0, 1, 2, 3

Numbers and their operations or Arithmetic have emerged out of the need to count and record. So Numbers can be regarded as the starting point of Mathematics from a historical point of view at the least. This point of view puts Number Theory at the foundation of Mathematics. After all, numbers are the purest objects of Mathematics. Lets call this step as One.

Mathematical Logic and Set Theory are closely related yet separate fields. They have always been there in Mathematics but it took time to establish them as distinct fields. Lets call this step as Two.

When One and Two are defined like this, Three becomes obvious; Analysis, Algebra and Geometry. 

This implies an obvious question; What would be the ground that is Zero? It exists in every mathematical field in different forms and as '0' in Number Theory and null set '{}' in Set Theory. It took time to invent the number Zero and it has not been obvious that it simplifies Arithmetic. Even the ancient Greeks and Romans failed to invent or discover it. Counting seems to be the foundational step Zero. Even animals must count. There is Counting Theory also called Combinatorics.

0. Counting,

1. Number Theory,

2. Mathematical Logic, Set Theory,

3. Analysis, Algebra, Geometry.

This (0, 1, 2, 3) classification of Mathematics rings an interesting bell; Hurwitz's (1, 2, 4, 8) theorem. 

12 aspects of economy

In line with the recent 'The Ternary Classification' post, another quartet can be added to the two quartets of the 'Eight Aspects of Economy' post of 2010;

(((Gather, Hunt), (Farm, Manufacture)),

((Information, Intelligence), (Imagine, Create)),

((Barter, Trade), (Commerce, Finance)))

as the Application layer to complete the classification. I renamed the (Spiritual, Artistic) duality to (Imagine, Create). This way, the first quartet becomes the Measure, and the second the Theory of the ternary foundational classification. In the economics context, the first and third element can be called Production and Exchange.

The Ternary Classification

The classifications based on the stack of 12 for various fields I have attempted until now have the dominant symmetry 3*4. The outer aspect is ternary and the inner is quaternary or rather dual binary. For example for Mathematics;

(((Number,Function),(Operation,Equation)),
((Set/Logic,Analysis),(Algebra,Geometry)),
((Probability,Statistics),(Computation,Information)))

and for Physics;

((((Energy,Time),(Momentum,Distance)),((Angular-Momentum,Area),(Spin,Volume))),

(((Mass,Gravity),(Electric-Charge,Electric-Field)),((Weak-Charge,Weak-Field),(Strong-Charge,Strong-Field))),

(((Higgs,Atom),(Boson,Planet)),((Lepton,Star),(Quark,Black-Hole))))

 

The one for Physics has the symmetry of 3*4*2=24, and its outer shape is still 3.

 

If 3 words are assigned for the elements of this ternary symmetry they would probably be;

 

(Measure, Theory, Application) 

Previously I have mentioned the number triplet (0,1,∞) which also has ternary symmetry and this suggests Zero and Infinity have Binary symmetry. They are inversions of each other around unity anyway.

Why 3 dominates over 2 is open for investigation.

Science fractal

Consider the mapping;

SM0=1 ... Higgs ... Higgs ... Mathematics
SM1=1/2 ... Light ... Atom ... Physics
SM2=1/2 ... Weak ... Cell ... Chemistry
SM3=3/2 ... Quarks ... Planet ... Biology
SM4=18 ... Leptons ... Star ... Psychology
SM5=1080 ... Time ... Galaxy ... Intelligence
SM6=12150 ... Gravity ... Universe ... Aesthetics

Biology corresponds to Planet and sits in the middle. Implying that the universe is a biological form, or that all life obeys M6. Also displaying the analytical/emotional split of a brain. Chemistry/Psychology coupling shows that chemicals make the mood, like Star light make Cell. Physics/Intelligence coupling shows that Physics (electricity or light) make Intelligence, since our chemical and biological brain support the electrical action that make Intelligence. Since light speed is needed to go beyond the chemical reaction speeds to make a hyper-structure like brain. The most interesting is Mathematics/Aesthetics coupling. Aesthetics is beyond intelligence. It is beyond religion, beyond everything, but it needs all of them.

