The discreteness
of things
and the magical number 7
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This paper considers the importance of discreteness to description using types and numbers.
It reviews how the number 7 has been used to divide descriptions of the universe into discrete things.
And draws some conclusions about reality and description, memory and communication.
What is atomic or indivisible)?
An elementary or fundamental particle is a particle whose substructure is unknown.
In other words, it is unknown whether it is composed of other particles.
Matter particles
At one time, the elementary particles of matter were considered to be atoms.
By 1914, experiments by physicists Ernest Rutherford, Henry Moseley, James Franck and Gustav Hertz had largely established the structure of an atom as a dense nucleus of positive charge surrounded by lower-mass electrons.
By 1920, Rutherford divided the nucleus into protons
Later, the atom was divided into electrons, protons and neutrons
In the 1960s, protons and neutrons were divided into quarks, which include electrons and neutrinos (the tiniest quantity of reality imagined).
Energy particles
In the 18th century, light was seen as particles; in the 19th century (after Maxwell) light was seen as waves.
In 1905, Einstein pointed out many anomalous experiments could be explained if the energy of a light wave were localized into point-like quanta that move independently.
Even though the wave itself is spread continuously over space.
So electromagnetic radiation was divided into photon particles.
Yet each particle could seemingly span a field (as would a wave); a paradox still eluding satisfactory explanation.
Physicists’ latest “standard model” divides matter into particles called fermions (quarks, leptons, antiquarks, and antileptons).
And divides forces between fermions into particles called bosons (gauge bosons and the Higgs boson).
The discreteness of things
Some things seem naturally discrete:
· An elementary particle (as above)
· A body of matter in one phase (solid, liquid or gas)
· A living thing (animal or plant).
But when does the day end and the night begin?
Where is the line between a mountain and a valley? Where is the edge of a hurricane?
Generally, the end of one thing and the start of another, the boundaries of things we discuss and deal with in the world, are drawn in descriptions we impose on the world.
And this is true of most if not all business systems.
Between 12 thousand and 4 thousand years ago, in Mesopotamia (Iraq and Syria today), humans achieved many things.
Firsts included urbanization, the wheel, writing, astronomy, mathematics, and wind power.
Women enjoyed nearly equal rights and could own land, file for divorce, own their own businesses, and make contracts in trade.
Mesopotamians named celestial bodies and the days of the week.
7 planets {Sun,
Moon, Mars, Mercury, Jupiter, Venus, Saturn}
In ancient times, people observed the 7 physically discrete celestial bodies enumerated above.
They believed these entities revolved around Earth and influenced its events.
Nowadays, the total number of celestial bodies depends on how astronomers choose to define “planet”.
The human concept/idea/notion of “planet” is used as an example in discussion of type definition below.
7 days of the week {Sunday,
Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
The Mesopotamians divided the passage of time into 7-day weeks.
They named the 7 days after the 7 celestial bodies; think Sun-day, Moon-day and Saturn-day.
Later, Greeks sophists believed the number 7 related these to colours of the rainbow and notes of the musical scale.
7 regions of radiation {gamma rays, x-rays, ultraviolet, visible light, infrared, microwaves, radio waves}
The continuous electromagnetic spectrum has been divided by scientists into the 7 regions named above.
The division into 7 regions represents our human view of the different effects and uses of radiation in different bands.
The divisions are uneven; the visible light spectrum, so important to us, is only one ten trillionth of complete spectrum.
7
colours of the rainbow {red, orange, yellow, green, blue, indigo, violet}.
The sun (the primary source of energy for life on Earth) radiates a small amount of ultraviolet, all visible light, and some infrared.
A point source of white light, when refracted through a glass prism, yields a continuous spectrum of wavelengths, with no bands.
Newton originally (1672) divided this continuous spectrum into 5 main colours {red, yellow, green, blue, violet}.
Later, he added orange and indigo to the set, giving 7 main colours, in accord with the belief of the Greeks sophists.
Colours do not exist out there; they are entirely a matter of perception.
