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Full Idea
The modern extensional notion of function is just an arbitrary correspondence between collections.
Gist of Idea
A function is just an arbitrary correspondence between collections
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 1)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.24
A Reaction
Shapiro links this with the idea that a set is just an arbitrary collection. These minimalist concepts seem like a reaction to a general failure to come up with a more useful and common sense definition.
| 8490 | First-level functions have objects as arguments; second-level functions take functions as arguments [Frege] |
| 21566 | 'Propositional functions' are ambiguous until the variable is given a value [Russell] |
| 18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |
| 13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
| 13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| 13700 | A 'total' function must always produce an output for a given domain [Sider] |
| 15105 | F(x) walked into a bar. The barman said.. [Sommers,W] |