You may want to review Russell's A History of Western Philosophy. Bertrand pretty effectively shredded most of the fantastical precepts Platonism relies on in his analysis in the chapter on Theory of Ideas (15 in the Audible audio book).

The basic gist is that the basis of Platonism is metaphysical more than logical. Mysticism is the key bedrock of the form of the Good. Plato's ideal of Ultimate Truth is self-contradictory when reconciled with Geometry. Plato's lack of subtlety with syntax causes a lot of his arguments to fall flat, such as "beauty is beautiful." The analysis of the Parmenides is beautiful, highly recommended.