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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism

[reasons for doubting the existence of maths entities]

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