What is the connection between Platos theory of Forms & the platonic realm in the philosophy of mathematics

Question

From what I understand about Platos theory of Forms, it seems little to do with mathematical concepts, although he uses geometry as an example, for example the form of the Triangle, he also introduces other Forms, such as the Good. It appears that what abstract mathematical entities live are and where they live is not his main concern here.

In the philosophy of mathematics one is introduced to the Platonic realm where mathematical concepts like the real line and the number one lives - but nothing else lives there.

What are the connection between these two ideas? Is it purely historical, in the sense that Platos theory of Forms inspired the one for the philosophy of mathematics. If so, how far can we trace this idea as a distinct one in the philosophy of mathematics; or is there a proper relationship between them in the sense the Platonic realm is Platos theory of Forms restricted to mathematics - but then Why were the other Forms left out?

Answer 1

The connection between the idea of the Good and the idea of the Triangle is that, in the physical world, there exist neither things that are perfectly good nor things that are perfect triangles. Even if you help someone, you probably do it because it gives you a pleasant feeling, or because it's just a habit, so it's not pure good. Similarly, the triangle drawn in my book is not a perfect triangle: it is ever so slightly deformed by the structure of the paper and the inaccuracy of whatever drew it.

I could say, "I have this figure here in my book that looks like it has three angles, but it's ever so slightly off; I will refer to it by describing the little quirks in its outline that come closest to its actual shape". Alternatively, I could simply say, "granted, it is an imperfect triangle; but we all know what a perfect triangle would be in theory, and it's more efficient to refer to this perfect triangle when discussing mathematics rather than this imperfect drawing in my book, so I will just call it a triangle, keeping this perfect theoretical triangle in mind". The same can be applied to a good action.

The concept of an ideal, perfect triangle or the perfect good can be helpful and efficient in our daily tasks. But Plato went farther: he held that their perfection makes them in a way divine. They are different from physical triangles and good behaviour in our daily lives; they're something of a higher order. That's why he postulated that these idea(l)s must exist in some way in a higher plane of existence, that they must transcend their imperfect physical copies.

The fact that we, humans, could in some way touch on or even comprehend these ideas must mean that we, too, must be or have been connected with the divine. For how can we have a perfect triangle in mind if we have never seen one? A being in no way connected to the divine surely could not distil a perfect triangle out of the imperfect physical things in the temporal world.

Answer 2

In a nutshell: to promote his political agenda Plato invented a new rhetoric which relies on imitating mathematics. Aristotle was able to 'abstract' logic from discourse while Plato offered mostly analogies.

Geometry does not have a scale, so any circle is the unit circle or all circles are indeed the same. Two circles are already numerically different and, further, they are spatially different (differing by places, and/or size). Same for squares, cubes etc. It is the absence of differences that makes them 'perfect'. What fascinated Plato is the generality of mathematics, e.g. that for any triangle there is a circle passing through its vertices. Analogies are easy to construct: for any man there should be a compelling idea of justice, of a King etc.

The trick does not work smoothly but its weakness was transformed into strength: there is no such things as perfect dirt or greed. Negatives entities are not Good, of course. The exemple offered by Plato in his Republic borders on farce as he entertains (seriously?) the idea of (perfect) bed.

There are a serious difficulties when discussing Plato in English. Just because native speakers take 'ideas' to be mostly subjective, they are called instead 'forms' (a latin word) and for 'forms', such as the ones exhibited by mathematics, a hazy conception about some kind of 'intermediates' is supplied. The famous analogy of the divided line (Rep.) exposes how Plato sought to place his imitation as the pinnacle of philosophy.