Mathematics without numbers...
In another series of pages called "Life on Earth" I made the point that the behavior most human was to introduce novelty in the universe. It occurs to me that there is another activity many humans share. That is the making and enjoyment of music. My experience with music began in my sixth year with my father and his friends practicing at our house when their five piece dance band was reunited after World War II had split them up. By the time I was sixteen I was playing piano every Saturday night with Father's band at a local dance hall. This experience left me with a sound track in my head that I can start up anytime I need to put my mind in neutral for a while. I suspect that most people respond to the music of their culture.
There is a lot of agreement that music is sound but less agreement about the boundary in sound between music and noise. There is a poem, written by one of my favorite poets, John G. Saxe, on the subject. It was recited by my grandfather Delmar Small when I was young.
The Musician and the Ass
Near Litchfield town, one summer's day
A strong lunged ass was heard to bray.
The green hills echoed back his voice;
To hear it made his heart rejoice.
"What a pity," said the ass.
"That I should have to live on grass.
My lungs are strong; my voice is loud;
At dances I would draw a crowd.".
"Hear my music. How it fills
The valleys lying 'mongst the hills.
It is sweet, I know, for see what
Great ears for music I have got!
A great musician heard the din
While passing with his violin.
He stopped a while upon his way
And bid the old ass cease to bray.
"My long eared friend," the fiddler said,
This neighborhood must wish you dead;
For worse than any sounding brass,
Is your braying, Mr. Ass."
"If you want music cease your din
And listen to my violin."
He rubbed the resin on the bow.
He tried the notes both high and low.
Using a large stone for a chair
He played a grand soul-stirring air.
Before the fiddler ceased to play
The ass again began to bray.
No violin or song of bird
Could for a moment then be heard.
At last the old ass dropped his head
And unto the musician said:
"Music is sound, my friend, you see.
Therefore all sound must music be.
Of mine the world will be the proudest,
Because mine, my friend, it is the loudest."
What more could the musician say?
What further do but let him bray?
He wandered off through twilight dim:
Ass wisdom was too much for him.
There is a pretty good Wikipedia article that has a lot to say about music. The following quote from that article applies to the point of the poem.
Common sayings such as "the harmony of the spheres" and "it is music to my ears" point to the notion that music is often ordered and pleasant to listen to. However, 20th-century composer John Cage thought that any sound can be music, saying, for example, "There is no noise, only sound."
It has been my experience that most teenagers experience their parent's music as noise and vice versa - noise being in the ear of the beholder, so to speak. I have had another experience with noise. In my youth I was a lobster catcher up in Casco Bay on the Maine coast. At the age of fifteen or so I was hauling traps by hand from a flat bottomed skiff powered by an outboard motor. During the run from one trap to the next I would crank it up to full speed, not making any money between traps. The motor wasn't especially loud but it produce a sort of broad band sound mostly in the lower frequencies. I found that by concentrating on a piece of music that I had previously heard, I could sort of extract the tune from the noise of the motor. It worked best with voices rather than instrumentals so I used it to recreate some of the hymns the church choir sang. My most spectacular success was with the Mormon Tabernacle Choir doing "The Battle Hymn of the Republic". In such ways a solitary young fellow entertains himself with music.
In general I think most of us want our music to be "ordered and pleasant to listen to" as opposed to John Cage's liberal view that any old noise will do. I further suspect that it is the well ordered aspect of music that makes it pleasant to listen to. That notion leads me to the connection between music and mathematics. Another Wikipedia article provides a lot of detail from several points of view on this connection. The emphasis in the article is on the use of mathematics in analyzing music. More interesting to me is the idea reflected in this quote from the Wikipedia article, "the Pythagoreans of ancient Greece are the first researchers known to have investigated the expression of musical scales in terms of numerical ratios, particularly the ratios of small integers."
Rather than applying mathematics to music, it appears to me that the Pythagoreans used the study of music to access modern mathematics, extending the counting numbers into the idea of ratios, hence rational numbers. The study of rational numbers led to the development mathematical proofs and the notion of irrational numbers. Irrational numbers were adjoined to the rational and counting numbers, negative numbers and zero to produce the real numbers. Real numbers proved incomplete, missing square roots of negatives, so imaginary numbers came along... and vectors, tensors, and so on and so forth to include mathematical entities that boggle the imagination. One wonders if, in the absence of human fascination with music, we would have any of the benefits of 21st century science.
There is something eminently satisfying about hearing a piece of music that touches your soul. My own high points are: "Ave" Maria sung by "Beverly Sills", the symphonic poem "Les Preludes" composed by Franz Lizst and the fiddle tune "Orange Blossom Special" composed by Ervin Rouse. On every hearing I get the sense that I have experienced something special. I get that same sense of satisfaction when I finally wrap my mind around some mathematical abstraction I have been struggling with. Suddenly it comes into focus and I see it in all its scope and magnificence. This purely subjective experience of music and mathematics is, for me, an indication that they are deeply connected. That is as may be, but the enjoyment of both music and mathematics are clearly mental activities, bringing us to consideration of the human mind or consciousness.
In the "Life on Earth" series of web pages I argued that the brain by itself cannot account for human consciousness. If you have not read the "Life on Earth" series, at least read the four pages linked below, in the order shown:
The 'something else' other than activity of the brain that goes into human consciousness remains a mystery but I am suggesting that our appreciation of music, that has no clear evolutionary advantage, may have its origin in that 'something else'.
Imagine yourself in the position of a creator wanting to include an element of self exploration in the universe, as envisioned in "Why is There Life". What better mechanism than the inclusion of willful creatures? Willful behavior begins with the simple, instinctive will to survive and multiply. Even at that basic level there is some impact on the future state of the universe but the effects are localized to the immediate neighborhood or at most to the planet where the creatures live. To create more far reaching effects requires creatures that can operate in the mental realm that we call mathematics. To make their first foray into that unnatural realm they need two characteristics - they need to be curious and they need to appreciate order. Evolutionary pressure favors curiosity. The appreciation of order begins with music.