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Duality principles
1. Duality principles
2. Duality principles
There are areas of study where there is more than one way to view things.
In ancient China, the yin and yang symbol represents a philosophy of how opposite ideas may be complementary or in some way connected.
The literal meaning of the Chinese symbol for yin yang is "
陰陽" and is composed of "
dark light".
Some of the same ideas or found in the writings of Greek philosophers including Aristotle.
A modern expression of this idea is a duality principle. A duality principle is two (or more) ways of looking at the same reality.
3. Aristotle: Topica
English: ... for example, in the case of clear and dim, for sound is said to be clear and dim and so is colour. Now there is no difference in the terms used, but the variation in kind is immediately obvious in their use; for clear is not used in the same sense as applied to colour and as applied to sound. (Loeb#391, p. 311)
Greek: … οἷον ἐπὶ τοῦ λευκοῦ καὶ μέλανος. Φωνὴ γὰρ λευκὴ καὶ μέλαινα λέγεται, ὁμοίως δὲ καὶ χρῶμα. Τοῖς μὲν οὖν ὀνόμασιν οὐδὲν διαφωνεῖ· τῷ δ´ εἴδει κατάδηλος ἐν αὐτοῖς εὐθέως ἡ διαφορά· οὐ γὰρ ὁμοίως τό τε χρῶμα λευκὸν λέγεται καὶ ἡ φωνή. Aristotle: Topica [106a]
Aristotle explains how words that are often translated as
"white" and
"black" have different meanings in different contexts.
"λευκός" ≈ "bright, light, pale, happy" and, if the context is color, "white" or "light".
"μέλας" ≈ "dark, dim, indistinct" and, if the context is color, "black" or "dark".
For color, it is "light" and "dark", which could be "white" and "black".
For sound, as in a musical sound, it is "clear" and "dim" or "indistinct".
4. Unclear, dark and black
*G3189 *3 μέλας (mel'-as) : apparently a primary word; black:--black.
 |
Words: μελαιναν=1 μελας=2
|
The ancient Greek word
"μέλας" ≈ "dark, dim, indistinct" and, if the
context is
color,
"black". It the source of the English word
"melanoma" (dark skin spots),
"melancholy" (dark bile) and
"Melanie" which was a female name from ancient Greek.
Since this word appears only three times in the
GNT (Greek New Testament) and
once in this form, by Jesus, a play on words may be
in play five verses later.
"μελαιναν" ≈ "indistinct, dark, black" (sound or color)
"μίλιον" ≈ "mile" and "ἑν" ≈ "one"
"μυλινον" ≈ "millstone" as used in Revelation 18:21.
Greek: ... μελαιναν (dark, dim, indistinct)
Greek: ... μιλιον εν (mile one)
Greek: ... μυλιον εν (millstone one)
Both have the same consonants but somewhat similar vowel sounds. Both verses have to do with "
communication" and "
messages".
5. Matthew 5:36 Hairs of the head
Matthew 5:36 Neither shalt thou swear by thy head, because thou canst not make one hair white or black. [kjv]
μητε εν τη κεφαλη σου ομοσης οτι ου δυνασαι μιαν τριχα λευκην ποιησαι η μελαιναν [gnt]
"λευκός" ≈ "bright, light, pale, happy" and, if the context is color, "white" or "light".
"μέλας" ≈ "dark, dim, indistinct" and, if the context is color, "black" or "dark".
6. Perspective
Alan Kay (American computer scientist) . Talk at Creative Think seminar, 20 July 1982.
A change in perspective is worth 80 IQ points.
Perspective is worth 80 IQ points.
Point of view is worth 80 IQ points
Kay uses the example of solving problems using polar coordinates rather than Cartesian coordinates. Each works better in certain circumstances. Kay developed (invented) object-oriented and user interface ideas that changed computer science and led to the modern window interface with icons, the mouse, etc.
Principles of duality in many fields allow the same problem to be addressed from two seemingly different points of view.
7. Duality in computation
There are two ways of looking at computation.
Operational view: Turing machine - how computation works (bottom-up view)
Declarative view: Church lambda calculus - what computation does mathematically/functionally (top-down view)
The good computer scientist needs to understand both views and how to switch between them as needed.
