I was a bit curious about von Neumann's work in quantum mechanics and found this. There are a number of online sources (pdf) if you want to search for it. Hil

I was a bit curious about von Neumann's work in quantum mechanics and found this. There are a number of online sources (pdf) if you want to search for it. Hilbert Space Theory and Applications in Basic Quantum Mechanics by Matthew R. Gagne > "Ultimately it was Von Neumann, while a student of Hilbert, who realized the equivalence of wave mechanics and matrix mechanics and proposed the current formulation. However, the argument at the time was largely heuristic and unconcerned with mathematical detail. In [CMMC] he argues that the bulk of the equivalence proof were and have been historically taken for granted. The reader can consult [CMMC] for more on the technicalities of equivalence proofs. While the theoretical and conceptual leaps that made the formulation of quantum mechanics possible came largely from Bohr, Heisenberg, and Schrodinger, Von Neumann is responsible for the last unifying mental leap and also for many of the technical mathematical achievements that accompanied it. We discuss Von Neumann and the mathematical perspective of these events next." If you ever take a intro QM course or even a modern chemistry course, you'll get introduced to the Schrodinger Wave Equation and how it explains the hydrogen atom. However, on a computer, basically all QM calculations are done with matrix mechanics. JvN, as I read it (no QM physicist expert am I), mathematically tied these two conceptions together utilizing a complex abstraction called Hilbert space: > "Hilbert space gives a means by which one can consider functions as points belonging to an inÖnite dimensional space. The utility of this perspective can be found in our ability to generalize notions of orthogonality and length to collections of objects (i.e. functions) which don't naturally suggest the consideration of these properties." Clearly that's a major achievement, in a head-spinningly complex area. However, in looking at several other areas of JvN's legacy - the invention of computation in its current form, and economic game theory - it's all a bit more curious and iffy. With computation, there really seems to be a lot of controversy over where the credit for the theories and practice really belongs, i.e. Alan Turing on one hand and some of the people who actually engineered the first working American vacuum-tube computers on the other. Some of the literary efforts at cementing JvN's legacy in this area have been questionable, i.e. "Turing's Cathedral" which isn't really about Turing at all, it's a cloaked JvN hagiography. As far as game theory as a mathematical basis of economics and 'rational nuclear war', it honestly seems to be, at this point, discredited on multiple grounds, such as humans not being rational actors. Neoclassical economic model projections based on such theories have failed spectacularly time and again (NAFTA outcomes, failure to foresee the 2008 crash, etc.) for example. Of course if this basic structure is pulled out from under the feet of neoclassical economists today, there's not much left... Could it be they're trying to build a statue to their hero to avoid that fate? Furthermore, JvN made some predictions that were entirely off base in the area of climate, as he believed long-term forecasts were possible, meaning that he thought tickling the climate system at just the right point could cause a hurricane to hit a particular location. The discovery of chaos, aka sensitive dependence on initial conditions, by Ed Lorenz about a decade later, put the end to such dreams of climate warfare. He certainly was a brilliant mathematician, but his other contributions are a bit more questionable. The concept of the 'universal genius' doesn't really seem to apply in practice. And, hagiography is bad bibliographic practice.