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When philosophers study mankind's knowledge, it's more typical to study the things mankind would ideally know at the end of all time assuming mankind could cont
by xamuel 9y ago
When philosophers study mankind's knowledge, it's more typical to study the things mankind would ideally know at the end of all time assuming mankind could continue forever.
(Formally: the knowledge predicate is usually assumed to satisfy modus ponens: if mankind knows "A implies B" and mankind knows "A", then mankind knows "B")
Under this abstraction, if mankind knows the axioms of logic and Peano arithmetic, then mankind [ideally eventually] knows the primality of all primes.
Contrast: Which Turing machines will mankind [ideally eventually] know are never-halting? This is a much harder question and the answer is probably not "all never-halting Turing machines"
- schoen 9y agoIt's a little funny that mankind gets to use a full-fledged infinite-tape Turing machine in order to compute arbitrarily large computations, since no such machine would fit in our universe. https://en.wikipedia.org/wiki/Limits_of_computation https://en.wikipedia.org/wiki/Limits_of_computation https://en.wikipedia.org/wiki/Transcomputational_problem https://en.wikipedia.org/wiki/Transcomputational_problem
- deleted 9y ago[deleted]
- fooker 9y agoThe description of a Turing machine that can do arbitrarily large computations can be finite. See: Kolmogorov complexity.
- schoen 9y agoSure, but it feels a little funny in one way to say that people "know" all of its output, although I'd agree that for other purposes being able to write a program to generate something is a relatively good description of what it means to understand it.
- JackFr 9y ago> then mankind [ideally eventually] knows the primality of all primes We already know the primality of all primes. They're prime. All numbers though . . .
- deleted 9y ago[deleted]