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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)

#81

Post by taejo »

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 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.

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

#82

Post by constantcrying »

k92mueller wrote: post #43

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…

>we are limited to the unit square (0Every symbol has measure zero and so every symbol is identical.

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

#85

Post by oefrha »

nyc111 wrote: post #84

Neither finite nor infinite is defined. I don't know what those words mean. So the assumption of an infinite alphabet doesn't make sense to me.

Finite and infinite are well defined in every set theory. You shouldn’t approach a topological proof without acquainting yourself with the most basic set theoretic foundation.

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

#86

Post by colinwilyb »

"I am assuming that the reader is familiar with the terms metric, metric space, topological space, and compact set."

On behalf of the math-declined folks, I wish math articles had subtitles, references, and definitions built-in.

I wonder if a browser extension for Greek letter definitions would be possible?

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

#87

Post by namanyayg »

flir wrote: post #77
daedalus_j wrote: post #63

Earlier quoted context omitted.

The Minds from The Culture scifi series have something like this... The top level glyph is the base meaning and you can encode essentially infinite depth of meaning beyond that... Of course you have to go multidimensional at some point, so you can't really print it out beyond a few layers of resolution...

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

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

#88

Post by ycombinete »

colinwilyb wrote: post #86

"I am assuming that the reader is familiar with the terms metric, metric space, topological space, and compact set." On behalf of the math-declined folks, I wish math articles had subtitles, references, and definitions built-in. I wonder if a browser extension for Greek letter definitions would be possible?

The problem is that the definitions of in such footnotes would be full of terms that one also needs defined.

For example here's the first few sentences from the Wikipedia page on Compact Sets:

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.[1] The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact.

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

#89

Post by passion__desire »

kmoser wrote: post #37
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.

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, and blue thereafter." All emeralds examined before time t are both green and grue. The riddle asks: Why is it rational to predict that future emeralds will be green rather than grue? This paradox highlights the problem of choosing between competing hypotheses that equally fit past observations. It questions our basis for preferring "natural" predicates (like green) over "artificial" ones (like grue) in inductive reasoning. Goodman's riddle challenges the idea that induction is based solely on observed regularities. It suggests that our choice of predicates in forming hypotheses is influenced by factors beyond mere observation, such as simplicity, familiarity, or projectibility. The New Riddle of Induction has significant implications for philosophy of science, epistemology, and artificial intelligence. It raises questions about the foundations of scientific prediction, the nature of natural kinds, and the role of language in shaping our understanding of the world.

Related :

https://x.com/eshear/status/1812926436623413285

There are crystal structures that simply won't form anymore, even though they did a few decades ago. Real life Ice-9? Could make an incredible sci-fi book.

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

#90

Post by FredPret »

flir wrote: post #77
daedalus_j wrote: post #63

Earlier quoted context omitted.

The Minds from The Culture scifi series have something like this... The top level glyph is the base meaning and you can encode essentially infinite depth of meaning beyond that... Of course you have to go multidimensional at some point, so you can't really print it out beyond a few layers of resolution...

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…

I think arcade games worked like that. Looked it up and oscilloscopes do, too:

https://en.m.wikipedia.org/wiki/Vector_monitor

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