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    Benacerraf's identification problem shows that definition... — Carmelics
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    Challenges→Sort logic is an alternative way of looking at mathematics where definition rather than construction is the focus

    Benacerraf's identification problem shows that definitional equivalence across structures underdetermines mathematical ontology, undermining definition as a foundational focus.

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    Reasons For

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    • 1.Multiple isomorphic structures can satisfy identical mathematical definitions, creating indeterminacy about which structure constitutes 'the' natural numbers.
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    • 2.If definition alone cannot pin down ontology, then definition cannot serve as a foundational anchor for mathematical existence claims.
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    • 3.Structural properties are all mathematics genuinely cares about; ontological identity beyond structure is metaphysically idle.
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    Reasons Against

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    • 1.Isomorphism demonstrates equivalence for mathematical purposes; ontological indeterminacy doesn't undermine definitions' foundational utility.
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    • 2.Definition can succeed by specifying structural role without determining unique concrete instantiation—these are distinct foundational tasks.
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    • 3.Benacerraf's problem assumes platonism; constructive or nominalist frameworks sidestep identification problems entirely without abandoning definition.
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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Benacerraf's problem assumes platonism; constructive or nominalist frameworks si...Definition can succeed by specifying structural role without determining unique ...If definition alone cannot pin down ontology, then definition cannot serve as a ...Isomorphism demonstrates equivalence for mathematical purposes; ontological inde...
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    Multiple isomorphic structures can satisfy identical mathematical definitions, c...Sort logic is an alternative way of looking at mathematics where definition rath...Structural properties are all mathematics genuinely cares about; ontological ide...

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