In a way rationalism, idealism and existentialism are three sides of the same coin (maybe using some Riemannian geometry). I would like to say that Rene Descartes work in writing Meditations on a Method with the cogito ergo sum ‘sound bite’ sort of approach to examining mind and reality (of subjective perceptions) with first principles (with-oneself,from oneself, for-oneself) is the foundation of rationalism. Yet one knows that reason and cogitation about objects and/or perceptions is a rational activity. Descartes simply was the first philosopher to apply disciplined rational principles to examine the phenomenon of mind at its source. Jean Paul Sartre followed Descartes's path of investigation three centuries later.
Before getting to Sartre and his tome ‘Being and Nothingness’ I should detour to a bit of sight-seeing to Bishop Berkeley and his ‘invention’ of idealism in his ‘Three Dialogues’ (of Hylas and Philonous). Berkeley named his work ‘ideaism' and that morphed into idealism. Idealism of course has a more popular application entirely unrelated to the meaning of ideaism. Idealism in the popular use refers to lofty ideals that one pursues regardless of the crude political facts of life (wherein lofty ideals are showpieces in politics rather than things actually applied). I am considering Berkeley’s ideaism here rather than the political slum of debasements of democracy that are anathema to idealists).
Berkeley seems to have applied the rationalism or existentialism of Descartes ( I should have described that relationship before that of Berkeley’s ideaism for purposes of clarity. Yet I would need to have simultaneous dual threads more suitable for a flowchart) theologically. Because I can’t use retro-causality in the human history of ideas without fiction, I can’t say that Berkeley applied Sartre’s existentialism with theological filters. If he had one might say that Berkeley examined Sartre’s existential, rational perspective of the ‘empirical world’ objectively, considering its causality Berkeley speculated that the entire experience of reality could be produced by God for the experience of human minds.
Berkeley’s ideaist paradigm is quite comparable to that of The Matrix. Except instead of a benevolent God creating and exciting and challenging experience for humanity to exist in; one in which their character in part serves to write some of the script of reality, evil artificial intelligence using machines and technology have made a fake world that humans are deceived in. Berkeley’s ideaist parameters converted to the usual worst case human trajectory of evil and adversity following and consistent with original sin.
Sartre’s modern rationalist paradigm described in ‘Being and Nothingness’ advanced Descartes's verificationist method of considering reality based entirely in subjective epistemology to the empirical world. Rather than re-litigating Descartes's verificationist approach to determining if one’s apparatus of thought and perception was valid or true, Sartre seems to have implicitly stipulated those parameters and taken the practical approach of describing how and what one experiences in the self-evident empirical world that provides experience and content to the perceiver. Sartre’s paradigm was named by moderns Existentialism.
Existentialism utilizes Bishop Berkeley's adaptation of Descartes' rationalism that Berkeley had transferred to the empirical world providing immersive experience for the perceiver. Berkeley considered the complete complex of compresence (Russell’s term) of experience and speculated upon its metaphysical supports. Sartre considers the complete complex of experience with an explanatory application of mechanics; differentiating and inventorying categories of content of experience without the metaphysics of Berkeley or the metaphysical doubt of Descartes. Sartre considers the mechanics or content of objective perceptions; not the neurological, epistemological or metaphysical.
That raises one final point concerning Plato and Aristotle. Following the work of pre-Socratic predecessors like Parmenides and Heraclitus, Plato and Aristotle considered the empirical world. In the day, at least outside the Jewish nation with their cosmological description given in the book of Genesis, philosophers were required to invent their own cosmologies. Some simpler cosmologies were innovated with inferences from pagans such as greater, superhuman beings (like Thor) need to exist to manufacture all of the phenomena of the Universe of experience. Philosophers and even pagan religiosity extrapolated causes from the observed phenomena of the empirical world.
Reasonable observations would see recurrent patterns and forms in nature. Plato experiencing recurrent forms and patterns in nature inferred the existence of a metaphysical library of forms, or patent office of forms, from which everything materially existing was a copy (although imperfect copies because the library or patent office office forms lasted eternally while copies in the world broke down, aged, died etc.). Aristotle began the work of classifying the kinds of forms that exist making observations of nature inventing scientific classification of biological species. He also invented formal logic and syllogisms. Followers believed that syllogistic reasoning about material objects and forms was entirely integrated and accurate. Modern investigations demonstrated that words including logic are labels applied to the existential experience. Yet of course science has also conducted empirical investigations into the mechanics of the field of objects and fields experienced (i.e. quantum mechanics).
That was the final interesting point to consider; Plato’s metaphysical inferences about the support for observed forms is comparable to string theory’s metaphysical inferences with mathematics and extra-dimensions to explain observed forms and problems in the quantum world (such as gluons, quarks, hadrons, gravity etc.). Although string theorists haven’t yet reached Aristotle’s level of classifying observed forms (as votaries of the Copenhagen Interpretation (classical quantum physics) have accomplished with the zoo of atomic and sub-atomic particles, the work so far is intriguing and stimulated a range of new metaphysical speculation concerning the nature of the Universe presented to experience experience for the mind, body and soul (perhaps a soul is the unique data comprising a human beings life).
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