“How do you mean?
I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions.
That is very true.
Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, –what would they answer?
— Plato, Chapter 7. “The Republic”
The one thing I ponder over with endless fascination is how we learn, specifically, how we connect unknown abstract concepts to real-world objects. This is a riddle as old as the history of philosophy itself. Plato was one of the first to identify and compound this problem, many contemporary thinkers (including linguists) now think he was very, very naughty to have divorced specific objects from their ‘eidos’ or ‘essences’. In thought, at the very least 😉
The example traditionally dished out to explain theory of universals and particulars in every Philosophy 101 class typically goes something like this:
”Platonic form can be illustrated by contrasting a material triangle with an ideal triangle. The Platonic form is the ideal triangle — a figure with perfectly drawn lines whose angles add to 180 degrees. Any form of triangle that we experience will be an imperfect representation of the ideal triangle. Regardless of how precise your measuring and drawing tools you will never be able to recreate this perfect shape. Even drawn to the point where our senses cannot perceive a defect, in its essence the shape will still be imperfect; forever unable to match the ideal triangle.”
Source: Wikipedia (!)
Triangles, along with other geometrical concepts, abstract concepts like beauty and truth as well as numbers are all, according to Plato, universals or forms, that exist eternally, in all their perfection, beyond space and time. That is to say, regular human beings like us will never be able to experience these forms in our everyday lives- the triangles and beauty we experience in our world of particulars, are only poor imitations, shadows, in which the forms inhere to a little extent. This so called ‘inherence’ itself is very intangible and therefore, debatable. There is therefore, according to Plato, no number ‘6’ in our world, it only inheres temporarily in, say a group of objects that can be quantified as 6.
While this may sound wacko, there is some truth in Plato’s problematization of universals and particulars. In Saussurean signification, the sign is comprised of a signifier and signified. Plato’s universals, are in a way, Saussure’s signifieds.
What takes my breath away is that my kindergarteners, the little tabulae rasae that they are, easily and quickly grasp universals from particulars. A considerable part of our classroom transactions are spent giving these children examples of particulars. We were learning about the number ‘6’ last week, and I was thrilled and amazed when a child spotted the number (manifesting as a different particular, :p) in a book they were reading in D.E.A.R. later. It is nothing short of mind-boggling how quantifying ‘6’ objects finds its way to connect with the visual representation of ‘6’ and how that, may ultimately be derived from a universal form. What is admirable about the IB Primary Years Programme, is how sensitive they are to these nuances of learning- the emphasis given to conceptual understanding, and to form itself, as a key concept, continues to inspire and move me.
It fascinates me how we don’t ever come across these elusive forms or universals in the classroom or in the world outside and how we all yet, implicitly understand them. There’s something mysterious in that leap of understanding that I can’t wait to explore further.