Plato's argument for premise 1: The world that we perceive with the senses often deceives us. This would not be so if the world and objects that we perceive with the senses were the real objects.
It seems that all the objects we perceive with the senses are simply images or experiences in our mind. They are only subjective points of views on the real objects. For example, the world appears radically differently to a color blind person than it does to us. The objects that we perceive as colored, then, must not be the real objects, but just our experience of these objects that is determined by my particular subjective point of view and perceptual apparatus.
Once one sees that the world that we perceive through the senses is not the real world but just an image of it, it becomes difficult to determine at what level of description we get in touch with the real objects that make up the world.
In general, we assume that the more objective the concept or description, the more real the object it describes is. For example, when we see a person far away, we automatically follow our objective concept of humans as being about 6 feet tall and see the person as normal sized, even though the subjective image we have is of a very tiny person. In general, we form a more objective concept by combining different points of view into a more objective description that takes into account what all the other views had in common. This process is called dialectic: the back and forth discourse between different points of view that leads to their combination or synthesis into a more objective conception that takes into account both points of view.
Plato's argument for the second premise: What then are the real objects? They cannot be the subjective images we perceive. These often deceive us.
What about the everyday material objects, like chairs, tables, rocks, and trees, that we think our subjective perceptions of things refer to? The concepts we from of these are slightly more objective than subjective images. They combine and take into account all of the subjective images we can form of a single object, such as this particular asterisk * . Yet, there are reasons to avoid taking this as the real object as well:
1. We only contact these objects through subjective images. We never perceive them directly.
2. These objects contain a number of properties that are mixed together. Any description of the object that doesn't separate out these properties cannot explain what makes the object act the way it does. For example, If all you know about the asterisk above is that it is the particular thing it is, you will not know as much as if you know that it is black, star-shaped, made of ink on paper, etc.
3. These objects are always changing, taking up different properties from moment to moment, and going in and out of existence.
For these reasons, it seems that the only level at which things really exist must be the level of single properties separated from particular objects. These are the forms:
1. Our concepts of these are more objective than our perceptions of material objects. For example, my concept of blackness takes into account all the points of view anyone can have on any black object, while my perception of the asterisk above only takes into account those views you can have of this object. The object that my concept refers to, the form, must be more real than a material object.
2. The forms explain why an object is the way it is. It is the fact that an object has the properties that it does that makes it what it is, not that it is the particular thing it is.
3. The forms never change.
B. The argument from mathematics:
The most certain knowledge we have, the knowledge of mathematics, could not have come from sense perception:
1. In geometry we have access to perfect squares and circles, but no such objects exist in the material world.
2. We can know truths such as 2+2=4 without having to check our experience of the material world.
The objects that we think about in mathematics must be real, since they are most certain. Since they could not exist in the material world, there must be another realm in which they exist that is even more real, the realm of forms.