Religion replaces art in broken spirits. Spirits are broken, mostly early in childhood, by power dynamics of society. Spirit of an individual is broken to dominate the person, like breaking horses. But a shepherd is needed to keep the herd, so preachers are invented by the rich and powerful. Revolutionary preachers are eliminated, if not then adopted, like Romans adopted Christianity and Moors adopted Islam. A monotheistic religion naturally reflects the politics of an empire. This is not to say that there are no spirits. Just replace Art with Spirit. But it belongs to the bearer only. When we learn to go beyond the L&S+B&D+S&M+V&M (The Modular Group of monkey culture; Love and Sex, Bondage and Discipline, Sadism and Masochism, Violence and Murder) culture of our monkey past, we will know that a spirit is the most valuable entity of a society, and that it should not be broken by the proxies of the bullies.

 

EDIT 10-6-2026:  First published in 2006. Now some typos corrected. Seeing that the post got some attention lately, I had a reread. It still looks good but the first column of the particle physics classification looks wrong. I had better ones since this post, for example 'Universe in a Box' and '24 Aspects of the Universe'.

Capturability of the mainstream

Why is it that the human mainstream is so easy to capture?

In Cronulla race riots in Australia recently, a large number of youth were captured by a few well organised extreme racists. Since the Iraq invasion and the subsequent bombing in Bali, racial harmony suffered greatly. Racist Anglos were ready to be captured it seemed (most Anglos are decent good people I must add). Many of them were drunken and wrapped in Australian flags. They displayed a disgraceful scene, chasing and beating any dark skinned person who happened to be there, including women. The police chief later explained it as crowds being "captured". The oragnisers of the event would have been happy with the results, but I am sure they were not too happy with being seen as a mob of drunken losers.

Didn't Hitler capture an entire nation? He took so many of them to the grave. It seemed capturing humans was as easy as capturing sheep, maybe easier. Hitler was not alone of course. The Europian aristoctarcy and corporations were desperate to counter the Soviet revolution. They supported despots all over Europe to suppress local revolutions. Franco did not reach the efficiency of Hitler, but he also killed a lot of people. Mussolini was of the same kind. They keep showing docos on how bad were the dictators, but they never mention the fact that they were the ones behind the curtain. Later Hitler went rogue, as they do, and became an embarrasment. But Franco stayed a darling until the end.

Neo-cons also tried to capture the American mainstream and succeeded to a great degree. If they were successful in the invasion and holding of Iraq, we would have quite a different world order now. It was amazing to watch how easily they did it to the Americans. Few low life religious preachers were enough to mesmerise a large section of the society. As happened before in Europe, we all know who was behind the curtain. Greedy and ignorant corporate bully. The story of US is still unfolding. Democrats are putting a show but not too effective. Most democrats are also sponsored puppets.

Humans are easy to capture in all scales, from one person to large crowds. You just use slightly different methods. Controlling the media helps greatly. Internet must be quite a nuisance to these bastards. Falling in love is a consensual capture, supported by chemistry. Finding friends also falls into capturing. In the end it comes to reprogrammability. The mainstream provides a programmable power base for the marginal elements. It is a good representation of the concept of nothingness. It slumbers in the center to provide stability with it's size and power. We should study and understand this fact about ourselves and bring it to attention, if we want a peaceful future.

EDIT: 2-5-2026, I am republishing this as it was 20 years ago.

A geometric proof of the Pythagorean Theorem

 

Ulam spiral on triangle on hexagonal grid

Ulam spiral on triangle on hexagonal grid transformed by 12 symmetries and added. Black is no primes, white is 8 primes. Note the many stars embedded in the hexagon.

Ulam spiral on triangle on hexagonal grid

Ulam spiral on triangle on hexagonal grid transformed through 12 symmetries and added, and a color scheme chosen to represent the number of primes. White is 9 primes, yellow 8, purple 7...

Ulam spiral on a triangle on hexagonal grid is transformed through 6 symmetries and added. Black is no primes, white is 5 primes. When the remaining 6 symmetries are added, the curl should disappear.

Ulam spiral on triangle. I think this is the largest one on the net, with 7503 primes. It is interesting that the spiral brings out the symmetries of triangle, as it does in the square case.

Ulam spiral transformed through the 8 symmetries of the square and added. Black is no primes, white is 8 primes. It is interesting that the primes are highly correlated wrt to the symmetries of the square. Note the resemblance to Buddhist sand paintings.

