Is the frequent occurrence of seven and the octave merely an obsession with that number, or does it reflect something in nature as well?
First we must ask is whether seven deserves extra attention, if it is a special number, or if its occurrence in various contexts is just what we could expect of any number. Without discussing that particular aspect here and now, I think we can say that it is indeed special. Let us at least assume so for this discussion.
Then, why is it special? There are three possibilities (which do not have to exclude one another):
It is a once turned up idea, kept alive by the force of intellectual tradition.
It is a part or result of the structure of our nervous system and/or our mental organisation.
It is an essential part of the fundamental structure of physical reality.
Seven Wonders of the World, Seven Seas, Seven Vices or Virtues, etc. are all based on an intellectual idea, kept alive by tradition. Still, no tradition survives for millennia unless it contains something essential. Why is the human intellect so fond of forming sets of seven? Can it have something to do with our way to think, with the structure of our mind? Perhaps.
In the 19th century, Scottish philosopher William Hamilton noted that one can observe seven distinctive items without moving the focus of the consciousness.
Later studies and observations have shown that seven seems to be the limit for the consciousness. George A. Miller has expressed this with proper scientific caution as seven plus or minus two.
If you make an effort, you can keep seven distinctive impressions, words, numbers, thoughts, sounds, whatever in your mind at the same time, but hardly more.
You can make the experiment to throw a number of something - matches, playing cards, or something else - on the surface of a table. Unless they are structured in some sort of group symmetry, you can see how many they are by a glance, without counting them - up to seven. If they are more you have to count them individually, which you do by switching the focus of your consciousness gradually.
Is this "limit of seven" inherent in our nervous system? If it is, perhaps sets of seven (of something) are optimal for our intellectual capacity; for perception, memory, and thinking. Perhaps we ought to use a mathematical notation with seven as the base instead of ten?
The primary colours of spectrum are: red, orange, yellow, green, blue, indigo, violet. Although the change of colour (wavelength) is gradual and continuous, we tend to see seven colours. (Six, if we do not count indigo as a separate colour. But never more than seven).
Colour is the way a certain wavelength of light is represented in our mind. What determines which wavelengths we can perceive, or how we see them? Physical reality or our mind?
Diatonic scales contain seven tones, arranged in octaves. Is that a merely intellectual invention or does it reflect a profound structure of our mind - or even, as the Pythagoreans believed, of cosmos (physical reality)?
My last example is the Periodic Table.
John Newlands (1837–1898) formulated an intuitive 'law of octaves' to explain why the Elements - as the tones of a scale - were repeating similar properties at every eighth step, when they were ordered according to their atomic mass. At the time he was ridiculed, but take a look at a modern Periodic Table where the Elements are arranged after atomic numbers! Then you understand why it is "periodic". You probably need some basic knowledge of chemistry and the Periodic Table to appreciate the harmony of it.
The table is an intellectual construct, but what about the octave-like structure on which it is based? Does it come naturally from the properties of the elements or is it just how we perceive them?
By now it should be obvious that there is one question we cannot answer: what is a part of the subject, and what is a part of the object. That is to say, what is a part of our minds (the observer, the subject), and what is a part of physical reality (the observed, the object).
Even if we assume that seven is a part of our mind, our nervous system, the subject, we do not know whether it is also a part of the object; or if it is, to what extent it is so.
In his "Critique of Pure Reason" [Kritik der reinen Vernuft], Immanuel Kant (1724-1804) turned the then prevailing philosophical tradition upside down, when he claimed that we cannot know anything about the thing as such [Ding an sich], but our experiences and impressions are formed by the mind and its way to organise them. This was, according to Kant, a Copernican revolution of the epistemology [Erkenntnistheorie].
When he said that the object of our knowledge is not the "real" object, but our idea of it as it is formed by the mind - the mind itself (the subject) in a sense became the object of our knowledge. So in this context we cannot make a clear distinction between subject and object.
This thought was not as new as Kant believed. Within the tradition of Alchemy it had long been well established that all processes of nature and reality are influenced by, and dependent on subjectivity.
After Kant, it has continued to influence and occupy thinkers: from Arthur Schopenhauer (1785-1860), who took Kant's idea as a starting-point when he created his own pessimistic philosophy - to Alfred North Whitehead (1861-1947), who created an "organismic philosophy" to solve the problem with what he called the "bifurcation of nature" (the dualism subject-object; or perception-fact).
Another way to express that the only thing we know about something is the impression it has made on our minds, is represented by this excerpt from: The Book of Law, Introduction, privately issued by the O.T.O. in 1938:
"...the object that you see is never the same as the one I see; we infer that it is the same because your experience tallies with mine on so many points that the actual differences of our observations are negligible. For instance, if a friend is walking between us, you see only his left side, I his right; but we agree that it is the same man, although we may differ not only as to what we may see of his body but as to what we know of his qualities. This conviction of identity grows stronger as we see him more often and get to know him better. Yet all the time neither of us can know anything of him at all beyond the total impression made on our respective minds."
In science, relativity and quantum theory reach a point where distinction between subjectivity and objectivity is questioned. We will not go into scientific detail here, it would take us too far; but it is interesting to note that it has been shown in quantum mechanics that, to appear, minute particles (on a subatomic level) are dependent on the observation act.
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