If the nominalist can paraphrase away apparent reference to mathematical entities without explanatory loss, the enhanced argument's inference to Platonism is blocked by an equally good rival hypothesis.
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apparent reference(as used in philosophy of language and metaphysics)
When language seems to be talking about or pointing to something (like when we say 'the number 7 is prime'), even if we're not sure that thing really exists.
explanatory loss(as used in epistemology and philosophy of science)
A situation where simplifying or changing your explanation makes it worse or less able to account for what we observe and experience.
paraphrase away(Technical usage in the context of ontological commitment and paraphrase nominalism)
To show that a singular term in a sentence can be eliminated by replacing the sentence with an equivalent sentence that does not contain that singular term
rival hypothesis(as used in epistemology and philosophy of science)
An alternative explanation or theory that competes with another explanation to account for the same facts or observations.