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6. Mathematics / C. Sources of Mathematics / 7. Formalism

[maths is the consequences of a set of symbols]

20 ideas
Formalism misunderstands applications, metatheory, and infinity [Dummett on Frege]
Only applicability raises arithmetic from a game to a science [Frege]
Formalism fails to recognise types of symbols, and also meta-games [Brown,JR on Frege]
The subject matter of mathematics is immediate and clear concrete symbols [Hilbert]
Hilbert substituted a syntactic for a semantic account of consistency [George/Velleman on Hilbert]
The grounding of mathematics is 'in the beginning was the sign' [Hilbert]
Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess [Russell]
Formalism can't apply numbers to reality, so it is an evasion [Russell]
Formalism is hopeless, because it focuses on propositions and ignores concepts [Ramsey]
Formalism says maths is built of meaningless notations; these build into rules which have meaning [Quine]
Formalism is a bulwark of logical positivism [Musgrave]
Formalism seems to exclude all creative, growing mathematics [Musgrave]
Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
The most brilliant formalist was Hilbert [Brown,JR]
For nomalists there are no numbers, only numerals [Brown,JR]
Does some mathematics depend entirely on notation? [Brown,JR]
Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics [Feferman/Feferman]
Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]