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Models and reality
1. Models and reality
A model is an abstraction of reality.
Essentially, all models are wrong, but some are useful. George Box, Statistician.
The best material model of a cat is another, or preferably the same, cat. Norbert Wiener (and A. Rosenblueth).
A model is a useful fiction.
2. Model fit
Whenever one is presented with a logical model, one should take the following steps.
1. Does the logical model fit? How well does it fit? What does it explain and what does it not explain?
2. If the logical model fits, what are the implications and consequences in terms of reality?
Example: Albert Einstein discovered and jump-started the field of quantum mechanics. He never liked the idea. The model fit and he knew it fit.
3. Models
What is a model?
A model is an abstraction of reality.
As a representation of reality,
models are often used to answer or predict specific questions about that reality.
The purpose of data science, for example, is insight.
4. Models: simple
One goal is to create models of what was said that, in a sense, minimize assumptions of what was said while not assuming things that might have been meant. Here is a simple way to think about a model.
A
model is an abstract representation of the real world with a postulated
mapping between the real world and the model (and between the model and the real world).
5. Models: refined
Here is a more refined way to think about a model. A model is an abstraction of reality.
Essentially, all models are wrong, but some are useful. George Box, Statistician.
The best material model of a cat is another, or preferably the same, cat. Norbert Wiener (and A. Rosenblueth).
A model is a useful fiction. George Box, Statistician.
6. George Box
Essentially, all models are wrong, but some are useful. 1976, 1978. George Box, Statistician.
A model is a useful fiction.
7. Norbert Wiener
The best material model of a cat is another, or preferably the same, cat. Norbert Wiener (and A. Rosenblueth).
8. Actors and pastors
It can be a mixed blessing to have a pastor with an acting background.
Actors speak of things imaginary as if they were real, while you preachers too often speak of things real as if they were imaginary. Thomas Betterton (English actor and theater manager during Restoration England) (1635-1710)
Actors (tend to) take an imaginary world and make it seem real.
Pastors (tend to) take a real world and make it seem imaginary.
With a pastor with an acting background, it can be hard at times to separate reality from fiction.
9. Mathematics and reality
Do whole numbers exist?
Do integer numbers exist?
Do real numbers exist?
Does infinity exist?
10. Infinite things
Two things are infinite: the universe and human stupidity; and I'm not sure about the universe. Albert Einstein's (Physicist)
11. Reality
Is God a mathematician?
Is mathematics real or just in the imagination (of man)?
Albert Einstein (English): "As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality". (Albert Einstein, 1879-1955).
Albert Einstein (German): "Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nict sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit" (Albert Einstein, 1879-1955).
12. Map of the world
Is a one to one (1 to 1) model useful? What would make a good 1 to 1 map of the world?
When the map (model) does not match the world (reality) what do you change?
Change the map to fit the reality of the world.
Change your understanding of the world to fit the map.
Some churches and pastors adapt to the reality that will
fill the seats (to support their belly).
13. Implications of a model
A model is said to faithfully reflect the real world if
implications of the model (usually derived via mathematical calculations), when mapped back into the real world, are a sufficient approximation of truth in the real world to be useful.
Models can be deceptive.
14. Day the universe changed
James Burke, science historian, did an interesting video series in the 1980's entitled "
The Day the Universe Changed: A Personal View by James Burke".
The title comes from the philosophical idea that the universe essentially only exists as you perceive it through what you know; therefore, if you change your perception of the universe with new knowledge, you have essentially changed the universe itself. Wikipedia,
http://en.wikipedia.org/wiki/The_Day_the_Universe_Changed
15. Average ocean depth
A model is a simplified representation of something (e.g., a reality).
Models can be deceptive. Take, for example, the depth of the ocean.
16. Average ocean depth
The average ocean depth is over
2 miles deep (
12,200 feet).
On a model globe, the oceans (and highest mountains) would be about the thickness of a piece of paper.
17. On the earth
To see this, consider the following calculations. On the earth:
12,000 feet ocean depth
--------------------------------- = 0.000284 (depth/diameter)
(5,280 feet/mile) * (8,000 miles)
18. On a globe
On a 12-inch globe, this thickness would be as follows.
0.000284 * 12 = 0.0034 inches
19. Ream of paper
Now consider a ream of paper and the thickness of 500 sheets of paper.
(1.5 inches/ream)
----------------- = 0.003 inches/sheet
(500 sheets/ream)
So on a model globe, the oceans (and highest mountains) would be about the thickness of a piece of paper.
20. Descriptive or predictive
Models can be classified as descriptive or predictive.
21. Descriptive models
To describe something is to talk about the attributes/properties of something.
Marketing models tend to be more descriptive than predictive.
22. Predictive models
A
descriptive model is a model that talks about the attributes/properties of something.
To
predict something is to claim that something will happen before it actually happens.
If you predict and you are wrong, you lose credibility.
23. Predicting the future
A
predictive model is a model that can claim what will happen in that model before it actually happens.
The best way to predict the future is to invent it. Alan Kay (American computer scientist)
24. Assumptions
A predictive model should clearly state the assumptions under which the model correctly predicts what will happen.
Engineering models tend to be more predictive than just descriptive.
25. Black box models
An
assumption is something that is to be true for the desired conclusions to be drawn.
When presented with a model, you might have questions about the inner workings of the model.