8. Duality in computer science
| Computer science viewpoints |
| Machine |
Human |
| Turing machine |
Lambda calculus |
| How it works |
What it does |
| Bricklayer |
Architect |
| Engineering |
Mathematics |
| Efficiency |
Correctness |
| Run faster |
Run better |
| Save space |
Reduce redundancy |
| Small details |
Big picture |
| Programming language |
Pseudo‑code |
| Flow charts |
Pseudo‑code |
| Control flow |
Data flow |
| Code‑driven programming |
Data‑driven programming |
| Operational |
Denotational |
| Command |
Declarative |
| Imperative |
Functional |
| Procedures |
Functions |
| Bottom‑up |
Top‑down |
| Up the tree |
Down the tree |
| Synthesized |
Inherited |
| Inductive |
Deductive |
| Forward‑chaining |
Backward‑chaining |
There are (at least) two ways of thinking about programming. A good computer scientist needs to use
both ways.
Important developmental skills: cooking from recipes, making things in shop class, building with Lego's, playing from musical notations.
9. Colors and color models
Two
simple models (there are others), useful for depicting digital ideas, are the
RGB and
CMYK color models. Each model is correct. One way may be easier to work with for certain purposes.
10. Views of the same thing
Glass half-empty or half-full
Type 1 and type 2 errors in statistics
Trinitarianism and Modalism
11. Optimism
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)
12. Duality in statistics
Statistics has two correct ways of looking at reality. Both are correct. One may work better in a given situation.
Frequentist statistics (null hypothesis, confidence intervals, etc.).
Bayesian statistics (inverse probability, probability of causes, etc.).
Many statisticians disagree over both frequentist and Bayesian statistics being correct ways of looking at reality.
13. Problem solving
In problem solving, a top-down design approach can be contrasted with a bottom-up design approach.
14. Counting
The English word
"dozen" comes from French which comes from the Latin word
"duodecim" ≈ "twelve" which means, literally,
"two ten" or "
two" plus "
ten".
The German word
"dutzend" ≈ "dozen" comes from Latin as does the Russian word
"дюжина" (du-schi-na) ≈ "dozen".
Two ways of looking at the same thing.
Six of one
Half dozen of the other.
By the time I tell you 12 boring jokes about eggs you will be a
"dozen".
15. Nothing but a future shadow
Colossians 2:17 Which are a shadow of things to come; but the body is of Christ. [kjv]
α εστιν σκια των μελλοντων το δε σωμα του χριστου [gnt]
Examples of "
nothing" or "
holes".
Keyhole (place to insert a key).
Donut hole (idea turned into an actual product).
Electrons and current as "holes".
Beer with "bubbles" going up and "beer" going down.
A
"shadow" is like a "
hole" in that it cannot exist without some physical object and some "
true light" source. A "
false light" is "
darkness". A
"shadow" has philosophical connections with Plato.
The Cat Stevens song "
Moon shadow" comes indirectly from the sun.
16. Heat and cold
In temperature as a measure of energy, one can feel heat going out or in or could going out or in.
17. Electricity
Electricity can be viewed as electrons going one way or holes (no electron) going the other way.
Benjamin Franklin established the electrical conventions of negative and positive.
18. Longest common subsequence
|
nematode-knowledge
empty-bottle
emt-ole
|
The
LCS = Longest Common Subsequence problem is a dual problem of the
SED = Shortest Edit Distance problem.
The solution to these problems are used in open source file comparison tools such as WinMerge and DiffMerge.
In 1974, Hirschberg published a reasonably space and time efficient solution to these problems.
19. Colors and color models
 |
RGB additive model
light added to black/screen
black + red + green+ blue = white
|
 |
CMYK subtractive model
pigments subtracted from white/paper
white - cyan - magenta- yellow = black
|
Two
simple models (there are others), useful for depicting digital ideas, are the
RGB and
CMYK color models. Each model is correct. One way may be easier to work with for certain purposes.
20. Wave particle duality in physics
Physics has two correct ways of looking at reality. Both are correct. One may work better in a given situation.
wave model
particle model
Heisenberg Uncertainty Principle applies. A good physicist needs to know and use either.
In programming,
Heisenbugs are bugs in programs that do not appear when looking.
Electrons act as both waves and particles. It appears that only by going to a higher dimension can one resolve the paradox.
[Nobel Prize]
21. Waves and particles
Einstein's theories could not explain aspects of quantum mechanics that included electrons acting as both waves and particles, the forces of gravity, etc.
[photoelectric effect]
22. Particles and waves
If there are two slits open, single electrons exhibit a wave-like behavior.