Universe in a box.

24 ASPECTS OF THE UNIVERSE

A basic classification of the universe can be derived from the Ideal Gas equation;

P*V = k*N*T

where k is the Boltzmann constant. It is one of the most fundamental constants of physics. The 4 thermodynamic variables can be mapped to the most universal categories of physics;

P: (Energy,Moomentum)
V: (Time, Spaace)
N: (Mass,Charge)
T: (Gravity,Boson)

Where;

Moomentum = (Momentum,Angular-Momentum,Spin)
Spaace = (Distance,Area,Volume)
Charge = (Electric,Weak,Strong)
Boson = (Light,Weak,Gluon)

The overall 4ness is reflected within each subcategory giving 4*4 = 16 subcategories. It is a mixture of (1,3) and (2,2) symmetries, inside and outside.

Where are the Higgs Boson, Dark Matter and Dark Energy? I think they are in k. They are the fields that the rest float on. They may even make a quartet, predicting another field;

(Higgs,?,DE,DM)

to make 5 categories and 20 subcategories in all.

Where are the Black Hole and the rest? I think they make another quartet;

(Atom,Planet,Star,Black-Hole)

Thus making 6 categories and 24 subcategories. I had a yarn before about the specialness of the number 6. 24 is also a very special number. It is one of John Baez's favorite numbers.

http://en.wikipedia.org/wiki/6_(number)
http://en.wikipedia.org/wiki/24_(number)
math.ucr.edu/home/baez/numbers/24.pdf

If all listed together in a certain order, the symmetry simplifies somehow;

(Higgs,?,DE,DM)
(Time, Distance,Area,Volume)
(Energy,Momentum,Angular-Momentum,Spin)
(Gravity,Light,Weak,Gluon)
(Mass,Electric,Weak,Strong)
(Atom,Planet,Star,Black-Hole)

There is deep symmetry to it with many aspects, each reflecting a mathematical property of 24, starting with the simple arithmetic ones like;

4*6 = 2*2*2*3 = 2*12 = 3*8

BWV-999 analysis continued (3)

There are 43 bars of 3/4 time signature. The last bar is the conclusion and stands apart. The remaining 42 are divided into 7 parts, each with 6 bars. Bach's music, and most music anyway, obey numbers in their organisation. Bach takes it to more extremes.

The harmonic signature (the tonality) is C-minor (Cm) in the original lute version. It has been transcribed to Dm when played on the classical guitar. Dm suits the guitar very well, in that it uses the open strings maximally.

In a way, the prelude is a warm up for the player and the audience and an introduction to the tonality of the instrument and the suites that follow. It is an arpeggio anyway. Its local rhythmic structure is frozen to the arpeggio chosen.

It is divided into melody (temporal) and harmony (spatial). Each supporting the other. The melody is the spine of the piece, and it is made of the even notes, if the 12*16th notes of the bars are numbered 1 to 12. They do not change as quickly, and make a melodic tremolo that cuts through the vertically arranged chords. The arpeggio piece is in fact a melody embellished by the chords that also make up a rhythm with the changes.

The (arpeggio) chords divide into two; treble and bass. They reflect each other. For example the notes (2,3,4) of the 1st bar, are reflected by (9,11,13). 13th note is the 1st of the next bar. This is musical linking, or chaining, that I described before in the link below. The bass chord next to the treble is sometimes an octave above when things resolve and settle down.

Bach used two main chord types here; triads and diminished triads. They are linked by intermediate forms made of inversions (rotations). The treble and bass chords meet at the start of the bars to make the dominant chords made of 4 notes. Eg Dm of the 1st bar; (D,2D,2F,2B). The (1,3) symmetry like that of the Quaternions.

The diminished triads sometimes meet and make a large chord with all notes spaced by a minor 3rd interval. This is the most unstable chord, since it want to be a minor and a major at the same time. Bach uses them as the changing points between the stable triads, which also meet occasionally to make a large chord.

The piece is also divided into two (2*21) with the peak happening in between the 21st and 22nd bars with the EM chord.

Bach builds the harmony around the open strings that are related by being the 4ths (the reverse 5ths) of each other. The guitar has an amazingly harmonic structure in the arrangement of the open strings. More amazingly, the number of the fingers suit this arrangement. 


