All that is gold does not glitter,
Not all those who wander are lost;
The old that is strong does not wither,
Deep roots are not reached by the frost.-- J.R.R. Tolkien, The Fellowship of the Ring
Symbols
Since we are born, and even before that, we interact with the world through collision and violence. For me to live, something must die, be it a plant or an animal. For me to stand, the floor must push me. For me to see, light must crash into my eyes. For me to speak, I must shape the air. And yet, on the inside, we live in a dream, from the violence we create a world, a universe, in our mind. Our mind projects the reality inside of itself. And since each of our minds is uniquely shaped by violence, I can only interact with you through symbols. Symbolic language is hundreds of thousands years old, and it is possibly our greatest creation.
In this chapter I will try to explain what are symbols, how they transform and
evolve, how does it feel to do symbolic execution, and what is computation.
This is the Eye of Horus, the left wedjat eye, it is an ancient Egyptian symbol, more than 5000 years old. The very first time you see it, it will speak to you. You will try to explain it, examine it; without reason.
Horus lost his left eye in a battle with Set, the god of chaos. Later restored by Hermes Trismegistus, Thrice-Great Hermes, also known as Thoth, the god of wisdom, its restoration is considered a triumph of order over chaos. The left wedjat is the symbol of the moon. And since it was healed from wisdom, it became a symbol of healing and renewal. You might notice today on some medications or recipes symbol Rx (℞), it originates from the Eye of Horus, you can see the shape of the R, later it became the symbol of Jupiter, and then the first letter of the Latin word 'Recipere'.
How much cultural experience is packed in this symbol? 5000 years of hope, hundreds of millions of people praying to it every day, teaching their children how to use it, how to draw it..
This is the alchemist symbol of the philosopher's stone. The second Adam.
It represents the evolution of a whole culture, whole societies have been violently transformed because of it. The philosopher's stone, some say, is able to transform any metal into gold. You might think it is manifestation of the infinite human greed, but others believe it is the transformation of the soul. The expression of Anima Mundi, the soul of the world. The world, Plato says, has soul and reason.
Now, pause for a bit and think, is the symbol changing our culture, or our culture changing the symbol.
To understand one symbol means to understand everything.
The word 'sun' is only 3 symbols, and it itself is a symbol, ⬤ is only 1 symbol, however their interpretation is up to you. When you read them, what do you see? I see a sunrise, cycling to work, passing the lake, a burning star, I hear the sound of the wind, I can even smell the air. You might see a sunset, or feel the nnheat, or might even see the moon on a cold night. Information lives in two worlds, outside as a symbol and inside as a dream. Neither world is more real than the other.
A symbol is not merely a group of dots, a sound wave, or a shape. When I write
the symbol for one: l, I am not just making a mark, I am creating a bridge
between the physical and the abstract, or in some cases between two abstract
worlds. This bridge works in both directions: the physical symbol shapes our
mental concept of 'oneness', while our understanding of 'oneness' gives meaning
to the symbol. Also that was not the symbol for one (1), that was small letter
L: l, you made it into one when you thought about it being a number.
The symbols change us and we change their meaning. We interact with symbols in two ways, we can interpret them or evaluate them.
Interpretation is giving meaning to the symbol, for example reading black cat,
you interpret it and imagine a black cat, unless you have aphantasia, in which
case you just think of a black cat without an image.
Evaluation is the process of giving life to symbols. When you see 2 + 2, your
mind doesn't just read characters, it gives them meaning, as it iterates through
ideas and experiences, it produces new symbol: 4, without you even wanting to
do it, I dare you, try to not do it, try to read 2 + 2 and not think of 4.
The symbol's meaning and the process it invokes in you, exists neither in the
symbol nor in your mind, but in their interraction and transformation.
Evaluation of symbols is to execute the symbol, let is live and act, according
to its relationship with everything else.
I am very interested in this particular relationship between the symbols and their observer, or evaluator, especially when the evaluator is symbolic as well.
There is a famous example from Gödel, Escher, Bach: can a record player play all possible records? What about the record that produces vibrations that damage the record player? Can a human think all possible thoughts? What about thoughts that make you inhuman?
