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Full Idea
The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms.
Gist of Idea
Deductivism says mathematics is logical consequences of uninterpreted axioms
Source
Stewart Shapiro (Thinking About Mathematics [2000], 6.2)
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.149
A Reaction
[Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right.
Related Ideas
Idea 8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
Idea 8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
Idea 10061 The If-thenist view only seems to work for the axiomatised portions of mathematics [Musgrave]
| 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] |