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Single Idea 12211
[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
]
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'!
The
21 ideas
with the same theme
[reasons for doubting the existence of maths entities]:
12339
|
Aristotle removes ontology from mathematics, and replaces the true with the beautiful
[Aristotle, by Badiou]
|
12556
|
Mathematics is just about ideas, so whether circles exist is irrelevant
[Locke]
|
14162
|
Mathematics doesn't care whether its entities exist
[Russell]
|
17881
|
Mathematician want performable operations, not propositions about objects
[Skolem]
|
3663
|
How can you contemplate Platonic entities without causal transactions with them?
[Putnam]
|
17927
|
Realists have semantics without epistemology, anti-realists epistemology but bad semantics
[Benacerraf, by Colyvan]
|
9936
|
The platonist view of mathematics doesn't fit our epistemology very well
[Benacerraf]
|
9910
|
Number-as-objects works wholesale, but fails utterly object by object
[Benacerraf]
|
2521
|
'Real' maths objects have no causal role, no determinate reference, and no abstract/concrete distinction
[Katz]
|
9963
|
If we all intuited mathematical objects, platonism would be agreed
[Jubien]
|
9962
|
How can pure abstract entities give models to serve as interpretations?
[Jubien]
|
9964
|
Since mathematical objects are essentially relational, they can't be picked out on their own
[Jubien]
|
12211
|
It is plausible that x^2 = -1 had no solutions before complex numbers were 'introduced'
[Fine,K]
|
17823
|
If mathematical objects exist, how can we know them, and which objects are they?
[Maddy]
|
9934
|
Number words became nouns around the time of Plato
[Burgess/Rosen]
|
8310
|
Does the existence of numbers matter, in the way space, time and persons do?
[Lowe]
|
17454
|
Children can use numbers, without a concept of them as countable objects
[Heck]
|
10089
|
Talk of 'abstract entities' is more a label for the problem than a solution to it
[George/Velleman]
|
10008
|
Arithmetic is not about a domain of entities, as the quantifiers are purely inferential
[Hofweber]
|
15935
|
Modern mathematics works up to isomorphism, and doesn't care what things 'really are'
[Lavine]
|
8681
|
The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts?
[Friend]
|