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 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 floor.
(Anyone know what I'm talking about? I assume it was obsoleted by digital solutions, but I'd like to know if there were any serious flaws in the design).
Yeah, in my memory the pattern was specifically a Hilbert Curve but rotated 45deg (that can't be right, can it? That would just make things harder). I must have seen it between 1987 and 1997.
That works if you're fixed to a grid, but the argument is much more general: it allows symbols to be of arbitrary size (as long as they're still within the unit square), in any orientation, formed by continuous or (not-too-pathological) discontinuous penstrokes, etc.
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.
We can play a lot of games with "what is a symbol", but compactness pervades many of the models that we use to describe reality. The crux of the argument is not necessarily that the symbols themselves are compact as sets, but that the *space of possible descriptions* is compact. In the article, the space of descriptions is (compact) subsets of a (compact) two-dimensional space, which (delightfully) is compact in the appropriate topology.
In your example, the symbols themselves could instead be modeled as a function f:[0,1]^2 -> [0,1] which are "upper semicontinuous", which when appropriately topologized is seen to be compact; in particular, every infinite sequence must have a subsequence that converges to another upper semicontinuous function.
Much of the fun here comes from the Tychonoff theorem, which says that arbitrary products of compact spaces is compact. Since the *measurement space* is compact, the topology of the domain is not as important, as long as the product topology on the function space is the appropriate one. (Mystically, it almost always is.)
The proof defines F(X) as the set of compact subsets of X (here the unit square). The compactness of F(X) follows from (necessary?) the compactness of its elements, so we need to find a topology where all our allowable symbols are compact, and this is not the standard topology if you allow the spiral. If you take another topology (equivalently space of symbol descriptions) then you again must show that this compactness still corresponds to the desired notion of "distinguishability".
My topology is rusty and I don't genuinely doubt the validity of the argument, but I'm having fun trying to poke holes.
Your example centered on a symbol which, if viewed as a subset of the plane, is not compact. I tried to argue that the set of symbols that you describe (ink of varying levels of intensity in the unit square) still is a compact set, even though the symbols themselves are no longer represented by compact sets.
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.
For the proof given in TFA they use the assumption that the symbol is compact and hence closed. I argue that the symbol needn't be closed, it may still be measurable.
The proof doesn't assume that they are formed by not-too-pathological penstrokes.
<wrong>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.)</wrong>
I think that every horrible-totally-pathological ink splash has a nice indistinguishable version, but my real analysts is a bit rusty, and there are a few horrible things that I may have forgotten.
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).
I made a comment elsewhere on this thread that explains that symbols themselves being compact isn't so important, but that the set of descriptions of the symbols must be compact. For example, if the description of the symbol is not the symbol itself as a set, but a map f:[0,1]^2 -> [0,1] that describes the "intensity" of ink at each point, then the natural conclusion is that the description of a symbol must be upper semicontinuous, which makes the set of descriptions compact.
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 what it was representing as its ink changed (or didn't).
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.
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.
If the observation time is finite, and the time resolution of the observation is limited, then you'd still only have a finite number of distinguishable symbols, wouldn't you?
That's true if the time resolution is limited. Theoretically there could be no limit. It would not make for a very practical alphabet, but it would still be an unlimited alphabet.
Just as Rorschach's mask always changes in Watchmen. We would have books with changing alphabet. In the library of babel, Harry Potter would be located in one section at a given time T1 but maybe at a different section at time T2.
nyquist's sampling theorem is from 01915. western union started offering wirephoto service in 01921, and the associated press started mass distribution of news photos in 01935. these aren't discrete pixels but they did demonstrate that nyquist's sampling theorem applies to images too
If you want to use that sampling theorem, you have to bring up frequencies. Explaining those and how they apply to printer's ink is arguably more complicated than what Turing did; especially for his contemporary target audience.
The argument from pixels is more natural to us these days, that's why we think it's simpler.
(I had similar reactions to many other older style proofs when I was studying math.)
You also have to argue for finite color resolution, or else a color noise floor that masks the difference between similar colors.
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.)
I totally agree with your argument. However just to throw out an idea for a possible counterexample (as mathematicians like to do):
What if you used aperiodic Penrose tiles and could detect intensity? Would it be possible to encode anything in that? There would be no repetition but when would you hit limits of discernability with all the overwriting?
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)
The number of notes per bar is not really finite given you can make any a:b polyrhythm and you can make any subdivision you like. For example, see the famous falling coin rhythm based on the Fibonacci series in Bartok's "Music for Strings, Percussion and Celeste"[1]
[1] If you want a look at what "Pyramids on Mars"-style musical analysis is like, Erno Lendvai's book about Bartok is pretty awesome. Since Bartok never really wrote or said anything about his compositional methods it's very hard to prove anything conclusively.
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.