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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.

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    The "intelligible world", a.k.a. Platonic realm, is not in one's mind according to Plato, it is an objective eternal world of ideal entities, same for all. In his more mythical moments (Meno, Phaedo) Plato describes how our souls reside there before birth, forget everything when thrown into the sensible world at birth, and then unforget it when prompted by the sensible "shadows", which "imitate" their ideal prototypes. Myth aside, Plato claims that there is a separate world of mathematical objects, mathematicians access it via a kind of mindsight, and the sensible world is its imperfect copy. – Conifold Sep 14 '17 at 23:52
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    Useful: F.M. Cornford, Mathematics and Dialectic in the Republic VI.-VII. (1932). – Mauro ALLEGRANZA Sep 15 '17 at 7:10
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    One of the central discussion in Plato's Dialogues is the "divided line" one: see Republic, Bk.6, 509d. For comments: N.Pappas, Plato's Republic (2013), page 169 and S.Rosen, Plato's Republic: A Study (2005), page 263. – Mauro ALLEGRANZA Sep 15 '17 at 7:29
  • @Conifold For the intelligible world substantially existing, you are absolutely correct, in my drowsiness I failed to word my thoughts correctly, I'll amend it for future visitors, thanks for the expansion. For the second point, could you provide an explicit textual reference? – Anish Gupta Sep 15 '17 at 19:05
  • Plato's concept of the ideal objective real world, beyond the fallible material world, is motivated mostly by mathematical thinking. Things like the abstract notion of a circle are the really real things that our poor attempts at creating in the physical world never accomplish exactly. The mathematical object is more real than the approximation in a drawing or scuplture. – Mitch Sep 15 '17 at 20:16
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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?

  • Thank you for your answer, you've certainly provided useful clarifications. However, fundamentally you can formulate all of mathematics (or maybe excluding certain highly abstract fields, which could be rooted in the study of logic?) in set theory. Thus, I don't see why we wouldn't have a hierarchy everywhere. In particular, I would like to hear your views on what the Platonic structure of mathematics is, is it a tree as I'm suggesting, or is it more like a web? – Anish Gupta Sep 23 '17 at 21:15
  • @Anish gupta: its more like a web, see the extract above by Thurston (though trees are important, too); Category theory by the way provides an alternative foundation to mathematics other than set theory. – Mozibur Ullah Sep 24 '17 at 16:14
  • Thurston seems to be describing what a working mathematician's mental structure of mathematics should be like, perhaps for optimal learning, or to prove new theorems. I don't see why it is an answer for what the Platonic structure of mathematics really is, as he does not provide an argument for why it is a web rather than say a tree of Ideas of varying abstractness. – Anish Gupta Sep 24 '17 at 18:19
  • @anish gupta: he doesn't describe it as his 'mental structure' but how mathematics itself is structured; he says its highly 'interconnected' - hence why its more like a web. – Mozibur Ullah Sep 24 '17 at 21:21
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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.

  • Plato shows 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).. I understand you are trying to present what you think Plato's views are, but surely because you can make theorems about all triangles, "in all triangles, the sum of the angles is 180°", and specific triangles, "The triangle with side lengths of 1-√3-2, has angles 30° 60° 90°"; it looks like it is obviously false, or perhaps you could clarify what is going on here. – Anish Gupta Sep 23 '17 at 21:25
  • @AnishGupta Well I can only add, at the moment, that Plato's Forms are not exactly what were later called "universals". Not every universal term represents a Platonic Form. What it does represent instead, according to Plato, I cannot establish at the monent. – Ram Tobolski Sep 23 '17 at 22:36

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