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Single Idea 12459
[filed under theme 6. Mathematics / C. Sources of Mathematics / 7. Formalism
]
Full Idea
The subject matter of mathematics is the concrete symbols themselves whose structure is immediately clear and recognisable.
Gist of Idea
The subject matter of mathematics is immediate and clear concrete symbols
Source
David Hilbert (On the Infinite [1925], p.192)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.192
A Reaction
I don't think many people will agree with Hilbert here. Does he mean token-symbols or type-symbols? You can do maths in your head, or with different symbols. If type-symbols, you have to explain what a type is.
The
24 ideas
with the same theme
[maths is the consequences of a set of symbols]:
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9887
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Formalism misunderstands applications, metatheory, and infinity
[Frege, by Dummett]
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8751
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Only applicability raises arithmetic from a game to a science
[Frege]
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9631
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Formalism fails to recognise types of symbols, and also meta-games
[Frege, by Brown,JR]
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22293
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Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency
[Hilbert, by Potter]
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12459
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The subject matter of mathematics is immediate and clear concrete symbols
[Hilbert]
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10113
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The grounding of mathematics is 'in the beginning was the sign'
[Hilbert]
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10115
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Hilbert substituted a syntactic for a semantic account of consistency
[Hilbert, by George/Velleman]
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21570
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Numbers are just verbal conveniences, which can be analysed away
[Russell]
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6424
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Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess
[Russell]
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6425
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Formalism can't apply numbers to reality, so it is an evasion
[Russell]
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13425
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Formalism is hopeless, because it focuses on propositions and ignores concepts
[Ramsey]
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10154
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Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics
[Feferman/Feferman on Tarski]
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1616
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Formalism says maths is built of meaningless notations; these build into rules which have meaning
[Quine]
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10062
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Formalism seems to exclude all creative, growing mathematics
[Musgrave]
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10063
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Formalism is a bulwark of logical positivism
[Musgrave]
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8749
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Term Formalism says mathematics is just about symbols - but real numbers have no names
[Shapiro]
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8750
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Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
[Shapiro]
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8752
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Deductivism says mathematics is logical consequences of uninterpreted axioms
[Shapiro]
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9629
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For nomalists there are no numbers, only numerals
[Brown,JR]
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9630
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The most brilliant formalist was Hilbert
[Brown,JR]
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9639
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Does some mathematics depend entirely on notation?
[Brown,JR]
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22310
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The formalist defence against Gödel is to reject his metalinguistic concept of truth
[Potter]
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23442
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Game Formalism has no semantics, and Term Formalism reduces the semantics
[Linnebo]
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8716
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Formalism is unconstrained, so cannot indicate importance, or directions for research
[Friend]
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