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Reflexive fixed points in the Bible
1. Reflexive fixed points in the Bible
2. Animal crackers
3. Constraint logic
A constraint logic system is a system of constraints whose solution needs to satisfy those constraints.
A unifier unifies the constraints without contradictions.
A
MGU (Most General Unifier) unifies the constraints without contradictions and has a minimal frontier in the lattice of constraints, a
LFP (Least Fixed Point).
If there are multiple solutions, a constraint logic system will attempt to find all
MGU constraints.
4. Fixed point puzzle
Consider the following puzzle (origin unknown, found Summer 1991) consisting of a
self-referential sentence.
In this sentence, the number of occurrences
of 0 is ________,
of 1 is ________,
of 2 is ________,
of 3 is ________,
of 4 is ________,
of 5 is ________,
of 6 is ________,
of 7 is ________,
of 8 is ________, and
of 9 is ________.
Fill in the blanks with the appropriate numbers so that the sentence is true.
For example, if you put in 1 for the number of times 1 appears, then 1 now appears 2 times.
This puzzle is a simple constraint logic puzzle.
You are to find a MGU that is a LFP.
5. Document fixed point
Whenever you write a paper, you are trying to achieve a fixed point in paper writing space. That is, you want to write a paper such that when you proof the entire paper, there are not changes to be made.
You paper is done either,
when you reach a fixed point in paper writing space,
or you give up and turn in what you have.
6. Reflexive property
A relation
R on set
A is
reflexive if for every
x in
A,
x R x.
This can be written in mathematical form as follows.
This is read as "
for all x in (set) A, (condition) x R x (is true)"
7. Reflexive laughing
An example of a
reflexive rule is the following.
It is good to be able to laugh at oneself.
Have you heard that being able to laugh at yourself may help
lengthen your life?
Here, the "
laugh at" relation is applied
reflexively to itself. That is, relating "
laugh at" from "
you" to "
you".
8. Not reflexive laughing
Have you heard that laughing at your spouse may help
shorten your life?
Here, the "
laugh at" relation is not applied reflexively.
9. Fixed point combinator
A
fixed point is a
x for function
f (taking parameter
x) such that
f(x) = x.
Book:
Introduction to combinators and lambda calculus. J. R. Hindley, J F Seldin. 978-0521318396.
A fixed point
combinator, usually called
Y (in lambda calculus, originally by Curry), is a higher order function that takes a function as an argument and returns a fixed point for the argument (if it exists). A
combinator is a closed lambda expression in that it has no free variables. In
lambda calculus terms, the fixed point combinator can be defined as follows.
Y = λ f . ( λ x . f ( x x )) ( λ x . f ( x x ))
If such a fixed point exists, then
Y(f) = f(Y f), and
Y(f) = f(Y f) = f(f(Y(f))), and so on (infinite regression).
A text formatting system that can format itself can model this behavior (to any limit desired).
10. Exponential constant
e(0.0) = e0.0 = 1.0000...
e(1.0) = e1.0 = 2.7182...
e(2.0) = e2.0 = 7.3890...
...
|
The exponential constant e, Euler's number, discovered by Jacob Bernoulli in 1683, is defined such that the slope (first derivative) of the function e(x) is e(x) (i.e., as a fixed-point). The value of e = e(1.0) = e1.0 is approximately 2.718281828459.
Numbers such as e are transcendental and, like irrational numbers, have no exact representation and can only be approximated.
|
11. Nouns and verbs
A reflexive use of a noun and verb is to have a noun use a verb to describe itself. This becomes interesting if the verb is an action form of the noun itself.
The word "verb" is a "noun".
The word "nounify" is a "verb".
If you are not sure, when what's the "
ask"?
Some examples where the
verb matches the
noun.
Healer (physician) healing the healer.
Savior saving the savior.
12. Aristotle: Nouns and verbs
English: A noun is a sound having meaning established by convention alone but no reference whatever to time, while no part of it has any meaning, considered apart from the whole. (Loeb#325, p. 117)
Greek: Ὄνομα μὲν οὖν ἐστὶ φωνὴ σημαντικὴ κατὰ συνθήκην ἄνευ χρόνου, ἧς μηδὲν μέρος ἐστὶ σημαντικὸν κεχωρισμένον· Aristotle: On Interpretation [16a]
"ὄνομα" ≈ "noun".
"φωνὴ" ≈ "voice, sound"
"σημαντικὴ" ≈ "meaning" and is the source of the English word "semantics".
