Benedetto Croce (1866-1952) (I) distinguishes two pure theoretical forms in the cognitive spirit : intuition and concept (concrete universal); while the first gives rise to art, the second, when expressed in the form of a definition, constitutes philosophy properly so-called, and when expressed in the form of individual judgments constitutes history. Besides these Croce distinguishes two forms of practical elaboration of knowledge, that is : the formation of empirical classificatory pseudo-concepts, and of abstract numerative and measurative pseudo-concepts ; the first are concrete but not universal and make up the natural sciences ; the second are universal but not concrete and constitute the mathematical sciences. Thus, according to Croce, the mathematical concepts (pseudo-concepts to him) have no reality (2) : 'there is never a geometrical triangle in reality, because in reality there are no straight lines, right angles, and sums of angles equal to two right angles .... A thought which has nothing real for its object is not a thought: and these concepts are not concepts but conceptual fictions.' 'The geometrical triangle is no use either to fantasy or to thought ... but is indispensable to a man measuring a field.' Croce mentions Bertrand Russell's definition of mathematics which we have already examined and comments (3): 'A science that does not affirm anything does not
belong to the theoretical spirit, it is not even poetry: and a science that does not refer to anything is not even an empirical science, which always refers to a determinate group of representations ... and ... it is impossible to believe that the principles of mathematics are real. In fact, strictly speaking, they are all entirely false. The numerical series is obtained by starting from unity and always adding another unit ; but in reality there is nothing that can function as first term of a series and no way of generating a discontinuous series .... '