Ishan Mukherjee

“The young von Neumann made an instant impact on his new tutors. His first mentor, Gábor Szego˝, who would later lead Stanford University’s maths department, was moved to tears after their first meeting.”
― Ananyo Bhattacharya, The Man from the Future: The Visionary Ideas of John von Neumann

“[Von Neumann's childhood home's] library’s centrepiece [was] the Allgemeine Geschichte, a massive history of the world edited by the German historian Wilhelm Oncken, which began in Ancient Egypt and concluded with a biography of Wilhelm I, the first German emperor, commissioned by the Kaiser himself. When von Neumann became embroiled in American politics after he emigrated, he would sometimes avoid arguments that were threatening to become too heated by citing (sometimes word for word) the outcome of some obscurely related affair in antiquity that he had read about in Oncken as a child.”
― Ananyo Bhattacharya, The Man from the Future: The Visionary Ideas of John von Neumann

“Russell’s paradox threatened to deal a far more serious blow to set theory than earlier ideological objections. The problem was this: consider a set of objects – all possible types of cheesecake, say. This set may include any number of different cheesecakes (New York cheesecake, German Käsekuchen, lemon ricotta, etc.) but, because a set is not literally a cheesecake, the set of all cheesecakes is not a member of itself. The set of all things that are not cheesecakes, on the other hand, is a member of itself.
But what, Russell wondered, about the set of all sets that are not members of themselves. If this is not a member of itself, then, by definition, it should be (because its members do not include itself). Conversely, if it is a member of itself, then it should not be (because it does). This was Russell’s paradox in a nutshell. His analysis of the paradox revealed it to be similar in form to several others, including the liar’s paradox (‘this statement is a lie’). ‘It seemed unworthy of a grown man to spend time on such trivialities,’ he complained, desperate for a solution, ‘but what was I to do?”
― Ananyo Bhattacharya, The Man from the Future: The Visionary Ideas of John von Neumann

“[Euclid's] second postulate, for example, says that any line segment may be extended indefinitely. That is difficult for even the most querulous to argue with. The fifth, on the other hand, states that if two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side (labelled a and b in the diagram below) is less than two right angles (i.e. 180°), then the two lines inevitably must intersect each other on that side if extended far enough. If, on the other hand, a and b do add up to 180°, the two lines never meet so are said to be parallel.
To mathematicians, that looks less like a postulate and more like a theorem in need of proving.”
― Ananyo Bhattacharya, The Man from the Future: The Visionary Ideas of John von Neumann