Editing John von Neumann (section)

=== Mathematical range ===

The mathematician [[Jean Dieudonné]] said that von Neumann "may have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions".{{sfn|Dieudonné|2008|p=90}} According to Dieudonné, his specific genius was in analysis and "combinatorics", with combinatorics being understood in a very wide sense that described his ability to organize and axiomize complex works that previously seemed to have little connection with mathematics. His style in analysis followed the German school, based on foundations in [[linear algebra]] and [[general topology]]. While von Neumann had an encyclopedic background, his range in pure mathematics was not as wide as [[Henri Poincaré|Poincaré]], [[David Hilbert|Hilbert]] or even [[Hermann Weyl|Weyl]]: von Neumann never did significant work in [[number theory]], [[algebraic topology]], [[algebraic geometry]] or [[differential geometry]]. However, in applied mathematics his work equalled that of [[Carl Friedrich Gauss|Gauss]], [[Augustin-Louis Cauchy|Cauchy]] or [[Henri Poincaré|Poincaré]].{{sfn|Dieudonné|2008}}

According to Wigner, "Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique."{{sfn|Dyson|1998|p=77}} Halmos noted that while von Neumann knew lots of mathematics, the most notable gaps were in algebraic topology and number theory; he recalled an incident where von Neumann failed to recognize the topological definition of a [[torus]].{{sfn|Halmos|1973|p=389}} Von Neumann admitted to Herman Goldstine that he had no facility at all in topology and he was never comfortable with it, with Goldstine later bringing this up when comparing him to [[Hermann Weyl]], who he thought was deeper and broader.{{sfn|Goldstine|1985|p=16}}

In his biography of von Neumann, [[Salomon Bochner]] wrote that much of von Neumann's works in pure mathematics involved finite and infinite dimensional [[vector space]]s, which at the time, covered much of the total area of mathematics. However he pointed out this still did not cover an important part of the mathematical landscape, in particular, anything that involved geometry "in the global sense", topics such as [[topology]], [[differential geometry]] and [[Hodge theory|harmonic integrals]], [[algebraic geometry]] and other such fields. Von Neumann rarely worked in these fields and, as Bochner saw it, had little affinity for them.{{sfn|Bochner|1958|p=441}}

In one of von Neumann's last articles, he lamented that pure mathematicians could no longer attain deep knowledge of even a fraction of the field.{{sfn|Ulam|1958|p=8}} In the early 1940s, Ulam had concocted for him a doctoral-style examination to find weaknesses in his knowledge; von Neumann was unable to answer satisfactorily a question each in differential geometry, number theory, and algebra. They concluded that doctoral exams might have "little permanent meaning". However, when Weyl turned down an offer to write a history of mathematics of the 20th century, arguing that no one person could do it, Ulam thought von Neumann could have aspired to do so.{{sfn|Ulam|1976|p=291}}