The previous;
http://a-theory-of-nothingness.blogspot.com.au/2007/02/analysis-of-symmetry-in-bwv-999.html

12 aspects of being

Anahata

(((Number,Function),(Operation,Equation)),
((Set/Logic,Analysis),(Algebra,Geometry)),
((Probability,Statistics),(Computation,Information)))

(((Money,Trade),(Finance,Economics)),
((Emotion,Relationship),(Religion,Politics)),
((Communications,Technology),(Philosophy,Science)))

(((Nucleotide,Nucleic-Acid),(Amino-Acid,Protein)),
((Muscle,Neural),(Connective,Epithelial)),
((Inner-Sense,Outer-Sense),(Inner-Action,Outer-Action)))

(((Sound,Syllable),(Word,Sentence)),
((Definition,Explanation),(Description,Conclusion)),
((Expression,Conversation),(Meeting, Conference)))

(((Note,Interval),(Measure,Theme)),
((Tempo,Rhythm),(Melody,Harmony)),
((Solo,Duet),(Chamber,Orchestra)))

(((Higgs?,?),(Dark-Matter?,Dark-Energy?)),
((Position,Momentum),(Charge,Field)),
((Matter,Planet),(Star,Black-Hole)))

I have finally come back to 12, after being stuck on 4 for a while. This time though I reckon I found the symmetry I have been after for a long time. It has the numbers 1, 2, 3, 4, 6 included in the overall symmetry, indicating that the classification is in fact a computer. 12 is a very popular number in religion and science. The lists speak for themselves.

The most important outcome is the predictive power of this classification. It places the scalar elements of the universe nicely into the 1st quartet, and predicts an unknown element besides Higgs and the already theorized dark ones. Perhaps Higgs splits into two.

The symmetry has two other less dominant representations;

1. (((,),(,)), ((,),(,)), ((,),(,))) ... 3*4
2. (((,),(,),(,)), ((,),(,),(,))) ... 2*6
3. (((,,),(,,)), ((,,),(,,))) ... 4*3

Following the commutation and association property of multiplication of naturals;

3*4 = 4*3 = 2*6 = 12

Does not seem to have the 6*2 symmetry. It is not surprising that the overall symmetry imitates the small natural numbers, since they are the simplest and purest forms. The pure symmetry of the naturals has to be broken though, to have complexity. 3*4 seems to be the dominant symmetry. I think this is based on the triplet symmetry of numbers;

(0,1,Infinity)

The 3*4 symmetry is in fact 3*2*2 = 3!!, the double factorial of 3. The double factorial has a self reference; it reiterates itself.

An interesting thing that seems to be related to this classification is the Kleiber's Law. It relates the metabolic rate of animals to their masses;

Metabolic Rate ~ m^(3/4)

It's explanation involves the self similarity and the fractal structure of the metabolism.

------------------------------------------------

EDIT 25-5-2026:

The Universe classification above is wrong. A better one is included in the later post "Universe in a box";

((((Energy,Time),(Momentum,Distance)),((Angular-Momentum,Area),(Spin,Volume))),

(((Mass,Gravity),(Electric-Charge,Electric-Field)),((Weak-Charge,Weak-Field),(Strong-Charge,Strong-Field))),

(((Higgs,Atom),(Boson,Planet)),((Lepton,Star),(Quark,Black-Hole))))

This one resonates much better with the other classifications like the musical one for example. The first musical one is Note and corresponds to (Energy,Time) in the Universal one. Since Note expresses itself as energy in time, this one is spot on.

