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Full Idea
It is not implausible that before the 'introduction' of complex numbers, it would have been incorrect for mathematicians to claim that there was a solution to the equation 'x^2 = -1' under a completely unrestricted understanding of 'there are'.
Gist of Idea
It is plausible that x^2 = -1 had no solutions before complex numbers were 'introduced'
Source
Kit Fine (The Question of Ontology [2009])
Book Ref
'Metametaphysics', ed/tr. Chalmers/Manley/Wasserman [OUP 2009], p.163
A Reaction
I have adopted this as the crucial test question for anyone's attitude to platonism in mathematics. I take it as obvious that complex numbers were simply invented so that such equations could be dealt with. They weren't 'discovered'!
| 12211 | It is plausible that x^2 = -1 had no solutions before complex numbers were 'introduced' [Fine,K] |
| 12209 | The indispensability argument shows that nature is non-numerical, not the denial of numbers [Fine,K] |
| 12212 | Just as we introduced complex numbers, so we introduced sums and temporal parts [Fine,K] |
| 12214 | 'Exists' is a predicate, not a quantifier; 'electrons exist' is like 'electrons spin' [Fine,K] |
| 12213 | Ontological claims are often universal, and not a matter of existential quantification [Fine,K] |
| 12215 | The existence of numbers is not a matter of identities, but of constituents of the world [Fine,K] |
| 12216 | Real objects are those which figure in the facts that constitute reality [Fine,K] |
| 12218 | Being real and being fundamental are separate; Thales's water might be real and divisible [Fine,K] |
| 12217 | For ontology we need, not internal or external views, but a view from outside reality [Fine,K] |