Euler made a crucial observation: if a path through this network is going to cross every link exactly once, then each node within the path must have an even number of links attached to it. That’s because whenever you enter the node by one link, you need to leave it by another, so the node needs two links if you visit it once, four if you visit it twice, and so on. The only nodes that can have an odd number of links attached to them are the nodes where the walk starts and ends (if they are distinct).
This immediately tells us that the path we are looking for doesn’t exist in the Königsberg problem. All nodes have an odd number of links attached to them. The beauty of this argument, as Euler suggested in his letter, is that it works for any network, no matter how large or complex. A path that crosses every link exactly once only exists if all, or all but two, nodes have an even number of links attached to them. The converse is also true (though Euler didn’t deliver a rigorous proof for this): if all, or all but two, nodes have an even number of links attached to them, then a path that crosses every link exactly once exists.
Euler’s thoughts about the Königsberg problem marked the beginning of an area of maths called graph theory, which you might also call network theory.
Waiting for Gödel
the theorem proved, using mathematics, that mathematics could not prove all of mathematics. Of course, it has a proper and technically precise formulation, but the late logician Verena Huber-Dyson paraphrased it for me as follows: “There is more to truth than can be caught by proof.” Or, as the British novelist Zia Haider Rahman put it in his award-winning début, “In the Light of What We Know,” “Within any given system, there are claims which are true but which cannot be proven to be true.”
.. Famously, Gödel informed the judge at his U.S. citizenship hearing about an inconsistency that he had discovered in the Constitution, which would allow a dictator to rise in America.
.. included a computer scientist obsessed with recursion (that is, self-referential things, like Russian nesting dolls or Escher’s drawing of a hand drawing a hand);
.. In the nineteenth century, the German mathematician Georg Cantor launched an investigation into various sizes of infinities, thereby inventing set theory, which became an overarching paradigm for mathematics. “A set is a Many that allows itself to be thought of as a One,” Cantor said. But, from nearby realms of logic, paradoxes emerged, such as the Russell set, the set of all sets that do not contain themselves.
.. For any consistent axiomatic formal system that can express facts about basic arithmetic:
1) There are true statements that are unprovable within the system.
2) The system’s consistency cannot be proven within the system.
.. “Gödel’s theorem has a major impact on what all computer scientists do,” he told me. “It puts a fundamental limit on questions we can answer with computers. It tells us to go for approximation—more approximate solutions, which find many right answers, but not all right answers. That’s a positive, because it constrains me from trying to do stupid things, trying to do impossible things.”
The number of American teens who excel at advanced math has surged. Why?
You wouldn’t see it in most classrooms, you wouldn’t know it by looking at slumping national test-score averages, but a cadre of American teenagers are reaching world-class heights in math—more of them, more regularly, than ever before. The phenomenon extends well beyond the handful of hopefuls for the Math Olympiad. The students are being produced by a new pedagogical ecosystem—almost entirely extracurricular—that has developed online and in the country’s rich coastal cities and tech meccas. In these places, accelerated students are learning more and learning faster than they were 10 years ago—tackling more-complex material than many people in the advanced-math community had thought possible.
.. Parents of students in the accelerated-math community, many of whom make their living in stem fields, have enrolled their children in one or more of these programs to supplement or replace what they see as the shallow and often confused math instruction offered by public schools, especially during the late-elementary and middle-school years.
.. The roots of this failure can usually be traced back to second or third grade ..
.. Her children, who attended public school in affluent Newton, Massachusetts, were being taught to solve problems by memorizing rules and then following them like steps in a recipe, without understanding the bigger picture.
Italian mathematician taken off flight after fellow passenger alarmed by his notepad calculations
An Italian maths professor was escorted of an American Airlines flight after a fellow passenger feared his mysterious scribbling on a notepad was evidence that he was a terrorist.
In fact Guido Menzio was working on an equation connected with a presentation on price–setting.
.. Mr Menzio, an economics professor at the University of Pennsylvania was asked to come to the front of the aircraft where he was spoken to first by the pilot and then an official.
Mr Menzio, a highly respected academic who has also had spells at Princeton and Stanford universities, succeeded in convincing the authorities that his doodles were an equation.
He was allowed to board the flight, which left two hours late, without the woman whose suspicions caused the delay.