How do mathematical objects fit into Plato's theory of Ideas?

Question

Say if I describe a triangle ABC, with all sides equal - and I describe another triangle EDF with all sides unequal.

These triangles I have just described do not exist in the visible world, but in the intelligible world according to my understanding. We identify both ABC and EDF as triangles, but we identify ABC as an equilateral triangle and EDF as a scalene triangle. I've identified a few Ideas: namely the Idea of Triangle, the Idea of Equilateral Triangle, and the Idea of Scalene Triangle.

I'm struggling to understand how Plato's theory of Ideas works for mathematical objects, considering they do not at all exist in the world of material things, but the intelligible world, accessible to humans only through reason. When I talk of a triangle with specific side lengths, am I describing a specific Idea with those side lengths? Clearly, there is some sort of hierarchy, perhaps like this:

Idea of Triangle -> Idea of Scalene Triangle -> Idea of 3-4-5 triangle -> (Geometer's conception of 3-4-5 triangle)?

What I am looking for is some clarification on how mathematical objects fit in to Plato's theory of Ideas, I am under the impression that Plato regarded geometers highly, so perhaps there may be some specific commentary on this, (I understand this has something to do with Dianoia)? However, I am also interested in learning how hierarchy in the Theory of Ideas in general, for non-mathematical objects.

Answer 1

Plato used the notion of mathematical forms as a kind of stepping stone to his theory of forms themselves; they weren't to be seen as an end in themselves.

According to some later developments of Platos philosophy their is a complex inter-relationship and hierarchy of ideas; and this is apparent too in the purely mathematical realm. Though 'locally' we might see a simple hierarchy as you've drawn, in its fullness it's far more intricate than that; for example, Thurston, a famous modern geometer wrote:

Mathematics is a huge and highly inter-connected structure. It is not linear. As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of structure, not just familiarity with a particular your through the structure

Three-dimensional Geometry & Topology, Vol 1

One could add that the development that you've outlined is 'familiarity with a particular tour' - that of triangles!

He adds:

Think of a tinker-toy. The key is the pieces which have holes allowing you to join them with rods to form interesting and highly inter-connected structures. No interesting mathematical topic is self-contained or complete: rather it is full of "holes" or natural questions and ideas not readily answered by techniques native to the topic. These holes often give rise to connections between the given topic and other topics that seem at first unrelated. Mathematical exposition often conceals these holes for the sake of smoothness; but what good is a tinker-you if the holes are all filled in with modelling clay?

Answer 2

Plato discusses mathematical objects in the analogy of the divided line, at book 6 of the Republic. What the analogy indicates, I will argue, is that mathematical objects are not genuine entities in Plato's ontology. Only the Forms (Ideas) are genuine entities in Plato's ontology. Mathematical thinking is indeed said to require much purer reflection on the Forms, compared to thinking about material objects (*). Still, mathematical thinking is not a reflection on abstract mathematical entities, according to Plato (**).

The divided line expresses of the following classification of objects:

1. concrete objects
   1.1 material objects
   1.2 visual shadows and reflections
2. abstract objects
   2.1 the Forms = Ideas
   2.2 mathematical objects

Structurally, the mathematical objects are compared to visual shadows and reflections. This gives us the first indication that mathematical objects are not genuine entities. Because it is unlikely that Plato considered visual shadows and reflections as genuine entities. Shadows and reflections are phenomena, they are objects of the mind, but quite surely not genuine entities, for Plato. Hence, so are mathematical objects.

Second, Plato does hold that when e.g. we draw concrete triangles on paper, we are really trying to think about abstract entities. Plato's actual examples, however, show us that the abstract entities that we think about are always abstract Forms, Ideas (such as the Form of a triangle), but not abstract mathematical entities (such as abstract particular triangles).

And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on ... (emphasis mine)

Third, Plato claimed that mathematics cannot - in principle - be thought purely in the abstract. Mathematical thinking necessarily requires concrete objects, images. These images he also calls the concrete "hypotheses" of mathematical thinking, hypotheses that mathematics cannot do without. But if mathematical thinking were based on abstract mathematical entities, why couldn't it be done, at least in principle, purely in the abstract? Hence, mathematical thinking is not based on abstract mathematical entities, according to Plato. it is rather based on abstract Forms plus concrete, material images.

[In mathematics] the soul is compelled to use hypotheses; not ascending to a first principle, because she is unable to rise above the region of hypothesis, but employing the objects ... as images.

By contrast, Plato holds that the "dialectical", philosophical reflection on the Ideas can be done completely in the abstract.

And when I speak of the other division of the intelligible, you will understand me to speak of that other sort of knowledge which reason herself attains by the power of dialectic ... by successive steps she descends again without the aid of any sensible object, from ideas, through ideas, and in ideas she ends.

In summary, the analogy of the divided line tells us that for Plato, at least at the time when he wrote the Republic, only the Forms, the Ideas, were genuine entities. Particular mathematical objects, in turn, were not believed by him to be genuine entities. There were no abstract particulars entities for Plato. Mathematical objects, concrete objects, visual shadows and reflections, were all, ultimately, just different grades of shadows, reflections and imitations of the Ideas.

(*) For the purposes of this interpretation I distinguish between "objects" and "entities". I use "objects" for what we appear to think about, for apparent entities. There are then mathematical objects, that is mathematical things we (apparently) think about. But they are not genuine, self-standing entities, according to Plato. Our thinking is constituted differently from what it appears.

(**) I identify here the views expressed by the character Socrates in book 6 of the Republic as Plato's views.