

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

Full Idea

Frege defines numbers in terms of 'equinumerosity', which says two concepts are equinumerous if the items falling under one of them can be placed in one-to-one correspondence with the items falling under the other.

Gist of Idea

Numbers are definable in terms of mapping items which fall under concepts

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Roger Scruton - Short History of Modern Philosophy Ch.17

Book Ref

Scruton,Roger: 'A Short History of Modern Philosophy' [ARK 1985], p.246

A Reaction

This doesn't sound quite enough. What is the difference between three and four? The extensions of items generate separate sets, but why does one follow the other, and how do you count the items to get the one-to-one correspondence?

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]
7739 Arithmetic is analytic [Frege, by Weiner]
9945 Logicism shows that no empirical truths are needed to justify arithmetic [Frege, by George/Velleman]
16905 Arithmetic must be based on logic, because of its total generality [Frege, by Jeshion]
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]
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]
10025 Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes]
10037 'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead]
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]