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Single Idea 13663
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
]
Full Idea
Some authors (Poincaré and Russell, for example) were disposed to reject properties that are not definable, or are definable only impredicatively.
Clarification
'Impredicative' definitions tend towards circularity
Gist of Idea
Some reject formal properties if they are not defined, or defined impredicatively
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.174
A Reaction
I take Quine to be the culmination of this line of thought, with his general rejection of 'attributes' in logic and in metaphysics.
The
19 ideas
with the same theme
[maths entities only allowed if freshly defined]:
18203
|
Avoid non-predicative classifications and definitions
[Poincaré]
|
8746
|
To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it
[Russell/Whitehead, by Shapiro]
|
21568
|
A one-variable function is only 'predicative' if it is one order above its arguments
[Russell]
|
18126
|
A set does not exist unless at least one of its specifications is predicative
[Russell, by Bostock]
|
18128
|
Russell is a conceptualist here, saying some abstracta only exist because definitions create them
[Russell, by Bostock]
|
18124
|
Vicious Circle says if it is expressed using the whole collection, it can't be in the collection
[Russell, by Bostock]
|
21559
|
We need rules for deciding which norms are predicative (unless none of them are)
[Russell]
|
21558
|
'Predicative' norms are those which define a class
[Russell]
|
8747
|
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities
[Gödel, by Shapiro]
|
10045
|
Impredicative definitions are admitted into ordinary mathematics
[Gödel]
|
18131
|
If abstracta only exist if they are expressible, there can only be denumerably many of them
[Bostock]
|
18134
|
Predicativism makes theories of huge cardinals impossible
[Bostock]
|
18135
|
If mathematics rests on science, predicativism may be the best approach
[Bostock]
|
18136
|
If we can only think of what we can describe, predicativism may be implied
[Bostock]
|
18133
|
The usual definitions of identity and of natural numbers are impredicative
[Bostock]
|
18132
|
The predicativity restriction makes a difference with the real numbers
[Bostock]
|
13663
|
Some reject formal properties if they are not defined, or defined impredicatively
[Shapiro]
|
8730
|
'Impredicative' definitions refer to the thing being described
[Shapiro]
|
15370
|
Predicativism says mathematical definitions must not include the thing being defined
[Horsten]
|