Beauty exists as a thing in itself
Beauty is a Form, a perfect mind-indepndent idea, which is replicated in material objects that we percieve through our senses. Beauty is unchanging and so in no way can it be in the eye of the beholder.
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For Plato, beauty is a Form. A Form is essentially a mind-independent entity that captures the perfect representation of some quality or object. They are mind-independent and unchanging evaluative concepts that objectively represent ideas. That said, our beliefs and judgements do not influence their nature of existence.  A form is a distinct singular thing that can manifest in multiple ways, such as in different objects but is not part of the object. Every object has a form as objects are essentially copies of this conceptual perfection. With the above said, beauty exists independent of particular beautiful things and independent of those who cast aesthetic judgements. Beauty can be understood on a conceptual level without appealing to objects. Throughout the course of our lives, we will see many pleasant and beautiful things, and they may indeed differ. This is what leads people to believe that beauty is in the eye of the beholder. Yet, there exists in another realm a singular perfect Form of beauty that makes the multiple instances of beauty within our realm possible. Our perception of beautiful things entirely comes from its existence in the Form.  So, beauty is not in the eye of the beholder because beauty is a mind-independent concept that can be understood in its own right and without reference to particular instances of it.
To believe that beauty is objective and exists as a mind-independent concept, we must first accept Plato‘s Theory of Forms. The principal issue with this argument is that we can raise doubt concerning the existence of Forms, given that they are of an idealist nature. As an Idealist, Plato has asserted that abstractions of material objects are more real than the objects themselves which is problematic. What's more, these perfect abstractions exist in a realm that we cannot even access. Given that we can doubt the credibility of Forms, we can reject the idea of beauty being objective and thus not in the eye of the beholder. This argument subsequently crumbles.