"ρῆμα" ≈ "verb, spoken word" and is the source of the English word "rhetoric".
English: A verb is a sound which not only conveys a particular meaning but has a time-reference also. No part of itself has a meaning. (Loeb#325, p. 119)
Greek: Ῥῆμα δέ ἐστι τὸ προσσημαῖνον χρόνον, οὗ μέρος οὐδὲν σημαίνει χωρίς· Aristotle: On Interpretation [16b]
13. Code and data
A deep result of programming language and computational theory (first by Von Neumann) is that "
code" and "
data" can be interchanged and depend on the context or viewpoint used.
The interchange of "
data" and "
code" is much like how "
nouns" and "
verbs" are (by ending change, etc.) interchangeable.
The same appears to be true of statistics "
data" (collected) and statistics "
code" (assumed interdependency of parts).
The computer science field of
programming languages is concerned with
programs that (read and) write programs.
14. Reflexive fixed points in the Bible
A
reflexive relationship is a relationship that refers to itself.
A
fixed point, also called an invariant point, is a function where an argument supplied to the function returns or maps to itself.
Fixed point semantics is an important idea in computation and programming language theory.
Here we look at the idea of reflexive fixed points in the
GNT (Greek New Testament).
15. Links
There are many simple reflexive relationships in the Bible such as the "
blind leading the blind".
Reflexive relationships in the Bible
I like reflexive verbs that please me
Third person reflexive references
It is "
blind leading the blind" and
not the "
blind blinding the blind" which is the subject of this page.
Here we look only at relationships where the reflexive action is the same as the entity being acted on such as "
salt salting the salt", "
treasurizing the treasure", etc.
To get started, we need some foundational ideas of computer programming and (simplified) programming language theory.
16. Users running programs
Simplified
user actions:
1. Prepare
input data (keyboard typing, mouse movement and clicks, input files, etc.).
2. Run the executable
program binary code (on a computer, laptop, phone, etc.).
3. Obtain the
output results (screen output, audio output, output files, etc.).
From where does the
user get the executable
program binary code?
17. Programmers writing code
From where does the
user get the executable
program binary code?
Simplified
coder (programmer) actions.
1. Write
program text.
2. Compile it using the executable
compiler binary code.
3. Obtain the output as executable
program binary code for the user.
From where does the
coder (programmer) get the executable
compiler binary code?
18. Teaching objective
I often have students in beginning programming do the following. In an advanced course, a different language is required for each step.
Write a program to output their name.
Write a program to write a program to output their name.
Write a program to write a program to write a program to output their name.
... and so on ...
19. Programs that write programs
Here is a progression of programs that write programs in the
Python programming language.
Program
output
|
Hello World
|
Program that
writes a program.
|
print("Hello world")
|
Program
that writes a program
that writes a program.
|
print("print(\"Hello world\")")
|
Program
that writes a program
that writes a program
that writes a program.
|
print("print(\"print(\\\"Hello world\\\")\")")
|
Most people will do this to some level. A computer scientist who recognizes the (inductive) pattern will look for a fixed-point in program-writing space. That is, a finite description of a potentially infinite object (series of programs).
20. Programs that write programs
|
list1 = [
"list1 = [",
"x",
"\t]",
"for item1 in list1:",
"\tif item1 != \"x\":",
"\t\tprint(item1)",
"\telse:",
"\t\tfor item2 in list1:",
"\t\t\ts1 = \"\"",
"\t\t\tfor ch1 in item2:",
"\t\t\t\tif ch1 == \"\\t\":",
"\t\t\t\t\ts1 = s1 + \"\\\\t\"",
"\t\t\t\telif ch1 == \"\\\"\":",
"\t\t\t\t\ts1 = s1 + \"\\\\\\\"\"",
"\t\t\t\telif ch1 == \"\\\\\":",
"\t\t\t\t\ts1 = s1 + \"\\\\\\\\\"",
"\t\t\t\telse:",
"\t\t\t\t\ts1 = s1 + ch1",
"\t\t\tprint(\"\\\"\" + s1 + \"\\\",\")",
]
for item1 in list1:
if item1 != "x":
print(item1)
else:
for item2 in list1:
s1 = ""
for ch1 in item2:
if ch1 == "\t":
s1 = s1 + "\\t"
elif ch1 == "\"":
s1 = s1 + "\\\""
elif ch1 == "\\":
s1 = s1 + "\\\\"
else:
s1 = s1 + ch1
print("\"" + s1 + "\",")
|
Such a sequence happens in even professional programming where the programmer does not recognize the pattern.
