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    An inductive proof enables us to perceive that a direct p... — Carmelics
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    An inductive proof enables us to perceive that a direct proof of any particular proposition can be constructed, even though it cannot prove the infinite possibility of application.

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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.An inductive proof proves the inductive base φ(1) and the inductive step φ(n) → φ(n+1).
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    • 2.Once the inductive base and inductive step are proved, we need not reiterate modus ponens m-1 times to prove a particular proposition φ(m).
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    • 3.The proxy statement φ(m) is an eliminable pseudo-proposition standing proxy for the proved inductive base and inductive step, not a proposition that asserts its own generality.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Mathematical induction proves a universal generalization ∀n φ(n) as a single logical act, not merely a procedure for constructing finite proofs.
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    • 2.Frege and Russell demonstrated that the logical content of '∀n φ(n)' is irreducible to any finite conjunction or constructive recipe for particular instances.
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    • 3.If induction only licenses perception of constructibility without proving the infinite generalization, then transfinite arithmetic and completed infinities lose any rigorous foundation.
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    Reason against 2 of 2
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    • 1.Wittgenstein's eliminability thesis treats φ(m) as a pseudo-proposition, but this collapses the distinction between a proof and a decision procedure for checking proofs.
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    • 2.Gödel's incompleteness results demonstrate that syntactic provability within a system cannot be identified with the semantic truth of the proposition proven, undermining any purely procedural account of what induction establishes.
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    Key Terms

    Inductive proof(as used in logic and mathematics)
    A method of proving something by showing it works for a few cases, then arguing it must work for all similar cases following the same pattern.
    Infinite possibility of application(as used in logic and mathematics)
    The idea that a rule or proof could work for an endless, unlimited number of cases or situations.
    direct proof(Used in Prawitz's proof-theoretic semantics to define the standard for elimination rule justification.)
    A proof that proceeds by application of introduction rules, serving as the canonical form of proof against which elimination rules are justified.
    proposition(Used in the context of a semantic theory sensitive to differences in subject matter.)
    The content expressed by a sentence, individuated at least in part by the subject matter of the sentence and the contents of its subsentential expressions.

    Connections

    2 topics

    Proof of definition segments1 linkedPhilosophy of Language1 linked

    Related

    An inductive proof proves the inductive base φ(1) and the inductive step φ(n) → ...Frege and Russell demonstrated that the logical content of '∀n φ(n)' is irreduci...

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
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    Here the ‘conclusion’ of an inductive proof [i.e., “what is to be proved” (PR §164)] uses ‘\(m\)’ rather than ‘\(n\)’ to indicate that ‘\(m\)’ stands for any particular number, while ‘\(n\)’ stands for any arbitrary number. For Wittgenstein, the proxy statement “\(\phi(m)\)” is not a mathematical proposition that “assert[s] its generality” (PR §168), it is an eliminable pseudo-proposition standing proxy for the proved inductive base and inductive step. Though an inductive proof cannot prove “the
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    Details

    Gödel's incompleteness results demonstrate that syntactic provability within a s...
    If induction only licenses perception of constructibility without proving the in...
    +4 moreShow less
    Mathematical induction proves a universal generalization ∀n φ(n) as a single log...Once the inductive base and inductive step are proved, we need not reiterate mod...The proxy statement φ(m) is an eliminable pseudo-proposition standing proxy for ...Wittgenstein's eliminability thesis treats φ(m) as a pseudo-proposition, but thi...

    Similar

    A proof that P ≠ NP would validate existing inductive evidence for the...80%A proof that P ≠ NP would validate existing inductive evidence for P ≠...80%A proof that P ≠ NP would validate existing inductive evidence for P ≠...79%A general proposition is senseless prior to an inductive proof78%
    Type
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    Perspectives
    3 (1 for, 2 against)
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    1 edit