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I searched Turing's paper and the word "alphabet" does not appear. I guess the author interprets Turing's use of "symbol" as an "alphabet". Can anyone familiar with Turing's paper clarify?
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.
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?
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.
Earlier quoted context omitted.
The article ends by noting that even with colored ink, the number of distinguishable symbols is finite. Indeed, if there is a limit to the number of values held by each of the variables, that would limit what you could write. If we're looking for theoretical ways around that constraint, you could allow inks that change over time, so you would have to observe each character for a certain amount of time in order to see…
For some reason, I am thinking about the green-blue and grue-bleen paradox. https://en.wikipedia.org/wiki/New_riddle_of_induction The "New Riddle of Induction," proposed by Nelson Goodman in 1955, challenges our understanding of inductive reasoning and the formation of scientific hypotheses. Goodman introduced the concept through his famous "grue" example. Imagine a property "grue," defined as "green up to time t, an…
Earlier quoted context omitted.
It doesn't, however, allow for non-closed symbols. I can imagine a spiralling brushstroke that gets fainter as it approaches the centre. Maybe the proof can be strengthened, and it certainly won't pass the distinguishability criterion, but we must be rigorous, here if anywhere.
Why should it not allow for your spiralling brushstroke? You can divide the spiral into the strokes corresponding to each turn around the center. We can assume that each of these strokes is measurable (otherwise we would not need the construction of the spiral). Then the entire spiral is just a countable union of those measurable strokes, thus it is measurable itself.
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.
I’ve thought about melodies in music too. Obviously, the number of melodies possible is extremely vast. But in 4:4 timing, there must be a limit?
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.
I’ve thought about melodies in music too. Obviously, the number of melodies possible is extremely vast. But in 4:4 timing, there must be a limit?
Earlier quoted context omitted.
This reminds me of a design for analogue TV from the late 80s/early 90s. Instead of the electron gun scanning in horizontal lines, it follows a space-filling, non-crossing, self-similar curve. Want to upgrade the resolution? Just build a TV that renders the curve one generation deeper, and stuff more data into the analogue broadcast signal. Older TVs can still render the new signal, maybe with a slightly higher noise…
The concept you're talking about is a Hilbert Curve. But this is the first time I've heard it applied to analogue tv. https://en.m.wikipedia.org/wiki/Space-filling_curve
Ah! Found this: https://www.ripcorddesigns.com/blog-news/who-needs-rastas-an... - there's even a patent! I was searching on space-filling curve, not Hilbert curve.
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 Turing machine has a memory, it can absolutely cope with symbols meaning something different depending on what appears before them. The trouble is that it can't distinguish between infinitely many distinct symbols on the paper tape. Turing here shows that a human can't do that either, so that is not a way in which humans have more computational power than a Turing machine.