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    Putnam and Benacerraf's arguments about mathematical redu... — Carmelics
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    Challenges→The class F provides a machine-independent characterization of the complexity class FP

    Putnam and Benacerraf's arguments about mathematical reduction show that multiple incompatible formalisms can be co-extensional, meaning equivalence underdetermines which characterization is the 'intrinsic' one.

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    Key Terms

    Benacerraf(in philosophy of mathematics and ethics)
    Paul Benacerraf, a philosopher who argued that if abstract objects (like numbers or moral facts) don't exist in the physical world, we have a serious problem explaining how we could ever know anything about them.
    Co-extensional(describing when two things match up but aren't necessarily connected in a meaningful way)
    Having exactly the same members or applying to exactly the same things, even if for completely different reasons.
    Equivalence underdetermines(as the logical problem created by co-extensional formalisms)
    When two things work equally well and produce the same results, it doesn't give us enough information to figure out which one is actually correct or real.
    Formalisms(as the different mathematical systems being compared)
    Different formal systems or symbolic languages used to express mathematical ideas—think of them as different ways of writing out math rules.

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    Intrinsic(describing the kind of continuities that ground identity)
    Something that belongs to or is part of something by its very nature, rather than coming from outside or being relational.
    Mathematical reduction(as the central topic being discussed)
    The attempt to explain what math really is by reducing it to something more basic, like sets or logical rules.
    Putnam
    # Putnam "Putnam" most commonly refers to **Hilary Putnam** (1926-2016), an influential American philosopher who made major contributions to philosophy of mind, language, and science. He is famous for thought experiments like the "brain in a vat" scenario, which explores questions about reality and how we know what's real. His work fundamentally changed how philosophers think about the relationship between our minds, language, and the external world.

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    Proof of definition segments1 linkedTruth & Knowledge1 linked

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    The class F provides a machine-independent characterization of the complexity cl...

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