…or so I’ve heard.
…or so I’ve heard.
The original meaning of the word calculator was one-who-calculates. Before machines did it, scientists would get people to do the tedious work of routine number crunching. They were still in use on big government projects like the Manhattan Project, where, I have heard, certain idiot savants, people of mediocre or impaired intellect who possessed uncanny abilities in math, e.g., extracting the square-root of enormous numbers, were employed. The NYU Institute for the Study of the Ancient World is hosting a nice display of artefacts of the Ur-calculators, the scribe/engineer/mathematicians of the ancient Mesopotamian world.
The exhibit is remarkable for the evidence it presents that the Pythagorean Theorem was known to these mud scribblers 1500 years before Pythagoras himself. The scholar who edited and interpreted these pieces was adamant in his belief that much of Greek mathematics had been imported by them from elsewhere.
The header picture, from the NYTimes site, makes the tablet look like it has a very beautiful bronze-like color. It doesn’t, but it’s still pretty remarkable! It’s a list of pythagorean triples, i.e. lengths of the sides of right triangles
A toy I developed when I was learning to use some GIS software tools. This is a video of a sequence of Mira Fractals. If you would like the toy for your own use, drop me a line, and I’ll email it to you. It’s a small file, very easy to use, and fascinating to watch. This is only one of many fractal sequences possible, plus you can make your own!
In his Lives of the Artists, Giorgio Vasari writes of Filippo Brunelleschi, the architect of the great dome on the cathedral in Florence,
…we should never turn up our noses when we meet people who in their physical appearance do not possess the initial grace and beauty that Nature should bestow upon skillful artisans when they come into the world…
Filippo was a great genius, but not all that good looking. Note the use of the word should in the phrase “…that Nature should bestow upon skillful artisans.” How can Nature do anything that it should not do? Just whose rules does Nature obey, if not its own? The idea here, that Nature has done something wrong, made a mistake, had a little hiccough, in making a great genius an ugly man, or at least, insignificant in appearance, may be common to Italy, or to Renaissance thinkers, but it is also part of an immensely deep and broad current of thought in western culture since the Ancients. Outward beauty reflects inward perfection. Personal beauty is a manifestation of the soul’s purity.
“Beauty is truth, truth beauty,” – that is all
Ye know on earth, and all ye need to know.
And if beauty is the emanation of the soul, why should an artistic genius not be beautiful? How could it happen? It’s a violation of the nature of the universe, the ordered universe in which truths are manifest in the order and lovliness of things. And the most beauteous things of all are the pure things, the mathematical entities, the Pure Forms of Plato, the Ideas.
Historians of ideas (Man of Roma included) agree that this great torrent of intellectual traditions has its source with Pythagoras, the student of Thales, and a predecessor of Plato and Socrates. He was a brilliant thinker, a mystic, an analyst, a mathematician, and the founder of a cult that has lived on to our day in various forms. In the wonderful collection of brief mathematical lives, Gems of Calculus, he is referred to as 3/5 genius, 2/5 sheer fudge . Bertrand Russell appraises him similarly, and that is no small compliment!
From his followers’ mystical preoccupation with Number, and awestruck encounters with the order of the universe, Plato developed his metaphysical notions, Platonists mixed Plato with eastern cults, early Christians mixed Plato with Christ, later Christians mixed it all up into neo-Platonism, the Renaissance rediscovered paganism, Platonism, and mysticism allied to the beauty of art, and secular and mystical philosophers of the succeeding ages remained in thrall to the notions of:
Just a listing of these notions evokes so many associations, it’s clear Mr. P. was onto something big. Did he invent these ideas? Probably not. But he was the first to articulate them in a way that had sticking power in the Western tradition. I would guess that these notions have their roots far deeper, in the human organisms evolution as an information processing being. The intellectual excitement of these ideas is a refined form of the fundamental “Aha!” feeling that comes with discovery…of food…of the lever…of the power of fire….