The Fifth Element

Vajra cross

Map the four variables of the ideal gas law to the categories of the universe;

((P,V),(N,T)) --->
(((Energy,Momentum),(Time,Space)),((Mass,Charge),(Gravity,Boson)))

Where;

Charge = (Electric,Weak,Color)
Boson = (Light,Weak,Gluon)

Rearrange the categories to expose an implied symmetry;

(((Energy,Time),(Mass,Gravity)),((Momentum,Space),(Charge,Boson)))

Compare it to the categories of life;

(Tissue,Organ) =
(((Muscle,Neural),(Connective,Epithelial)),((Isense,Osense),(Iaction,Oaction)))

Where;

Organ = (Head,Body) = (Sense,Action)

Isense = Inner-sense = ((Pain,Pleasure),(Temperature,Pressure),(Rhythm,Harmony))
Osense = Outer-sense = ((Touch,Balance),(Taste,Smell),(Hear,See))

Iaction = Inner-action = ((Lungs,Kidneys),(Stomach,Penis-Vagina),(Heart,Brain))
Oaction = Outer-action = ((Arms,Legs),(Head,Torso-Tail),(Feet,Hands))

Rearrange the categories of life and compare it to the ideal gas and the universe;

((P,V),(N,T))
(((Energy,Momentum),(Time,Space)),((Mass,Charge),(Gravity,Boson)))
(((Muscle,Isense),(Neural,Osense)),((Connective,Iaction),(Epithelial,Oaction)))

Why does the ideal gas law underlie both classifications?

Perhaps because the primordial soup of life can be approximated as a gas of chemicals dissolved in water. The same argument must apply to the universe. Since Dark Matter can be considered as a quantum liquid, the fifth categories for the universe and life are implied;

(Higgs,DarkMatter)
(Biochemical,Water)

The fifth element in the case of the ideal gas must be the Boltzmann constant;

P*V = k*N*T

Thus the overall symmetry is that of the Vajra cross;

((P,V),{k},(N,T))
(((Energy,Momentum),(Time,Space)),{Higgs,DarkMatter},((Mass,Charge),(Gravity,Boson)))
(((Muscle,Isense),(Neural,Osense)),{Biochemical,Water},((Connective,Iaction),(Epithelial,Oaction)))

64 Aspects of the Universe

The ideal gas law;

P*V = k*N*T

k; Bolzmann constant.
P; Pressure
V; Volume
N; Number of particles
T; Temperature

Map the four thermodynamic variables to physical categories;

(Energy,Momentum)
(Time,Space)
(Mass,Charge)
(Gravity,Boson)

The symmetries reflect each other;

((P,V),(N,T))
(((Energy,Momentum),(Time,Space)),((Mass,Charge),(Gravity,Boson)))

The choice of the physical categories is also supported by the common (1,3) symmetry;

Momentum; (Px,Py,Pz)
Space; (x,y,z)
Charge; (Electric,Weak,Color)
Boson; (Light,Weak,Gluon)

Mass and Energy are scalars, Time and Gravity are vectors.

The eight physical categories are fully meshed; each is related to the other seven, although the indicated relationships are the strongest. Thus an 8*8 matrix is needed to contain all relationships.

Thirteen Aspects of Mathematics

My previous post "64 Aspects of Mathematics" assumed 8 elementary theories and predicted higher order theories through a non-commutative multiplication of the octet by itself.

This post is a speculation on the resemblance of the octet to the major scale of the 12 tone equally tempered music. The octet classification has some imperfections that may be addressed better in a stack of 13 (12+1).

The major scale has a modular symmetry when the octave of the root is appended (2C gives a feeling of completion);

(((C,D),(E,F)),((G,A),(B,2C)))

This symmetry can be heard easily by trained ears. The Stern-Brocot Tree easily explains this modularity. Modular Group, SL(2,C), is mentioned in many papers dealing with the music theory.

The equal tempered music is a marriage of two symmetries; the modular group and the logarithm. The cycle of fifths almost meets the octaves in 7 octaves;

((3/2)^12)/2^7 = 1.0136

This is the miracle that makes the equal temperament possible. I think this is not an accident.

The 13 tone stack is;

C,#C,D,#D,E,F,#F,G,#G,A,#A,B,2C

The mathematics octet can similarly be expanded by filling in the missing theories;

Set
-Logic
Analysis
-DifferentialG ... ?
Geometry
Algebra
-Combinatorics ... ?
Probability ... QM
-StochasticP ... QFT
Statistics ... SM
-Inference ... Td
Computation ... SR
Information ... GR

Logic for #C and so on. The most interesting part of this comparison is the relationship of the first to the last;

(Set,Information) ... (C,2C)

Obviously there is a very simple relationship between C and 2C. Is there such a thing for the Set and the Information Theories? There are some studies on this subject. A notable one is by Gregory J. Chaitin, "Gödel's Theorem and Information";

http://www.cs.auckland.ac.nz/~chaitin/georgia.html

I have come across a few studies (proofs?) of Riemann's Hypothesis via the concept of entropy. But as far as I know these are not taken to be the last word yet.

