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Single Idea 8752

[filed under theme 6. Mathematics / C. Sources of Mathematics / 7. Formalism ]

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]

The 24 ideas with the same theme [maths is the consequences of a set of symbols]:

9887 Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett]
8751 Only applicability raises arithmetic from a game to a science [Frege]
9631 Formalism fails to recognise types of symbols, and also meta-games [Frege, by Brown,JR]
22293 Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Hilbert, by Potter]
12459 The subject matter of mathematics is immediate and clear concrete symbols [Hilbert]
10113 The grounding of mathematics is 'in the beginning was the sign' [Hilbert]
10115 Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman]
21570 Numbers are just verbal conveniences, which can be analysed away [Russell]
6424 Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess [Russell]
6425 Formalism can't apply numbers to reality, so it is an evasion [Russell]
13425 Formalism is hopeless, because it focuses on propositions and ignores concepts [Ramsey]
10154 Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics [Feferman/Feferman on Tarski]
1616 Formalism says maths is built of meaningless notations; these build into rules which have meaning [Quine]
10062 Formalism seems to exclude all creative, growing mathematics [Musgrave]
10063 Formalism is a bulwark of logical positivism [Musgrave]
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]
9629 For nomalists there are no numbers, only numerals [Brown,JR]
9630 The most brilliant formalist was Hilbert [Brown,JR]
9639 Does some mathematics depend entirely on notation? [Brown,JR]
22310 The formalist defence against Gödel is to reject his metalinguistic concept of truth [Potter]
23442 Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo]
8716 Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]