The Man from the Future: The Visionary Life of John von Neumann

Operators can, for example, be added, subtracted and multiplied and so are said to ‘form an algebra’. Operators connected to each other by similar algebraic relationships are dubbed ‘rings’.

His breathtaking discovery was that the language of mathematics could be used to make meta-statements about mathematics.

When he did so, he discovered a strange property of his arrays: multiplying one array, A, by another, B often gave a different answer from multiplying B by A.

Schrödinger, however, realized what was missing from de Broglie’s work was an equation describing how matter waves snaked through space and time, similar to those derived for light

Multiplying position by momentum or, conversely, momentum by position gives slightly different results.

After listening to the scientists expound, Von Neumann broke in: ‘Well, gentlemen, suppose you tell me exactly what the problem is?’ For the next two hours the men at Rand lectured, scribbled on blackboards, and brought charts and tables back and forth. Von Neumann sat with his head buried in his hands. When the presentation was completed, he scrib

He discovered three irreducible types of operator ring that he called ‘factors’.

Pondering the physical meaning of noncommutativity led Heisenberg in 1927 to an extraordinary new law of nature, which stated that the position and momentum of a particle cannot both ever have exact values at the same time.

Gale-Shapley ‘deferred acceptance’ algorithm,