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Turing's argument says "conditionally compact" but the article talks only about "compact". They are not the same; "conditionally compact" means every sequence has a Cauchy subsequence, while (sequentially) compact means every sequence has a convergent subsequence. If the metric space Turing mentions is complete, then I guess they're the same. Is that obvious?
Interesting argument. This assumes that cognition must also be happening on a compact manifold which seems like a reasonable assumption but the conclusion is somewhat counterintuitive because it means there are only finitely many personality types and ways of thinking.
Earlier quoted context omitted.
I guess that means reincarnation is real.
Assuming you're not trolling: nope, that would be an empirical question of how many brain states ever actually arise in the world during the time humans exist, and how much space and time it takes to maintain a given brain state. The universe is currently expected to stop being able to support human life at some point in the future, so the pigeonhole principle argument requires some physical parameters to be known be…
That may appear every (10^10)^50 years according to
Earlier quoted context omitted.
Is that counterintuitive? There's a finite number of electrons in a human brain (which fits in a space of size 1m^3), and information takes time to propagate across any distance, and humans die before 150 years; this all gestures at there being not only finitely many personality types and ways of thinking, but finitely many human mind states .
But there may yet be more brain states than number of possible thoughts by the finite humans.
Interesting argument. This assumes that cognition must also be happening on a compact manifold which seems like a reasonable assumption but the conclusion is somewhat counterintuitive because it means there are only finitely many personality types and ways of thinking.
Interesting argument. This assumes that cognition must also be happening on a compact manifold which seems like a reasonable assumption but the conclusion is somewhat counterintuitive because it means there are only finitely many personality types and ways of thinking.
Well, this is true only if you think cognition and thought as a pure biological process totally dependent on state of brain.
For example, in 1937 the third glyph is number three, but in ВАЗ, the third glyph is Cyrillic letter Z. They're not different in writing.
It would not be hard to devise a symbolic system where new symbols are context dependent, and there are infinitely many of these depending on the context. The context will be governed by a finite number of rules. The glyphs will not change meaning in a specific text.
Human language is an example of such system. If you treat words as a whole as symbols, you can have infinitely many different words as they become defined via the specific text, but obviously you can't have infinitely many different words in a single text because they will stop being mutually intelligible at some point.
Interesting argument. This assumes that cognition must also be happening on a compact manifold which seems like a reasonable assumption but the conclusion is somewhat counterintuitive because it means there are only finitely many personality types and ways of thinking.
Who said this had anything to do with cognition at all? I think Turing's argument goes through with humans replaced by automata and eyes by cameras. It's just that perception has finite resolution.
Turing's argument says "conditionally compact" but the article talks only about "compact". They are not the same; "conditionally compact" means every sequence has a Cauchy subsequence, while (sequentially) compact means every sequence has a convergent subsequence. If the metric space Turing mentions is complete, then I guess they're the same. Is that obvious?
Convergent sequences are always Cauchy; for metric spaces, compactness and sequential compactness are the same.
> Turing says that a certain space, the space of all compact subsets of [0,1]^2 endowed with the metric "integral of minimal distance required to transform {1 ink at each point of P1} u {infinite amount of ink at (2, 0)} into {1 ink at each point of P2} u {infinite amount of ink at (2,0)}", is conditionally-compact. How is that related to the article's argument?
This is not obvious, I think. The article has moved away from Turing's "integral of the distance we have to transfer ink", instead using "maximum distance we have to transfer any ink", and I don't have a great intuition for whether this is a legit transformation of the argument. (I'm sure both proofs are correct, but it's not obvious to me that they are the same proof.)
I don't agree. What if the next symbol's meaning conditionally depends on the previous ones? For example, in 1937 the third glyph is number three, but in ВАЗ, the third glyph is Cyrillic letter Z. They're not different in writing. It would not be hard to devise a symbolic system where new symbols are context dependent, and there are infinitely many of these depending on the context. The context will be governed by a…