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    Infinite mathematical objects, such as the completed set ... — Carmelics
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    Home/Modality & Possibility
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    Infinite mathematical objects, such as the completed set of all natural numbers and arbitrary irrational numbers represented by Dedekind cuts, do not exist.

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Only finite mathematical objects (like numbers and sentences) exist in whatever sense mathematical objects exist.
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    • 2.Completed infinities are not finite objects.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Dedekind cuts and the set of natural numbers are presupposed by classical analysis, which underwrites physical theories whose predictive success we have no rival explanation for.
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    • 2.Quine and Putnam's indispensability argument holds that ontological commitment is owed to any entity whose posit is ineliminable from our best confirmed scientific theories.
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    • 3.The finitist's restriction to finite objects cannot recover the full continuum required by differential equations, leaving a systematic explanatory gap the claim does not resolve.
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    Reason against 2 of 2
    ?
    • 1.Cantor's transfinite arithmetic demonstrates that completed infinities obey consistent, non-contradictory formal laws, satisfying the existence criterion Hilbert called 'consistency entails existence'.
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    • 2.If a mathematical structure is internally consistent and theoretically indispensable, denying its existence requires a stricter criterion of existence that the finitist has not independently justified.
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    Related

    Cantor's transfinite arithmetic demonstrates that completed infinities obey cons...Completed infinities are not finite objects.Dedekind cuts and the set of natural numbers are presupposed by classical analys...If a mathematical structure is internally consistent and theoretically indispens...
    +3 moreShow less
    Only finite mathematical objects (like numbers and sentences) exist in whatever ...Quine and Putnam's indispensability argument holds that ontological commitment i...The finitist's restriction to finite objects cannot recover the full continuum r...

    Similar

    Only finite mathematical objects (like numbers and sentences) exist in...80%Finite objects are mathematically unproblematic77%Therefore, lawless irrationals must also be introduced to complete the...77%If no reduction of natural numbers to sets succeeds, then natural numb...76%

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    Some mathematicians and philosophers have adopted finitism not merely as a methodological viewpoint, but also as a metaphysical one. Finite objects, like numbers and sentences, exist (in whatever sense mathematical objects exist), but infinite objects (like the completed set of all the natural numbers, or even arbitrary irrational numbers represented by Dedekind cuts) don’t. Versions of this view are often attributed to the 19th century number theorist and algebraist, Leopold Kronecker, who is q
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    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit