On finding a palette, pt. 4
Returning full circle… the possibility of total freedom can actually be its own cage, like waking up and realizing you have the whole day free but then can’t decide what to do. That type of freedom can actually be paralyzing. In this regards, limits may actually lead to a type of freedom that is guided by rules.
One way to illustrate this is by looking at the ancient Geometric problem of “squaring the circle”. This is the challenge of taking any given circle and constructing a square with the same area as that circle. While this problem dates back as early as the ancient Egyptians around 3400 BCE it was the Greeks who first limited the method of trying to solve the problem by only using a compass and straight edge. For the Greeks, this method was one of pure Geometry whereby the goal was to derive a set of postulates and theorems and excluded any use of numbers.
The Egyptians approach to this problem used numbers as evidenced in the Ahmes (Rhind) Papyrus where multiplication, divisions, fractions and other numerical methods were used to solve the problem including even a value for pi (π). The Greeks, who no doubt were aware of the marvels of Egyptian engineering and mathematics, however, chose a different approach. By limiting themselves to only a compass and ruler and devoid of any actual numbers they entered a different type of theoretical space that was separate from any real world scenario of measurement in feet or inches. Instead, the compass and ruler opened as a way of thinking purely spatially in the mind.
It is no wonder that the first Greek on record to have attempted to square the circle with these means was Anaxagoras. Living around 500-428 BCE, he was one of the first Greek philosophers who postulated that a Mind (nous) permeated the Cosmos and caused the world to move, disrupting a primordial stasis, and resulting in the fragmentation and separation of a unified whole into the variegated world today.
As he says in one of his surviving writings (Fragment 13):
And when Mind (Nous) began to cause motion, separating off proceeded to occur from all that was moved,
and all that Mind moved was separated apart,
and as things were being moved and separated apart, the rotation caused much more separating apart to occur.1
Anaxagoras was a major influence on Socrates, Plato and Aristotle for his philosophy of Mind although they thought he didn’t take it far enough but no doubt opened a pathway in Greek thought. It cant be forgotten that Plato would later put a quote above his academy that stated “Let none but geometers enter through this door.”
This problem of squaring the circle persisted throughout the course of Greek mathematics taken up by such luminaries as Archimedes, Euclid and Eudoxos and in fact was one of the key drivers of discovering new theorems in Geometry. Similar to the scientific method, it was the problem that was the driving force behind new discoveries, new sciences, new ancillary problems.
The solution to squaring the circle with limited tools was not discovered by the Greeks and was left open into the modern age by later mathematicians. It wasn’t until the late 19th century, around 1882 when it was discovered that squaring the circle by only using a compass was mathematically impossible.
So was this endeavor of looking for the impossible a waste of time? If numbers had just been introduced then this problem would have been solved a thousand years ago with the advancement of decimal numbers. But then would the field of Geometry and theoretical thinking ever evolved? Or would engineering just have saved the day and removed any hypothetical questions about parallel lines in infinite planes?
In some regards the limitation of thinking through a compass and ruler enabled the search for an impossible solution and thus turning Geometry into a kind of Theology. There is no doubt that theoretical Physics can also claim to be descended from this tradition although we’ve opened the playing field to a solution by any means necessary in modern science. Perhaps limiting things to a few rules, or working from a certain palette, can be the key to opening up new horizons and new ways of thinking about the world we live in.
Richard D. McKirahan, Jr., Philosophy Before Socrates (1994) Indianapolis: Hacket Publishing Company, Inc., p. 199.