Johhny proves what he wants

Arguably the most intelligent man to walk the planet, Hungarian American mathematician John von Neumann was born on December 28, 1903 into a wealthy non-observant Jewish family. He made immense contributions to many fields of mathematics including functional analysis, game theory, measure theory, ergodic
theory and several more.

His work in applied mathematics influenced quantum theory, economics and defense planning. He was also responsible, along with Claude Shannon and Alan Turing, for the invention of the stored program digital computer. A colleague of his, Paul Halmos describes him as follows: “You could practically imagine Johnny with a checklist before him as he was going down through various human disciplines, ‘mathematics, physics, chemistry, economics… I’ve done this… I’ve done that…’ He was always looking for green fields to conquer. I don’t know whether it was personal ambition or intellectual curiosity. I think, to some extent at least, it was that he wanted to leave his mark in everything. He was striving to be a universalist which is a hard thing to be in the 20th century. He came close.”

He is known as ‘the last representative of the great mathematicians who excelled in
both pure and applied math.’ Nobel laureate in physics, Hans Bethe said about Johnny, “I have sometimes wondered whether a brain like von Neumann’s does not indicate a species superior to that of man.” Mathematician Rózsa Péter’s assessment of von Neumann’s abilities is also rather astounding: “Other mathematicians prove what they can, von Neumann proves what he wants.” Once, German physicist Max Born presented the following puzzle, that got somewhat popular in 1920s, to von Neumann:

Two cyclists cycle towards each other from the 2 ends of a 40-mile-track at 20 miles per hour each. At the same time, a fly that travels at 50 miles an hour, starts
from the front wheel of one cycle, touches the front wheel of the second, then comes back to the first and then back to the second again. It continues in this manner till it is crushed between the 2 front wheels. How much distance does the
fly travel?

It is not too hard to arrive at the solution if one notices that two cyclists will meet in one hour because they are 40 miles apart and have speed of 20 miles per hour. Therefore, the fly travels 50 miles in this time since its speed is 50 miles an hour. Not many people could see this answer at first and nor did Johnny. Yet, when Sam finished the question, he was ready with the answer and said, “50 miles of course!” Born was amazed and said, “You are the first one of my scientist friends who saw the solution at once.” Johnny replied, “Well, I can’t understand that. It is a simple infinite geometrical series.” Apparently, von Neumann had summed up the lengths of the infinitely many distances traversed by the fly, by the time Sam finished speaking. This anecdote just illustrates the outstanding calculating powers that Johnny possessed.

George Polya, who had taught von Neumann, called him the only student he was ever intimidated by. He was so quick. There was a seminar for advanced students in Zurich that he was teaching and von Neumann was in the class. He came to a certain theorem and said it is not proved and it might be difficult.
Von Neumann didn’t say anything but after five minutes he raised his hand. When George called on him, he went to the board and proceeded to write down the proof. “After that,” he said, “I was afraid of von Neumann.”

Newton Vs Leibniz: The Fight Over Calculus

There’s nothing like a classic fight of the egos, and the battle over who first discovered calculus was quite vicious. Calculus is a method for determining the relationship between quantities rather than the quantities themselves. That concept had been brewing long before Newton or Leibniz were born. They did, however, individually write a complete calculus system. The conflict happened at the turn of the 18th century between Isaac Newton and Gottfried Leibnitz. Newton claimed to have begun working on a form of calculus (which he called “the method of fluxions and fluents”) in 1666, at the age of 23 but did not publish it except as a minor annotation at the back of one of his publications decades later. Newton, though he explained his (geometrical) form of calculus in Section I of Book I of the Principia of 1687, he did not explain his eventual fluxional notation for the calculus in print until 1693 (in part) and 1704 (in full). Gottfried Leibnitz began working on his variant of calculus in 1674, and in 1684 published his first paper employing it, “Nova Methodus pro-Maximis et Minimis”(A new method for maxima and minima). Leibnitz continued to publish results on the new calculus in the Acta Eruditorum and began to explore his ideas in extensive correspondence with other scholars. Within a few years, he had gathered a group of researchers to help him spread his techniques, including the brothers Johann Bernoulli and Jakob Bernoulli in Basel, as well as the priests Pierre Varignon and Guillaume-François- Antoine de L’Hospital in Paris.

Leibnitz’s zealous support for the new calculus, the instructional tone of his publications, and his ability to assemble a community of researchers- all led to his immense effect on later mathematics. In contrast, Newton’s tardy publication and personal reluctance resulted in a diminished presence in European mathematics. While formally, Leibnitz was the first to publish his theory of infinitesimal calculus, in retrospect, Newton and his supporters claimed that Leibnitz had plagiarized Newton’s ideas from an earlier manuscript. So, the claim that Leibnitz invented calculus independently of Newton rested on the basis that Leibnitz published a description of his method some years before Newton printed anything on fluxions.

Leibnitz always alluded to the discovery as being his own invention (this statement went unchallenged for some years), enjoyed the strong the presumption that he acted in good faith, and demonstrated in his private papers his development of the ideas of calculus in a manner independent of the path taken by Newton. But according to Leibnitz’s detractors, the fact that Leibnitz’s claim went unchallenged for some years is immaterial. The arguments for their side were mainly based on their claims that Leibnitz saw some of Newton’s papers on the subject in or before 1675 or at least 1677, and obtained the fundamental ideas of calculus from those
papers. That Leibnitz saw some of Newton’s manuscripts had always been likely. At that time there was no direct evidence that Leibnitz had seen Newton’s manuscript before it was printed in 1704; hence Newton’s conjecture was not published. But Gerhardt’s (a Mathematics historian) discovery of a copy made by Leibniz tends to confirm its accuracy. Those who question Leibnitz’s good faith argue that to a man of his ability, the manuscript along with Newton’s letter sent to another mathematician on 10 December 1672 who was recorded working together with Leibnitz, was enough to give him a clue as to the methods of the calculus used by
Newton. Because Newton’s work used fluxional notation, anybody expanding on it would have to establish a notation, however, some contradict this.

As with most things in life, those with greater political power typically get their way. So in 1711, at the peak of the battle, the Royal Society came to a conclusion and accused Leibnitz of plagiarism. (Newton was the Royal Society’s president, thus this came as no surprise), a charge that falls apart when you trace the details. In the end, Newton’s campaign was effective and damaging. He emerged with the credit. But when people like Leonard Euler and the Bernoullis erected the field of applied analysis, they used Leibnitz’s calculus. So one can see, there is a bit of justice to the story for Team Leibnitz because nowadays, Leibnitz’s cleaner notation is what is most universally used. The prevailing opinion in the 18th century was against Leibnitz (in Britain, not in the German-speaking world). Today the consensus is that Leibnitz and Newton independently invented and described calculus in Europe in the 17th century. Leibnitz died poor and dishonored, while Newton was given a state funeral. Yet history validates Leibnitz, as the power of Newton’s onslaught diminishes with time and Leibnitz steadily establishes himself as one of the great mathematicians of all time.