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Single Idea 10113
[filed under theme 6. Mathematics / C. Sources of Mathematics / 7. Formalism
]
Full Idea
The solid philosophical attitude that I think is required for the grounding of pure mathematics is this: In the beginning was the sign.
Gist of Idea
The grounding of mathematics is 'in the beginning was the sign'
Source
David Hilbert (works [1900]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.148
A Reaction
Why did people invent those particular signs? Presumably they were meant to designate something, in the world or in our experience.
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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