The number of colours that the human eye is able to distinguish in the spectrum is in the order of 100 (c.f. the Munsell colour system).
The apparent discreteness of colours is an artefact of human perception; and the exact number of colours is a somewhat arbitrary choice.
You could as well say the rainbow contains infinite colours, not only infinitely small divisions but also infinite overlapping bands.
7 notes in the
diatonic musical scale {do, re, mi, fa, so, la, ti}
Of many different musical scales the best known is the 7 note scale taught in a song from the film “The Sound Of Music”.
Using the just intonation tuning system, the frequencies of notes in the scale are related by simple ratios to the base note.
Read the appendix for more about music scales, and my own naïve experiment.
7 numbers in the
Fibonacci series {0, 1, 1, 2, 3, 5, 8}
The ratios found in the first 7 numbers of the Fibonacci series are related to frequencies of musical notes (though not to a recognised musical scale).
7 as a limit in
short-term memory (Miller’s law)
“The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information” was published in 1956 by the cognitive psychologist George A. Miller.
It is often interpreted to argue that the number of objects an average human can hold in
working memory is 7 ± 2.
If 7 (or 111 in binary) is an average limit to human short term memory.
Then that, along with the number of visible planets, might account for the common appearance of 7 in human discourse.
7 as a rule of thumb
for enterprise architects
Miller’s law is quoted in advice given to enterprise and system
architects on the optimal granularity of system decomposition.
One is advised to decompose a system into about 7 subsystems, then decompose each subsystem in the same way, and so on.
Decomposition to 7 levels results in not far from one million subsystems (still well short of all the elementary actions in the software systems a business uses).
Roger Penrose contends that the philosophy of mathematics can't be understood other than by assuming numbers have always existed.
"mathematical truth is absolute, external and eternal, and not based on man-made criteria ... mathematical objects have a timeless existence of their own..."
However, this contention depends how the term "existence" is interpreted.
A philosopher may argue that mathematical objects are eternal in an ethereal or logical sense.
The position here is that the term “existence” implies instantiation in a material or physical sense.
In this sense, mathematical objects (types and instances) have existed in minds (and the products of minds) only since they were conceived by life forms who use them to understand and predict realities.
In other words: before
life, there was no description.
It is difficult to explain why a reality has no property (qualitative or quantitative) without an observer.
Because we can’t describe a reality without using words that give names to qualities of that reality! But here goes.
Before life as we know it: were there bodies in space? Yes.
Were the 7 “celestial bodies” visible from earth differentiated from other bodies orbiting the sun? No.
Was white light diffracted through water droplets into a spectrum from infra red to ultra violet? Yes.
Were the 7 “colours of the rainbow” distinguished? No.
Did two sounds separated by an octave create concordant harmonic vibrations? Yes.
Were they regarded as a "smooth chord” that bounds a scale with 7 notes? No.
As a realist, I believe the stuff in what we call Mount Fuji exists.
As an idealist, I believe that Mount Fuji does not – independent of minds - have the qualities of height, width, beauty or symmetry.
Because all values given to those qualities are measurements made by observers, according to their chosen definition of those qualities.
Moreover, there is no reason to assume the descriptive qualities of height, width, beauty and symmetry existed before life.
If you postulate they did exist, then you have the challenge of locating them in time and space.
Where was the number 7, out there, waiting to be discovered?
The number 7 is a quality attached to a collection of things that have been grouped because they are similar in some way.
But before life there
was no perception and grouping of similar things, so there could be no counting
and no quantities.
Visibility, colour, musical smoothness, height, width, beauty and symmetry and are qualities that are perceived.
These qualities do exist in the perceptions, memories and communications of living actors.
And once in symbolised in memory (by inheritance or learning) a quality of a new thing can be matched to quality perceived in an old thing.
But outside of perceptions, memories and communications made living actors, there is no abstract quality.
Cats, bees and humans
An actor cannot perceive a quality of a thing unless it has sensory organs.
It cannot recognise a quality of thing unless it already has (by inheritance or learning) a biological encoding of that quality.