When the experiment part is compared to a classification of physics, IT corresponds to General Relativity. Perhaps this is why the Black Holes have a simple entropy equation. The Modular Group and General Relativity relationship is also all over the net.

The Thermodynamics (Td) and Special Relativity (SR) neighborhood is also interesting;

http://arxiv.org/abs/0712.3793
"On the Relationship between Thermodynamics and Special Relativity"

64 Aspects of Mathematics

This is an attempt to classify mathematics in an 8*8 matrix. Most items below can be found in the Internet, eg search for Algebraic Geometry or Geometric Algebra. The ones not found are predicted by the matrix. All 8 items are theories in themselves, eg Information Theory or Theory of Computation. When it repeats like Analysis-Analysis, it stands for Elementary Analysis. The model demonstrates the non-commutativity of mathematical classification.

The model also predicts the 3rd and higher iteration theories, like Algebra-Geometry-Analysis. Thus, while Mathematics unites, it also diverges, reminding its paradoxical nature.

1.1. LogicSet-LogicSet

1.2. Analysis-LogicSet

1.3. Algebra-LogicSet

1.4. Geometry-LogicSet

1.5. Probability-LogicSet

1.6. Statistics-LogicSet

1.7. Computation-LogicSet

1.8. Information-LogicSet

2.1. LogicSet-Analysis

2.2. Analysis-Analysis

2.3. Algebra-Analysis

2.4. Geometry-Analysis

2.5. Probability-Analysis

2.6. Statistics-Analysis

2.7. Computation-Analysis

2.8. Information-Analysis

3.1. LogicSet-Algebra

3.2. Analysis-Algebra

3.3. Algebra-Algebra

3.4. Geometry-Algebra

3.5. Probability-Algebra

3.6. Statistics-Algebra

3.7. Computation-Algebra

3.8. Information-Algebra

4.1. LogicSet-Geometry

4.2. Analysis-Geometry

4.3. Algebra-Geometry

4.4. Geometry-Geometry

4.5. Probability-Geometry

4.6. Statistics-Geometry

4.7. Computation-Geometry

4.8. Information-Geometry

5.1. LogicSet-Probability

5.2. Analysis-Probability

5.3. Algebra-Probability

5.4. Geometry-Probability

5.5. Probability-Probability

5.6. Statistics-Probability

5.7. Computation-Probability

5.8. Information-Probability

6.1. LogicSet-Statistics

6.2. Analysis-Statistics

6.3. Algebra-Statistics

6.4. Geometry-Statistics

6.5. Probability-Statistics

6.6. Statistics-Statistics

6.7. Computation-Statistics

6.8. Information-Statistics

7.1. LogicSet-Computation

7.2. Analysis-Computation

7.3. Algebra-Computation

7.4. Geometry-Computation

7.5. Probability-Computation

7.6. Statistics-Computation

7.7. Computation-Computation

7.8. Information-Computation

8.1. LogicSet-Information

8.2. Analysis-Information

8.3. Algebra-Information

8.4. Geometry-Information

8.5. Probability-Information

8.6. Statistics-Information

8.7. Computation-Information

8.8. Information-Information

Eight aspects of mathematics

Mathematics can be classified in 4 dualities;

Logic/Set-Theory, Analysis

Algebra, Geometry

Probability, Statistics

Computation, Information

Logic and Set Theory are considered to be the foundations. Information stands for Information Theory. Computation is not the computer science, it is the theory of computation.

The symmetry is modular like that of the major scale;

( ((c,d),(e,f)), ((g,a),(b,2c)) )

This symmetry also applies to my previous post, Eight Aspects of Economy.

Curiously enough, the second quartet of the universe classification is missing;

Frame, Celestial

Quantum, Material

As I suggested in a previous post, the 2nd quartet must be the 4 levels of a multiverse theory;

Level-1, Level-2

Level-3, Level-4

The endpoints of the major scale is an octave apart, (c,2c), when completed with 2c. I wonder of there is such a thing in the mathematics and the universe categories.