The generalized pattern is to achieve a fixed point by having a program output an exact copy of itself.
One such example is on the left. The need to store a copy of itself and use some special markers to distinguish the data from the code provides a very simplistic idea of some of the requirements for DNA (Deoxyribonucleic Acid) to work.
|
21. Meta-coder writing compilers
From where does the
coder (programmer) get the executable
compiler binary code?
Simplified
meta-coder actions.
1. Write
compiler text.
2. Compile it using the executable
compiler binary code to get the
compiler binary data.
3. The
compiler binary data is the
compiler binary code for the
coder (programmer).
A primary aspect of the field of programming languages and program language theory is that of programs that write programs that write programs, etc.
How can this process obtain a
reflexive fixed point?
22. Reflexive fixed point
Simplified
meta-coder actions as
reflexive fixed point.
1. Write
compiler text.
2. Compile it using the executable
compiler binary code.
3. Obtain the output as executable
compiler binary code for the coder (programmer).
This
compiler binary as output should be the exact same executable binary code as the
compiler binary itself.
To achieve this, one can use what is called an iterative
boot-strap process.
23. Reflexive fixed point build
1 Program run
2 Program compile
3 Compiler compile
4 Self-compile
Here is a review of the steps.
To achieve this reflexive fixed point (last step), one can use what is called an iterative boot-strap process.
The field of programming language theory is largely that of programs that write programs (including writing itself).
24. Boot-strap formatter build
A
formatter is a program that takes
text as
input and outputs modified or transformed
text as
output (which could be
code).
... more to be added ...
25. Production formatter data flow
26. EBNF in EBNF
Here is the same grammar in textual form.
Syntax = { Production } .
Production = "Variable" "=" Expression "." .
Expression = Term { "|" Term } .
Term = Factor { Factor } .
Factor = "Terminal" | "Variable" | "[" Expression "]" | "{" Expression "}" | "(" Expression ")" .
27. BNF textual grammar
28. Matthew 5:13 Pigs and sheep
Matthew 5:13 Ye are the salt of the earth: but if the salt have lost his savour, wherewith shall it be salted? it is thenceforth good for nothing, but to be cast out, and to be trodden under foot of men. [kjv]
υμεις εστε το αλας της γης εαν δε το αλας μωρανθη εν τινι αλισθησεται εις ουδεν ισχυει ετι ει μη βληθεν εξω καταπατεισθαι υπο των ανθρωπων [gnt]
 |
 |
salt
fruit
|
blocking
the way
|
"Salting" "salt" with
"salt" is a
reflexive process.
13:
You are the salt of the earth. If the salt is false salt, how shall the salt be salted (
pigs).
If you use false salt as in false fruit or works, it is no better than if it had been thrown on the road as a false way (enticement) for men or pigs to follow (as false "fruit").
Sheep: Instead, the "
Holy Spirit" helps create the
true "
fruit" or
"salt" of the "
spirit" to be
"cast" on the
true "way" on "
earth" to "
heaven"
"before" men (not "
pigs").
29. Matthew 5:14 Fish and sheep
Matthew 5:14 Ye are the light of the world. A city that is set on an hill cannot be hid. [kjv]
υμεις εστε το φως του κοσμου ου δυναται πολις κρυβηναι επανω ορους κειμενη [gnt]
 |
 |
light
light
|
swimming
in sin
|
"Lighting" "light" with
"light" is a
reflexive process.
14:
You are the light of the world. If the light is false light, how shall the light be lighted (
fish).
If you use false light, it is no better than the many who make a city on a man-made mountain and have false salt (fruit or works). That false light and false salt (fruit or works) cannot be hid and the many cannot remain/abide there.
Sheep: Instead, the "
Son" is the true
"light" and "
rock" on which the many should build their "
house" in (among the "
people" or "
fish" of) the "
world".
30. Matthew 5:15 Birds and sheep
Matthew 5:15 Neither do men light a candle, and put it under a bushel, but on a candlestick; and it giveth light unto all that are in the house. [kjv]
ουδε καιουσιν λυχνον και τιθεασιν αυτον υπο τον μοδιον αλλ επι την λυχνιαν και λαμπει πασιν τοις εν τη οικια [gnt]
 |
 |
candle
kingdom
|
preying
on others
|
"Candling" a
"candle" with
"candle" as in a
"candlestick" or
"kingdom" is a
reflexive process.
