The two modes of thought do not mix well

From A Jacques Barzun Reader by Michael Murray, the essay History as Counter-Method and Anti-Abstraction:

The reason lies in the difference between two orientations of the human mind: the intuitive and the scientific. Pascal, who possessed the genius for both, gave of them a definitive account in his Pensees. Whoever wants to understand the difference between, say, Carlyle's French Revolution and Crane Brinton's Jacobins - why one is a history and the other not - should turn to Pascal's first chapter and assimilate the series of distinctions set forth there between the esprit de finesses and the esprit de geometrie. Neither esprit is higher or deeper or better than the other. They are only radically divergent modes of conceiving and working with reality.

A compressed paraphrase could run as follows: in science (the geometrical mind), the elements and definitions are clear, abstract, and unchangeable, but stand outside the ordinary ways of thought and speech. Because of this clarity and fixity, it is easy to use these concepts correctly, once their strange artificiality has been firmly grasped; it is then but the application of a method. In the opposite realm of intuitive thought (finesse), the elements come out of the common stock and are known by common names, which elude definition. Hence it is hard to reason justly with them because they are so numerous, mixed, and confusing: there is no method.

From the dissimilarity it follows that genius in science consists in adding to the stock of such defined entities and showing their place and meaning within the whole system of science and number; whereas genius in the realm of intuition consists in discerning pattern and significance in the uncontrollable confusion of life and embodying the discovery in intelligible form.

Obviously the two modes of thought do not mix well: there are no natural transititions from one to the other, the movement of the mind in each goes counter to the other. Once understood, the opposition resolves many puzzles and conflicts in contemporary culture, which is torn and racked by the imperialistic demands of each "mind" simultaneously.