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    Under strict constructivism in the tradition of Brouwer a... — Carmelics
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    Challenges→At least one of the inclusions L ⊆ NL, NL ⊆ P, P ⊆ NP, NP ⊆ PSPACE must be proper

    Under strict constructivism in the tradition of Brouwer and Bishop, a proof that not-all-are-equalities does not constructively exhibit which specific inclusion is proper, leaving the disjunction epistemically inert.

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    Reasons For

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    Reason for
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    • 1.Constructive proofs require explicit algorithms; negation-of-universal statements lack computational content about which case holds.
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    • 2.Epistemically inert means unusable for further reasoning; a disjunction without instantiation cannot guide mathematical construction.
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    • 3.Brouwer-Bishop tradition prioritizes intuitionistic validity where existence means constructible; abstract negations violate this principle.
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    Reasons Against

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    Reason against
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    • 1.Proof by contradiction of ¬(all-equal) does yield information: it eliminates the all-equal case, restricting the logical space constructively.
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    • 2.Some disjunctions gain epistemic force through constraint elimination alone; knowing what's *not* true enables bounded search and reasoning.
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    • 3.Modern constructivism (e.g., CZF) accepts some classical logical principles; dismissing all non-intuitive proofs overstates Brouwer-Bishop strictness.
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    Key Terms

    Bishop(history of mathematics)
    Errett Bishop was an American mathematician who developed constructive analysis, a stricter version of math that only accepts proofs you can actually build or compute.
    Brouwer
    Brouwer was a Dutch mathematician and philosopher (1881-1966) who fundamentally changed how mathematicians think about proof and logic. He argued that math shouldn't just accept something as true because it follows logically; instead, mathematicians should be able to actually construct or demonstrate mathematical objects to prove they exist. His ideas challenged the traditional approach to mathematics and influenced debates about the foundations of mathematical reasoning that continue today.
    Constructively exhibit(constructivist philosophy)
    To actually show or demonstrate something in a concrete way, rather than just proving it must exist somewhere.
    Epistemically inert(epistemology (theory of knowledge))
    Unable to tell you anything useful or give you actual knowledge—it just sits there without helping you understand or know anything.
    constructivism(Philosophy of medicine)
    The view that diseases or disorders are classified as pathological due to social values rather than purely scientific or natural evidence
    disjunction(as used in formal logic)
    A logical 'or' statement; 'a or b' (written a∨b) is true when at least one of the two parts is true.
    epistemology(Contrasted with purely descriptive scientific inquiry)
    A normative enterprise that tells us how we ought to reason from evidence and how we ought to justify our beliefs, as distinct from merely describing how we do reason or justify beliefs
    proof(Frege's formal system; the definition still used by logicians today)
    Any finite sequence of statements such that each statement is either an axiom of the formal system or follows from previous members of the sequence by a valid rule of inference.

    Connections

    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    At least one of the inclusions L ⊆ NL, NL ⊆ P, P ⊆ NP, NP ⊆ PSPACE must be prope...Brouwer-Bishop tradition prioritizes intuitionistic validity where existence mea...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Constructive proofs require explicit algorithms; negation-of-universal statement...
    Epistemically inert means unusable for further reasoning; a disjunction without ...
    +3 moreShow less
    Modern constructivism (e.g., CZF) accepts some classical logical principles; dis...Proof by contradiction of ¬(all-equal) does yield information: it eliminates the...Some disjunctions gain epistemic force through constraint elimination alone; kno...