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Full Idea
We seem to understand some statements about all sets (e.g. 'for every set x and every set y, there is a set z which is the union of x and y').
Gist of Idea
We understand some statements about all sets
Source
Hilary Putnam (Mathematics without Foundations [1967], p.308)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.308
A Reaction
His example is the Axiom of Choice. Presumably this is why the collection of all sets must be referred to as a 'class', since we can talk about it, but cannot define it.
| 9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
| 9941 | Science requires more than consistency of mathematics [Putnam] |
| 9943 | You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam] |
| 9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
| 9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
| 9944 | We understand some statements about all sets [Putnam] |