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.
Useful: F. One of the central discussion in Plato's Dialogues is the "divided line" one: see Republic, Bk. For comments: N. Pappas, Plato's Republic , page and S. Rosen, Plato's Republic: A Study , page 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?
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.
Show 3 more comments. Active Oldest Votes. 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.
He adds: Think of a tinker-toy. Improve this answer. Community Bot 1. Mozibur Ullah Mozibur Ullah 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?
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: 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.
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.
Add a comment. The divided line expresses of the following classification of objects: concrete objects 1. Ram Tobolski Ram Tobolski 7, 1 1 gold badge 10 10 silver badges 20 20 bronze badges.
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.
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.
Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. If we can place this theory into its historical and cultural context perhaps it will begin to make a little more sense. Plato was born somewhere in B. He belonged to a wealthy and aristocratic family. He died some time between B.
What was obvious to many of the early Greek philosophers was that we live in a world which is not an easy source of true, ie, eternal, unchanging knowledge. The world is constantly undergoing change. The seasons reflect change. Nothing is ever permanent: buildings crumble, people, animals and trees live, and then die.
Even the present is deceiving: our senses of sight, touch and taste can let us down from time to time. What looks to be water on the desert horizon is in fact a mirage. Or what I think of as sweet at one time may seem sour the next. Heraclitus, a pre-Socratic philosopher, claimed that we can never step into the same river twice. In his Socratic dialogues Plato argues through Socrates that because the material world is changeable it is also unreliable.
But Plato also believed that this is not the whole story. Behind this unreliable world of appearances is a world of permanence and reliability. But what is a Platonic Form or Idea? Take for example a perfect triangle, as it might be described by a mathematician. 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?
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. 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.
Same for squares, cubes etc. It is the absence of differences that makes them 'perfect'. What fascinated Plato is the generality of mathematics, e.
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?
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