In order to continue, I must explain what evaluation is, and what computation is, in its deepest sense, since we, humans, can evaluate symbols, I will try to make you experience symbolic evaluation and transformation.
Lets start with the following sentence:
I am what I was plus what I was before I was.
Before I began, I was nothing.
When I began, I was one.
While reading the words you interpret them, you asign them meaning and understand them. Now lets evaluate them, but I will rewrite the riddle in a different way, even though it means the same thing, it will be a bit easier to write down the process.
F(n) = F(n-1) + F(n-2)
F(0) = 0
F(1) = 1
Surprise! It is the Fibonacci sequence.
Now, lets evaluate it in our head:
0 | 0: Before I began I was nothing
1 | 1: When I began I was one
2 | 1 = 1 + 0 I am what I was plus what I was before I was.
3 | 2 = 1 + 1 I am what I was plus what I was before I was.
4 | 3 = 2 + 1 I am what I was plus what I was before I was.
5 | 5 = 3 + 2 I am what I was plus what I was before I was.
6 | 8 = 5 + 3 I am what I was plus what I was before I was.
7 | 13 = 8 + 5 ...
8 | 21 = 13 + 8 ...
... | ...
50 | 12586269025 = 4807526976 + 7778742049
... | ...
250 | 7896325826131730509282738943634332893686268675876375 = ...
... | ...
Try another one:
This sentence is false.
The previous sentence is true.
You might feel physical pain while evaluating it, if you keep cycling between the statements, deeper and deeper into confusion. Kind of like this optical illusion, white dots appear and disappear, they are there, but they are not.
Experiencing true infinity by just evaluating few symbols. But the infinity is made by you both having vocabulary, and applying the English grammar rules.
Lets deconstruct their grammar:
This sentence is false.
- Main clause: "This sentence is false."
- Subject: "This sentence" (a noun phrase: determiner "this" + noun "sentence")
- Verb: "is" (copula)
- Complement (predicate adjective): "false" (adjective describing the subject)
This is a simple linking structure: Subject + Linking Verb + Adjective.
The previous sentence is true.
- Main clause: "The previous sentence is true."
- Subject: "The previous sentence" (a noun phrase: determiner "the" + adjective "previous" + noun "sentence")
- Verb: "is" (copula)
- Complement (predicate adjective): "true" (adjective describing the subject)
Again, a simple linking verb pattern: Subject + Linking Verb + Adjective.
This sentence is false. The previous sentence is true.
When taken together, these two sentences form an infinite loop:
First sentence: Subject ("This sentence"), Copula ("is"), Complement ("false" - adjective). Second sentence: Subject ("The previous sentence"), Copula ("is"), Complement ("true").
What is a subject, what is a linking verb, what is a noun:
- Subject: The doer or main focus of the sentence.
- Verb: The action word, or in the case of a "linking verb," a state-of-being word (e.g., "is," "are," "was," "were").
- Complement: Information that follows a linking verb and describes or renames the subject. This can be an adjective (predicate adjective) or a noun (predicate nominative).
ChatGPT did the grammar deconstruction, I know almost nothing of English grammar.
Deconstructing the vocabulary:
- "this" - demonstrative determiner/adjective pointing to the current sentence
- "sentence" - noun referring to a grammatically complete unit of language
- "is" - present tense form of "to be", functioning as a linking verb
- "false" - adjective describing a statement that is not true
- "the" - definite article specifying a particular thing
- "previous" - adjective describing something that came before
- "true" - adjective describing a statement that is factual/correct
But where do we stop?
- "demonstrative" from Latin "demonstrativus" meaning "pointing out", "demonstrare" = de- (completely) + monstrare (to show) a word that directly indicates which thing is being referenced
- "determiner" from Latin "determinare" = de- (completely) + terminare (to bound, limit) a word that introduces or modifies a noun
- "adjective" from Latin "adjectivum" = ad- (to) + jacere (to throw) a word that describes or modifies a noun..
- ...
