Rationalism and idealism are compatible
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Artist-idealist-rationalist philosophers use both in their work.
Socrates, Plato and Schopenhauer all used idealist-rationalist philosophy to critique art.
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The Argument
Socrates, Plato and Schopenhauer all directly and critically incorporated the analysis of artworks into their idealist philosophy. Their critique of particular artists and artworks are fundamental, even inseparable, from their idealist-rationalist philosophies. Not only did they offer unparalleled insights into the nature of art and the artistic temperament; they also philosophized in an artistic style so magnificent as to qualify them as among the greatest (and most beautiful) stylists in the history of literature and oral presentation.
These three philosophers might be called "form" idealists, who clearly and consistently demonstrate profound artistic insights and understanding of individual artists and artworks. This is a class apart from "Content" philosophers, who lack an intimate relationship to individual artists and artworks. Content idealist philosophers, such as Berkeley, Hume, and Kant, may discuss individual artists and works of art obliquely and in general terms but are incapable of offering serious and original insights and lack the artistic style available only to Form philosophers.
Rationalist idealists represent a special class of pure philosophers for whom truth, morality, and beauty are primary: these qualities of their philosophy will never be subordinate to any other standards. These structural differences can never be overcome and that explains why Berkeley, Hume, and Kant would never have even attempted a serious discussion or critique of art in this very specific and special sense.