Contentual number theory is Hilbert's term for the elementary, finitistic part of number theory in which numerals
are treated as concrete finite strings of strokes and
reasoning concerns directly surveyable operations on them. For example, a numeral
such as 
represents three by its displayed construction rather than by appeal to an abstract
completed totality.
In Hilbert's program, contentual reasoning was intended to supply the secure metamathematical standpoint from which proofs involving ideal objects could be studied. The distinction is between contentual (inhaltlich) statements with an immediate finitistic interpretation and ideal statements introduced to simplify and systematize mathematics; it is not a distinct modern branch of number theory (Zach 2007).
