more from this thinker
|
more from this text
Single Idea 9631
[filed under theme 6. Mathematics / C. Sources of Mathematics / 7. Formalism
]
Full Idea
Early formalism (Thomae etc) was crushed by Frege: first, mathematics must be about classes of symbols (abstract types), not the symbols themselves (the tokens); second, games may be meaningless, but meta-games are not.
Gist of Idea
Formalism fails to recognise types of symbols, and also meta-games
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by James Robert Brown - Philosophy of Mathematics Ch.5
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.64
A Reaction
Brown goes on to show how Hilbert revived the formalist project. A really austere formalist view of mathematics clearly seems to be missing something basic, either in physical nature, or in the world of ideas.
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]
|