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Earlier quoted context omitted.
Next week on Show HN: I made a fractal font with unlimited scaling
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
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.
I have no idea about topology, so sorry if my question is stupid: Assume we are limited to the unit square (0<=x<=1 and 0<=y<=1). Now I map every real number in [0,1] to a symbol by simply defining the symbol for the number r as x=r and y=r. This results in a uncountable infinite amount of symbols. Now it's clearly not true that it's possible to describe any of those new symbols with a finite amount of symbols from a…
Earlier quoted context omitted.
Next week on Show HN: I made a fractal font with unlimited scaling
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
That leads to a simple information-theory based argument. There is a limited spatial resolution and noise floor, and so the amount of information that can be coded is finite.
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.
If we have 300 dpi, we can still code unlimited amounts of information if the color space is continuous, and noise-free.
(The article mentions the human eye limitations, but we could use technological instrumentation to extract info from a print; the human eye doesn't limit what we can code on a BluRay disc, e.g.)
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.
Next week on Show HN: I made a fractal font with unlimited scaling
Earlier quoted context omitted.
The proof doesn't assume that they are formed by not-too-pathological penstrokes. The idea is that you should measure the amount of black and white ink to change a symbol into another simbol, and if the total amount of ink is less than ε then they indistinguishable. (Where ε is some constant you must choose for the whole system.) I think that every horrible-totally-pathological ink splash has a nice indistinguishable…
By "not-too-pathological" I intended at least to include the requirement "measurable".
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).
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.
That's how the math works out in general unless you think some kind of soul exists.
Once you die it decomposes and goes on its way, and forms something new.
Its not a "soul" in the same sense but still pretty trippy