(By which I mean encoding in a bio-electro-chemical code that is mysterious to us.)
And it cannot describe a quality of a thing unless it has the ability to translate its biological code into communicable symbols.
Cats inherit their ability to recognise and chase anything resembling a mouse's tail.
They surely inherit a biological encoding of qualities we verbalise as "stringy" and "wriggly".
Their brain surely matches a new perception to this stored symbolic representation.
Honey bees inherit their ability to recognise and communicate the locations of pollen sources.
They surely inherit a biological encoding of a thing we verbalise as a pollen source.
They inherit also their ability to encode its direction and distance qualities.
And their ability to translate and communicate these qualities into the symbols of dance movements.
They can (research suggests) recognise a quantity up to 4, and communicate it thus.
Humans learn and remember numbers in a different way from other animals, verbally, through speech and writing.
Surely, our recollections of the number 7 are mostly sense memories of our spoken and written symbols for it?
Surely, the number 7 did not appear in the universe before life, and will disappear when all communications and memories of life forms are lost?
Qualities have to be symbolised in some form or another if they are to be perceived, remembered or communicated,
Surely, the beginning of symbolisation was biological encoding of perceptions?
Verbalisation in speech, and then in writing, were great leaps forward.
But still, qualities and quantities are abstractions from reality, which only emerged through biological evolution.
They have a practical value in helping life forms to predict the future behaviour of the universe; they are not the universe itself.
The number 7 is not special; other
numbers make a variety of appearances in human discourse.
Which comes first, numbers or the
things that we discuss and describe?
This section presents a short
story of where numbers come from.
Physicists consider our world to be embedded in a four-dimensional space-time continuum.
It is called a “continuum” because it is assumed that space and time can be subdivided without any limit to size or duration.
Dividing the space-time continuum
We can’t perceive or describe the contents and life of the universe all at once.
Luckily for us,
matter and energy is unevenly distributed in the space-time continuum.
The substance of
the universe varies across space and changes over time (nothing lives forever).
In the physical world, there are somewhat solid chunks of matter in space/fluids, and somewhat rapid movements and changes.
We naturally divide the universe in space where there is a noticeable change in substance, and in time when a noticeable change occurs.
Perceiving things as discrete
The somewhat fuzzy discreteness of things in reality is made sharply discrete in our perceptions and descriptions of reality.
We perceive there to be discrete trees in a wood, footsteps along a path, colours in rainbow and notes in melody.
Where there is a continuum in reality, psychologists use the term a “just noticeable difference” (JND) to explain how we distinguish one position from another.
Remembering discrete things
Memory is a product of biological evolution that enables an animal to recognise news things as similar to old things.
Being able to recognise similar things (food items, friends, enemies) helps an animal survive.
Recognising types of things
To begin with, animals must have recognised only fuzzy family resemblances.
There is no universal agreement about the notion of “family resemblances” (referred to in Wittgenstein’s philosophy).
Here, we associate the notion with the concept of a polythetic type.
That is, a broad set of criteria that are neither necessary nor sufficient to identify an instance.
Each instance of the type must possess some of the defining characteristics, but need not possess all.
Later, humans developed more rigid classifiers or types.
An Aristotelian or monothetic type is a set of characteristics that are both necessary and sufficient in order to identify instances of that type.
E.g. The necessary and sufficient
description of a prime number is that that is divisible only by 1 and itself.
Using symbols to communicate
Animals that live in a group can
benefit from sharing facts about things the world around them.
To communicate, they use symbols
to represent things and the qualities they possess.
A symbol is formed by arranging
matter and energy (smell, gesture, sound, mark, whatever) to match an element
in a vocabulary.
Abstraction of perceived types into verbal descriptions
Obviously, humans have the advantage of being able to use words as symbols to describe things and their qualities.
A type is an abstraction from things we perceive or describe as being similar.
It gives us least an idea, model or description of a discrete thing we are interested in.
