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According to Plato...
You have an image in your mind when you see a chair. The image is that of the form of a chair. The form is an archetype (a perfect model) of that chair, not the chair itself!
Form had a simple meaning until the birth of the Greek schools of philosophy. It was related to the appearance of an object. Mainly its shape. But then objects never had a consistent shape, and they kept changing with time. This made people think about what caused this change, and why did the object change shape at all?

The fact that the object is matter, which changes shape under different circumstances gave way for the question of the actual and original shape of any object. This is how the theory of form and matter came into existence, and since Plato proclaimed the theory of form, it is known as the Platonic forms.
Platonic Forms
Plato calls 'forms' the supra-sensible entities (that which cannot be sensed). Anything on earth, be it physical (house), or non-physical (honesty) is an impure copy of its ideal and pure 'form'. According to Plato, an eternal force/energy (God) has created all the ideal forms, from which we derive everything to make copies of it.
Ex. When a person thinks a girl is beautiful, how does he/she know if what he is seeing is beauty or something else. It's because he compares what he is seeing with the ideal form of beauty.

Ex. What is the shape of a ball? It's round, we all know that, otherwise it wouldn't be a ball. How do we know what we are calling round is really round? Again, we compare it with the real form of it. A ball, or any other round object, is only an imitation of its form.

Ex. Let's take a bigger example. Like the ball was an imitation of a form, everything else that we see is only a copy of the real form. So, are we all the copies of a real non-physical entity which cannot be seen through our eyes?
A Second Reality
Plato says, a form has only one property, and hence it is pure. For example, the form 'round' has only one property―roundness―but a ball (an imitation), will have several other properties, like weight, texture, etc.
All the things that are derived from forms are a reality that we see. What we cannot see, yet know about, are forms. So forms, a second reality, are pure and ideal, exist as non-physical entities and are truer than physical entities (things and ideas). As Plato says, intellectual truth is more true than physical truth, forms are the cause of all the material that is created around us.
When we talk about reality, there's a different reality for every single living being on the earth.
Ex. Let's consider, a normal healthy person, a colorblind person, and a dog.
There is a difference in what these three perceive through their sense of sight. A normal person would see all the colors in the human color spectrum. A colorblind person would see only a few colors, and a dog would see colors according to its color spectrum. So, which of these is real? Which color is the real color?
Infrared radiation
Ex. Humans cannot see infrared radiations, because they are beyond our vision spectrum. But because now we have the technology to see them through devices, we know they exists. Likewise, many things might be existing that we haven't discovered yet.
This makes it clear, that there is another reality that exists beyond the reception of our senses. Forms are a part of that reality.

But without experiencing forms, we still know them. If what we have seen are only the imitations of real forms, then ideally we shouldn't have known what the real forms look like, but surprisingly, we somehow know. By this, Plato concluded that our souls must have known these forms before they started to live in us, i.e., before our birth, and hence, the belief that souls are immortal.
Five Levels of Forms
Forms have a network of connections with each other and with the materials derived from them. It's like a hierarchy, where according to Plato, the form of 'goodness' is at the highest level. From there, everything proceeds. So to attain the level of goodness, a man must cross its sub-levels of forms and materials. There are five levels of forms.
Levels of Platonic forms
Levels of Platonic forms
Six Types of Forms
Transcendent: These forms are a-spatial and a-temporal, meaning independent of space and time. Ex. redness does not exist in a particular space or time.
Pure: These forms have only one property. Its copies have a mixture of properties and hence they are impure.
Archetypes: These forms are archetypes of the one property that they have. Objects are created following the perfect models - forms.
Ultimately Real: The ultimately real entities are forms and not the material objects. The reality of objects depends upon the reality of their forms.
Causes: The cause behind everything that we see that exists today is forms.
Systematically Interconnected: The network of connection between forms and their copies forms the levels of the hierarchy starting from the form of goodness.
Arguments
The arguments from objectivity
▶ The more objective is the concept, the more real thing it represents.

What we perceive through our senses, can deceive us. This is because, one moment we perceive a thing, the next moment it changes. The world is at a constant pace of change. As Heraclitus, a philosopher who lived around one hundred years before Plato, said 'It is not possible to step into the same river twice', Plato believed that a thing that keeps changing cannot be pure and perfect. So, the things that we perceive are nothing but images and experiences of our mind. The first reality is forms, the copy is the objects, and the second reality is what we perceive. What looks like red to me is my reality, what looks like gray to a colorblind person is his reality! But what we are perceiving is highly subjective, based on our senses.
▶ Forms are more objective than material objects.

If what we see is only a copy, what then is the real thing? You contact an object through your subjective perception, not directly. When you perceive an object, it is the subjective perception, which is impure. The thing you refer it to (form), is not subjective. Ex. If you see a red flower, you will see only those things related to that flower as a whole : red, shape, color, smell, etc. That's subjective perception. If you relate it to the color red, you will imagine all the things related to red. That's objective. Hence, forms are more objective.
The argument from mathematics
In mathematics, things are certain. Such certainty could not have come through our sense perception. It couldn't have come from our experiences or images in the mind.

▶ We talk about perfect squares in mathematics. But do these exist physically?

▶ We know 3 + 3 = 6. This didn't come from our experience of the physical world.

These entities used in mathematics are so certain and constant that they must be real. But they aren't in the physical world. So, they must surely be in the other world of reality, i.e. forms.
Criticism
Self- criticism
▶ Plato realized that what if the form that he believes is the real, and pure entity is a copy of another form that is actually real? So then, there can be a possibility of an infinite loop of forms where no one knows which is the ultimately real one.

▶ Taking the same point into consideration for objects; if the objects we see are only a representation of themselves, then we don't observe the objects, but their representations. In this way, the real form will never be known, because we first need to know what object the representations are representing.
Aristotelian
▶ Plato talks about the real (forms) and the non-real (material). Aristotle stated that, for Plato, everything that science has known has form. Also, Plato considered material to have form. Here, there is contradiction in his own theory. Forms exist as objects of science, but they don't exist as materials.
▶ Also, Aristotle's theory of perception states that, our senses receive the form of an object, and that is how we perceive them. Thus, the form would be a mix of colors, textures, flavors, and not just shapes.