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.
> The human eye cannot tell two symbols apart when they are too similar.
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.
BTW I noticed that the clickhouse CLI has an interesting visualization trick: wide numbers have each third digit rendered with an underline style. That's as effective as a thousands separator "," but you can copy paste the number into another program without having to edit the commas away
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.
I dunno, somehow the conclusion that an infinite alphabet requires symbols to grow arbitrarily large to allow them to be distinguished from the others doesn't seem quite as interesting as the conclusion that all written alphabets are finite.
I mean clearly you can design an infinite alphabet by just using numbers for all symbols and writing the numbers in whatever system you want. Sure some symbols will require more paper or their details become too small to see, but it is still an infinite written alphabet.
The more obvious argument is that an infinite alphabet is pointless because nobody can remember an infinite list of symbols (hence any practical example of such an alphabet must have some rules to reduce the symbols to a finite set of atoms). There's nothing theoretically impossible about one existing.
> somehow the conclusion that an infinite alphabet requires symbols to grow arbitrarily large to allow them to be distinguished from the others doesn't seem quite as interesting as the conclusion that all written alphabets are finite.
Well yeah, I only brought that up to try to steelman the comment I was responding to.
> I mean clearly you can design an infinite alphabet by just using numbers for all symbols and writing the numbers in whatever system you want. Sure some symbols will require more paper or their details become too small to see, but it is still an infinite written alphabet.
That's all explicitly covered in the article. In particular, not having details too small to see is basically the definition of an alphabet for the purposes of the theorem. And as soon as you're using "numbers", you aren't really using an "infinite alphabet", you're using whichever numeral system with an extra layer of interpretation. Turing separately covered the case of simulating an infinite alphabet with countably infinite strings in a finite alphabet.
A different but to my taste more mathematically conventional way to make this argument would be instead to say that a symbol is a function from the square to [0, 1], that is, a grayscale image. A symbol, because of limitations of either its writer/printer or its reader/viewer, should have some "regularity": as you move around the image, there should be some quantitative restrictions on how rapidly the darkness of the gray color should change. In this kind of setup, the space of symbols is compact by a version of the Arzela-Ascoli theorem, which I think was fairly well-known by the time Turing was working. This also has the advantage of being straightforward to generalize to things like colored images by just changing [0, 1] to, say, [0, 1]^3 to represent RGB space, or whatever you want, as some others are mentioning in the comments.
As an aside, this is an interesting companion read:
As a working mathematician, I can say that this kind of argument has become totally routine and, were I reading Turing's paper carefully, after seeing "epsilon" and "compact" I would think "makes sense" and move on. But, historically speaking, it's interesting to realize how recent the development of the abstract idea of compactness was when Turing was writing---the time between Frechet and Hausdorff's work on compactness (see the pedagogical history) and Turing is about the time between Google being founded and today.
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.
Another way of stating this, which I think is rather non-obvious to people who have not studied some amount of pure math, is that technical mathematical definitions are almost never helpful absent a substantial amount of additional context, because their purpose is generally to capture and formalize a much more intuitively-understood idea or motivating phenomena. A professor of mine liked to joke that MathWorld was a website devoted to collecting definitions which were as technically correct as possible while also being useful to nobody[1], and I think the quoted wikipedia passage follows in that tradition beautifully.
[1] I don't know if this is still the case, but it certainly was in the late '00s.
I'd like to see a series of sites or interactive ebooks that work as described below, which would address that problem. I can't actually build such a site or ebook because (1) I'm not sufficiently mathematically sophisticated and (2) my front end skills aren't up to it.
Each one would be devoted to one interesting major mathematical theorem, such as the prime number theorem. The initial view would be a presentation of the theorem as it would be presented if it were a new discover being published in an appropriate journal for the relevant field.
At any point in that view you can pick a term or a step in the proof and ask for it to be expanded. There are two expansion options.
1. More detail. You use this when you understand what is being said but just don't see how they got from A to B. It adds in the missing details.
2. Background. You use a background expansion when you need to know more about some term or theorem that is being used.
I'll give an example of how these might work later.
The details or background you get from either of those expansions can themselves be expanded. Background expansion should work all the way down to pre-college mathematics.
Ideally if you started at the initial view of the prime number theorem site without having had any college mathematics and did background expansions all the way down you would end up learning the necessary calculus and complex analysis and number theory to understand the initial journal-level proof.
You would not learn all of the calculus or complex analysis or number theory one would normally learn in those courses. You would just learn the parts necessary for the top level proof.
I think it would be possible to choose a selection of interesting major theorems to do these sites for such that their combined background expansions would include everything that you'd get in a normal college mathematics degree program.