How much vocabulary is needed for the infinity to occur? How much grammar is needed? How can the language's gramatical rules be written in the very language they describe? What about the grammar rule: "A sentence must end with a period.", is it gramatically correct? What if it was "A sentence must end with a period" without the period?
In the same time, when you are reading the sentences you are not thinking about the grammar at all, nor about the vocabulary, nor about the words even. Almost instantly confusion arises from the paradox. I am not even sure you and I are reading the sentence in the same way. This is quite strange is it not? Most people can read this without any trouble:Tihs scnetnee is flase. The perivuos scnetnee is ture., and get instantly into confusion. Somehow words are still readable if the first and last letter are correct. But if we read scnetnee as sentence, then what is actually the symbol of sentence?
I have tricked you a bit. This sentence is false is already a paradox in
itself. If the sentence is false then it must be true, since it claims to be
false, but in that case, it must be false because that is its statement, true,
false, true, false.. Epimenides declared: all Cretans are liars, and he
himeself was a Cretan, and people say he always tells the truth. This paradox is
even in the Bible, Titus 1:12 12 One of Crete's own prophets has said it:
"Cretans are always liars..", but it does not say if whoever declared the
statement is a liar or not. However Crete's own prophet must be Cretan as well.
Now lets try something that requires more steps, so that you can experience the application of logic rules:
S1: The next sentence is true.
S2: The fourth sentence is false, if the next sentence is true.
S3: The previous sentence is true.
S4: The first sentence is false.
We will rewrite it so it is easier to evaluate
S1 → claims S2 is true
S2 → claims (if S3 is true then S4 is false)
S3 → claims S2 is true
S4 → claims S1 is false
If S1 is true:
- Then S2 must be true (by S1)
- If S2 is true and S3 is true, then S4 must be false (by S2)
- S3 confirms S2 is true
- But if S4 is false, it means S1 is true
If S1 is false:
- Then S4 is true (since S4 claims S1 is false)
- If S3 is true, then S2 must be true
- If S2 is true and S3 is true, then S4 must be false
- But we started by assuming S4 is true
Now we are one layer above the grammar and its rules, the sentences themselves have rules, in our case S4 must be false, in order for S1 to be true, which leads to contradiction. But what is the transformation here? The sentences are the same, written on the page, what is being transformed? It is your thought. You are transforming each sentence, from true to false and so on, which is itself changing the rules, since the sentences are their own rules.
This process of evaluating information and allowing it to transform itself is the act of computation.
I am not trying to say that you are a computer, I am trying to show what it means to experience computation. The fact that your brain can compute statements, that does not make you a computer, just as your heart pumping blood, does not make you a pump.
This duality of existence of information, both as its state and as its transformation, both as the actor, and the play, this duality is what we will investigate in this book. The painter and the painting.
Now try to evaluate this Zen Koan:
Yamaoka Tesshu, as a young student of Zen, visited one master after another. He called upon Dokuon of Shokoku.
Desiring to show his attainment, he said: "The mind, Buddha, and sentient beings, after all, do not exist. The true nature of phenomena is emptiness. There is no realization, no delusion, no sage, no mediocrity. There is no giving and nothing to be received."
Dokuon, who was smoking quietly, said nothing. Suddenly he whacked Yamaoka with his bamboo pipe. This made the youth quite angry.
"If nothing exists," inquired Dokuon, "where did this anger come from?"
This is what computation is, the process that gives life to information, allowing it to transform itself. A program is a sequence of computations, and it itself is information.
Notice that in this definition, symbols are not required for computation, but in order for us to manipulate or understand computation, symbols are required.
I read what I write.
Each reading changes what I write next.
Each writing changes what I read next.
By now, you have intuition about what evaluation is, or at least how it "feels" when you are evaluating symbols, however, you were doing it unconsciously. Now we will create a former rule that we want to apply, step by step.
I will show you the most amazing game you have seen, you will not be the
player, you will be the board. Start by writing the following numbers on paper
0 0 0 1 0 0 0.
0 1 2 3 4 5 6 (column indexes, so that I can reference them)
-------------
0 0 0 1 0 0 0
Each round, you write a new row, applying the following rules to each cell.