If the electron is a particle, how does it know whether there are one or two slits open?
http://nanoatlas.ifs.hr/double_slit.html (as of 1972)
23. Kaluza
Einstein received a letter from Theodr Kaluza in 1919 suggesting that some of the problems would be resolved if another space dimension were added, making four space dimensions and one time dimension.
As more dimensions were added, the concept of tiny vibrating strings was introduced. The matrix today consists of at least ten dimensions.
Much of this work culminated with the Yang-Mills theory (1954).
Yang: Nobel prize (1957)
24. Objective function
To provide value, a business needs an
objective function that can be maximized or minimized.
A simple objective function, such as with linear constraints, has a feasible region for which an optimum value of the objective function needs to be determined.
What might the objective function be for the following?
(medical) hospital
(medical) insurance company
educational institution
social work (government) group
church
What happens if the objective function of the person doing the work does not match the objective function of the institution?
25. Optimization
Not every constraint can be maximized or minimized at the same time.
The idea of "
optimization" means finding a solution that maximizes or minimizes the combined constraints.
It is
not all or nothing for each part, but both are needed.
☐ Value: Love God. Love God's rules. Hate the sin.
☐ Value: Love neighbor
as yourself. Love the sinner.
How well can you succeed if you provide just
one value?
26. The primal problem
|
Primal problem
|
Dual problem
|
linear program
var
x1 "Galaxy bars (lots)"
x2 "Continental bars (lots)"
max
2 x1 + 3 x2 "profit ($)"
st
3 x1 + 1 x2 <= 18 "milk (tons)"
1 x1 + 2 x2 <= 12 "cocoa (tons)"
3 x1 + 3 x2 <= 21 "sugar (tons)"
end
|
linear program
var
y1 "marginal value of milk"
y2 "marginal value of cocoa"
y3 "marginal value of sugar"
min
18 y1 + 12 y2 + 21 y3 "cost"
st
3 y1 + 1 y2 + 3 y3 >= 2 "profit of Galaxy bars (lot)"
1 y1 + 2 y2 + 3 y3 >= 3 "profit of Continental bars (lot)"
end
|
Objective: maximize
|
Objective: minimize
|
Every primal linear programming has a dual problem.
27. The dual problem
The dual of this problem is as follows.
Note the following.
The objective of the primal is to maximize, the objective of the dual is to minimize.
The "<=" constraints in the primal are "<=" constraints in the dual.
The column and row coefficients in the primal constraints are transposed in the dual.
In intuitive terms, the primal approaches the optimum solution from one direction while the dual approaches the optimum solution from the other direction. It may be the case that one of them is easier to solve than the other. In this case, since the primal has only two variables, the primal can be easily solved graphically, while, since the dual has three variables, the dual is not easily solved graphically.
28. Statistics
Two ways of looking at statistics:
frequentist model (human use for static decisions due to historical factors)
Bayesian model (computer use for dynamic decisions due to it works)
Each model is correct. In many cases, one way may be easier to work with or use than the other model.
29. The Art of Statistics
The good news is that the Bayesian approach opens fine new possibilities for making the most of complex data. The bad news is that it means putting aside almost everything you may have learned in this book and elsewhere about estimation, confidence intervals, P-values, hypothesis testing, and so on. David Spiegelhalter,
The Art of Statistics: How To Learn From Data (p. 305)
Note: This comment is made on page 305 of a very interesting and popular book on statistics.
30. Business
The Bayesian approach is used, behind the scenes, by many top companies.
Those top companies do not talk about it much.
It is a competitive advantage, so, as someone once said (I have heard this from several people in different industries about cutting-edge techniques), If they tell you about it, it means that they are not using it any more.
31. Bible examples of duality
| Good |
Not good |
|
| Kingdom of Heaven |
Kingdom of Air |
(same Greek words) |
| King |
queen (harlot) |
|
| Father |
mother (harlot) |
|
| vineyard |
vineyard |
(same word, need context) |
| wide area |
scattered path |
switched by the early church |
The telling of jokes uses duality in that there are (at least) two ways to take what is said.
[literal meanings, Jesus]
[physical and logical abstraction]
32. Logical and physical views
Many ideas have both a logical and a physical view.
The logical view is more abstract and is concerned about what is done, not how it is done.
The physical view is more concrete and is concerned about how it is done and not so much what is done.
Note: In more complex systems, there can be more than one logical view, but there is essentially only one physical view (that is, reality).
33. End of page
