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Is this the case just because it's common for these arguments to be structured as contradiction proofs? Since contradiction proofs are arguments that operate i

Is this the case just because it's common for these arguments to be structured as contradiction proofs? Since contradiction proofs are arguments that operate in connection with the structure of proofs in general rather than exclusively in terms of the specific problem, then the entire argument may collapse on a technicality like "very large" vs "infinite". It's almost like the truth status of the argument is sharply discontinuous: you can't modulate the argument slightly and arrive at sensibly related assertions. Whereas if the argument were made solely in terms of the problem itself (without the added meta-layer argument from the contradiction proof), but still used the halting problem, then we might consider the fact that in reality our computers have finite state so are technically exempt, but the argument still holds for all intents and purposes because of how large the finite state is. Or is there another reason these arguments are generally bogus?

AnHonestComment 6y ago
You actually noticed something deeper than your original question — “continuity of logic”, in the sense of “continuous function”. If you imagine you have a lot of inputs to a predicate, then the topology of their truthiness tells you something — and we can talk about “smooth” logic models, where being a little wrong in our assumption means we’re a little wrong in our conclusion. We can then apply tools from analysis, like bifurcation theory. The “halting problem schema” has a bifurcation at the finite/infinite boundary, which isn’t particularly uncommon for high level functions. But it’s important to know, when writing proofs. And generally speaking, places bifurcations can happen are regimes where your model is going to struggle. In the business world, knowing the “logic faults” of your model and keeping yourself in a “smooth regime” is important.
blonde_ocean 6y ago
How else do you prove something using the halting problem other than a proof of contradiction?

AnHonestComment 6y ago

You actually noticed something deeper than your original question — “continuity of logic”, in the sense of “continuous function”. If you imagine you have a lot of inputs to a predicate, then the topology of their truthiness tells you something — and we can talk about “smooth” logic models, where being a little wrong in our assumption means we’re a little wrong in our conclusion. We can then apply tools from analysis, like bifurcation theory. The “halting problem schema” has a bifurcation at the finite/infinite boundary, which isn’t particularly uncommon for high level functions. But it’s important to know, when writing proofs. And generally speaking, places bifurcations can happen are regimes where your model is going to struggle. In the business world, knowing the “logic faults” of your model and keeping yourself in a “smooth regime” is important.

blonde_ocean 6y ago

How else do you prove something using the halting problem other than a proof of contradiction?