Comment on Theories on Theories

Ziglin@lemmy.world ⁨5⁩ ⁨weeks⁩ ago

• sparkyshocks@lemmy.zip ⁨5⁩ ⁨weeks⁩ ago

  Exactly.

  HERE’S A THEOREM: IF IT’S PROVEN, IT’S TRUE EVERYWHERE, FOREVER

  But at the same time, even if it’s true everywhere forever, it might still not be provable, because Gödel.

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  • yetAnotherUser@discuss.tchncs.de ⁨5⁩ ⁨weeks⁩ ago

    Worse: If the chosen axioms are contradictory, then the theorem is effectively worthless.

    And it is impossible to know whether axioms are consistent. You can only prove that they are not.

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    • ytg@sopuli.xyz ⁨5⁩ ⁨weeks⁩ ago

      You can go deeper. To prove anything, including the consistency or inconsistency of a theory, you need to work within a different system of axioms, and assume that it is consistent, etc.

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  • ytg@sopuli.xyz ⁨5⁩ ⁨weeks⁩ ago

    even if it’s true everywhere forever, it might still not be provable, because Gödel.

    No. Gödel’s completeness theorem says that if something is true in every model of a (first-order) theory, it must be provable. Gödel’s incompleteness theorem says that there exists statements that are true sometimes, and these can’t be provable.

    The key word is “everywhere”.

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  • technocrit@lemmy.dbzer0.com ⁨5⁩ ⁨weeks⁩ ago

    I think saying that a theorem is true presumes the axioms upon which it is based and so the entire system is “true everywhere forever”.

    I often find it helpful to think of chess as my axiomatic system. When we say the king is in checkmate, it presumes that we accept all the underlying rules of chess. And these pieces that theoretically form a checkmate will always do so forever… Assuming the regular rules of chess, etc.

    When you put things in terms of chess, these “deep” statements about “math” often become banal.

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  • pfried@reddthat.com ⁨5⁩ ⁨weeks⁩ ago

    But that’s math.

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    • sparkyshocks@lemmy.zip ⁨5⁩ ⁨weeks⁩ ago

      I see philosophy as a place to make nonrigorous arguments.

      Wait do you think Bertrand Russell and Alan Turing and Kurt Gödel weren’t making philosophical arguments?

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      • pfried@reddthat.com ⁨5⁩ ⁨weeks⁩ ago

        They are clearly mathematical. Starting with definitions and axioms and deriving from there using mathematical statements.

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    • lemonwood@lemmy.ml ⁨5⁩ ⁨weeks⁩ ago

      I see philosophy as a place to make nonrigorous arguments.

      It’s the other way around: math is where your just ignore questions about what makes sense, what knowledge is, what truth is, what a proof is, how scientific consensus is reached, what the scientific method should be, and so on. Instead, you just handwave and assume it will all work out somehow.

      Philosophy of mathematics is were three questions are treated rigorously.

      Of course, serious mathematicians are often philosophers at the same time.

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      • pfried@reddthat.com ⁨5⁩ ⁨weeks⁩ ago

        You’re just covering my third paragraph. Yes, everybody is a philosopher because we don’t have the tools to do away with philosophical arguments entirely yet.

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• HexesofVexes@lemmy.world ⁨5⁩ ⁨weeks⁩ ago

  Ehh…

  Gödel basically showed we can never know which “mathematics” is the “correct one”.

  “Proven true assuming my axioms are true” is closer to reality.

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