A new classification of quantum mechanical quantities according to their measurability is discussed. We have three different types of quantities: first, exactly measurable quantities which commute with all the conserved quantities; second, approximately measurable quantities which do not commute with all additive conserved quantities; and finally nonmeasurable quantities which are conjugate to the absolute invariants. The epistemological implications of the measurement process are discussed and it is pointed out that the quantum mechanical description of this process does not furnish an adequate explanation of the acquisition of knowledge.