I think many people would find that a more interesting way to learn the material from a college mathematics degree program than the normal series of courses that each covers one subject. By getting it via background expansions everything you are learning you are learning for a specific application which can be more motivating.
Here's an example of what expansions might do.
There's a theorem from Liouville that says that if a is a real root of an irreducible polynomial with integer coefficiants of degree v >= 2, and p and q are any integers (q != 0) then there is a constant C > 0 which does not depend on p and q such that
|a-p/q| > C/q^v
In Gelfond's book "Transcendental and Algebraic Numbers" he says (I'm changing the names of some of his variables because I don't want to deal with typing greek letters and some of his terminology for clarity):
> The proof of this is quite straightforward. Suppose a is a real root of an irreducible equation
f(x) = a0 x^v + ... + av = 0,
> where all of the ai (i = 0, 1, ..., v) are integers. Then, using the mean value theorem, we get
|f(p/q)| = |a-p/q| |f'(z)| >= 1/q^v; z = a + t(p/q-a),
|t| <= 1,
> from which the Liouville theorem follows directly.
Directly for Gelfond maybe. It certainly does not follow directly for me! I understand everything he's saying, but the steps between some of the things is a little too big for me. I'd need to use a details expansion (maybe more than once). What that should give is something along the lines of how Wikipedia proves that theorem, which is the first lemma in the "Liouville numbers and transcendence" section of their article on Liouville numbers [1].
If I didn't know what the mean value theorem is (or the extreme value theorem, which shows up after expansion to a Wikipedia level of detail) that would be time for a background expansion.
You would probably end up writing copies of such books, since same proof will be included multiple times. And more copies written by humans means more errors.
Automated theorem proverbs can probably solve this problem though..
I had a professor who derived Riemann metric from just area of triangle and limits over the course of a semester. So I see what you're getting at, but such books will probably he too long for most people to read anyway.
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?
> Turing says that a certain space, the space of all compact subsets of [0,1]^2 endowed with the metric "integral of minimal distance required to transform {1 ink at each point of P1} u {infinite amount of ink at (2, 0)} into {1 ink at each point of P2} u {infinite amount of ink at (2,0)}", is conditionally-compact. How is that related to the article's argument?
This is not obvious, I think. The article has moved away from Turing's "integral of the distance we have to transfer ink", instead using "maximum distance we have to transfer any ink", and I don't have a great intuition for whether this is a legit transformation of the argument. (I'm sure both proofs are correct, but it's not obvious to me that they are the same proof.)
Given any compact metric space $(M, d)$ (which in the article is the closed unit square), the set of non-empty compact subsets of $M$ under the Hausdorff metric $h$ is also a compact metric space.
Yeah, seems like Turing defines a symbol to be a compact subset of the unit square. Whether that makes sense or not to you is up to interpretation I guess, but like you say the closure should be more or less the same as the symbol for "reasonable" symbols.
You then need to assume that each symbol has some variation in how it's written, I think, which you can think of as an open set in the symbol-space (which is compact under $h$ as per my previous comment). The collection of all these opens forms a cover of the symbol-space, which by compactness must have a finite sub-cover (Turing's "alphabet").
From a quick reading of the footnote, it seems that Turing uses optimal transport distance rather than the Hausdorff distance. Convergence in the latter implies convergence in the former, but not the other way round.
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.
> 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.
On the contrary, coming up with good general abstractions reduces the number of concepts you have to keep in your head and makes it much more possible to search for novel ideas. Imagine trying to figure out whether a given computing device was different to other computing devices without having the general concept of a Church-Turing computer - you wouldn't even get started.
Math is all about taking a notion from life that you might consider “common sense” and using it to solve a problem. You have to prove it works carefully though.
This is Turing’s works where he is translating the idea of a human “computer” into a mathematical machine.
> is there ever any concern we are making too many redundant concepts
I need to read the contextual work, but I think it's actually the opposite here. Turing is a mathematician and is transferring a "foreign" notion into common mathematical concepts. It is a recasting of the concept so that it can be used. It isn't done just to do it.
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 finite number of rules. The glyphs will not change meaning in a specific text.
Human language is an example of such system. If you treat words as a whole as symbols, you can have infinitely many different words as they become defined via the specific text, but obviously you can't have infinitely many different words in a single text because they will stop being mutually intelligible at some point.
Of course you can have infinitely many different semantics, but this is a question about available syntaxes. (Trivial proof: for each n, take the semantics S_n which assigns to the symbol "X" the meaning "integer n", just as English usually assigns to the symbol "3" the meaning "integer 3".)
I'm not sure I follow. Natural languages usually have very simple syntax. The Greeks had great literature while still in "boustrophedon with no whitespace" phase.