15:
You are the kingdom of heaven/air. If the kingdom is a false kingdom, how shall the kingdom be kingdomed (
birds).
If you use a false kingdoms by taking false light with false salt (fruit or works), holding up for everyone in the house built on sand to see and glorify, that house cannot stand and the many cannot exist there.
Instead, the "Kingdom" of the "Father" or "God" is the true "house" or "kingdom" built on the "rock".
These ideas might be better understood in a table.
31. Matthew 6:19 Treasurize
Matthew 6:19 Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt, and where thieves break through and steal: [kjv]
μη θησαυριζετε υμιν θησαυρους επι της γης οπου σης και βρωσις αφανιζει και οπου κλεπται διορυσσουσιν και κλεπτουσιν [gnt]
Let us look at the Greek for verse 19 in more detail. The
KJV (King James Version) translates as
"lay up" what in the Greek is a verb form of
"treasure" as in the made-up English word of
"treasurize". Jesus often appears to use play-on-word double-meanings (in Greek).
How does one
"treasurize" a
"treasure"? This is a
reflexive (self-referential) relationship.
The word
"treasure" appears to be a code-word with one or more meanings - all of which should make sense.
32. Distributed knowledge
The following is part of reasoning about
distributed knowledge systems.
It is not enough to "know" something, like when the light turns red, stop! (Or, three more cars can go through).
One must "know" that "everyone knows". This is not enough.
One must "know" that "everyone knows" that "everyone knows", ..., for the number of people involved in the "knows".
One can argue that "
common sense" among a group of people is the fixed point of the "
knows" relation applied to that group of people.
33. Ignorance of agnostic lack of belief
Ignorance has to do with not knowing something with an emphasis on "
not knowing that one does not know".
As soon as you can say "
I do not know X" then "
you know that you do not know X".
For computer scientists and others dealing with and reasoning about knowledge, an important concept is "
knowing that you know" something. John understands this.
1 John 2:3 And hereby we do know that we know him, if we keep his commandments. [kjv]
και εν τουτω γινωσκομεν οτι εγνωκαμεν αυτον εαν τας εντολας αυτου τηρωμεν [gnt]
34. Descartes
What do you really know?
"I think therefore I am" (1644) : René Descartes (French philosopher, mathematician and statistician)
The Latin words "Cogito ergo sum" ≈ "I think therefore I am".
Is the logic and implication of what Descartes says clear?
An optimist may see a light where there is none, but why must the pessimist always run to blow it out? René Descartes (French philosopher, mathematician and statistician)
35. Reflexive fixed point: Know that we know
1 John 2:3 And hereby we do know that we know him, if we keep his commandments. [kjv]
και εν τουτω γινωσκομεν οτι εγνωκαμεν αυτον εαν τας εντολας αυτου τηρωμεν [gnt]
In John 2:3, John will, as usual, write as a computer scientist, this time in the field of intelligent distributed systems, when he writes that "
we know that we know".
How do you
"know" something?
I "know" something.
I "know" that I "know" something.
I "know" that I "know" that I "know" something.
... and so on.
When is the reflexive fixed point reached? Saying this twice is both (in most cases) necessary and (in all cases) sufficient (i.e., once a fixed point is reached).
36. 1 John 2:3
KJV: And hereby we do know that we know him, if we keep his commandments.
Greek: και εν τουτω γινωσκομεν οτι εγνωκαμεν αυτον εαν τας εντολας αυτου τηρωμεν
37. Aristotle: On Interpretation
The ancient Greek word
"λόγος" ≈ "rational explanation" but, in context, can mean
"sentence" or even
"word".
English: A sentence is significant speech. (Loeb#325, p. 121)
Greek: Λόγος δέ ἐστι φωνὴ σημαντική… Aristotle: On Interpretation [16b]
The "
spoken word" can be very powerful and has often has
"reasoning" behind it. God
speaks or
declares something and it is so.
Genesis 1:3 And God said, Let there be light: and there was light. [kjv]
και ειπεν ο θεος γενηθητω φως και εγενετο φως [lxx]
The ancient Greek word
"φωνή" ≈ "voice, sound" comes from the ancient Greek word
"φως" ≈ "light" and the ancient Greek word
"νους" ≈ "mind". That is, a
"sound" or
"voice" is "
light" into the "
mind".
