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Turing's topological proof that every written alphabet is finite (2010)

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Re: Turing's topological proof that every written alphabet is finite (2010)

#51

Post by refulgentis »

montecarl wrote: post #4

It is even easier to just say, given a fixed area of paper, and a finite printing resolution, there is a finite number of symbols possible? Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.

Print resolution is far higher than 300 DPI, floor for the cheapest is 1000, most will do 4K. (not a correction, I am only sharing this because I think of this often because it seems leverage-able for something unique, and HN is a good place to share that sort of thing, in case someone is inspired)

Re: Turing's topological proof that every written alphabet is finite (2010)

#52

Post by byteknight »

myhf wrote: post #45
perihelions wrote: post #36

Earlier quoted context omitted.

You can take a glyph from that font, cut up and rearrange it, and come out with two copies of the original size. Saves money on printer ink

assuming you are willing to spend an infinite amount of time doing the cutting

I smell a supertask

Re: Turing's topological proof that every written alphabet is finite (2010)

#53

Post by tgv »

thriftwy wrote: post #27

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…

The alphabet itself would still be finite, like our vocabulary.

Re: Turing's topological proof that every written alphabet is finite (2010)

#54

Post by tel »

gus_massa wrote: post #49
Smaug123 wrote: post #39

Earlier quoted context omitted.

By "not-too-pathological" I intended at least to include the requirement "measurable".

I just notice that I used the wrong metric. The article uses the Hausdorf metric and I used the measure of the symetric difference. And the article assume that the sets are compact, so they are measurable as you say. Anyway compact sets can be quite pathological (but not as pathological as non measurable sets).

It also suggests the argument generalizes to symbols as non-compact sets.

Re: Turing's topological proof that every written alphabet is finite (2010)

#56

Post by nairboon »

ainoobler wrote: post #26
aiauthoritydev wrote: post #25

Earlier quoted context omitted.

Well, this is true only if you think cognition and thought as a pure biological process totally dependent on state of brain.

I don't think that. I'm just explaining what follows logically if the cognitive manifold is compact.

What exactly do you mean with "cognitive manifold"?

Re: Turing's topological proof that every written alphabet is finite (2010)

#57

Post by ainoobler »

nairboon wrote: post #56
ainoobler wrote: post #26

Earlier quoted context omitted.

I don't think that. I'm just explaining what follows logically if the cognitive manifold is compact.

What exactly do you mean with "cognitive manifold"?

It's a made up concept and it includes all possible ways people can think anything. If you want something concrete then you can consider sheaves of finitely presented algebras on the nervous system considered as a topological space with the obvious covering relations, continuous transformations, and algebra morphisms.

Re: Turing's topological proof that every written alphabet is finite (2010)

#58

Post by andrewflnr »

nxobject wrote: post #3

Just a reading note to myself: the key assumption is that there is a "resolving limit" \epsilon such that symbols that are "closer and smaller" than \epsilon are indistinguishable. The implicit assumption there is that you can resize symbols while keeping them the same.

I'm not sure that assumption is necessary, at least for finitely sized alphabets. For finitely many sizes of symbol that are meant to be distinguishable, you can define the smaller ones as just taking up smaller portions of the square. If you try to scale down too far, they'll eventually hit the epsilon limit and be indistinguishable again. Maybe you could break the theorem by growing arbitrarily large, but that really stretches the word "alphabet". He was thinking of notebook pages, so encoding things in the size of arbitrarily large symbols wasn't top of mind.

Re: Turing's topological proof that every written alphabet is finite (2010)

#60

Post by meroes »

Isn’t this just taking a common sense concept and adapting it to the concepts and language of Turing machines and also topology?

While yes it shows these are good tools, capable of confirming/proving these intuitions and showing exactly how these specific systems achieve that, is there ever any concern we are making too many redundant concepts? That may lead to confusion and obfuscate the search for novel ideas, which I think won’t always happen through said tools. It’s not like these kind of intuitions were on shaky grounds or could be disproven.

I wonder if anyone shares these opinions?

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