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Single Idea 8744

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique ]

Full Idea

The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter.

Gist of Idea

Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own

Source

Stewart Shapiro (Thinking About Mathematics [2000], 5.1)

Book Ref

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.113

A Reaction

This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism.

The 15 ideas from 'Thinking About Mathematics'

8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
8730 'Impredicative' definitions refer to the thing being described [Shapiro]
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
8752 Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
8764 Categories are the best foundation for mathematics [Shapiro]
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]