38. Genesis 1:3
KJV: And God said, Let there be light: and there was light.
Hebrew: ויאמר אלהים יהי אור ויהי אור׃
Greek: και ειπεν ο θεος γενηθητω φως και εγενετο φως
39. John 1:1-2 In the beginning
John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God. [kjv]
εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος [gnt]
1:2 The same was in the beginning with God. [kjv]
ουτος ην εν αρχη προς τον θεον [gnt]
The ancient Greek word
"λόγος" ≈ "rational explanation" but, in context, can mean
"sentence" or even
"word" (but not the spoken word).
The ancient Greek word
"προς" ≈ "to, towards". In John 1:1, the translation of this Greek word has been problematic. Here, it is translated as
"with".
Let us use the definition of
"rational explanation" as a reflexive fixed point using the usual definition of
"to" or
"towards".
From the human point of view, the
"rational explanation" can be equated with "
God".
40. Reflexive fixed point: Reasoning of reasoning
John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God. [kjv]
εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος [gnt]
1:2 The same was in the beginning with God. [kjv]
ουτος ην εν αρχη προς τον θεον [gnt]
How do you
"reason" by way of
"rational explanations"?
I "reason" about something.
I "reason" about the "reasoning" of something.
I "reason" about the "reasoning" of the "reasoning" of something.
... and so on.
When is the
reflexive fixed-point reached?
Math notation: X → X (iterated function)
Math notation: reasoning → reasoning
41. Reflexive fixed point: Reasoning of reasoning
John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God. [kjv]
εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος [gnt]
1:2 The same was in the beginning with God. [kjv]
ουτος ην εν αρχη προς τον θεον [gnt]
 |
1. Goal: "reasoning" was "in the beginning" if
2. Relation: "reasoning" → "God" if
3. Equality: "reasoning" ≡ "God".
|
Translating John 1:1 using
"to" or
"towards" is in line with a
reflexive fixed point while the
KJV changes the word to
"with". The reasoning is top-down and backward-chaining.
Bottom-up thinking (most people): put equality first, then relation, then goal.
Top-down thinking (computer scientists, John): put goal first, then relation, then equality
42. Reflexive fixed point: Reasoning of reasoning
John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God. [kjv]
εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος [gnt]
1:2 The same was in the beginning with God. [kjv]
ουτος ην εν αρχη προς τον θεον [gnt]
One might conclude that, from our point of view,
"God" is the
reflexive fixed-point of the
"rational explanation" of everything (with itself). The fixed point provides a consistent universe of time and space as we know it. This is the starting point for this top-down introduction at the beginning of John.
Discuss: Why might theologians, commentators, pastors, etc.,
not have used "
to" or "
towards" and, instead, used "
with"? Consider: Computational thinking and programming language theory were not developed until the late 20th century.
Discuss: Why does John use many modern computational thinking and programming language theory ideas in his writings, including top-down backward-chaining thinking?
43. John 1:1
KJV: In the beginning was the Word, and the Word was with God, and the Word was God.
Greek: εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος
44. John 1:2
KJV: The same was in the beginning with God.
Greek: ουτος ην εν αρχη προς τον θεον
45. Translation
John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God. [kjv]
εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος [gnt]
1:2 The same was in the beginning with God. [kjv]
ουτος ην εν αρχη προς τον θεον [gnt]
Context is important.
The translation at the end of the verse as "
was God" would be "
was a god" if this phrase were considered
alone out of context.
In context, the ending "
was God" is better translated as "
was the God".
However (as discussed above) that the "
with God" is actually "
to/towards the God".
46. The New Testament Diglot
It is interesting when a modern Greek translation changes a word to a meaning used in, say, the
KJV even though the original Greek word in ancient times has the same meaning today. Thus, on the left is the original and on the right is the modern translation.
In both verses,
"to, towards" is changed to
"with".
GNT: John 1:1 εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος
Modern: John 1:1 απ ολἀ πριν υπηρχε ο λογος και λογος ητανε με τον θεο κι ηταν θεος ο λογος.
GNT: John 1:2 ουτος ην εν αρχη προς τον θεον
Modern: John 1:2 απ την αρχη ηταν αυτος με τον θεο.
From: Todays Greek Version of The New Testament by the Greek Bible Society.
Note: Accents removed and all lowercase used.
47. End of page
