One of the easiest examples which describes Plato’s Theory of Forms can be found in mathematics.

Think of the line as a mathematical object, beginning at zero and ending at, say, ten. Zero is an end point, and ten is the point at the opposite end of the line. In between, there are an infinite number of other points, which make up the line. We can name some of these points: one, for example, or two. We can find them on the line quite easily; one will be one-tenth of the way along the line, and two will be one-fifth of the way along. There are also points that have a decimal place, such as 2.5; we can find it a quarter of the way along the line. Indeed, we can locate the position of any point we care to name between zero and ten. We can even find a point with a long decimal place. There are some points with infinitely long decimal parts, such as, 3.33’ (3.3 recurring which means that the decimal part is represented by an infinite row of 3’s: Also the same as 1/3) or pi or the square root of two, but we can still, theoretically, locate them on the line. There are also some points with infinitely long decimal parts that we cannot even name though it is not unreasonable to assume that despite our inability to name them, they are still there.

There are also many points that we could name if only we could be bothered to, such as, a point with a billion decimal places. It would take some time to name one, but once we had written it out, we could be sure that it was on the line and that we could find its exact location. The idea is that we have a world of actually existing and known points, all of which can be located on the line. The line represents the Platonic realm of points. Prior to discovering or naming even the most boring point, we know that it already exists on the line.

The theory of forms, or ideas, marks the basis of Plato’s entire philosophy. The realm of ideas is the place in which generic existence can be found. Take, for example, the concept of a table. We are all familiar with a wide range of different types of tables. But is it possible to define what a table actually is? It is clear that we cannot include at all, as a definition, that a table is an object with four legs because we could quite easily point to tables that have more legs or fewer legs, which would effectively counter the definition of having four legs. Tables exist that have no legs, so to define a table as a thing that has legs or does not have legs is not to define the table at all. All things have legs or do not have legs, so such a definition would be vacuous. Citing the way that a table is used would be unhelpful because tables can be used for a wide range of purposes, and finding a single use to which all tables are put would be impossible.

So what is it about a table that makes it a table? Where, precisely, are we to find the essence of “table-ness”? Plato answers by saying that we know the essence of what a table is because we have access to the eternal realm of forms. We know, via this privileged access, what comprises the essence of a table, so that when we come across a table in the world that we experience, we are able to identify it as a particular table from the realm of all possible tables.

The realm of ideas, then, is a realm of perfection. These days, philosophers contrast particular objects with the notion of universals. A point is an actual position; the line is the universe of points. In the actual world, a particular is something that we can experience — for example, a specific shade of yellow.

When we experience something that is yellow, we know it to be yellow even though we may never have experienced that particular shade of yellow before. How is it possible to know the color of something when we have never experienced that color before? The doctrine of Plato’s forms claims that we bear knowledge of the universal notion of yellowness, and as such, we are able to recognize even those shades with which we have no prior experience.

The philosophy of the Greeks still offers many ideas relevant to the work of contemporary philosophers: the problems with which the Greeks were concerned continue to crop up over and over again in various guises. The notion of substance was a chief concern of the Greeks. It was also a chief concern of the philosophers of the seventeenth century. In later posts we shall look at how the notion of substance was developed or used during these two time periods.