Look up to the previous row, and check itself and neighbors, in our example on cell 2, on the left you have 0, in the middle is itself, with value 0 and on the right you have 1, Cell 6 has 0 on the left, and we get outside of the board on the right, so we assume 0, same for cell 0, on the left we assume 0, on the right is also 0 (cell 1 is 0 in our example).
The rule is the following:
left,middle,right 111 110 101 100 011 010 001 000
output 0 0 0 1 1 1 1 0
So in our example, if we evaluate the first row, and apply the rules
0 1 2 3 4 5 6
-------------
0 | 0 0 0 1 0 0 0
1 | 0 0 1 1 1 0 0
You can see on cell 2 when you look at row 0, cell 2, on the left it nas 0, on
the right it has 1, so we look in the rules and see 001 gives us 1. and on
cell 3 010 gives us 1. Lets do few more rounds.
0 1 2 3 4 5 6
-------------
0 | 0 0 0 1 0 0 0
1 | 0 0 1 1 1 0 0
2 | 0 1 1 0 0 1 0
3 | 1 1 0 1 1 1 0
The board is too small to see, but the pattern it creates, is actually amazing.
You can see the rules clearly and also the pattern they generate.
If you create enough columns it becomes this:
There are more games like this, that play themselves, they just need a board to
evaluate the rules. The one we just played is called rule30, and it generates
this interesting shape. The interesting thing is, if our first row is 0 0 0 0 0 0, applying the rules produces another empty row, because 000 outputs 0. So
when looking at an empty page, it might seem there is nothing going on, but underneath, this amazing pattern was hidden.
As I said, a program is a sequence of computations, but in this game, what is
the actuallty the program? Is it the rules, is it the process of applying them,
or the very first row 0 0 0 1 0 0 0? I would argue that the rules are the
program, and 0 0 0 1 0 0 0 is the initial condition, the application of the
rules is computation. But what about rule 110, the rules change just a tiny bit,
but it has profound consequence.
left,middle,right 111 110 101 100 011 010 001 000
output 0 1 1 0 1 1 1 0
If you run it by itself with 0 0 0 0 0 1 it creates this beautiful pattern
But, if you run it against an infinitely repeated specially crafted background pattern, then rule110 becomes a computer. It still amazes me, the relationship between the background, the rules, and their evaluation. And the process of abstract computation.
There are other zero player games that are computer, if you see them play you might notice how this might work.
Conway's Game of Life is a famous one, it is not one dimentional like rule30 or rule110, which operate row by row, but it is two dimensional, grid based. There are rules about how the cell evolves depending on its neighbours.
- Birth: A dead cell with exactly three living neighbors becomes alive in the next generation
- Survival: A living cell with two or three living neighbors stays alive
- Death by loneliness: A living cell with fewer than two living neighbors dies
- Death by overcrowding: A living cell with more than three living neighbors dies
Those games are real computers, and by that I mean it is a system that can transform information and let information transform itself as it is being evaluated. People are actually writing programs for the game, and I kid you not, this game can run any program that you can run on your computer, or on any other system that we call a computer.
We will get deeper into the topic of computation later. For now, I will leave
you with the confusion of the program that is a game that is a computer. Do a
Life in Life in Life search in youtube if you want to see how it looks.
It seems, the symbols, their interpretation, their evaluation, and their output, all live in separate worlds, and yet, their output can create new symbols, and the symbols can change their evaluation rules, as the rules are also symbolic.
It also seems, that incredibly simple rules, can create infinite complex systems. Including systems that can simmulate themselves, or simmulate worlds.
Now, in case of rule110, what is actually the program? Is it the background? Is it the initial condition? Is it the rule itself? What if we have rule110 written in the background of rule110, so that it evaluates the rules of rule110?
That is what Life in Life in Life does. It is a game of life, inside game of
life, inside game of life.
But you must think above, beyond the rules, beyond the evaluators, beyond the state, but in their relationship, as the rules can change the evaluators, who change the state, and the rules are state as well.
The world, Plato says, has soul and reason.