  In two papers entitled "On a new foundation [Neue Begr\"undung] of quantum
mechanics," Pascual Jordan (1927b,g) presented his version of what came to be
known as the Dirac-Jordan statistical transformation theory. As an alternative
that avoids the mathematical difficulties facing the approach of Jordan and
Paul A. M. Dirac (1927), John von Neumann (1927a) developed the modern Hilbert
space formalism of quantum mechanics. In this paper, we focus on Jordan and von
Neumann. Central to the formalisms of both are expressions for conditional
probabilities of finding some value for one quantity given the value of
another. Beyond that Jordan and von Neumann had very different views about the
appropriate formulation of problems in quantum mechanics. For Jordan, unable to
let go of the analogy to classical mechanics, the solution of such problems
required the identication of sets of canonically conjugate variables, i.e., p's
and q's. For von Neumann, not constrained by the analogy to classical
mechanics, it required only the identication of a maximal set of commuting
operators with simultaneous eigenstates. He had no need for p's and q's. Jordan
and von Neumann also stated the characteristic new rules for probabilities in
quantum mechanics somewhat differently. Jordan (1927b) was the first to state
those rules in full generality. Von Neumann (1927a) rephrased them and, in a
subsequent paper (von Neumann, 1927b), sought to derive them from more basic
considerations. In this paper we reconstruct the central arguments of these
1927 papers by Jordan and von Neumann and of a paper on Jordan's approach by
Hilbert, von Neumann, and Nordheim (1928). We highlight those elements in these
papers that bring out the gradual loosening of the ties between the new quantum
formalism and classical mechanics.
