An investigation is made of the mathematical foundations of mental test theory. The investigation culminates in the introduction of three axioms which define certain random variables called "tests." The existence of such variables is shown by means of an example. The axioms are then used to derive most of the classical results which (a) are independent of the internal structure of tests, and (b) do not require the use of sampling theory. The derivations tend to be terse, but most theorems are followed by a few interpretative remarks of more intuitive character. Furthermore, the paper contains several long passages of purely heuristic considerations. Whenever appropriate, examples or counter-examples are included.