In our own day, these ideas live on, certainly in religious rhetoric, but they are also increasingly problematic as I shall discuss later. Consider just Thomas Pynchon’s novel, V, in which a young woman, Esther, is having an affair with a plastic surgeon, Dr. Shale Shoenmaker (Dr. Shale Beauty-maker). He wants to give Esther a nose job, she is not sure why she should have one. It’s a popular comic theme from the 1950’s. The Mad Magazine song goes,
“I once knew a girl with such a big schnoz,
she couldn’t get a boyfriend, or a job!
So she got a nose job!
Yeah, yeah, yeah!
The good doctor tells Esther that he wants to bring out the true beauty within her, make her outward experience in harmony with the inward nature of her soul, rectify, improve on the work of nature. It is a pure Renaissance Neo-Platonic argument about art, truth, and beauty, but he was being ruthlessly satirical. And of course, in our Botoxed present, who can deny that we have gotten something a bit wrong with the beauty-truth-body equation?
How did this concept get going? It was the music. The Pythagoreans noticed that by causing a string to vibrate and sound a tone, they could create pleasing scales of tones by holding down the string at set increments of its length, effectively shortening the string. Thus, the musical intervals were codified, if not quite born. Simple ratios, pleasing scales. In the visual realm, pleasing proportions, the Golden Mean, which lives on in dimensions of our rooms and the size of a standard piece of writing paper.
Something always puzzled me about this, however, since I am not musical. How did they know the scales were pleasing, were right? They just heard it, but other people heard differently. Asian tonal scales are not the same as ours. The de-tuned scales of the blues and other genres are pleasing to their audiences, but hardly classical. My nom de plume, Lichanos (more in my By Way of Explanation), refers to a particular ancient scale. Was it a deviant one? Was there some fudging of the scales at the creation? Did they weed out the not-quite-right tones so that only the ones with “good” ratios remained? I await the response of the archaeo-musicologists amongst you!
I said earlier that this current of thought is not always good – it was very much an impediment to the development of science. Ideas that were not “beautiful” were discarded. Ideas not deduced from geometry and pure forms were considered suspect. Even in Newton’s day, he felt he must prove his theories twice: once as geometrical demonstrations that fill the pages of his Principia, and once in terms of argument that are derived from his laws of motion and his observations. In science, the truth is not always the beauty of Pythagoras and Plato. Avogadro’s Number, without which we cannot solve chemical equations, is an ugly number. Planck’s constant is not pretty either. The acceleration of gravity (32.2 feet per second per second) is not lovely. Don’t even mention the contant of universal gravitational attraction! Even so, the lure of number remains, as an obstacle and as a motivation for science.
This split between two modes of apprehending the universe is represented in Raphael’s famous image of the School of Athens. Plato gestures upward towards the empyrean realm. Aristotle points downward towards the earth.
Plato dominated western thought until the great resurrection of The Philosopher, as the schoolmen called Aristotle, in the 12th century Renaissance. He had his own scientific “issues” and the reign of Platonism was by then, in any case, well established.
The seduction of the geometric! The fact that geometry revealed incommensurable, irrational numbers only placed a slight speed bump in front of the onward rush of the Pythagorean fleet.
If a=1, and b=1, then c=square root of 2. Punch that into your calculator and see what a nice, beautiful number you get! Still, the Numbers as the final reach of truth, the ultimate ground, the thing in itself carried on. How was it that mathematics could tell us about this earthly realm? Fire a missile, from a canon or a slingshot, and it follows a parabolic arc (Gravity’s Rainbow) to its final resting place. We can predict its path precisely – why? The disjunction between experience and the pure realm of mathematics is bridged in physics, but how? Kant wrestled with this and concluded…well, another time.
Today, the debased form of this issue lives on, melded with religious fundamentalism, in the argument against Darwin from Intelligent Design. How could the world be anything but designed, according to plan? There is a whiff of the old pagan, Pythagoras, here.
For this, for everything, we are out of tune;
It moves us not. — Great God! I’d rather be
A Pagan suckled in a creed outworn;
So might I, standing on this pleasant lea,
Have glimpses that would make me less forlorn;
Have sight of Proteus rising from the sea;
Or hear old Triton blow his wreathed horn.
Postscript: Giotto and Perfect Circles