What is common between the rationalists and empiricists?

Where there is disagreement, as there is between rationalists and empiricists, there is always a background of agreement : there can only be disagreement if you at least agree on what there is to disagree about. So much by way of sage introduction !

▻ RATIONALISM

I'd offer a provisional characteristisation of rationalism as a philosophy for which reason is a source of knowledge of the physical world independent of and superior to empirical observation in the form of sense perception or introspection. 'Superior to' indicates that reason can supply beliefs which are immune from error. A rationalist accepts the possibility and fact of a priori knowledge, knowledge derived by reason independently of (prior to) experience. Reason may operate independently of experience by virtue of innate ideas but a rationalist is not as such committed to the existence or reliability of innate ideas.

There has historically been an assumption, or tendency to suppose, that knowledge of physical world takes a mathematical form.

▻ EMPIRICISM

For empiricism there is no knowledge independently of experience. The closest we can come to a priori knowledge is a knowledge of analytic truths - knowledge for instance of tautologies. This tells us nothing about reality, only at most some truths about language or concepts. All knowledge of the physical world derives from empirical observation in the form of sense perception or introspection.

Empiricism is not closely associated with the view that a knowledge of the physical world takes a mathematical form.

▻ GENERALISATION

I have only been able to offer a wide and approximate contrast between rationalism and empiricism. There is really not way round looking at individual thinkers and seeing what they agreed and differed on.

▻ COMMONALITIES

☛ Direct perception and certainty

The 'reason' on which rationalism relies is not necessarily or even usually deductive, inferential, 'discursive' reasoning - the deduction of a conclusion from premises. It is often, as we see clearly in Descartes, rational intuition or insight. Rationalists use deduction but Descartes' 'Meditations' and to an extent his 'Rules for the Direction of the Mind' run on intuitions - the products of mental acts by which we perceive directly and with absolute objective certainty*, without the aid of empirical observation, some particular truth.

[* Certainty is ambiguous between (psychological) 'absolute conviction' and (epistemological)'immunity from error'. I use the term in the second sense.]

Direct perception also plays its part in empiricism but not perception of the intuitive kind. If, for instance, it appears to me that I am looking at a tree, then it is certain that that is how things appear to me. Even if I am hallucinating, it is still certain that it appears to me that I am looking at a tree. Without further analysis, Descartes regarded such empirical certainty as too thin to support the edifice of knowledge; Meditation I shows how sense-based certainty can go along with epistemological unreliability. How things appear, whatever the fact of our experience, is no sure guide to how they really are.

☛ Individualism

In Descartes on the one hand on the rationalist side, and on the other Locke and Hume on the empiricist, there is a kind of epistemological individualism. Locke is careful, or tries to be, to admit no truth which does not recommend itself to his own mind. Descartes, equally, commits himself to not accepting as true anything which he does not clearly apprehend to be such. Hume keeps closely to beliefs which he can justify in terms of his own 'impressions and ideas'.

☛ Rationalism does not exclude the empirical totally

Observation, sensation, and experiment - which clearly belong to the world of experience - are by no means excluded from the Cartesian search for knowledge. Descartes never says that we or he can build an adequate body of knowledge without any reference to the senses. It is just that the contribution of the senses is either unnecessary or highly unreliable at the deepest, foundational level of knowledge. The foundations of knowledge are supplied by clear and distinct ideas derived from intuition working in the 'natural light' (lumen naturale) of reason. There is plenty of room for observation, sensation, and experiment but only insofar as their results can be justified by derivation from or consistency with foundational clear and distinct ideas.