The problem of the one and many in Greek philosophy was initiated by the dialectic of Zeno. He endeavored to prove that the existence of one is absolute. He asserts that if we admit existing of many we shall face as many contradictions as when we accept the existence of one.
Correlation among many real object defined by one general appellation developed in Plato's Theory of Ideas. Oneness is embodied in the concept of the Idea. The theory is an essential part of Plato's philosophy; he devotes several chapters of The Republic (V-VII) to discussion of this theory. The theory of Ideas is the belief in unchanging, universal absolutes. These absolutes exist independently of the physical world and human mind. This means that "Idea" does not depend on the concepts in the mind of a man because it exists separately. The ideas are eternal; it means that they existed always.
Also they are unchanging; unlike everything which exist in physical world, and can be felt, seen, experienced sooner or later will change the ideas remain the same.
And finally, the Ideas are universal, in other words, they are general in nature. To understand this latter feature it is relevant to oppose ideas with their particulars (concept of many, individuals). An individual is a specific, concrete object, such as a cup, a potato in a pot, neighbor's cat.
In Plato's view there were two worlds, two domains of reality – one was the physical world and the other – the world of Ideas. According to Plato there are Ideas of everything which we perceive in the physical world. A cup, a potato and a cat have their Ideas. Apart from physical objects all abstract notions like ugliness, beauty, lie, good, etc. also have the ideas.
But there is only one Idea for each of these notions as well as physical objects while in that other world the physical world there are many of all of these. We know that in the physical world we can find thousands and millions of cups, or cats but in the world of Ideas there is only one Idea of cup and only one idea of cat. The world of idea is not dependent on the physical world, that means that any changes and transformations with the particulars will never change their ideas (as we know ideas are unchangeable). To illustrate this Plato says that the death of the cat cannot alter or destroy the idea of cat. Neither can the birth of a new cat do it.
Plato's theory first derived from the problem of language and knowledge. The philosopher knew that our language and our knowledge are full of universal notions. For example, sentence "Human life must be highly treasured," is made of universal words and notions. The question is where these universals come from. According to Plato it's impossible for them to come from the physical world as long as it is inhabited solely with particular individuals. To look around, we can see only this computer, this pencil, this cup of coffee. But we never see the universal concepts of computer, pencil or coffee.
Plato suggested as a solution to this riddle that another world must exist, the world of Ideas. Accordingly to Plato the reason for being able to apply a universal notion, such as computer, to a particular object in our world, this new, with LCD monitor, high-speed computer, is because there is a universal prototype of a computer in the world of Ideas. Thus the Ideas in the world of Ideas serve as models or prototypes. They are the perfect types, for everything in the world of our experience.
Mathematical concepts were of special concern for Plato. Plato wonders why it is possible to use in mathematics such concepts as perfect circle or equality even though we never find perfect circle and a perfect equality in our physical world. And again his solution is the world of Ideas. The Idea of circle and equality in the world of Ideas explains why human being is able to exploit the idea of circularity and equality in this world. Thus, mathematics and abstract notions which it uses belong to the world of Ideas. Generally the knowledge about world of Ideas is obtained with the help of "opinion" and "understanding".
Opinion stands above the understanding, as it deals with pure ideas. The understanding is that kind of mind which is used by mathematics, it is lower than opinion as it operates hypothesis which could not be proved. In the physical world there are no straight lines, right angles, regular triangles so if mathematics wants to own something more than hypothetical truth, we must find the proof for existence of idea of straight lines in the world of Ideas. This can not be reached by understanding but according to Plato it can be reached by opinion which shows that there are straight lines in the sky, geometrical properties of which can be established categorically and not hypothetically. However, this approach has some trouble, for instance, if we speak about two circles that cross, evidently there should be two ideas - one for each circle – in the world of Ideas. Therefore, each mathematical object must have infinite number of prototypes in the world of Ideas, which is impossible, as long as the idea for each particular can be only one. (Russell, 1972)
The next fundamental reasoning for Plato's theory is that there is a world of Ideas was based on the necessity for a foundation for morality. He was deeply convinced that if there is no foundation for morality, people shall live in political and social disorder. He believed that if it is possible to be a moral person there must be an objective, firm, universal ground for morality. And of course for Plato it was impossible to find such a foundation in the physical world of imperfection. That is why, he concluded, there should be another world, the world of Ideas. This world contains ideal models or prototypes of moral concepts as well as all the other concepts. The possibility of being kind, for example, requires that in the world of Ideas there is an Idea of kindness, the ideal model of kindness. The same should happen with all other moral concepts. This argument draws forth another feature of Plato's theory of Ideas, which implies that the world of Ideas is more real and more perfect in comparison to the physical world. As it is obvious for Plato the physical world is a kind of second-rate world. And on the contrary, the world of Ideas is the most perfect world.
Moreover, Plato concludes that the world of sensible things is not real at all. For instance, he argues that something what is beautiful from one side is ugly from the other side. Plato all the time encounters difficulties in argumentation due to his lack of understanding of relative terms. He supposes that if A is bigger than B and smaller that C, then A is simultaneously both big and small and it looks for him very contradictory. Such difficulties Bertrand Russell calls "infantine sickness of philosophy". (Russell, 1972)
The problem of the one and many has been deprived of its chief interest and bewilderment for modern people. We can easily admit that a unit has many parts, that the continuous is the divisible and that similar principle may be applied to exclusively intellectual notions. Each time we comprehend the meaning of the words, we recognize that two contradictory statements present in this meaning are true. The antinomy is very familiar to us so we scarcely observe it. Hegel regarded the coexistence of antitheses in the unity of the idea as the supreme principle of philosophy. The law, which affirms the coexistence of contradictories which form imperfect parts of the truth, replaced the law of contradiction, which logicians asserted to be a basic principle of the human mind. So we may observe that all endeavors to adjust antinomies to each other originated in the old Platonic problem of the one and many.
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