It describes what is true of an instance, by defining one or more properties to be found in a member of a set.
|
This thing |
is an instance that |
this type |
|
A planet |
manifests the idea of |
Planet: a massive body that orbits a star. |
|
A note |
embodies the concept of |
Note: a pitched sound, matching one of several in a musical scale. |
Every type is a description, it is an abstract definition of a set member.
Conversely, as discussed in related papers, every description is a type.
Every description is an abstraction that can potentially be realised in several things.
Architectural system descriptions are polythetic types.
An instance of an operational system should match most of an architectural description, but not necessarily all.
Describing things using
numbers - abstraction of quantities from qualities
There is stuff out there; and it is divisible into things that share family resemblances.
The discreteness of things in perception and memory was biological/psychological before it was mathematical.
After discriminating one thing from another, intelligent actors can group things and classify things.
And after classifying or typifying some things, actors can enumerate and count things of that class/type.
A mathematician might say that notes in the just intonation
scale are based on mathematical ratios
It seems more fitting to say the notes are based on natural harmonics
created by the ratios of frequencies.
Expressed as numbers, each mathematical ratio is a model/description of that physical reality.
Numbers are a device that intelligent actors (who observe, envisage and describe reality) find useful to describe the world in terms of discrete things.
Honey bees can communicate numbers up to 4 by gesturing up to up to 4 times.
Many birds and mammals can hold small numbers in mind.
Humans use various number systems (binary, octal, decimal); and there are infinite possible number systems.
Types and numbers lead to mathematics and hard scientific knowledge about the world.
With no intelligent actors to observe, envisage and describe reality, there are no types or numbers; there is only stuff out there.
With no enterprise architecture description of any kind, there is no enterprise system, there is only business being done without roles or rules.
Once discrete elements have been observed, envisaged or described, they can be classified and counted.
Alternatively, you can pick a number and use it divide the universe into discrete elements (as was done with the number 7 above).
Of many different musical scales (http://vaczy.dk/htm/scales2.htm) the best known is the 7 note scale taught in a song from the film “The Sound Of Music”.
Using the just intonation tuning system, the frequencies of notes in the scale are related by simple ratios to the base note.
|
Two note chord |
Re to Do |
Mi to Do |
Fa to Do |
So to Do |
La to Do |
Ti to Do |
Do to Do |
|
Gap between notes |
2nd |
3rd |
4th |
5th |
6th |
7th |
Octave |
|
Ratio to the base note |
9:8 |
5:4 |
4:3 |
3:2 |
5:3 |
15:8 |
2 |
Other notes, such as minor thirds and minor sevenths are often used in popular music.
There are two minor seventh ratios: the ordinary one is 9:5, the “sweeter” harmonic seventh is 7:4.
“The smaller the numbers in an interval's ratio, the more consonant (sweet-sounding) it is.” (http://www.kylegann.com/tuning.html )
This table orders two-note chords from higher number ratios (more dissonant) to lower number ratios (more consonant).
|
More dissonant |
Major 7th |
2nd |
Minor 7th |
Minor 6th |
Aug 4th |
Minor 3rd |
Harmonic 7th |
Major 3rd |
6th |
4th |
5th |
Octave |
More consonant |
|
Ratio to the base note |
15:8 |
9:8 |
9:5 |
8:5 |
7:5 |
6:5 |
7:4 |
5:4 |
5:3 |
4:3 |
3:2 |
2 |
|
Singers, when unaccompanied, naturally sing notes according to the harmonic ratios in the just intonation scale above.
Trouble is, tuning instruments to the just intonation scale makes changing key dischordant.
For this reason, most musical instruments are nowadays tuned to a different 12-tone equal temperament scale.
The size of most equal-tempered intervals cannot be expressed by small-integer ratios; so each chord is highly complex as a mathematical ratio.
And if music was primarily about appreciating simple mathematical ratios, the chord should be heard as a screeching, jangling dischord.
I heard that the pianist Percy Grainger - because of this – said he didn't like the piano.)
But in practice, only the keenest of musical ears is able to detect the difference.
My own undergraduate experiment suggested non-musical people discriminate dissonant chords from consonant chords less, but like all chords more, than musical people.
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