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In two papers entitled "On a new foundation [Neue Begr\"undung] of quantummechanics," Pascual Jordan (1927b,g) presented his version of what came to beknown as the Dirac-Jordan statistical transformation theory. As an alternativethat avoids the mathematical difficulties facing the approach of Jordan andPaul A. M. Dirac (1927), John von Neumann (1927a) developed the modern Hilbertspace formalism of quantum mechanics. In this paper, we focus on Jordan and vonNeumann. Central to the formalisms of both are expressions for conditionalprobabilities of finding some value for one quantity given the value ofanother. Beyond that Jordan and von Neumann had very different views about theappropriate formulation of problems in quantum mechanics. For Jordan, unable tolet go of the analogy to classical mechanics, the solution of such problemsrequired the identication of sets of canonically conjugate variables, i.e., p'sand q's. For von Neumann, not constrained by the analogy to classicalmechanics, it required only the identication of a maximal set of commutingoperators with simultaneous eigenstates. He had no need for p's and q's. Jordanand von Neumann also stated the characteristic new rules for probabilities inquantum mechanics somewhat differently. Jordan (1927b) was the first to statethose rules in full generality. Von Neumann (1927a) rephrased them and, in asubsequent paper (von Neumann, 1927b), sought to derive them from more basicconsiderations. In this paper we reconstruct the central arguments of these1927 papers by Jordan and von Neumann and of a paper on Jordan's approach byHilbert, von Neumann, and Nordheim (1928). We highlight those elements in thesepapers that bring out the gradual loosening of the ties between the new quantumformalism and classical mechanics.

(Never) Mind your p's and q's: Von Neumann versus Jordan on the Foundations of Quantum Theory