The number of possible symbol meanings may be unbound. And there may be symbol definition rules so that any effective symbol set used to write a text is still plausibly an alphabet. Chinese actually does fit the bill if you are allowed to define new valid characters according to best practices.
You're talking entirely about the meaning of symbols, right. That's not a question Turing addressed at all, and the answer is trivially "there are infinitely many possible semantics". Turing only addressed the question of how many physically distinct symbols there could be, and came up with the result "finitely many".
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.
Not being to able to distinguish infinitely many symbols on a single tape does not mean there is a bound on the number of meanings that may be assigned to symbols.
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.
Is that counterintuitive? There's a finite number of electrons in a human brain (which fits in a space of size 1m^3), and information takes time to propagate across any distance, and humans die before 150 years; this all gestures at there being not only finitely many personality types and ways of thinking, but finitely many human mind states.
Oh, 100%, it seems extremely unlikely that multiple brain states in this sense don't code for the same thought! If nothing else, you could only avoid that with an absurdly precise definition of "brain"; any practical definition of "brain" is very likely to include matter that is physically irrelevant to at least some thoughts. I hedged with "1m^3" because the argument goes through even if you take the entire body to be the brain.
Assuming you're not trolling: nope, that would be an empirical question of how many brain states ever actually arise in the world during the time humans exist, and how much space and time it takes to maintain a given brain state. The universe is currently expected to stop being able to support human life at some point in the future, so the pigeonhole principle argument requires some physical parameters to be known before you can apply it; and even if the pigeonhole principle did apply, you still can't know by that argument whether any given brain-state is repeated (and if so, how many times). Things which you can deduce using only mathematics, and no physical data, are necessarily extremely weak statements about reality.
A very strong empirical statement about the nature of reality, and a strong statement about the nature of mathematics, neither of which everyone agrees with! (What will your reply be if we discover a halting oracle at the centre of the galaxy? Mathematics doesn't forbid that: the Church-Turing thesis is an empirical statement, and it's the empirical scientific law of induction which is what tells us that we're extremely unlikely to find a halting oracle.)
But you've missed the point: even assuming that mathematics can perfectly model the universe, you cannot use weak physical hypotheses ("human brains are finite, there is an injection from human mind-states into human brains") and fully general mathematics (the pigeonhole principle) to derive such specific strong truths as "a particular human mind-state is repeated in a predictable way in the universe". You can at best derive general truths such as "at least one human mind-state is repeated at least once, somewhere", if the universe happens to have its physical parameters set in such a way that the pigeonhole principle holds.
Who said this had anything to do with cognition at all? I think Turing's argument goes through with humans replaced by automata and eyes by cameras. It's just that perception has finite resolution.
At this point I kind of do. The matter that goes into forming your body and brain is somewhat special, having accumulated the properties it has experiencing billions of years of traveling through the universe.
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
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.
Why do we need the property that X is compact? Say we define X (the square) to exclude the boundary: X = (0,1)x(0,1). This is no longer a compact set, but it seems silly for the proof to now fail.
Note that $A_k = \left[\frac{1}{k}, \frac{2}{k}\right] \times \left[\frac{1}{k}, \frac{2}{k}\right] \to \{(0, 0)\}$ in $[0,1]\times [0, 1]$ with the Hausdoff metric, so it is a Cauchy sequence in $X = (0,1) \times (0, 1)$ but without convergent subsequence, i.e., $F(X)$ is not complete, hence not compact so the finite subcover argument fails.
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 finite alphabet (because otherwise this would also be possible for real numbers). Where is my intuition wrong?
How can you tell the difference between (r,r) and (r+ε,r+ε)? The argument in the article assumes that there is an ε such that it is impossible to tell the difference between r and r+ε. So this means that the entire square can be covered by squares of side length ε and since the unit square is compact this means that there are only finitely many ε squares required to cover the entire unit square. Variation within the ε squares is imperceptible so this means there are only finitely many symbols that can be perceived to be different.
>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.
Every symbol has measure zero and so every symbol is identical.
"In this paper Turing gives a topological argument that every written alphabet must be finite."
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?
non-anything question: Is this text/sheet an analogy for a thing's relationship to its substrate, or does the proof define a relationship or category of "thing-and-substrate?"
Turing's goal here is to justify why a Turing machine captures the "intuitive notion of a computation". To his audience, a "computer" is a person with a stack of paper and a pen, following an arbitrary procedure. Turing shows here that, even though a person can potentially draw infinitely many drawings on a sheet of paper, that doesn't give them any more computational power then writing symbols from a finite alphabet into a grid of squares.
The machine he introduces also writes symbols onto paper with a pen and reads them. So he really is talking about pens and sheets of paper, it's not an analogy for something else.
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.
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.