26. Black box models
1 Black box
2 White box
3 Glass box
A black box model only shows the input and/or output of the model, not the inner workings. A white box or glass box model allows the internals to be known.
An example of a black box model is quantum mechanics. How it works (functionally) is very well understood but why it works is unknown.
27. Functional behavior
A black box model shows the functional behavior of the model. That is, only from the outside without knowing how it works inside.
It acts as a mathematical function where one provides and input and gets an output.
28. Quantum mechanics
Nobody understands quantum theory.
Richard Feynman (American theoretical physicist)
He was including himself.
29. Quantum mechanics as a black box
The famous physicist Richard Feynman, the most famous physicist between the times of Albert Einstein and Steven Hawking, often stated that quantum mechanics is unquestionably correct in that accurate numbers to great precision can be computed from the theory.
On the other hand, according to Feynman, no one understands why quantum mechanics works or how it works inside, but we have a model that can be used to get useful results from the model.
A model such as quantum mechanics can explain what it does but no one is sure how it works, let alone why it works.
30. Referential transparency
Mathematical functions can be considered black boxes that, given an
input, provide an
output. This is much like a computer program. In a
black box function, you cannot see inside. You should not "
judge" or "
separate" based on the internal workings of the box - which you cannot see nor fully understand. Instead, you should "
judge" or "
separate" based on the external behavior of the box.
The principle of
referential transparency can be stated as follows.
If f(x) is equal to g(x) for all x in the domain of concern, then function f is equal to function g.
That is, if
f and
g are provided input
x and both produce output
y, for the domain of concern they can be considered the same.
The Greek word for "
judge" or "
separate" is related to the Greek word for "
barley".
31. Declarative model of creation
The creation account in Genesis, in terms of creation days, provides a declarative model of the creation.
A declarative model of a system is a model that describes the what of a system but not the how (or why) of a system. This term is used in computer science and software engineering (as black box functional testing). At another interesting level, each day of the creation is associated with a declarative statement as God says or speaks (or declares) what is to be done.
32. Black box model for the Trinity
Some people confuse a "
dimension" as a way to look at something with a "
mode" as a heresy. For those, each "
view" from one side of the box may help.
Does it make any difference in practice if we know the exact nature and inner workings of each aspect of the Trinity?
1 Corinthians 13:12 For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known. [kjv]
Each side uses scripture to justify their view. What do we know?
33. Thinking as a black box
A human cannot know exactly what someone else is thinking. Others are like a "
black box" from the outside. The Bible says this. Jesus, however, could know their thoughts.
 |
Details are left as a future topic.
|
... more to be added ...
34. Puzzle pieces and models
Each insight at one place to what Jesus appears to mean from what he says, in context, acts as a puzzle piece to determine what a similar phrase means in another discourse.
Presented by Jesus and recorded by Matthew: Declarative, sometimes top-down backward-chaining, and distributed fault-tolerant and redundant spread-spectrum constraint logic (code) word (and meaning) puzzle.
Inference method: Bottom-up nondeterministic model parse.
Each underlined word has a deep and technical meaning in the field of programming language theory and in computation theory as part of the general field of computer science.
Discuss:
How does a model approach contrast with an opinion-based approach to inferring what is meant in the Bible?
Provide historical examples of both model-based approaches and opinion-based approaches to inferring what is meant in the Bible.
35. Interpretations and models
Precisely defining models can confuse some people.
Definition: An
interpretation of a
first order language L consists of the following.
a. A non-empty set D, called the domain of the
interpretation.
b. For each constant in L, the assignment of an element in D.
c. For each n-ary function in L, the assignment of a mapping form D
n to D.
d. For each n-ary predicate in L, the assignment of a mapping from D
n into { true , false } (or, equivalently, a relation in D
n).
[Aristotle quote]
36. Interpretations and models
Definition: Let I be an
interpretation of a
first order language L and let F be a closed formula of L. Then I is a
model for F if the truth value of F with respect to I is true.
Definition: Let T be a
first order theory and let L be the
language of T. A
model for T is an
interpretation for L which is a
model for each axiom of T.
Some precise ways of dealing with symbols, languages, interpretations, models, etc., can be found in the field of logic programming. The above definitions are on pages 12-13 of: Lloyd, J. (1984).
Foundations of logic programming. Berlin: Springer-Verlag. This book (about 125 pages) was used in a graduate computer science course in logic programming that I took years ago.
Here is a start at a more precise code word model.
Here is a start at a more precise code word model.
37. Code word model
A logical variable represents a semantic concept. Predicates can select the part of the meaning needed for that part of a sentence fragment. Example:
Let X be a code word logical variable.
verb(X) is logical variable X as a verb. Mode, tense, etc., are omitted for simplicity.
noun(X) is logical variable X as a noun. Gender, etc., are omitted for simplicity.
adjective(X) is logical variable X as an adjective.
(and so on)
Suppose
X is the idea of "
salt".
noun(X) is "salt".
verb(X) is "salted".
adjective(X) is "salty".
Given the use of the logical variable for "
salt", an assignment of a value to this logical variable would make sense in each place where that logical variable is used.
Note that in an actual logical programming language (think of Prolog), the implementation would be somewhat different.
Simplified syntax model: noun(X)
Logic programming syntax: X, noun(X)
38. End of page
