  The question of how exactly Hilbert understood the numerals is difficult to answer. They are not physical objects (actual strokes on paper, for instance), since it must always be possible to extend a numeral by adding another stroke (and, as Hilbert also argues in “On the infinite”  (1926),  it is doubtful that the physical universe is infinite). According to  Hilbert (1922b, 202),  their “shape can be generally and certainly recognized by us—independently of space and time, of the special conditions of the production of the sign, and of the insignificant differences in the finished product.” They are not mental constructions, since their properties are objective, yet their existence is dependent on their intuitive construction (see  Bernays 1923,  226). What is clear in any case is that they are logically primitive, i.e., they are neither concepts (as Frege’s numbers are) nor sets. What is important here is not primarily their metaphysical status (abstract versus concrete in the current sense of these terms), but that they do not enter into logical relations, e.g., they cannot be predicated of anything. In Bernays’s most mature presentations of finitism  (Hilbert and Bernays, 1939;  Bernays, 1930),  the objects of finitism are characterized as formal objects which are recursively generated by a process of repetition; the stroke symbols are then concrete representations of these formal objects.