How do we make the leap from mundane to metaphysical?
This part will investigate the historical (greek) relationship to math, and lead up to Part III, where we relate I and II to proceed onward.
While geometry (notice: geo-earth, metros-measure) likely originated in Egypt, it quickly travelled to Greece and became entrenched in the teaching of their first mystic: Pythagoras. From there, a combination of Plato’s theory of Forms and the masterful representation of mathematics by Euclid leads us to our first realization of “the connection” between mathematics and the sublime.
Geometry and Pythagoras
As far as we can tell, geometry became necessary when Egyptian property taxes were tied to the amount of arable land someone owned. In most Egyptian towns, some of the land would wash away every time the Nile flooded. The tax rate would be adjusted accordingly, so the tax collectors developed methods to measure the missing land, which became geometry. It is also ensconced in their early construction methods, with evidence that known pythagorean triples were retained as knots in a rope: 3 knots on one side, 4 on another, 5 on the third, and you get a right triangle, which is very good if one is trying to create squared framing for construction.
Little is known about the transfer of this knowledge to Attica or other regions, but it is known that some greeks saw in this study a much broader application and metaphysical meaning. The biggest contender: Pythagoras.
Pythagoras was known mostly through satire and derision of him written by other thinkers and authors in his time, such as Heraclitus (who, in turn, died buried up to his neck in dung) and various famous playwrights.
A short and sweet biography will be enough: Pythagoras left his hometown to start a cult, where his followers were kept from eating meat and beans. When the square root of two was found to be irrational, they killed the discoverer. He died when his cult was sacked by angry townsfolk from the nearest city. It is unclear whether he was killed in the attack, whether they escaped and starved, or Pythagoras willingly died by not running through the bean patch.
However tenuous the man’s relationship with beans, his mathematics has a special note to it still.
Pythagoras’ Weird Beliefs
That is to say, it is still indistinguishable from his mysticism!
Pythagoras’ weird beliefs included a special form of reincarnation called metempsychosis, and came with it attendant complications about symbol and number. As far as we can tell, it was tied into his conception of The Universal Music, better known as the harmony of the spheres.
This came partially from Pythagoras’ discovery of a relationship between the length of a string and the note it played when taut. It seems that he realized that there were integer numbers of visible vibrational modes on a string that were associated with a given harmonic. This led him to a final set of weird beliefs, the one that is most obvious for math: numerology (and via it, divination and prophecy).
Assigning numeric values to items in life was, at the time, magical. Most humans could not count to 365, so knowing exactly the number of days in a year, or a season, was maddening to them. They knew there was a number, but not what the number was.
Pythagoras, and his accolades, could easily spend time assigning mystical value to numbers via poetry and symbol, and in doing so, add extra layers to the arcane knowledge they possessed.
And of course, we are left with his named theorem.
Chronologically, our next stop is Plato, but he may be the more important of the two remaining Greeks. Plato had a theory of forms, which can be boiled down (very roughly) to the idea that there is a “true form”, “essence”, or “ideal” associated with every physical object. It is roughly analogous to a soul, in that it contains the set of all essential attributes something needs to be, well… what it is.
Plato did not just think of this lightly, and he connected it directly to mathematics. In Plato’s Meno, his mouthpiece, Socrates, uses the “recollection” of mathematics by deduction, and equates the leading questions that he asks with a way to help a young boy “recollect” mathematical knowledge he had in former lives (or before existence as it is now). This is more than a passing resemblance to a connection to the geometric “form”.
Our final man to set the stage for Part III, Euclid. Little is known about Euclid, except that he was a talented mathematician who wrote the Elements, which was cited and used forever after, beginning with Archimedes. In fact, Elements stood as the best-selling secular book of all time until the past ten years, when it became unclear that this title was still applicable. It certainly sold as many copies as the Bible up to the middle of the 20th century.
What Euclid did was outline a complete, axiomatic method to derive most of classical mathematics. One of his students, Apollodorus, extended his work through the Conics, and those together made up geometry in near entirety until calculus and differential geometry were invoked in the 17th century for the first time (and even then, haltingly).
Relating the theory of forms to the propositions, axioms, and proofs of Euclid is a small step. The idea is that there is a way to access these forms, namely via correctly determining a set of axioms and definitions, and following them to their logical conclusions.
In book VII (7), Euclid defines, and then derives, much of what we now consider number and elementary number theory (primes, evens, odds, etc). He does so, astoundingly, using the geometric constructions in earlier books of the Elements.
That is, Euclid continued to think of number as distance, and in distance in turn was understood only as the ratio between radii of circles. It all comes back to the compass and straightedge, but once we have it, we have done something astounding.
Why it matters
This is our first sort of brush with the truth of the matter: a number does not actually exist in the world. Numbers are defined by Euclid to be
(ii) A multitude of units
and a unit is
(i) that by virtue of which each of the things that exist is called one.
THIS IS BIG! Existence, to Euclid, Plato, and (from what we know) Pythagoras, is one. Not a single substance, but rather for a “form” or “essence” to exist, there has to be at least one in the world. From that, we get number, which is a plural of unit. It is plainly spelled out: each of the things that exist is called one. To break this into a more modern reduction,
existence → quantity → number, and possibly (which we will discuss later)
number → quantity → existence.
This can be said in English that number is necessary for quantity, and quantity is necessary for existence; and possibly that the converse is also true (existence is necessary for quantity and quantity necessary for existence).
What is important is that Plato’s theory of forms posits that forms exist. Not here, but some “where”. Numbers themselves may not exist, but they are a necessary attribute of those things which do exist. Thus, we see something interesting implied: there is one form for each “thing”.
It’s ok if this isn’t so clear now, we will discuss it more later on. Suffice it to say that number and existence, at least to the greeks and the mystics, are from here on out very closely related, though where the lines are drawn is not always so clear.
Remember this. It will be important later, when we start Part III with mathematical realism.